Download 97407357 Vibration of Continuous Systems by Singiresu S Rao PDF

Title97407357 Vibration of Continuous Systems by Singiresu S Rao
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Table of Contents
                            page1
		Vibration of Continuous
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		. ...,
		10987654321 ~<\.\~.....::. •••• F.j
		,\~ ~~ \§.
		, '''': 6-",p";
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		Contents
page6
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		______________________________________ - ---.....".....,,--~-m-.A
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		______________ ~-~--...--IIIIIIIIIIIIII!-----,- ... ,.,~
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		Preface
		.. , ~~.~c"j
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		HaJ
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		Symbols
		A,B, C,D
		C1, C2, C3, C4
		EA
		EI
		1
		1
		10
		I(x, t)
		Fo
		GJ
		I
		R
		xix
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		L
		L-1
		s
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		x
		a,f3
		o
		~
		n
		2 a2 a2
		'V = ax2 + ii;!
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		1
		Introduction: Basic Concepts
		1.1 CONCEPT OF VIBRATION
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		-x
		x
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		o
		~~
		~
		~
		~~~~~~~~~~~""~
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		1.2 IMPORTANCE OF VffiRATION
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		1.3 ORIGINS AND DEVELOPMENTS IN MECHANICS
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		I
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		1.4 HISTORY OF VIBRATION OF CONTINUOUS SYSTEMS
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		1.5 DISCRETE AND CONTINUOUS SYSTEMS
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		T
		o
		" I
		/ I
		" I
		" I
		" I
		" I
		I I /
		,. <4:..- - ,.i A'"
		\ -' - --,!"", - 0
		I- ~r------i---- alum
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		-
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		I
		T
		T
		T
		1.6
		vmRATION PROBLEMS
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		1.7 VIBRATION ANALYSIS
		. (
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		1.8
		EXCITATIONS
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		1.9 HARMONIC FUNCTIONS
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		- .. -
		!
		"""""""
		i = a + ib
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		.~ 1- a
		a
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		o ." .... ~. "
		------------ ----- .• ..::. .
		.. -.
		,
		. .,
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		f----
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		~\ \\ )),
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		ao " .
		ao 2 . b' 2
		1.10 PERIODIC FUNCTIONS AND FOURIER SERIES
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		11"!"
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		71
		11T/2 .
		2"
		o
		1.11 NONPERIODIC FUNCTIONS AND FOURIER INTEGRALS
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		w
		fo Iwl
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		o
		10
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		1.12 LITERATURE ON VIBRATION OF CONTINUOUS SYSTEMS
		REFERENCES
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		30
		1~_"""'-~I'a:il;\rO"·ll>~~~ __ ~;"""''"'''''''.''.'.",",'''.''''''i'''\'""","~_~~~~_'-~"".l><~w"""'~~",,,L
page51
		PROBLEMS
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		2100
		L
		o
		o
		f( ) = {O, t < 0
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		.•...
		Vibration of Discrete Systems:
		Brief Review
		2.1 VmRATION OF A SINGLE-DEGREE-OF -FREEDOM SYSTEM
		mx + kx = 0
		Xo .
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		mx + ex +kx = f(t)
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		s----
		Cc = 2mwn = 2..j"f;;
		m
		(2.11)
		( Xo + S wriXo. )
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		mx + kx = fo cos wt
		Cj = XOWn (~ + ~) + Xo
		k - mw2 1 - (w/wn)2
		X 1
		-=
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		_!
		i
		,
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		mx + ex + kx = fo cos wt
		o
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		--"
		---
		---
		---
		---
		---
		/0
		o
		X 1
		-
		{-------
		- ec - 2,JT;k - 2mwn
		<P = tan -1 ew = tan -1 .l:S.!:.-
		k - mw2 1 - r2
		--
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		o
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		mx + eX + kx = loeiwt
		(2.40)
		X= 10
		k - mw2 + icw
		kX I
		- = H(iw) ==
		xp(t) = 10IH(iw)lei(wt-t/J)
		\kXI 1
		IH(iw)1 = 10 = [(1 - r2)2 + (2~r)2]l/2
		k - mw2 1 - r2
		(2.46)
		2.1.3 Forced Vibration under General Force
		L[x(t)] = x(s)
		L[x(t)] = sx(s) - x (0)
		L[l(t)] = s2x(s) - sx(O) - X(O)
		[s2x(s) - sx(O) - x(O)] + 2~wn[sx(s) - x(O)] + w;x(s) = F(s) (2.51)
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		1-
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		2.2 VffiRATION OF MULTIDEGREE-OF -FREEDOM SYSTEMS
		em]; + [c]i + [k]x = j
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		2.2.1 Eigenvalue Problem
		[m],i + [k]x = 0
		[[k] - w2[m]]X = 0 (2.71)
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		I[k] - w2[m]1 = 0
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		•....
		I
		, ' , ,_ ' ' .0 , ' ." '" •• '!'~:"""""""f'''"'' ""., " I!' I
		i=l=j
		X(j)T [k]X<;) = 0, i =1= j
		[Xl'[m][~] = [I] = [: 1
		(2.85)
		[X] = [x(1) X(2)
		(2.86)
		[m]i + [k]x = 6
		x(t) = LTJ;(t)X(;) = [X]ij(t)
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		x(t ,,0) = xo = . :
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		--
		: = ij(O) = [X]T[m]xo
		.. x.~ m, ~.,«,_X,)
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		{ (1») { }
		[ -w'mJ_~,kJ +k, -w'm'--:'k, H, Hi;} = m
		or
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		--
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		1 .
		h(t) = - sm Wit
		[X]T[m][X]ry + [X]T[k][X]ij = [X]T j (2.104)
		ij + [w;Jij = Q
		[m]i + [k]x = jet)
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		[c] = arm] + t3[k]
		em]; + (a[m] + {3[k]); + [k]x = j
		x(t) = [X]ry(t)
		ry + (a[I] + {3[wr])ry + [wr]ry = Q
		1
		(2. 1"23)
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		[AJY = AY
		[AJ = -[mJ-l[kJ -[mJ-1[cJ
		[BJ = [mr1
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		I[A] - A.(/]I == I[A]T - A[I]I = 0
		Z(j)T [A] = AjZ(j)T (2.137)
		i, j = , , ... , n
		i = , , ... , n
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		[Z]T[y] = [I]
		[Y] == Y ... y n
		yet) = b.rli (t)yCi) = [Y] ry(t)
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		em] = [~I ~J = [~ ~]
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		(E2.3.5)
		(E2.3.4)
		(E2.3.6)
		(E2.3.7)
		(E2.3.8)
		(E2.3.9)
		:.; {XI}
		X = Xz '
		y = [A]y + [E]l
		.. - I;~}
		y- .
		Xz
		.:. {XI}
		X = xz '
		[A] = -[m]-I[k] -[mrl[c]
		.. _ {Xl}
		Xz
		[kl + kz -kz] [30 -20]
		[k] = =
		-kz kz + k3 -20 25
		The equations of motion can be stated in state form as
		where
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		[A]Y = AY
		-2
		or
		[ ~ ~ -:? -:] {~~} = A {~~}
		2 5
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		. [0~5 ~ J {~} ~ ( ~g£t :_~og;~~f;: 1
		2.3 RECENT CONTRIBUTIONS
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		-
		[I].i(t) + [A]i(t) + [B]x(t) = jet)
		REFERENCES
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		--
		PROBLEMS
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page83
		...--
		5'
		o
		,
		I '
		i -r-",
		I \
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		.-.-.-.-.-.-.-.-.- ..
		aT"
		-.-.-.-.-.-.-.-.-.
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		~
		TT
		T
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		Derivation of Equations:
		Equilibrium Approach
		I:- d - -
		3.1 INTRODUCTION
		3.2 NEWTON'S SECOND LAW OF MOTION
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		d .
		a2u
		3.3 D' ALEMBERT'S PRINCIPLE
		3.4 EQUATION OF MOTION OF A BAR IN AXIAL VIBRATION
		au
		P = (J' A = EA- (3.4)
		ax
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page90
		.1
		---.
		r- dx -I I-_c'
		"------~-_.~._-
		t
		I
		I
		o
		a2u a2u
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page91
		au O' (
		at
		. au
		or
		3.5 EQUATION OF MOTION OF A BEAM IN TRANSVERSE
		a2w
		av aM
		ax ax
		I
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page92
		M tt rrt>t tt \
		-T(v8t~ )
		_._._.L._.~I~.L._._._._. __ x
		f--dx---J
		t
		O~-·_·~_·_·_·_·_·_I·
		I· ~ I ~I
		av a2w
		aM
		-(x, t) - vex, t) = 0
		a2M a2w
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		a4w a2w
		aw . (
		3.6 EQUATION OF MOTION OF A PLATE IN TRANSVERSE
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		~I
		Q,. ~"')
		;/
		,,+--" y
		/.
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		1h~ 1h~
		a2w
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		ow
		ow
		ay
		or
		2
		dx
		3.6.3 Strain-Displacement Relations
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		-
		t
		J ~-_.
		'--114- 1
		au
		ou ov
		exy = oy + ox
		exx = au = ~ (_zaw) = _zo2w
		ox ox ox ox2
		e = av = ~ (_zow) = _zo2w
		yy ay oy oy oy2
		ex = au + ov = ~ (_zaw) + ~ (_zaw) = -2z o2w
		y ay ox ay ax ox oy oxoy
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		-v -v
		Mx = -D --z + v--z
		My = -D --z + v--2
		Eh3
		D=
		Qx=-D- -+-
		Q =-D- -+_
		D -+2--+-- +ph-=O
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page99
		at
		a2w
		at
		at
		Mn = -D ['\12W - (1 - v) (~ ~: + ~:~)]
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page100
		3.7 ADDITIONAL CONTRIBUTIONS
		REFERENCES
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page101
		--
		ProoJems ,81
		PROBLEMS
		x .1
		\
		"
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page102
		1
		) (
		;
		~
		rx~
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page103
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page104
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		4
		Derivation of Equations:
		Variational Approach
		4.1 INTRODUCTION
		4.2 CALCULUS OF A SINGLE VARIABLE
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page106
		f'
		I
		\ ----
		--
		--
		--
		....
		..
		..
		,
		",' I
		--" I
		--
		dl
		- dl 1x2 (aj aj)
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page107
		Ie(O) = -'I + - - - -- T/dx = 0
		af d (af)
		4.4 VARIATION OPERATOR
		ox =0
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page108
		1X2 lx2
		OJ = of dx = 0
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page109
		4.5 FUNCTIONAL WITH mGHER-ORDER DERIVATIVES
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page110
		d/
		lx2 [af _!..... (~) + ~ (~)] TJdx
		[ af d ( af)] IX2
		l
		!
page111
		I = lx2 f(x, <p(0) , <p(1), <p(2), ... , <p(J») dx
		j = 1,2, ...
		~ ( ) dxn- j a<p(n- j)
		4.6 FUNCTIONAL WITH SEVERAL DEPENDENT VARIABLES
		(4A7)
		1= lx2 f(x, <Pl'~:"" <Pn, (<pdx, (</J2)x, ... , (<Pn)x) dx (4.50)
		4>i(X) as
		dl
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page112
		- - - --- TJI + ... + - - - --- TJ dx - 0
		~ - !!- (~) = 0,
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page113
		4.7 FUNCTIONAL WITH SEVERAL INDEPENDENT VARIABLES
		(j) dj ¢Ji (x)
		¢Ji = d .
		n . dn-j ( af )
		L ( ) dxn-j a4>~II-j) ,
		l= III f(x,y,z,¢J,4>x,4>y,¢Jz)dV
		v
		~(x, y, z) = 4>(x, y, z) + 8T/(X, y, z)
		T/(x, y, z) = 0
		7 = Iv f(x, y, z, ¢J + ST/, ¢Jx + ST/x, 4>y + ST/y, 4>z + ST/z) dV (4.68)
		dl
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page114
		- - -- + - -- + - -- TJdV
		af _ ~ (!L) _ ~ (!L) _ ~ (~) - 0
		4.8 EXTREMIZATION OF A FUNCTIONAL WITH CONSTRAINTS
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page115
		oL 1x2 (OF o4i of o4i ) 1x2 [OF of ]
		_ = -=_ + -=-_x dx = . -=17j + -=-(rlj)x dx,
		os j Xl o</J os j o</Jx os j Xl o</J o</Jx
		j=1,2
		(17j)x = dr/j(x)
		of I = 1x2 [OF 17j + of (17j)x] dx = 0, j = 1,2 (4.83)
		08j 81=82=0 Xl o</J o</Jx
		1:2 [~: - :x (:~)]17jdX=O,
		of _ ~ (OF) = 0
		o</J dx o</Jx·
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page116
		o
		~
		(X.I'YI) ::::::..--1 -r - -1"
		image1
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		image3
page117
		(dY)Z
		:x [(pgOY +).) C/i y; -)1 + y;)] =0
		(pgoY H) C/i y; -)1+ y;) = c,
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		image5
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page118
		4.9 BOUNDARY CONDITIONS
		lX2 [af _ ~ (!L) + ~ (~)] 1]dx + ~1] IX2
		+ [!L _ ~ (~)] 1JIX2 - 0
		[!L _ ~ (~)] 171x2 - 0
		,
page119
		(4.91)
		(4.92)
		..!.L IX2 - 0
		a¢XX Xl -
		.. af d ( af )
		o
		image1
		image2
		image3
page120
		2 10 dx2
		,,, E1" 2
		2
		image1
		image2
		table1
page121
		Elu = 2" 12 - '"3 + 2 + C4
		image1
		image2
		image3
		image4
page122
		4.10 VARIATIONAL METHODS IN SOLID MECHANICS
		a = [D]'£
		rr = ~ jjj,£T[D]'£dV
		image1
		image2
		image3
page123
		.. 'fff -T ff-T
		v v
		image1
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page124
		Jf = ! JJJ aT ([C]a + 2eo) dV
		v
		~ ~
		v ~
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		image5
page125
		v ~
		= III (uTi - !uT[C]u - 4/ u) dV - JJ uT4>dSZ - JJ(u - fr)Tq,dSI
		,,*'
		image1
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page126
		T = -'m-.-
		112 [d2r ... ]
		image1
		image2
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		image5
page127
		f = -VU
		_ au au au
		ax ay az
		L=T-U
		. 112
		aL d (aL)_o
		image1
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		image5
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page128
		1/2 1/2
		.. ""~~~""",,.-_~~.,
		1/2 3 [aT d (aT) ]
		1/2 1/2
		image1
		image2
page129
		--
		L = ~ III (p~T~ - gT[Dli + 2uT4;)dV + II ilT¢dS2
		image1
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		image5
		image6
page130
		, .... --------------------------------------------
		s s v
		eij = r ::: ::~ :::] == r ::: ::: :.::] (4.150)
		image1
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page131
		image1
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		image4
		image5
		image6
		image7
		image8
page132
		v v
		v
		= 2: P7itat V
		rr = I I I rro dV (4.162)
		v
		v - .. 52
		L=rr-T+W
		image1
page133
		4.11 APPLICATIONS OF HAMILTON'S PRINCIPLE
		4.11.1 Equation of Motion for Torsional Vibration of a Shaft (Free Vibration)
		1 {L (ao(x t»)2
		'I 10 2 10 at 2 10 ax
		,_,_,l,-!,-.-,_._. J~,_ •. x
		·1
		image1
		image2
page134
		112 [IlL (ae)2] 1'2 [ ae IL lL 0 (af)) ]
		o - GJ - dx dt = GJ-oe - - GJ- OfJdx dt
		II 2 0 ax ,) ax 0 0 ox· ax
		1/2jlL[O ( ae) a2e] ae 1(2)
		- GJ- - 10-2 oedx - GJ-oe dt = 0
		11 0 ax ax at ax 11
		a ( oe) a2e
		ox· GJ ax - 10 at2 = 0,
		( G J :~) of) = 0
		au a2w
		ex = - = -z-
		ax ax2
		. a2w
		image1
page135
		~
		o
		v A
		2 10 at
		image1
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page136
		image1
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page137
		•...
		a ( a2w)
		ax ax
		aw .
		a2w
		a ( a2w)
		- EI-2 =0 or ow =0
		ax ax
		E/~:~ = 0 or 0 (~:) = 0 at x = 0 and x = L (4.190)
		4.12 RECENT CONTRmUTIONS
		image1
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page138
		REFERENCES
		PROBLEMS
		image1
page139
		o .---.---.---.---.
		{L (dy)3
		1/ pA (au)2 1/ AE (au)2
		o 2 at 0 2 ax
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		image5
		image6
		image7
page140
		I [1 (d2W)2 [1
		(dW) (dW)
		+M! - -M2-
		yea) = A,
		[b /1 + (dy)2 = I
		W W
		•
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		image7
page141
		Derivation of Equations: Integral
		5.1 INTRODUCTION
		5.2 CLASSIFICATION OF INTEGRAL EQUATIONS
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page142
		image1
page143
		5.3 DERIVATION OF INTEGRAL EQUATIONS
		image1
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page144
		F
		at
		{ x(L - ~)
		L '
		r_J.i)_._J_._I/J. _._ x
		I·
		I
		o
		= P LlJ
		image1
		image2
		image3
		image4
page145
		w21L
		c =-
		w(O, t) = 0
		image1
		image2
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		image5
		image6
page146
		(5.21)
		K(x,O = I~(L - x),
		X(x)+'A (x-~)X(~)d~-'A- (L-OX(~)d~=O
		o L 0
		d2X
		X (0) = 0
		dX = _'A r X d~ + CI
		X(x) = -'Alx X(~) d~ lX dTJ + CIX + C2
		c} = ~ r\L - ~)X(~)d~
		X(X)='A1L K(x,~)X(~)d~
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page147
		•..
		.-,~ ... ~ ... ~~."",,", .. "
		'I I I
		5.3 Derivation oflntegral Equation:; '.:t29
		X(x) + A1x (x - ~)X(~)d~ - Axd = 0
		X(s) + A,"2X(s) - A"2 d = 0
		s s
		_ Ad
		'. X (s) = s2 + A
		d l\L -~)hsinh~d~ = dL
		(E5 .2.7)
		or
		L _ sin.J>::L = L
		.J>::
		sinhL = 0
		n = 1,2, ...
		image1
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page148
		x, t = -m x at2
		5.4 GENERAL FORMULATION OF THE EIGENVALUE PROBLEM
		5.4.1 One-Dimensional Systems
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page149
		-
		-'
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		image6
page150
		5.4.2 General Continuous Systems
		image1
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page151
		5.4.3 Orthogonality of Eigenfunctions
		~
		... ---- ~ .. , .....•..•.....
		¢i(X) = Ai Iv K(x, ~)¢i(~) d V(~)
		5.5 SOLUTION OF INTEGRAL EQUATIONS
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page152
		(C) + ~C2) +x (C) + ~C2) = -x + 1
		image1
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page153
		~
		I
		image1
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page154
		lim W(n-I\x) = CI WI (x)
		lim W(n)(x) = CI WI (x)
		. wt-I)(x)
		n-+oo wt)(x)
		W,(x) = lim wt)(x)
		or
		Iv m(x)Wil)(x) w] (x) dV(x)
		a, =
		Iv m(x)[W, (x)]2 dV(x)
		Iv m(x)[WI (x)f dV(x) = I
		Iv m(x)Wi') (x)W, (x) dV(x)
		= Iv m(x)Wi') (x)W, (x) dV(x) - a] Iv m(x)[W] (x)f dV(x) = 0 (5.56)
		image1
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		image5
page155
		W2(X) = lim win>(x)
		a a
		P-+P--=l
		~ L-~
		image1
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		image4
		image5
page156
		r----- x )r-L-X~
		~;---i-L-; ---- )
		.-.-.-.- .•. -.-.-.-.-+-.-.-.-.
		a=
		PL
		T'
		L -~ '
		~ < x
		image1
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page157
		1 12P L2
		.~
		2 12P
		image1
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page158
		d (aT) aT au
		dt a~k - aTJk + al]k = 0,
		k = 1,2, ... , n
		Lmk;ii; + P Lkk;l]; = 0,
		I
		,
		,
		j
		i
		,
		image1
		image2
page159
		-- L L
		image1
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		image5
		image6
page160
		=
		image1
page161
		X(l) = {1.OOOO}
		image1
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		image3
page162
		image1
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		image5
page163
		w(x) = I>i(X)l7i
		c(X) = w(x) - >.. Iv g(x, ~)m(~)w(~) dV(~)
		[m]ij = 5:.[k]ij
		kki = Iv g(Xk, ;)m(;)ui(;) dV(;)
		image1
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page164
		2
		lb b n
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		image5
page165
		.......- ' '
		image1
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page166
		REFERENCES
page167
		PROBLEMS
		o ~ x ~ 1 (5.1)
		K( !:) _ {x(l - ~),
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		image7
		image8
page168
		image1
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		image5
		table1
page169
		"'
		Solution Procedure: Eigenvalue
		X E V (6.1)
		5.4 6.1 INTRODUCTION
		6.2 GENERAL PROBLEM
		image1
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page170
		(6.2)
		L = ~ (Glp~)
		L=D(~+2~+~)
		image1
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		image3
page171
		Aj[w] = ABj[w],
		Ai[W] = 0,
		ot
		ot
		6.3 SOLUTION OF HOMOGENEOUS £QUATIONS:
		SEPARATION-OF -VARIABLES TECHNIQUE
		XeV
		ow
		Al w(O, t) + BI-(O, t) = 0
		ow
		xeG
		x eG
		image1
		image2
page172
		6.4 STURM-LIOUVILLE PROBLEM
		d [ dW]
		image1
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page173
		I
		-----~~~,
		image1
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page174
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page175
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page176
		c)= c) + C2
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page177
		am = ±m ·21l',
		Wm(X) = Cl cos2m.1l'x + C2 sin2m1l'x,
		m = 0, 1,2, ...
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		image2
page178
		6.4.2 Properties of Eigenvalues and Eigenfunctions
		m - IIwm(x)II'
		m=1,2, ...
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		image5
page179
		m =n
		I
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page180
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page181
		6.5 GENERAL EIGENVALUE PROBLEM
		at
		XEV
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page182
		EA ax2 +m at2 =
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page183
		aZ
		M=m
		1£ UI (x)L[Uz(x)] dx = 1£ Uz(x)L[UI (x)] dx
		{L {L dZ ( rr x )
		10 UI(x)L[Uz(x)] dx = 10 Clx(L - x)EA dxZ Cz sin T dx
		4CICzEAL
		1L 1L rrx dZ
		Uz(X)L[Ul (x)] dx = Cz sin -EA-z (ClxL - CIXZ) dx
		4CICzEAL
		L[Wd = AiM[Wd
		image1
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page184
		image1
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page185
		Cm = IIWm(X)1I2 '
		XeV
		6.6 SOLUTION OF NONHOMOGENEOUS EQUATIONS
		at . 1
		aw - -
		a;(X,O) = g(X)
		XeV
		A;[W(X)] = 0,
		-­
		image1
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page186
		image1
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page187
		1 it
		m = 1,2, ...
		6.7 FORCED RESPONSE OF VISCOUSLY DAMPED SYSTEMS
		image1
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page188
		(6.91)
		ow - -
		-(X, 0) = g(X)
		at
		Ai[W(X)] = 0, i = 1,2, ... , p (6.89)
		- a w(X t) a - -
		at at
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page189
		6.8 RECENT CONTRIBUTIONS
		image1
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page190
		REFERENCES
		image1
page191
		PROBLEMS
		+ (A(1 - x2) - n2]w = 0
		d2w
		Wj(x) = SIn -l-' i = 1,2, ...
		image1
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page192
		7
		Solution Procedure: Integral
		7.1 INTRODUCTION
		174"
		image1
page193
		7.2 FOURIER TRANSFORMS
		7.2.1 Fourier Series
		cos - = --------
		image1
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page194
		(7.6)
		(7.9)
		I JOO . JOO .
		image1
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		image5
		image6
page195
		2100
		2100
page196
		f(t) = - Lbn sinnt
		ao 200
		7.2.4 Finite Sine and Cosine Fourier Transforms
		7.2.3 Fourier Transform of Derivatives of Functions
		image1
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page197
		_ (ya)
		a- 1 2 00
		rr rr rr
		F(n) = f(t) cos - dt
		o a
		o a
		a a
		0, x> a
		image1
page198
		~F (;) ,
		1 foo .
		,j2ii -00
		= ..j2iC 10 ae dx =..j2iC -iw 0
		..j2iC -00 ..j2iC -00
		1 [00 .
		a,j2ii . -00
		image1
		image2
		image3
page199
		7.3 FREE VIBRATION OF A FINITE STRING
		2 a2w(x, t) a2w(x, t)
		c --- = ---'O<x <I
		1
		rr2c2 a2w(p, t) a2w(p, t)
		---=
		[2 ax2 ot2
		rr2c21" a2w 1" a2w
		-2- -a 2 sinnpdp = -2 sinnpdp
		lop 0 at
		1" a2w(p t) (OW ) I" 1"
		--2-' -sinnpdp = -sinnp -nwcosnp _n2 wsinnpdp
		o ap ap 0 0
		1" a2w(p t) 1"
		2' sinnpdp = _n2 wsinnpdp
		o ap 0
		n2rr2c21" a21"
		--2- wsinnpdp = -2 wsinnpdp
		W(n, t) = 1" w(p, t) sinnpdp
		d2W(n, t) rr2c2n2
		image1
page200
		aw .
		I I
		dW .
		. Jr 1/ .. nJr;
		C2 =-Wo(n)
		or
		or
		or
		image1
		image2
page201
		2 00 mret . 21 00 1 .. mret
		w(p, t) = - '" Wo(n)cos -- smnp + -z- '" - Wo(n) sm -- sinnp (7.63)
		rr~ 1 rre~n 1
		2 Loo nrr x nrr et it nrr ~
		I 1 1 0 1
		2 ~ 1 . nrrx . nrret it .. nrr~
		rre n 1 1 0 I
		7.4 FORCED VffiRATION OF A FINITE STRING
		aZw(x, t) 1-( a2w(x, t)
		P axz + x,t)=p atZ
		I( )_j(x,t)
		x,t - ---
		p
		p=-
		1
		rrz aZw(p, t) + I (lP t) = 2- aZw (7.69)
		1Z apz rr ' e2 atZ
		dZW(n, t) rrzeznz Z
		dtZ + 1Z Wen, t) = e F(n, t) (7.70)
		Wen, t) = in w(p, t) sinnpdp
		image1
		image2
		image3
page202
		image1
		image2
page203
		2 rr2c?n2
		7.S FREE VmRATION OF A BEAM
		a4w 1 a2w
		I
		.J
		image1
page204
		c =-
		pA
		I 0 ax2 I
		image1
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		image5
page205
		11 nrr x
		W(n, t) = w(x, t) sin -- dx
		() l
		w(x, t = 0) = wo(x)
		--.dw .
		dt (x, t = 0) = wo(x)
		W(n, t = 0) = Wo(n)
		-(n, t = 0) = Wo(n)
		11 nrrx
		Wo(n) = wo(X) sin - dx
		o I
		. 11 nrrx
		Wo(n) = wo(x) sin -- dx
		o l
		2 00 • nrrx
		w(n, t) = T L W(n, t) sm -l-
		image1
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page206
		If(t)1 ~ Ceat
		I fC+iOO
		7.6 LAPLACE TRANSFORMS
		image1
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page207
		100 100 C
		image1
page208
		image1
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page209
		F(s) = - = -- + -- + -- + ... + --
		1
		image1
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page210
		(s =--=--+---+ ... +- __
		+--+--+ ... +---
		image1
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page211
		7.6.3
		Inverse Transformation
		I
		i
		1
		I
		image1
page212
		1 la+i R
		1 la+ioo 1· .
		7.7 FREE VIBRATION OF A STRING OF FINITE LENGTH
		image1
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page213
		. t
		1
		(7. Ji38)
		I
		i
		1
		I
		1
		i
		i
		:
		image1
page214
		(E7.6.9)
		- - -U = 0
		ax2 at2
		image1
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		image4
page215
		7.8 FREE VmRATION OF A BEAM OF FINITE LENGTH
		c---+· =
		ox4 ot2
		2 EI
		c =-
		pA
		w2 w2 pA
		1
		I
		r
		j
		,
		I
		I
		image1
page216
		pA 1
		or
		7.9 FORCED VIBRATION OF A BEAM OF FINITE LENGTH
		image1
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		image4
		image5
page217
		P - P -
		1- 1-,
		w(I, t) = °
		ax2 (I, t) = 0
		W;(O, s) = 0
		W(I,s)=O
		W;(l, s) = 0
		I
		I
		!
		,
		)
page218
		1:-, 111/
		image1
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		image3
page219
		+ -----------~-------- (E7.7.21)
		I
		]
		I
		j
		j
		I
		I
		image1
page220
		REFERENCES
		image1
		image2
page221
		PROBLEMS
		I
		,
		j
		J
		I
		(a) f(t) = {~: ~: ~ < 2
page222
		au . (
		ot
		2 cPw cPw
		c--=--
		ox2 ot2
		image1
		image2
		image3
page223
		Transverse Vibration of Strings
		8.1 INTRODUCTION
		at
		aw o2w
		I
		I
		I
		j
		I
		I
		l
		I
page224
		I_ds-:
		~
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page225
		(8.9)
		8.2.2 Variational Approach
		[ 2] 1/2 [ 2] 1/2 [ 2]
		2 0 ax 2
		l
		!
		I
		image1
		image2
page226
		~ f [H' p(x) (~~)' dx - H' P(x) e~)' dx - ~klwZ(O.t)
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page227
		8.3 INITIAL AND BOUNDARY CONDITIONS
		aw
		ax
		ax
		j
		,
		I
		I
		I
		I
		j
page228
		t2:0
		ax x=o x=o
		8.4.1 Traveling- Wave Solution
		8.4 FREE vmRATION OF AN INFINITE STRING
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		image3
		image4
		image5
		image6
page229
		I
		I
		o
		a
		·-··-·-·--....x
		t
		I P
		I:
		~._._._._._._._._._.
		o
		aw
		aw
		oL'-'-'_'-'-'-'-'-'-'-'I'~'-'-'~x
		- at2' ax
		o
		I P
		cll-l._._._._._. ._L-;J c:_._._.~ x
		aw
		aw
		aw
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page230
		!\
		/\/\
		image1
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		image7
		image8
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		image10
page231
		---e x= -e -la -e x
		j
		j
		f
		1
		!
		image1
		image2
		image3
page232
		Wo 1.
		Cl = - + --Wo
		Wo 1.
		C2 = - - --Wo
		-J2ii -00
		-J2ii -00
		1 .. 1· .. ,
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		image2
		image3
		image4
		image5
page233
		100 d2 W . 1"'" 100 . 100 .
		-00 dx -':)0 -00 -00
		dW . 1+00 . +00 1+00
		image1
page234
		or
		or
		- sWo+Wo
		W=
		c2p2 + s2
		27r -00
		...:.... 1 1+00 .
		Wo(p) = r;:;-::: wo(x)e'pX dx
		27r -00
		W() 1 100 sWo(p) + Wo(p) -ipx d
		X,s = -- e x
		./iii'. ,-00 c2 p2 + s2
		L 2 2 2 = cos pet
		c p +s
		_] [ 1 ] 1,
		L 2 2 2 = - sm pet
		c p + s pc
		1 100 [- 1 ...:.... ].
		w(x,t)= r::;= Wo(p)cospct+-Wo(p)sinpct e-lpxdp
		wo(x)= r;:;-::: Wo(p)e-lpXdp
		27r -00
		wo(x) = ~ 100 Wo(p)e-ipX dp
		image1
		image2
		image3
		image4
page235
		./Ii J-oo
		I" "
		1" "
		8.5 FREE VmRATION OF A STRING OF FINITE LENGTH
		---W=O
		dx2 e2
		,
		I
page236
		W (0) = 0
		wx . wx
		-+-W=O
		8.5.1 Free Vibration of a String with Both Ends Fixed
		image1
		image2
		image3
		image4
page237
		Wn = --,
		l
		21r 2l
		1:1 = -.- = -
		j
		!
		I
		1
		!
		image1
page238
		·-·-·-·-x
		2 1/ .. mrx
		image1
		image2
		image3
		image4
		image5
		image6
page239
		t
		hl--------- _
		1 l ~
		~ . nrrx ncrrt
		21l . nrrx
		Cn = - wo(x) sm -- dx
		2 [ll/2 2hx nrr x jl 2h nrr x ]
		= - -sin--dx+ ~(l-x)sin-dx
		{ 8h . nrr
		o
		I
		j
		I
		~
		I
		~
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page240
		.•.........
		.•.........
		"
		-------_._.~-----_._-_.-
		~ .. - - ~
		o .... --.---.-.-.~----.-.-.-.~
		o
		o
		o
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		image9
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page241
		a L _
		o
		- -
		4ax
		4a (~-x)
		1 2 '
		1 1
		4 - - 2
		Wo(x) = 0
		1 1
		I
		I
		image1
page242
		._._._._._.~.
		o
		o
		o
		,
		o-'-'-'-'-'~
		n3c I I 4 I I
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page243
		I
		o
		aw
		aw
		J
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page244
		k]
		Yn = -­
		Notes
		sm-=
		image1
page245
		-
		Wn = 2Z
		8.6 FORCED VIBRATION
		j
		image1
page246
		n = 1,3,5, ...
		+-- ~ -:- sm -- Qn(T) sm dT
		= ---- sm ydy
		image1
		image2
		image3
		image4
		image5
page247
		/ /
		crr p n / 0
		o l 0 I
		image1
page248
		1,
		t:::O
		o I I
		image1
page249
		I
		crr P n Z 0 Z Z
		8.7 RECENT CONTRmUTIONS
		image1
		image2
page250
		REFERENCES
		image1
page251
		PROBLEMS
		l
		ow
		j
		image1
page252
		9
		Longitudinal Vibration of Bars
		9.1 INTRODUCTION
		9.2 EQUATION OF MOTION USING SIMPLE THEORY
		23.:f° .. ,.
		image1
page253
		_----Ic
		. t-:- dx -\
		w=o
		image1
		image2
		image3
page254
		1r = ~ t O'xxexxAdx = ~ tEA (au)2 dx
		2 Jo 2 Jo ax
		T = ~ t pA (au)2 dx
		2 Jo at
		(9.7)
		(9.12)
		(9.10)
		a ( ... au) a2u
		ax EA ax + f = pA at2
		EA au au 11 = 0
		ax 0
		au
		O'xx =EA- =0
		u=O
		9.3'" FREE VffiRATION SOLUTION AND NATURAL FREQUENCIES
		~ [EA(X) au(x, t)] = pA(x) a2u(x, t)
		ax ax at2
		image1
page255
		[E
		(9.16)
		u(x, t) = 11 (x - ct) + hex + ct)
		9.3.1 Solution Using Separation of Variables
		Vex, t) = V(x)T(t)
		image1
		image2
		image3
page256
		u=o
		image1
page257
		II
		II
		,..;
		~I-- ..:
		II
		II
		'-'I N
		II
		II
		..-.
		-
		+N
		'-'
		.S
		'"
		u
		o
		11 '"
		'"
		=
		.2
		OJ
		...
		:e
		a
		=
		....l
		.S
		o 0
		.-.. ..-.
		e:~
		:: ::
		II
		'"
		.g
		:e
		=
		U
		~
		239
		image1
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		image4
		image5
		image6
		image7
		image8
		image9
		image10
		image11
		image12
		image13
		image14
		image15
		image16
		image17
		image18
		image19
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		image21
		image22
page258
		24()'
		image1
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		image4
		image5
		image6
		image7
		image8
		image9
		table1
		table2
		table3
		table4
page259
		-
		au
		au .. ' ,au a2u
		AE-(O. t) = klU(O, t) + Cl-(O. t) + m1-a 2 (0, t)
		ax at t
		image1
		image2
page260
		~_~:E~P
		I,.
		EJ
		--~
		ax ~ c c
		at ~ c
		ax
		a2u
		at
		m = pAL
		image1
		image2
		image3
page261
		wi
		{3 _ pAL - !!!..
		U ( . WnX
		n x) = Bnsm-,
		I
		image1
		table1
		table2
page262
		au
		A E w wI
		t=J-
		au
		AE-(I. t) = -Ku(I, t)
		~E ~ . ~
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		image3
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		image5
		image6
page263
		.~\~"""""",,"''''h" .. "P'''"''''''''''''''''f,'I'<;I'''~'''\!.'''''''''''''.'.,,,.,,,.,,~,,,,,,,,,,,,"''''''''Y'>.~',n., .,.,,, .•.. ,, •.• ,"""~,.,,,~~~ .• ,..~r,~."""".,, •. C .•.. ,"'''r:c· '~~-'j
		I
		I
		m = pAL
		a cota = {3
		f3 = K
		~ __ ----------t
		image1
		image2
		image3
		image4
		table1
page264
		n = 1,2, ...
		i
		B =0
		au dU
		au dU
		a2u
		image1
		image2
		image3
		image4
		image5
page265
		o
		82 82
		Mw~ (Wj)2
		Aj = --j- = - 7
		image1
		image2
		image3
		image4
		image5
page266
		U!' = AiUi
		11 U!'Uj dx = Aj 1/ UjUj dx
		tU~U'dx=O
		image1
		image2
		image3
		image4
page267
		I
		9.3.3 Free Vibration Response due to Initial Excitation
		Mi = M 1/ u? dx
		(9.53)
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		image3
		image4
		image5
page268
		10 i=1 10
		10 i=1 10
		n7rX (n7rc n7rC )
		I I I
		image1
		image2
		image3
page269
		u(x, t) = ~ cos -z- Cn COS -z-t + Dn sin -z-t
		ou .
		ot
		o I I - m=n
		2'
		sol
page270
		. au
		4Eol L 1 nrr x nn ex
		, n2 n2 1 1
		~
		I- t -I- f -I
		image1
		image2
		image3
page271
		Fo
		\ Fox
		ex = --
		au .
		at
		. wZ
		Bsm- =0 or
		. wZ 0
		or
		image1
page272
		1 0 2EA 1 1 1/2 EA 1
		EAI n2rr2 0 1 1 1
		EA nrr 1/2 1 1
		---- --sm-- -_
		EAI n2rr2 1/2 1 1 1
		9.4 FORCED VffiRATION
		n2 1 1
		f=.£=1-
		Jo -
		image1
page273
		,.,-."'"","""~",.~"~."."""",""",",,,,,,p "M' '"W"';"'_"""l" •• ".".",~""""""",!~~"'"""",,","""·"~"'~~""""~"'"""~""".,~,, '," ,-,,,.,1"',,,",,,,,"'''' ~" •• -\
		10 -
		~:- - - - ...:'!: ~ ~~ - - - -I
		0F'-'-'~x --Fo .
		------------------------------,
		image1
page274
		/
		image1
		image2
page275
		9.5 RESPONSE OF A BAR SUBJECTED TO LONGITUDINAL
		a2u(x t) a2
		pA at2' - EA ax2 [u(x, t) - Ub(t)] = 0
		v(x, t) = u(x, t) - Ub(t)
		a2U a2v a2 p
		a2v(x, t) EA a2v(~, t) A a2 p(t)
		pA at2 - ax2 = -p ~
		a2v 2 a2v a2 p
		a2TJi 2 a2p t
		at2 +WiTJi = - at2 10 Ui(x)dx,
		i = 1,2, ...
		. 1 11 1t a2 p
		TJi(t) = -- Ui(x)dx 7T(r)sinwi(t - r)dr
		image1
		image2
page276
		00 u.(x) 1/ l' 82p
		9.6 RAYLEIGH THEORY
		9.6.1 Equation of Motion
		U = u(x, t),
		T=2Jo dx Jo pdA at +2Jo dx Jo pdA at + at
		image1
		image2
page277
		o (2 o3u) 0 (OU) o2u
		__ pv / -- - - EA- + pA- = f
		ox P oxot2 ox oxot2
		( OU 2 o3u) II
		ox oxot 0
		OU 2 o3u
		E/ ox +pv /poxot2 =0
		9.6.2 Natural Frequencies and Mode Shapes
		u(x, t) = U(x) coswt
		(_pv2/pw2 + EA) dx2 + pAw2U = 0
		U(x) = Cl COS px + C2 sin px
		(9.91)
		p=
		__ ----------t
		image1
		image2
page278
		pi = nn,
		n = 1,2, ...
		9.7 BISHOP'S THEORY
		av au
		ay ax
		aw au
		ezz = - = -v-,
		az ax
		.(au av) a2u (av aw)
		exy = ay + ax = -vy ax2' eyZ = az + a; = 0,
		ezx = (au + aw) = -vz a2u
		az ax ax2
		image1
		image2
page279
		=
		a2 ( a2u) a ( a3u) a ( au) aZu
		axZ p axZ ax axatZ ax ax atZ
		image1
		image2
		image3
		image4
		image5
		image6
		table1
page280
		d4 U d2 U
		PI = -P2 = Sj =.Ja + b,
		EA - pv2! w2
		a= p
		image1
		image2
		image3
		image4
page281
		I
		,
		szl = nrr,
		nZrrzE (AElZ + vZGl nZrrZ)
		n plz AElz + vZElpn2rr2
		image1
page282
		Mu + Lu = f
		image1
		image2
		image3
		image4
		image5
		image6
page283
		Hi-
		,- ,
		Mj
		{lOOt
		toot
		1 il
		Mj 0
page284
		w2 = -- p
		W· M·w· 0
		M, = l (M[U, (x )])U, (x) dx = l [ (PA - v' pIp a~' ) sin i" x ] sin in dx
		K, = l (L[U, (x)])U, (x) dx = l [ (v'C Ip a:: - E 1 a~' ) sin in ] sin i" x dx
		image1
		image2
		image3
		image4
page285
		fi(t) = 11 I(x, t)Uj(x)dx = 11 FoH(t)o (x -~) Uj(x)dx
		= FoH(t)Uj (x = ~) = FoH(t) sin i~l
		T/j(t) = _1_ t fi(r) sinwj(t - r) dr,
		1 1t 17 • irrl . d
		I1j(t) = -- roH(r) sm - smwj(t - r) r
		Mjwj 0 2
		Fo sin(irrlf2) 1t Fo sin(irrlf2)
		= H(r) sinwj(t - r)dr = 2 (1- COSWjt)
		M~ 0 M~
		u(x, t) = £...J 2 (1 - cos Wjt)
		j=l MjWj
		9.8 RECENT CONTRIBUTIONS
		image1
page286
		REFERENCES
		PROBLEMS
		image1
page287
		~--~~.:,
		I ~
		h-.--'--'--X _~~\_1( -.~
		image1
		image2
page288
		0~=FoSinnt
		) I
		I:-IA~'~ x
		/-
		o
		image1
		image2
		image3
		image4
page289
		1'0
		Torsional Vibration of Shafts
		10.1 INTRODUCTION
		10.2 ELEMENTARY THEORY: EQUATION OF MOTION
		(a Mt ) aZe
		ax at
		~
		I
page290
		0(J
		.-.-.-.-.>-.-1-.-.-.
		x-Idx!-
		tx
		image1
		image2
		image3
		image4
page291
		z
		t
		v(y, z) = -ze,
		w(y, z) = ye
		U(x, t) = 0
		vex, t) = -ze(x, t)
		image1
		image2
page292
		OU OV 00
		oy ox ox
		OU ow 00
		Oz ox ox
		[1 (00 )2 1 (00 )2]
		+ '2IIO at (0, t) + '2ho at(l, t)
		1 t ( 00 ) 2 [ 1 (00 ) 2 1 (00 ) 2]
		= '210 pIp at dx + '2IIO ar(O, t) + '2120 ar(/, t)
		(10.11)
		(10.12)
		,
		image1
		image2
		image3
page293
		81121 ~ t GIp (~e)Z dx + [~klleZ(O, t) + ~kt2ez(l. t)]
		_ ~ t pI (oe)2 dx _ [~IIO (ae (0, t))2 + ~Izo (ao (I, t))2]
		2 Jo p at 2 at 2 at
		(10.15)
		______ ~~I
		image1
		image2
page294
		f'2lGlp ao 0011 + k,IOoOlo
		+ 110 a2~ 00/ +k'200011 + ho a2~ 0(11) dt
		at 0 at
		+1'2 {ll[-a: (GIP:~)+Plp~:~ -mIJOOdx} dt=O (10.18)
		a20 a ( ao)
		10 at2 = ax Glp ax + m,(x, t) (10.19)
		( ao a20)
		-Glp ax + ktlO + ho at2 oe = 0
		( ao a20)
		Glp ax + k,20 + 120 at2 00 = 0 at x = 1 (10.20)
		( ao a20)
		Glpax -ktlO-hoat2 =0
		( ao a20 )
		Glp ax + ktlO + 110 at2 = 0 (10.22)
		a20 a20
		Glp ax2 (x, t) + mt(x, t) = 10 at2 (x, t)
		a20 a20
		ax at-
		10.3 FREE VffiRATION OF UNIFORM SHAFTS
		image1
		image2
		image3
page295
		d2T
		- + (liT(t) = 0
		image1
		image2
		image3
page296
		10.3.2 Natural Frequencies of a Shaft with Both Ends Free
		n = 1,2, ...
		e(x, t) = e(e)T(t) == A cos"7 + B sin"7 (C coswt + D sinwt)
		e (x, t) = sin wx (C' cos wt + D' sin wt)
		. wI
		wI
		de (0) = 0
		image1
		image2
page297
		. wI
		W - nJrc - nrrJ%
		I I p
		0(0, t) = 0
		ao
		o~_rG,p'I'_~,
		image1
		image2
page298
		a tan a =-fJ
		wI
		a=-,
		I
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page299
		image1
		table1
		table2
page300
		image1
		image2
		table1
page301
		CD
		----I
		or
		or
		-GJ-cos- +hwnsm-
		image1
		image2
		image3
		image4
		image5
page302
		anx an anx .
		= '" cos - - - sin - (Cn coswnt + Dn smwnt)
		I fh I - -
		B(x, t) = L Gn(x)(Cn coswnt + Dn sinwnt)
		( a~ _ 1) tan an = an (~+~)
		pJl 10
		fh =- = ­
		pJl 10
		h h
		3 G J wnl 4 . wnl 2 2 W~ . wnl w~ wnl
		-hw -cos-+I]hw sm--G J -sm--GJ-hcos-=O
		ne e n e c2 c c c
		Gn(X) = An (cos anx _ an sin anx)
		Notes
		image1
		image2
		image3
page303
		image1
page304
		1/ ejej dx = 0, i #- j
		11 1/ w21/
		e;ej 0 - e;e'.dx + --.!... ejejdx = 0
		11 1/ w?1/
		elej 0 - e;e'. dx + --.!... ejej dx = 0
		e;'(x) + -tej(x) = 0 (EIO.3.1)
		ej(x) + -fej(x) = 0
		c
		image1
		image2
		image3
		image4
		image5
page305
		10 1 1 10
		de-
		e' = _' (x = 0)
		de·
		e' = _, (x = 1)
		image1
page306
		Glp 1/ e;'ej dx + Glpe;oejo + Glpejoeio
		Glp 1/ ejei dx - Glpe;lejl - Glpejleil
		o
		Glpe;lejl - Glp 1/ e;ej dx + Glpejoeio
		o
		- Glpejoeio - Glp 1/ eje; dx - Glpe;lejl
		o
		Glp 1/ e;'ejdx - Glpe;oejo - Glpejoeio
		o
		image1
		image2
		image3
		image4
page307
		10.4 FREE VmRATION RESPONSE DUE TO INITIAL
		image1
		image2
		image3
page308
		ae .
		image1
		image2
		image3
		image4
		image5
		image6
page309
		10 i:l 10
		10 i:t 10
		I ~
		()
		---_t\
		image1
		image2
page310
		(EI0A.12)
		a20 a20
		10.5 FORCED VIBRATION OF A UNIFORM SHAFT: MODAL
		image1
		image2
		image3
		image4
page311
		image1
		image2
		image3
		image4
		image5
page312
		W = ~.= nn [G
		d2(f(t)
		1 .11 ao
		image1
		image2
		image3
page313
		8n(x)=YTcos-z-' n=I,2 ....
		h ~YT h
		lown Y T Wn
		e(x, t) = f: 2.. ~ cos ~ (~ ~ cosnrr) (t _1.. sinwnt)
		Wn Y T z [Own Y T Wn
		= L --2 COS-Z- t - -smwnt
		n=2.4 ....
		IIow2 1 W
		e(t) = -t + -1- L 2cos-Z- t - -sinwnt
		6/0 10 n=2,4 .... Wn Wn
		L 2cos-- t - -smwnt .
		10.6 TORSIONAL VmRATION OF NONCIRCULAR SHAFTS:
		SAINT· VENANT'S THEORY
		image1
page314
		.-.-(-
		.-.{-
		-.(--
		-·f--
		ao
		u = 1/f(y, z) ax (10.91)
		v = -zO(x, t)
		Ex = au + av = (a1/f _ z) ae
		y ay ax ay ax
		au aw (a1/f ) ao
		Exz = - + - - = - + Y _
		az ax az ax
		av aw
		az ay
		au
		ax
		au
		aw
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page315
		U = G (81/1 _,.) ae
		xv a" a '
		U • = G (81/f + Y) ae
		x. 8z 8x
		11/ If [(81/1 )2 (81/f )2]. (8e)2
		= 2 x=o G ay - z + a; + Y ax d A dx
		c= II G[(~~ -z)' +(~; +Y)']dA
		1T= ~ t C (8e)2 dx
		2 10 8x
		of /,' {lj ~G [(~; -z)' + (~; + Y )']dA} (:;)' dxdt
		_01'2 t ~plp (88)2 dxdt _01'2 t m,8dxdt =0
		1'2 t If 1 [(81/f )2 (81/1 )2] (80)2
		° ,[ 10 2G ay-Z + a;+Y 8x dAdxdt
		['21/ If [(81/1 )2 (81/f )2] 80 a(80)
		= G ---z + -+y ---dAdxdt
		'1 0 8y 8z 8x 8x
		+ t2 tlfG(aO)2[(81/f -z) 8(81/f) +(81/1 +Y) 8(81/1)] dAdxdt
		l't 10 ax 8y 8y 8z 8z
page316
		f [l G (:~)' dX] [- ij :Y (~~ - Z)NdYdZ + f. (~~ - z) 1,8'; d!
		- ij :Z (~~ + Y)8,;dYdZ + f. (~~ + Y) 1,8'; d!] dt = 0
		1121/ If (88)2 [(81/1) 8(01/1) (81/1 ) 8(01/1)]
		image1
		image2
		image3
		image4
		image5
		image6
page317
		10.7 TORSIONAL VmRATION OF NONCIRCULAR SHAFTS,
		INCLUDING AXIAL INERTIA
		T = H' 11 +V2<y,'l (:t~S +,2 (~~)' + Y' e~)'] dAdx
		2 0 ot
		112 112
		image1
		image2
		image3
page318
		\ oe
		v = -ze(x, t)
		ou o2e
		ov
		ow
		Cxy = ov + ou = (01/1 _ z) 00
		Exz =:~ +~; = (:~ + y) :~
		ov ow
		Er = - +- = -0 + e = 0
		~ oz oy
		{I (oe)2
		10.8 TORSIONAL VffiRATION OF NON CIRCULAR SHAFTS:
		TIMOSHENKO-GERE THEORY
		image1
		image2
		image3
page319
		axx = -------- ~ E1/1-
		=it+h
		11/ If ( 82a)2
		h=H~ff lG[(~~ -z) :~r +G[(~~ +Y) :~n dAd.
		image1
		image2
		image3
		table1
page320
		I1/t=!! 1/12dA
		'J, = f L!! E,"NG:~)' dAdxdt+ f [EI.:> G~) I~
		- o~ (EI1/t::~)oelb+ 11 0~2 (EI1/t::~)oedx] dt (10.140)
		02e 02 ( 02e ) 0 ( oe) 02 ( 02e)
		pIp ot2 - ot ox . pI1/t ox ot - ax C ox + ox2 EI1/t ox2 = m,(x, t)
		11 (00)2(021/1 021/1) .[11 (02°)211 (028)2 J
		G - dx -+- + p -- dx- E - dx 1/1=0
		o ox oy2 OZ2 0 ox at 0 ox2
		[coe + PI1/t~ _ ~ (EI1/t 02e)] 8e Ii = 0
		02e (00) Ii
		EI1/tox28 ox 0 =0
		(~~ - z) iy + (~~ + Y) iz = 0
		image1
		image2
		image3
		image4
		image5
page321
		10.9 TORSIONAL RIGIDITY OF NON CIRCULAR SHAFTS
		02t/f 021/1
		oy2 OZ2
		(~~ - z) ly + (~~ + y) lz = 0
		0<1>
		O'xy '=, -0 '
		0<1>
		O'xz = --
		oy
		oO'xx oO'xy oO'xz 0
		-+-+-=
		ox oy oz
		oO'xy oO'yy oO'yZ 0
		-+-+-=
		ox oy oz
		oO'xz oO'YZ oO'zz 0
		G0f) (ot/f -z) = 0<1> (10.151)
		ox oy oz
		G of) (ot/f + y) = _ 0<1> (10.152)
		ox ozoy
		02<1> 02<1>
		V2<1> = - + - = -2Gf3
		oy2 OZ2
		image1
		image2
page322
		t A. /
		1~-
		I
		act> act>
		az oy
		image1
		image2
		image3
		image4
		image5
page323
		= I dz (<I>yl~; -l~ <l>dY) = - I dz 1~2 <l>dy = - II <l>dA
		z
		t
		Ly
		L. . . .L--·
		image1
		image2
		image3
		image4
		image5
page324
		t
		t
		+
		--1-
		~a-+-a~
		If a¢ If a¢ I lp4 a¢
		image1
		image2
		image3
page325
		a2<t> a2<t> ( 1 1 )
		image1
		image2
page326
		10.10 PRANDTL'S MEMBRANE ANALOGY
		w=O
		w 4>
		image1
		image2
		image3
page327
		C = 2Gf3P
		- P
		-a ::: y ::: a, -b ::: z ::: b
		P
		aw aw
		-- ...•
		--x
		. ,
		image1
		image2
		image3
		image4
page328
		'-.1
		image1
		image2
		image3
		image4
		image5
		image6
page329
		4.808T
		13T
		image1
		image2
		image3
		image4
		image5
		image6
		table1
		table2
page330
		A· - --------
		image1
page331
		image1
		image2
page332
		REFERENCES
		image1
		image2
		image3
		table1
page333
		PROBLEMS
		image1
		image2
page334
		o
		,
		"""'"""'··~'~~~-"""·'~~-"'.ul.""~"'-'~ori"""_<t.~ .••• ~~~",,,,,~ •.... _". ''' •.• ,,_.
		image1
page335
		Ii
		Transverse Vibration of Beams
		11.1 INTRODUCTION
		11.2 EQUATION OF MOTION: EULER-BERNOULLI THEORY
page336
		.-----
		,
		''''
		,
		z
		ax
		image1
		image2
		image3
		image4
		image5
page337
		---
		-t-_~~I, {---_J_--~,
		au a2w
		a2w
		image1
		image2
page338
		1 t!!(aw)2 [1 (aw )2 1 (aw )2]
		T = 210 P ar dA dx + 2m) ar(O, t) + 2m2 ar(l, t)
		III (aw)2 [1 (aw )2 1 (aw )2]
		= - pA - dx + -m) -(0, t) + -m2 -(1, t)
		2 0 at 2 at 2 at
		W = il fwdx
		or 81'21~ t E/ (a2w)2 dx
		+ Uk, w'(O, t l+ ~k"C: (0, 1))' + ~ k, w' (/, t l+ ~k" (:: (/, 1) )']
		- H' pA (~~)' dx -[ ~m, (~~ (0,1))'
		+ ~m, (~~ (I,1))} l' fWdX} dt = 0 (11.7)
		1'2 III (a2w)2 1/2 [ a2w (aw)11 a ( a2w) II
		8 - E/ -2 dxdt= EI-28 - -_ EI-2 8w
		'1 2 0 ax /1 ax ax 0 ax ax 0
		+ t a22(E/ a2~) 8WdX] dt
		10 ax ax
		8 -k) w-(O, t) + -ktl -a (0, t) + -k2w (1, t) + -k'2 -(1, r) dt
		112 [ aw(O, t) (aw(o, t»)
		= k] w(O, t)8w(O, t) + ktl 8
		II ax ax
		aw(l, t) (aw(l, t»)]
		+ k2w(l, t)8w(l, t) + k'2 8 dt
		ax ax
		image1
		image2
		image3
page339
		11 ( aw 1(2) l' (1" a2w )
		image1
		image2
page340
		[ a2w awl
		EI-2 +kr2- = 0
		ax ax
		(_Ela2~ +ktlaw)o(aw)j =0 (11.14)
		ax ax ax x=o
		(E 1 a2~ + kr2 aw) 0 (aw) I = 0 (11.15)
		ax ax ax x=l
		[!.- (Ela2~) +kjw+mj a2~]owl =0 (11.16)
		ax ax at x=o
		[- aa (E 1 a2~) + k2w + m2 a2~] owl = 0 (11.17)
		x ax at x=l
		ax ax ax ax
		11:3 FREE VffiRATION EQUATIONS
		image1
		image2
		image3
page341
		....;
		.g
		323
		image1
		image2
		image3
		image4
		image5
		image6
		image7
		image8
		image9
		image10
		image11
		image12
		table1
		table2
page342
		324
		image1
		image2
		image3
		image4
		image5
		image6
		table1
		table2
page343
		a4w a2w
		11.4 FREE VffiRATION SOLUTION
		--------a-w
		,84 - - - -­
		image1
page344
		w = p2; El = (P/)2; E 1 (11.38)
		11.5 FREQUENCIES AND MODE SHAPES OF UNIFORM BEAMS
		11.5.1 Beam Simply Supported at Both Ends
		W (0) = 0
		image1
		image2
page345
		image1
		image2
page346
		W1(X) nfftn _
		o
		I"
		o /
		image1
		image2
page347
		image1
		image2
page348
		11.5.3 Beam Free at Both Ends
		---............ @ fJ21 = 7.8532
		o
		~ .. ~
		o
		W,«) @ --------~i' ... @ P,I.IO.9956
		o 1
		o 1
		image1
		image2
		table1
page349
		cos fJni - cosh fJni ..
		Wn(x) = (cosfJnx + coshfJnx) - . fJ I 'nhfJ i (smfJnx + slnhfJnX) (11.84)
		w(x, t) = L wn(x, t)
		~ [ cos fJni - cosh fJnI . ]
		= ~ (cosfJnx + coshfJnx) - sinfJni- sinhfJnI (smfJnx + sinhfJnx)
		11.5.4 Beam with One End Fixed and the Other Simply Supported
		image1
		image2
		image3
		image4
page350
		----
		---------1-
		--------l------
		r---
		o /
		o I
		o /
		dW/
		dx x=o
		image1
		image2
		image3
		image4
		image5
page351
		W(X) = CI (cos fJx - cosh fJx) + C2(sin fJx - sinh fJx)
		CI(cosfJ1- coshfJ1) + C2(sinfJ1- sinhfJl) = 0
		( E I ) 1/2
		Wn = (fJn1)2 pAl4 '
		cos fJnl - cosh fJnl
		[ cos fJnl - cosh fJnl ]
		Wn(x) = Cln' (cosfJnx - coshfJnx) - sinfJn1- sinhfJn1 (sinfJnx - sinhfJnx)
		image1
page352
		~
		W (0) = 0
		dx{O) = 0
		EI-(l) = 0 or
		~ ~ ~
		o /
		image1
		image2
		image3
		image4
		image5
		image6
		image7
		image8
		image9
		image10
		image11
page353
		. cos {3nl + cosh {3nl .
		Wn(x) = (cos{3nx - cosh {3nx) - . {3 l 'nh{3 l (sm{3nx - sinh{3nx) (11.110)
		sm n + SI n
		W (0) = 0
		dx (0) = 0
		d2 W d2 W
		E 1 dx2 (l) = 0 or dx2 (l) = 0
		a3w(l. t) a2w(l, t)
		E 1 ax3 = M ot2
		image1
		table1
page354
		image1
		image2
page355
		x=o
		at
		at
		image1
		image2
		image3
		image4
		table1
page356
		ax-
		x=o
		x=o
		a [ a2w(x,t)]
		a [ a2w(x,0]
		image1
		image2
		image3
		image4
		image5
page357
		at x=l
		a2W(x,t) aw(x,t)
		M(x, t) = E/(x) ax2 = -ktl-a-x-
		a2w(x, t) a3w(x, t)
		-Ctl axat - /01 axat2
		a [ a2w(x,t)]
		V(x, t) = ax E/(x) ax2 = 0
		. a2w(x, t) Ow(x, t)
		M(x, t) = E/(x) ox2 = kr2 ox
		o2w(x, t) l o3w(x, t)
		V(x, t) = ox E/(x) ox2 = 0
		11.6 ORTHOGONALITY OF NORMAL MODES
		-2 E/(x) 2 = w2pA(x)W(x)
		dx dx
		- -2 E/(x) 2 = wrpA(x)Wj(x)
		dx dx
		-2 E/(x) j2 = wj2pA(x) Wj (x)
		dx dx
		t Wj(x) d22 [E/(X)d2Wi;X)] dx =wr t pA(x)Wj(x)Wj(x)dx
		10 dx dx 10
		image1
page358
		o 0
		i = 1,2, ...
		image1
		image2
		image3
page359
		'-I
		~ .. _ { 0,
		i#j
		11.7 FREE VmRATION RESPONSE DUE TO INITIAL CONDITIONS
		image1
page360
		~
		at ~
		image1
		image2
page361
		i = 1,2, ...
		Wj (x) = C; sm -Z-, i = , , ...
		f;I; irrx
		c- = f2
		image1
		image2
page362
		11.8 FORCED VmRATION
		image1
		image2
page363
		w. = ;',,' / £1
		image1
		image2
page364
		1/ 1/ i1rX
		1
		~)
		.1 I~'"
		o
		\\!\ \\
		or
		C.= (2
		= /2 f'sin in Foo(x _ ,)dx = /2 Fosin in, (El1.4.6)
		image1
		image2
		image3
page365
		pAL Wi 10
		fl; . i7r~ 1
		__ 1_ -_
		I :
		·1
		image1
		image2
		image3
page366
		. nnx
		1 it Po it
		Fo 1:) 1 ( .•...• Q. )
		image1
		image2
		image3
page367
		Wi = I Jr Y pXi4
		. [ [
		-(x,O) =0
		I
		image1
		image2
page368
		for d - ~x =:: x =:: d + ~x
		w(x,O) = 0
		w(l, t) = 0
		f(x) = { ;~,
		1
		1014 . mr X( . W )
		11:9 RESPONSE OF BEAMS UNDER MOVING LOADS
		image1
		image2
		image3
		image4
		image5
		image6
page369
		,­
		2P nrr d
		I t
		1
		J
		image1
		image2
page370
		f(x) = -1- LS10 -1- sin -1-
		11.10 TRANSVERSE VffiRATION OF BEAMS SUBJECTED TO
		image1
page371
		·_.+.+._._._. __ ._.~x
		L I.
		x --jdxr-
		1._._._.-1._._._.-1._._._._._._ x
		T p r---=:---t---~: ~ "
		_I_._._._._._._._._._._l-.x
		1 (OW)2
		2 ox
		1 r' [OW(X, t)]2
		I
		image1
		image2
		image3
page372
		210 ox 10
		1 r' [o2W(X t)]2
		2 0 ot
		1, [ ow 112 fl2 0 (OW) ]
		f121' o2w
		image1
		image2
page373
		1" 1/2 [ aw (aw) 1/ ]
		o Wdt= -P-o - dx+ fowdx dt
		1/2 [ 11 aw a 1/ ]
		= - P--(ow)dx + fowdx dt
		/1 0 ax ax 0
		1/2 [ aw 1/ 1/ 0 ( aw) 11 ]
		= - P-ow + - P- owdx + fowdx dt
		/1 ax 0 0 ox ax 0
		-1" t [PA a2w + ~ (E1a2w) _ ~ (paw) - f]oWdt
		/1 10 ot2 ox2 ox2 ox ox
		_lhElo2~ o(aw)ll dt+1'2[~(Elo2~)_paow]owll dt=O
		t1 ax ax 0 /1 ox ox X 0
		~ (E102W) __ '°i.(pow) + pA a2w = f(x, t)
		ox2 ox2 ax ox ot2
		02W (ow) 1/
		ox ox 0
		[ 0 (02W) ow] 1/
		- E1- -p- 8w =0
		ax ox2 ox 0
		a (02w) ow
		- EI- -p-
		ax ox2 ax
		o4w a2w a2w
		E I ox4 + pA ot2 - P ox2 = 0
		image1
page374
		-2 (0) =0
		S --s ---=0
		EI EI
		2 2 P (P2 PA(2))/2
		s),s2 = 2EI ± 4E212 + EI
		image1
		image2
page375
		11.11 VmRATION OF A ROTATING BEAM
		I
		______________________ ~-------J
		image1
		image2
		image3
		image4
		image5
page376
		l--j
		image1
		image2
		image3
		image4
		image5
		image6
		image7
		image8
page377
		£1--4 - _pAn2(l2 - x2)--2 + pAn2/x- - pACliw = 0 (11.194)
		x=O
		x =/
		11.12 NATURAL FREQUENCIES OF CONTINUOUS BEAMS ON
		1 I 2
		~x II 4-x
		Wj(O) = 0
		I
		image1
		image2
		image3
		image4
		image5
page378
		----- = ---
		image1
		image2
		image3
		image4
page379
		Ellt dxz =
		dZ Wz (0)
		or dxz = 0
		image1
		image2
page380
		or
		dW1 (II> dWz(lz)
		-
		dx dx
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page381
		fJnI = nJr,
		2 2 [EI
		n = 1,2, ...
		BI = -B2
		WIn (x)
		( . sin fJnI. )
		= C2n sm fJnx - sinh fJnI sinh fJnx
		(. R sinfJnI inhR )
		-C2n sm pnX - sinh fJnI s pnX
		(sin fJn/ )
		image1
		image2
		image3
page382
		I
		I
		I
		I x
		11.13 BEAM ON AN ELASTIC FOUNDATION
		a4w a2w
		image1
		image2
		image3
		image4
		image5
page383
		d4W;(x) (PAW? kf) .
		4 ( k f w?)
		dx4 I I
		i=I.2.3 •...
		image1
		image2
		image3
page384
		d4Wn(x) = (PAW; _ kf) ~ X
		dx4 E 1 E 1 n ( )
		image1
		image2
		image3
		image4
		image5
page385
		11.13.3 Beam on an Elastic Foundation Subjected to a Moving Load
		d4w(z) d2w(z)
		w(z) = eJ.LZ
		image1
		image2
		image3
page386
		- 4E212 El
		J,Ll,Z = ±i/a - J"fi
		-EI-3 (z = 0+) + El- -3 (z = 0-) = Fo
		dz dz
		-iC2/a - J"fi - iC4/a + J"fi = 0
		iCz (/a - J"fiY + iC4 (/a + J"fiY = ~o
		image1
		image2
		table1
page387
		C2 = ------===
		i .4EI,JpJa -,Jp
		i . 4EI ,JpJa +,Jp
		i . 4EI ,JpJa -,Jp i· 4EI,JpJa +JP
		11.14 RAYLEIGH'S THEORY
		au o2w
		..... - - -z--
		, at - at ax
		1t2 1'2 11/ ( o2w )2
		Ia = 0 Ta dt = 0 -2 pI -~- dx dt
		1'21/ o2w (o2W)
		= pI --0 -- dxdt
		~ 0 at ax at ax
		1t21/ a3w (ow)
		Ia = - pI -2-- 0 - dxdt
		'\ 0 at ax ax
		1'2 [ a3w 1/ 1/ a ( a3w) ]
		Ia = -pI -2- ow + - pI -2- owdx dt
		'1 at ax 0 0 ax at ax
		image1
		image2
		image3
page388
		image1
		image2
page389
		a4w a2w a4w
		ax4 at2 ax2at2 ..
		. nrrx
		2 (n1r)4 2 ( n2rr2r2)_
		a -/- - wn 1 + /2 - 0
		n - 1 + (n2rr2j /2)r2'
		v =0,
		w = w(x,t)
		aw
		au aw aw
		Ezx = - + - = -
		az ax ax
		au av
		ax ay
		av aw
		au av
		Exy = ay + ax = 0,
		image1
		image2
page390
		o
		- ---ow
		"- ~-=-"l~: ox
		aw
		ax
		ow aws aWb
		aWb aw aws aw
		f/J = ax = ax - ax = ax - {3
		image1
		image2
page391
		z
		t
		~-._._-._-._.f-9-.o~._.+.~ x
		1----- X -+-j
		image1
		image2
		image3
		image4
		image5
		table1
page392
		1 r' [(a¢)2 (aw )2]
		= "2 Jo £1 ox + kAG ax - ¢ dx
		1 r' [(oW)2 (o¢)2]
		image1
		image2
		table1
page393
		image1
page394
		-=-----+--
		E I a2 f _ ~ a2 f __ 0
		w=o
		image1
		image2
page395
		pA
		I
		r2 =-
		ax
		w (0, t) = 0,
		-2 (I, t) = °
		image1
		image2
		image3
page396
		w(O, t) = 0,
		d2e1> dW Z
		dx2 dx n
		image1
		image2
page397
		kG EEl
		image1
		image2
		image3
page398
		11.16 COUPLED BENDING- TORSIONAL VffiRATION OF BEAMS
		a2b2t
		image1
		table1
page399
		~
		/
		f-b~
		t
		I
		o~·
		T
		--L
		drp d3rp
		image1
		image2
		image3
page400
		image1
		image2
		image3
		image4
		image5
		image6
page401
		j=I,2,3, ...
		image1
		table1
page402
		=0
		r =
		or
		image1
		image2
		image3
		image4
		image5
		image6
		image7
		image8
page403
		11.17 TRANSFORM METHODS: FREE VmRATION OF AN
		2 EI
		c =-
		aw 0" (
		at
		----e x= ---IP---P - lp e
		image1
		image2
page404
		-=- 1 100 .
		c P +s
		-J2ii -00 c2 p4 + s2
		image1
		image2
		image3
		image4
		image5
		image6
page405
		11.18 RECENT CONTRIBUTIONS
		image1
		image2
		image3
page406
		image1
		image2
page407
		'~,,~,References •. , .. J89
		REFERENCES
		image1
		image2
		image3
page408
		PROBLEMS
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page409
		T
		5 .
		5 .
		5 .
		5 .
		image1
		image2
		image3
		image4
page410
		T
		!~ '-. ------
		._._._._._._._._.~._.~
		1-- -- ~I
		image1
		image2
page411
		12
		Vibration of Circular Rings
		12.1 INTRODUCTION
		image1
page412
		\
		\
		o
		-t """"
		_._._~ec~
		P at2
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page413
		ap a2w
		p at2
		aMI + FR = 0
		aM2
		aMr
		-- - M2 + moR = 0
		image1
page414
		,p=ffadA
		M} = ffaXdA
		£ = 2. [-u + aw _ ~~ (w + au)]
		R ae R ae ae
		EA ( aw)
		M - EII~ (w au)
		I - R2 ae + ae
		image1
		image2
		image3
page415
		- = u (12.15)
		ao
		M2 = - Rn - -
		GJ (dV dn)
		image1
page416
		12.3 IN-PLANE FLEXURAL VffiRATIONS OF RINGS
		image1
page417
		..... ---::.­
		" "
		\ ,
		" "
		" "
		" "
		I I
		/ "
		.......... -=. __ n=l
		1 au W
		'1/1=--+-
		image1
		image2
page418
		/
		~ /
		~
		·'.t:.._L._
		1 (au )
		fi = - - + w - R¢J
		R ae
		F = kfiAG
		kAG (au )
		F=- -+w-R¢J
		R ae
		image1
		image2
		image3
page419
		aF a2u
		ap a2w
		aZF + F = pAR (~_ aZw) + R (p _ af)
		aez aeat2 atZ ae
		a6w a4w aZw
		= (PRZ PRZ) ~ (2PRZ _ pAR4 _ PR2) a4w
		E + kG ae4atZ + E E h kG ae2atZ
		(PRZ PAR4) aZw (PZR4) a4w _ (PZR4) a6w
		+ E + Eh atZ + kEG at4 kEG aeZat4
		pR4 aZ ( af) R4 ( af) RZ ( af)
		- kEAG atZ p - ae - Ell P - ae + kAG P - ae
		Ki(-nzSiS, - SiS,) + Kz(n4Sz + n4SzS1 - 2Sznz +nzSzS, + nZ + Sz + 1)
		E
		image1
		image2
		image3
page420
		a2Q 1 (E h a4v C a2v 3 a2v 3)
		ae2 = Eh + C R ae4 - R ae2 7- pAR at2 - R q
		12.4 FLEXURAL VIBRATIONS AT RIGHT ANGLES TO THE
		a6v a4v a2v pAR4 a4v pAR4 a2v
		R4 a2q R4 3 ( 1 1) a2mo
		- E h ae2 + Cq + R C + E h ae2 = 0
		image1
		image2
		image3
		image4
		image5
		image6
		table1
page421
		1 av
		R ae = a + f3 (12.41)
		( 1 av )
		Rae
		E h ( aa)
		M2 = Ii'" n - ae
		M = C (a + an)
		t R ae
		ae atZ
		aMt aZn
		ae atZ
		I
		. ......-J
		image1
page422
		a [ ( 1 av)] a2v
		- kAG ---ex +qR-pAR-=O
		ae R ae· at2
		a [Eh (aex)] (1 av ) c (an) a2ex
		- - n-- -kARG ---ex +- ex+- +phR-=O
		ae R ae R ae R ae at2
		a [C ( an)] Eh (aex) a2n
		- - ex+- -- n-- +moR-pJR-=O
		ae R ae R ae at2
		image1
		image2
		image3
		image4
page423
		ae6 + 2 ae4 + ae2 + E h ae2at2 - ---c- at2 - E h ae2
		R4 (1 1 ) a2mo
		+ Cq + R3 E h + c ae2 = 0
		image1
		image2
		image3
		image4
		image5
page424
		E
		SI =-,
		G
		+ T [SIS,S4 + S, +.' (1 + ~ - 2SIS,S. + S,S,) + .·S, (1 + SIS, + ~:)]
		1 [ 4E12 2C 2 (C 2)
		12.5 TORSIONAL VIBRATIONS
		image1
		image2
		image3
		image4
page425
		I
		_.--n I n--.
		::--,-'lf~_L -r----_C_~~<::-
		.- I
		I
		2 _ Cn2 + E/z
		pJR2
		12.6 EXTENSIONAL VIBRATIONS
		p = EA (8W _ u)
		image1
		table1
page426
		ap a2w
		P = EA (aw _ u) = PARa2u
		R de at2
		ap = EA (d2W _ au) =. ARa2w
		12.7 VIBRATION OF A CURVED BEAM WITH VARIABLE
		image1
		image2
		image3
		image4
page427
		\ \-dB
		__ .J_
		o
		w W
		aF
		ae - P + pFi = 0
		ap
		-+F+PPi =0
		aM
		ae - pF - pMi = 0
		P = EA [dW + w + _1_ (d2U + u)]
		p de Ap2 de2
		EI (d2U )
		M = -pi de2 + U
		</> = ~ (dU - w)
		p de
		image1
		image2
		image3
page428
		Pi = mw2U
		M. = mw2J (dU _ w)
		I Ap de
		image1
		image2
		image3
		image4
		image5
		table1
page429
		d4T} _ d3T} (" CP3Q2) d2Tj ( _ CP4QZ) dT}
		de4 - PI de3 + P~ + [4 I de2 + PI [4 I de
		+ (PS + P6Q~) U + (1- ':"Q2) P3 dW + CP4Q?W
		dez = P7 de2 + [4 Qi - 1 de + psT} + P9 de
		P2=~d2~ _.!(d~)2_2
		~ d()2 ~2 de
		2 d~
		2 d2~ 8 (d~)2 2 2
		ps = f de2 - ~2 de - [ ~ - 1
		ps = ~ d~ (1 + _1_)
		2 d~
		image1
		image2
page430
		4hx
		y=H-h+y
		image1
		image2
		image3
		image4
		image5
		image6
		image7
		image8
		table1
page431
		-1-'- -.-.-.- f
		Y , I I' /~~~ave
		. , ,
		1/ - I I \
		·~/~.I;- - -.- -'-1'-'-'-'- - ~ :i-=:j-'-
		where
		d=
		1 - sinae
		e=--·-
		1 +2a
		For the elliptic-shaped curved beapl shown in Fig. 12.8, the equation of the curved
		where
		h
		(12.116)
		(12.117)
		(12.118)
		f = g!!
		g =!(1 +2a)
		( ~-'-'-'-_'_~~-T.1h
		'-~.':~ ~
		j/' : A B: '\j
		._:. _:.a.
		, "
		---
		----------------------
		(12.119)
		(12.120)
		image1
		image2
page432
		aF
		- - P - pF:- = 0
		ae I
		ap
		p ae - F + Mi = 0
		y = .!. (au _ w)
		p ae
		.•
		table1
page433
		1 (au )
		p aB
		kAG (dU )
		image1
		image2
page434
		E
		12.8 RECENT CONTRIBUTIONS
		image1
		image2
		table1
page435
		i = 1
		i -"
		i = 3
page436
		! ....
		REFERENCES
		image1
page437
		PROBLEMS
		u dv N M Me
		d2u u M
		- + - =--
		image1
page438
		Vibration of Membranes
		13.1 INTRODUCTION
		(p ~:~ dx dY) and (p ~:~ dx dY)
		a2w
		'. ~ _~""',.,--< ,""""";~_ - _-"""""". "",·i"'.~.·,- •. ,",,,,,,,,,,.,,,,",,,,,,,,,,"-,,,,,,~~,, •.• ,,,,,,,.,,,,",,,,",.,,,.,,,;,_,.~."~"-'<WI~~_
		image1
		image2
page439
		ax2 ay2 at2
		r
		1
		o
		t
		image1
		image2
page440
		I
		b. i\
		\
		L_ I
		t;:::O
		image1
page441
		,,= Hf p W:)'+ e;)'] dA
		T=~ff p(oo~r dA
		w = f f fw dA
page442
		image1
page443
		ax
		ow
		p-=O
		oy
		ipow{,wdC=O
		Yc an
		13.3 WAVE SOLUTION
		ax
		OW3 = -ef'
		at
		image1
page444
		.-
		13.4 FREE VIBRATION OF RECTANGULAR MEMBRANES
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page445
		f3 =-r-a
		-+-=--
		image1
		image2
		image3
page446
		13.4.1 Membrane with Clamped Boundaries
		sinaa = 0
		n=1,2, ...
		or
		[ m 2 (n)2]1/2
		m=1,2, ... , n=1,2, ...
		image1
		image2
		image3
		image4
		table1
page447
		image1
		image2
		image3
page448
		Wo(x, y) = V08 (x - ~) 8 (y - ~)
		Bmn = _4_ r {b Voo (x _~) o(y _~) sin mrrx sin mfY dxdy
		4Vo "" 1 . mrrx . nrry . mrr . nrr .
		image1
page449
		image1
		image2
		image3
		image4
page450
		+
		+
		+
		-------+-------
		:
		+
		L------------'·_.x
		+
		+
		+
		+ +
		r-----r----,
		+ +
		L- --'. __ X L..- -'- --' __ X
		image1
		image2
		image3
		image4
		image5
page451
		. rrx . 2rry ( ,J"5rrct . ,J"5rrct)
		. 2rrx . rry ( ,J"5rrct ,J"5rrct)
		image1
		image2
		image3
page452
		00 W 00
		-=7r-- or x+y=a
		image1
		table1
page453
		a b a a b b
		a a b b
		2
		o
		"2
		..•
page454
		a a 2
		:l<W--"'~~ __ ""' __ """'''''''''_"''''"''''''";''''''~''''.'''.'''.''"''''''''''''''''~'''"''''''''''''''"'_i,' __ ... ., •..•.• ~ .• """'"'_",,,"""_~~"'"" .. , ...
page455
		ty
		'-
		3" T
		T -----------
		"3 ----------
		L- ---..l .. _
		o
		o
		'­
		F
		( I
		, "
		'--'"
		1.- --1."_
		-/5rr c
		w=-­
		~
		image1
		image2
		table1
		table2
page456
		13.5 FORCED VIBRATION OF RECTANGULAR MEMBRANES
		a b
		o 0 a b
		2
		Cmn = --
		m, n = 1,2, ...
		image1
		image2
		image3
		image4
		image5
		image6
page457
		image1
		image2
page458
		o
		1 it
		image1
		image2
		image3
		image4
		image5
		image6
		image7
		image8
page459
		=- -------+---+---
		w x, y, t = -- L...., L...., 2 2 2
		L...., L...., 2 2 2
		image1
		image2
page460
		o 0 a b
		o 0 a b
		•
		o p a b
		o 0 a b
		image1
		image2
page461
		,.
		o 0 a b
		x'=O y'=o a b
		x'=O y'=o a
		=-",,-----sm rrc -+- t
		image1
		image2
page462
		ot
		c + +_ =
		13.6 FREE VIBRATION OF CIRCULAR MEMBRANES
		V2 = _ + _
		image1
		image2
		image3
		image4
		image5
		image6
page463
		--+---+ w -- R(r)=
		image1
		image2
		image3
		image4
page464
		image1
		image2
		image3
		image4
		image5
page465
page466
		~~~~
		:0 @I'~J
		'-"
		1-20--1 1-20--1 I- 20--1 I- 2o-J
		,-10 '-to ®
		1-20--1 I- 2a--l I- 20--1 I- 20--1
		13.7 FORCED VIBRATION OF CIRCULAR MEMBRANES
		image1
		image2
		image3
		image4
		image5
page467
		W~~ ],J2 cos me }
		~-
		image1
page468
		1 1t
		image1
		image2
page469
		=
		image1
		image2
		image3
page470
		13.8 MEMBRANES WITH IRREGULAR SHAPES
		image1
page471
		o
		13.9 PARTIAL CIRCULAR MEMBRANES
		t3
		13.10 RECENT CONTRIBUTIONS
		image1
page472
		REFERENCES
		image1
page473
		PROBLEMS
		------------------------""""'-------_'!"'!"' ..... -------.,,- ... -.'~-.,,--~- ..
		image1
		image2
		image3
page474
		aw
		aw
		at' (x, y, 0) = 0
		aw (x, y, 0) = Wo
		image1
page475
		Transverse Vibration of Plates
		14.1 INTRODUCTION
		EQUATION OF MOTION: CLASSICAL PLATE THEORY
		457
		image1
page476
		D = ---­
		- + - + f(x,y,t) = ph- (14.7)
		au a2w
		image1
		image2
page477
		av a2W
		oz ay
		+ 2v-. -2 -2 +2(1- v) -- dA z2dz
		-- Z2 d z = -----:::-
		image1
		image2
page478
		ph if (aw)2
		1'2 112
		h = D t {ij V'wow dx d y + [[ V'w a~:) - ow a<::w) ] dC } dt (14.27)
		image1
		image2
page479
		hi = -----.- -----.
		hz=---------
		11 ax ay
		image1
page480
		o(8w) = o(8w) on + o(8w) os = o(8w) cosO _ o(8w) sinO (14.37)
		ox on ox os ox on os
		o(8w) = o(8w) on + o(8w) os = o(8w) sinO + o(8w) cosO (14.38)
		oy on oy os oy on os
		o2w [O(8W) O(8W).] o2w [O(8W) o(8w) ]
		oy2 on os ox oy on os
		o2w [O(8W). o(8w) ] o2w [O(8W) O(8W)]
		ox2 on os ox oy on os
		'1 J c on r ox ox oy
		Jc os ox oy ox oy
		1 o8w 1 og
		-~-g(x,y) dC = g(x,y)8wlc - 8w- dC
		c uS c os
		image1
		image2
		image3
page481
		1 aow 1 ag
		-g(x,y)dC = - ow-dC
		c as c os
		112 {1 a(ow)
		12 = -D(1- v) --
		(o2W a2w a2w )
		ay ax ax ay
		1 a [(a2w a2w) a2w ] }
		c as ox ay ax oy
		If 112 aw a(ow)
		h = -ph ----dxdydt
		11 at at
		If [ aw 112 112 a (aw) ]
		= -ph -ow - - - owdt dxdy
		at 1 11 at at
		14 = -0 i12 II fwdxdydt = _£12 II f owdxdydt
		------------------------_!!!!!!!!!!!!!!_-----_!!!!_-~.~~-~~ ~"._!!!!,,~!!!!!.-!!!!!!!!!!!!!!!!!!I
		image1
		image2
		image3
		image4
page482
		c ax x ay ay
		aaw dC
		_ D 1 {a\72w .;.. (1 _ v)~ [(a2w _ a2w) cos() sin()
		c an as ax2 ay2
		ax ay
		D\74w + phw - f = 0 in A (14.51)
		oX ox oy oy
		on
		on as ox2 oy2 ox oy
		w=o,
		w=o,
page483
		·-·--.x
		o~._._~x
		I
		o
		( a2w iJ2w)j
		14.3 BOUNDARY CONDITIONS
		wlx=a = 0
		awl
		ax x=a -
		image1
page484
		(()2w a2w)\ - 0
		V =_D[a3w +(2-v) a3w 1\
		(()2w iPW)\ - 0
		wlx=a = 0
		Mx=_D(02~+vo2~)1 =0
		ox- oy x=a
		0r'-'-'-'-'-'-'-'-'-'-'-' x
		o
		I
		o
		o
		I
		image1
		image2
		image3
		image4
		image5
		image6
		image7
		image8
		image9
		image10
		image11
page485
		o
		(iPW azW)1
		k awl
		z
		-'-'-
		z
		-'
		-'
		f \ I d
		. .--." .
		ay
		image1
		image2
		image3
		image4
		image5
		image6
page486
		awl =0
		a2w\
		aMxy
page487
		(a2w a2w) I
		Mx = -D --2 +v-z =0
		ax ay x=a
		aMx [a3w a3w JI
		ay ax axay x=a
		[a3w a3w J I
		-D -3 + (2 - v)--2 = -k2Wlx=a
		ax '-. ax ay .~=a
		Mxlx=a = -D (aa2~ + v aa2~) I = ktz aaw I (14.70)
		x y x=a X x=a
		Vxlx=a = (Qx + a:;y )Ix=a = -D [~:~ + (2 - v) a:3a:2 Jlx=a = 0 (14.71)
		image1
page488
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page489
		Mn =0
		14.4 FREE VIBRATION OF RECTANGULAR PLATES
		fJ 2 - .E...
		image1
		image2
page490
		image1
		image2
		image3
		image4
		image5
		image6
page491
		14.4.1 Solution for a Simply Supported Plate
		W(a. y) = 0, --2 + v-2- = 0,
		d2 W d2 W
		. mrr x . nrr y B '
		a b
		image1
page492
		image1
		image2
page493
		/
		/
		/
		t
		m~!~~---_---_-=-_---_-+-
		~m~~:1---~---fm:nt-
		{ SinamX}
		or
		m = 1,2, ...
		image1
		image2
		image3
page494
		,My(x,0)=-D--2 +v--2 =0
		image1
		image2
page495
		d2Y(b) = 0
		or
		nrr
		b
		~
		I
		image1
		image2
page496
		m, n = 1, 2, ...
		m,n = 1, 2, ...
		b
		Y (0) = 0
		- (0) = 0
		Y(b) = 0
		-(b) = 0
		mn = am + I-'n'
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page497
		14.5 FORCED VIBRATION OF RECTANGULAR PLATES
page498
		, ,
		uu:.
		.5
		'"
		~
		'"
		.g
		C
		image1
		image2
		image3
		image4
		image5
		image6
		image7
		image8
		image9
		image10
		image11
		image12
		image13
page499
		image1
		image2
		image3
page500
		2 1a1b m~x n~y
		image1
page501
		image1
		image2
		image3
		image4
		image5
page502
		o 10 .
		1 r (b
		B
		,y L...- L...- D(.l-.4 -.1-.4)
		ph L...- L...- \ w2 - n2
		image1
		image2
		image3
		image4
		image5
page503
		aMr + ! aMre + Mr - Me _ Qr = 0
		ar r ao r
		t
		image1
		image2
		image3
page504
		t
		ar r ae r
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page505
		ar x
		ao y sin 0
		a;:- = - r2 = --r-
		ar y .
		- = - = sm 0
		ao x cosO
		------
		- -
		ay r2 r
		aw aw ar aw ao aw 1 aw .
		ax ar ax ao ax ar r ao
		aw = aw ar + aw ao = aw sinO + aw cosO
		ay ar ay ao ay ar ao r
		aZw a (aw) a (aw) 1 a (aw) sinO
		axz = ax ax = ar ax cosO - ~ ao ax
		azw Z aZwsin20 awsinzO awsinZO aZwsinzO
		=-cos 0------+-· ---+---+---
		arz aoar r ar r ao r2 aoz rZ
		aZw = ~ (aw) = ~ (aw) sinO + ~ (aw) cosO
		ayZ ay ay ar ay ao ay r
		azw Z azw sin 20 aw cosz 0 aw sin 2B azw cosz 0
		=-sin 0+-----+- ------+---
		arz arao r ar r ao r2 aoz rZ
		a2w a (aw) a (aw) 1 a (aw) .
		ax ay = ax ay = ar ay cos 0 - ~ ao ay S10 0
		azw sinZO a2w cos20 aw cosZO aw sinZO azw sinZO
		= --- + ---- - - -- - --- - - --
		arz 2 arao r ao r2 ar Zr aoz Zr2
		image1
page506
		0(1 ow)
		= -D - - + - - + -- = -D -(V w)
		= -D - - . - + - - + - - = -D - -(V w)
		DV4w + ph ot2 = f
		14.6.3 Moment and Force Resultants
		image1
		image2
		image3
page507
		I aMrl} [ a 2 I - v a (1 a2w law) ]
		Vr = Qr + -;:--ae- = -D ar (Y' W) + -r- ae -;: ar J8 - r2 aii (14.189)
		VI} = QB + aMrl} = -D [~~(Y'2W) + (1 _ v)~ (~ a2w _ 2. aw) ]
		ar r ae ar r ar ae r2 ae
		14.6.4 Boundary Conditions
		w=o
		ar
		w=o
		Mr = -D [a2w + v (~aw + 2. a2w) ] - 0
		ar2 r ar r2 ae2 -
		I aMrB
		Vr = Qr + - -- = 0
		[a 2 1 - v a (I a2w 1 aw) ]
		-D -(Y' w)+--- ------ =0
		ar r ae r ar ae r2 af:)
		----------------------------------------------_ ...• ~
		image1
		image2
		image3
		image4
page508
		I
		14.7 FREE VIBRATION OF CIRCULAR PLATES
		at2
		D
		image1
		image2
		image3
		image4
page509
		~ [d2R(r) + ~ dR(r) ± 'A.2] = __ I_d2e = a2
		R(r) dr2 r dr e(O) d02
		d2 Rid R ( a2)
		_+ __ + ±'A.2 __ R=O
		dr2 r dr r2
		e(O) = A cosaO + B sinaO
		-+~-+ 'A. -- R=O
		d2 Rid R (2 a2) R -_ 0
		i
		image1
		image2
		image3
		image4
page510
		ar
		m = 0, 1, 2, ...
		d m
		( D )1/2
		image1
page511
		= ..:-.:.....:-::.:...:..--:.....~_~~-.:.::...:.-:...-~~ (14.233)
		image1
		image2
page512
		I
		/
		"
		~m.o".,
		/
		I
		~
		/
		image1
		image2
		image3
		image4
		image5
		image6
page513
		14.8 FORCED VIBRATION OF CIRCULAR PLATES
		r r or P
		D
		or2 r or
		at r=a
		image1
		image2
		image3
		table1
page514
		A =2=--
		[ d W ]a La
		image1
		image2
page515
		fa (d2 1 d )2
		14.8.2 General Forcing Function
		10 .
		d2W(1') _ 1 -
		----------------~~~~~j
		image1
		image2
		image3
page516
		ow
		at
		I YPh I
		10 b 1° 4bJ] ().ia)
		o a 0 a\
		w r, t = - ~ --- --- (14.264)
		image1
page517
		14.9 EFFECTS OF ROTARY INERTIA AND SHEAR DEFORMATION
		t
		I
		--.--.-.-.
		image1
		image2
		image3
		image4
page518
page519
		lh/2 lh/2 1"/2 ( aw)
		Qx = axzdz = Gcxzdz = G +CPx + - dz
		-h/2 -h/2 -"/2 ax
		= k2Gh (+CPX + ~:)
		lh/2 lh/2 1"/2 ( aw)
		Qy = ayz dz = GcyZ dz = G +CPy + -. - dz
		-h/2 -"/2 -h/2 oy
		= k2Gh (+cPy + ~~)
		lh/2 1h/2 E
		Mx = axxZ dz = -1--2 (cxx + Vcyy)Z dz
		-h/2 -h/2 - v
		= _E_lh/2 Z2 (aCPX + v acpy) dz = +D (aCPX + v acpy)
		1 - v2 -h/2 ax oy ax ay
		lh/2 1h/2 E
		My = ayyzdz = --2 (cyy + vcxx)zdz
		-h/2 -h/2 1 - v
		= _E_lh/2 Z2 (Orfjy + v OCPx) dz = +D (acpy + v OCPx)
		1 - v2 -h/2 ay '. ox . oy ax
		lh/21h/2 1h/2 2 (acpx acpy)
		Mxy = axyzdz = GSxyzdz = +G z -a- + -a- dz
		-h/2 -h/2 -h/2 Y x
		_ D(1 - v) (acpx acpy) _
		- 2 ay + ax - Myx
		image1
page520
		aMx
		ax I
		\ aMyx d
		or
		o2cp ph3 ,Pcp
		-Qy + ay + --a;- = 12 at2
		image1
		image2
page521
		dx
		-Qx + ~ + a;- = 12at2
		at
		image1
		image2
page522
		,
		ph3 02<1>
		( D ph2 02)
		ot
		image1
		image2
page523
		Cyy
		~ = [B]8
		[B] =
		o
		7r = ~ II I 8T[B]8dV
		v
		7r = - --2cXX + --2Cyy + Gcxy + k GcyZ + k Gczx dV
		2 I-v I-v
		v
		7r - ~ I dzlf ~Z2 (o<Px) + ~Z2 (o<PY) +2v_E_o<px O¢y
		- 2 1 - v ox 1 - v2 oy 1 - v2 ox oy
		+Gz2 (o¢x + O¢y)2 + k2G (¢y + ow)2 + k2G (¢x + OW)2] dA (14.300)
		oy ox oy ox
		image1
page524
		1 If I [(O¢x ot/Jy)2 (o¢x ot/Jy 1 (ot/Jx ot/Jy)2)]
		2 ox oy ox oy 4 oy ox
		w = f f wf dA
		D(1 - v) (ot/Jx o¢y) [O(O¢x) O(0t/Jy)]_k2G [( ow) (0 O(OW))
		2 ~ + ~ 0 + ~ h ¢x + ~ ¢x + ~
		( aw) (0 O(OW))] hOW o(ow) ph3 [ot/Jx o(o¢x)
		+ ¢y + oy ¢y + ay + p at at + 12 7it-a-t-
		+ O¢y O(O¢Y)] + fowl dAdt = 0 (14.306)
		ot at
		image1
		image2
		image3
page525
		image1
		image2
		image3
page526
		image1
		image2
page527
		image1
		image2
		image3
page528
		or
		c
		c
		image1
		image2
		image3
		table1
page529
		Mn =0,
		<Ps = 0,
		<Ps = 0,
		w=o
		Mn =0,
		Mn =0,
		w=o
		14.9.3 Free Vibration Solution
		<Px(x,y,t) = <t>Ax,y)eiwt
		<Py(x,y,t) = <t>y(x,y)eiwt
		D [(1- v)V2<t>x + (1 + v)~<P] - k2Gh (<t>x + 0'1W) + ph3w2 <t>x = 0 (14.338)
		D [(1 _ v)V2<t>y + (1 + V)~<P] _ k2Gh (<t>y + 0'1W) + ph3w2 <t>y = 0 (14.339)
		2 oy uy 12
		k2Gh(V2W + <P) + phw2W = 0 (14.340)
		- o<t>x o<t>y
		ax oy
		a1fr oH
		ox oy
		<t> - o1fr _ oH (14.343)
		y - oy ox
		image1
		image2
page530
		o [2 (4 1) W] 1 - v 0 2 2
		ox S S 2 oy
		oy S S 2 ox
		I h2
		D
		02 _ 2(Rkt - 1/ S)
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page531
		q,y = (tLl - l)ay + (tLz -l)ay - a; (14.362)
		2
page532
		~I
		2"
		I
		/ ..•
		~ I
		T
		I -
		L._._._._ J.._x
		T
		1
		I""
		image1
		image2
		image3
		image4
		image5
		image6
		image7
		image8
page533
		2 k (1 - v) 2 8k 4
		14.9.5 Circular Plates
		8r r 80 r 12 at2
		a;- + -;: ae + -; r(} - (} - 12 at2
		ar r 80 r at2
		image1
		image2
page534
		r
		z
		where the displacement components have been assumed to be of the fonn
		vCr, e, t) = z¢o(r, e, t)
		w(r, e, t) = w(r, e, t)
		image1
		image2
		image3
		image4
page535
		2 r ae ar
		a2w
		2 ae r ae 12
		image1
		image2
		image3
		image4
page536
		( 2) 1/2
		image1
		image2
page537
		M)"'(r, e) = D [~A}.' I (U; - 1) [J~ (I,r) + ;J~(I;r) - v~' J. (l,r)]I CDS me
		2 { [ v . vm2 ] }
		M:;' (r. e) = D(I - v) {~A}.' [ -7 J~ (I,r)+ ~ J.(I,r)] (UI - 1)
		Q$·'(r, e) = k'Gh { ~[A}.' U, J~ (I;r) + 81"' U; Y~(I,r)]
		image1
		image2
page538
		W = <t>e = Mr = 0
		W = Mr = Mre = 0
		image1
		image2
		image3
page539
		14.10 PLATE ON AN ELASTIC FOUNDATION
		- + 2-- + - = --- W = 0
		image1
		image2
		table1
page540
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page541
		14.11 TRANSVERSE VIBRATION OF PLATES SUBJECTED TO
		'" ( aNx) ,( aNyx) d el + e~
		ax ' . ay 2
		el +e'
		--
		image1
		image2
		image3
page542
		\'+- Y
		rdy
		Since the deflections are assumed to be smail, 01 will be smail. so that
		COSO} ~), cosO; ~ 1, cos -2-1 ~ 1 (14.438)
		In view of Eg. (14.438) and the fact that Nyx = Nxy, Eq. (14.437) can be simplified as
		aNx aNy
		- + -' = 0 (14.439)
		ax ay
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page543
		a82
		= 82 + - dy
		ay ax
		image1
		image2
page544
		aw
		aw
		( aNxy) -I _
		image1
		image2
		image3
		image4
		image5
page545
		111 ~ 111 ~ -, 11 1 ~ 11 1 + - d y ~ - + -- d y
		+ -+-- --+ --+- -=
		image1
page546
		).. =-­
		I (a2W CPW)
		- + - + - N1 - + N2 - =).. =--
		2 D [(mrr)2 (nrr )2]2 I [ (mrr)2 (nrr )2]
		image1
		image2
page547
		( )2 4 [ 2]2 [N ]
		14.12 VmRATION OF PLATES WITH VARIABLE TIDCKNESS
		image1
		image2
		image3
		image4
page548
		---------~---------
		,
		3 ---------~----------f---------
		2 ---------~----------+------
		, ,
		--------~----------+----------~----
		,
		image1
		image2
		image3
page549
		D -+2 +- +2-- --+- +2-- -+-
		+ -+- -+-
		14.12.2 Circular Plates
		or at
		image1
		image2
page550
		( aMr)
		aMr 1
		aMr 1
		Qr = -')- + -(Mr - M(j)
		(a2w \) aw)
		(1 aw 82W)
		image1
		image2
		image3
page551
		Qr = _ [D a3w + (aD + D) a2w + (!: aD _ D) aw] (14.484)
		ar3 ar r ar2 r ar r2 ar
		a (a2w laW) a [1 a (aw)]
		Qr = -D(r)- - + -- == -D(r)- -- r-
		ar ar2 r ar ar r ar ar
		1 a { a [1 a ( aw)]} a2w
		-- rD(r)- -- r- + ph- = fer, t)
		r ar ar r ar ar at2
		a { - a [1 a ( aw)]} a2w
		- rD(r)- -- r- = -phr-
		ar ar . r ar ar at2
		her) = hor
		Eh3
		image1
		image2
		image3
		image4
page552
		d2 ( d2W)
		p 3Eh3
		o
		z z z z
		I
		I
		------:=~lc=:------
		/ I
		rr=RI-1
		image1
		image2
		image3
page553
		=0
		---, (2)1/2
		14.13 RECENT CONTRIBUTIONS
		image1
page554
		image1
page555
		REFERENCES
		image1
page556
		image1
page557
		PROBLEMS
		image1
		image2
		image3
page558
		\
		\0(
		\
		\
		I
		yt----
		I
		I
		I
		I
		I
		I
		t
		o
		~=x-ytan8, 1]=--
		1 (a2 a2 a2 )
		cos2 8 a~2 a~ a1] a 1]2
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page559
		15
		Vibration of Shells
		15.1 INTRODUCTION AND SHELL COORDINATES
		Y = Y(a,{3)
		Z= Z(a,{3)
		-r = X(a, {3)7 + Y(a, {3)] + Z(a, {3)k
		a-r _
		a-r _
		image1
page560
		x
		-'--'- = cos Y
		image1
page561
		;,a . 'l.P = 0
		image1
		image2
page562
		•
		I ar I ..
		ar ar ....
		image1
		image2
		image3
		image4
		image5
page563
		a .-.-.-.L.-.-.- .•. y
		_---L--_
		~ -
		+
		t
		as = x = I r I
		lorl
		or or
		la'll
		image1
		image2
		image3
page564
		·_·_~z
		I
		7t-
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page565
		af) vo
		image1
		image2
page566
		"
		"
		"
		o
		a~ a~
		art - art -, ~
		--------
		-- = --- --- = --- (15.22)
		image1
		image2
		image3
page567
		Iz ani = zA. Iz an I = zB (15.24)
		aa Ra afJ Rfj
		2 __ Z2 A 2 2 Z2 B2 2
		Z dn· dn = -2 (da) + -2-(dfJ) (15.25)
		2z dF . dn = 2z (aF da + aF dfJ) . (an da + an dfJ)
		aa afJ aa afJ
		[ aF an 2 aF an aF an aF an 2]
		= 2z -·-(da) + -. -dadfJ+-· -dadfJ + -. -(d{3)
		aa aa a{3 aa aa a{3 a{3 a{3
		__ aF an 2 aF an 2
		2zdr· dn = 2z - . -(da) + 2z - . -(d{3)
		aa aa a{3 a{3
		or aii d .. ·)2 I ar II aii I (d)2 A2 (d 2
		- . -( a = - - a = - a)
		aa aa '. aa aa Ra
		or an 21 ar II an I 2 B2 2
		- . -(dfJ) = - - (dfJ) = -(d{3)
		ofJ a{3 a{3 a{3 R/3
		(ds')2 = A2 (1 + :a) 2 (da)2 + B2 (1 + :/3)2 (d{3)2 + (dZ)2
		(ds')2 = hll (a, fJ. z) (da)2 + h22(a. fJ. z) (dfJ)2 + h33(a, {3. z) (dZ)2 (15.30)
		hll(a. fJ. z) = A2 (1 + :aY (15.31)
		h22(a, fJ. z) = B2 ( 1 + :/3) 2 (15.32)
		h33(a, fJ. z) = 1 (15.33)
		image1
page568
		oa 0{3 oz
		image1
		image2
page569
		oa 0{3 az
		image1
page570
		0/3 oa
		OZ 0/3
		15.2 STRAIN-DISPLACEMENT RELATIONS
page571
		- I
		eii = 1 + ----+ ... -1 ~ ----.
		image1
		image2
		image3
page572
		. M2 M2
		image1
		image2
page573
		1 I 0 [ 2 ( Z )2] Ii
		a [( )2] - a [( )2]]
		+ aa A (1 + z/ Ra)
		!... [A (1 + ..:...)] = (1 + ..:...) aA
		!...[B (1 + ..:...)] = (1 +..:...) aB
		ow
		oz
		image1
		image2
page574
		z
		-« 1
		z
		-« 1
		El2 = Eap = B (1 + zl Rp) ofi A (1 + zl Ra)
		+ A(1+zIRa) aa B(I+zIRp)
		( z ) a [ v J 1 ow
		E -E -B 1+- - + _
		23 - pz - Rp az B (1 +zIRp) B (1 +zIRp) afi
		( Z ) 0 [ V] 1 ow
		E31 = Eza = A 1 + - - + _
		Ra az B (1 + zl Ra) A (1 + zl Ra) oa
		15.3 LOVE'S APPROXIMATIONS
		image1
		image2
		image3
		image4
page575
		82 = av(a, j3, z)
		alE aw
		833 = - = - = 0
		image1
		image2
		image3
page576
		u 1 ow
		v 1 ow
		£11 = - - (u + z8d + - - + -
		= A !... [u(a, fJ) + z8] ] +!.. ow(a, fJ)
		oz A (1 + z/ Ra) A oa
		1 ow
		+ A oa
		u 1 ow
		1 ow
		v 1 ow
		image1
		image2
page577
		CI2=; a~ (u+AZOI)+~ a: (V+BZO~)
		o 1 au v aA w
		cll = A act + AB afJ + Ra
		olav u aB w
		c22 = B afJ + AB act + RfJ
		o A a (U) B a (V)
		cl2 = BafJ A + A act B
		1 aOI 02 aA
		A act AB afJ
		1 a (h 01 a B
		B a fJ AB act
		B afJ A A act B
		image1
page578
		aw
		ax
		v I aw
		x ax
		olav w
		o av I au
		k 1 aOe I au 1 a2w
		ee = R ea = R2 ao - R2 ao2 (EI5.4.7)
		k aOe I aox 1 av 2 a2w
		xe = a; + R ea = R ax - R axao (E15.4.8)
		au a2w
		Sxx = s~x + zkxx = - - Z -2 (EI5.4.9)
		ax ax
		olav w z au z a2w
		See = eee + Zkee = R ae + R + R2 ae - R2 ae2 (EI5.4.1O)
		e - eO + k = au + 2. au + ~ av _ 2z a2w (EI5.4.H)
		xe - xe z xe ax R ae R ax R ax ae
		aw
		ax
		ee = --­
		1 aw
		image1
page579
		o au
		8 =­
		olav u w
		89 = ---- + - + ---
		o 1 au ov v
		8 =----+- --
		cosao ov 1 oZw 1 ow
		1 ov 2v 1 oZw 2 ow
		=--- - -+~---
		1 ( ow )
		09 = ..!. (v __ 1_ ow)
		image1
page580
		15.4 STRESS-STRAIN RELATIONS
		o 1 au w
		ESe = - + - - _
		1 au 1 a2w
		562
		image1
		image2
		image3
		table1
page581
		1
		. G
		E
		E
		E
		E 0
		15.5 FORCE AND MOMENT RESULTANTS
		image1
		image2
page582
		image1
		image2
		image3
		image4
		image5
		image6
page583
		total force = (TlIdsfJdz = (TuB 1 + - d{3dz
		NlI = (TlI 1 + - dz
		image1
		image2
		image3
page584
		E jh/2
		E [0 0 h/2 (Z) ] Eh 0 0
		C=--2
		I-v
		(I - V) 0
		image1
		image2
page585
		Eh3
		(1 - v)
		Notes
		______ ~~,~"'~_~ I
		image1
page586
		(au v av V)
		Nxx = C(£~x + v£211) = C ax + R ae + R W
		Nxll = Nllx = C -2- £xll = C -2- ax + Rae
		o 0 ( 1 av w au)
		NIIII = C(£IIII + v£xx) = C R ae + R + v ax
		(aex vaell)
		Mxx = D(kxx + vkllll) = D a:; + R &e
		D (a2w v av v a2w)
		= - ax2 + R2 ae - R2 ae2
		( 1 aee aex )
		Mell = D(kell + vkxx) = D -- + v-
		R ae ax
		( 1 av 1 a2w a2w)
		= D R2 ae - R2 ae2 - v ax2
		(1 - v) (1 - v) (aee 1 aex)
		Mxe = Mex = D -- kxll = D -- -- + --
		2 2 ax R ae
		= D (~) (~av _ ~~)
		2 R ax R axae
page587
		= C [_au + v (_1 __ av + ~ + _w_)]
		= C (_l_;_V) (xs~ao :~ + :~ -;)
		= C ( __ 1_ av +' ~ + __ w_ + v au)
		( cosao av 1 a2w 1 aw a2w)
		= D x2 sin2 ao ao - x2 sin2 ao a02 - -; ax -v ax2
		= D (_l_;_V) (-x-ta-~-a-o -:-~ - -x""'2-~-v-a-o--x si~ao ~:~ + x2 s~nao ~;)
		image1
		image2
page588
		A
		I \\
		I 1\
		I \\
		I \ \
		I \ \
		I \ \
		[1 au w (1 av u cot¢ W)]
		(I-V)
		N¢B = N9¢ = C -2- £¢9
		image1
page589
		[ 1 au 1 a2w ( 1 av 1 a2w
		= D R2 a4J - R2 a4J2 + v R2 sin 4J ae - R2 sin2 4J ae2
		+ R2 R2 a4J
		= D [ __ I __ av I a_2_w + _co_t_4Ju
		R2 sin4J ae R2 sin2 4J ae2 R2
		_ cottj) aw v (_I au __ I a2w)]
		R2 a4J + R2 a4J R2 a4J2
		= D -2- R2 sin4J ae - R2 sin4J a4J ae
		+ _I __ a_v _ 1 .a2 w + _c_os_4J __ a_w __ co_t_4J v + _c_o_t4J __ a_w)
		R2 a4J R2 sin 4J a4J ae R2 sin2 4J ae R2 R2 sin 4J ae
		15.6 STRAIN ENERGY, KINETIC ENERGY, AND WORK DONE
		15.6.1 Strain Energy
		rr = ~ f f f (0'11£11 + 0'22£22 + 0'33£33 + 0'\2£12 + 0'23£23 + 0'31£31) dV (15.148)
		v
		image1
		image2
page590
		( U lOW)}
		image1
		image2
		image3
page591
		-I
		V aw
		u aw)
		Ra oa
		..,
page592
		j
		Wd == L h (fau + !fJv + !zw) ds2 dsg
		= L h (fau + !fJv + !zw)AB da df3
		image1
		image2
		image3
page593
		15.7 EQUATIONS OF MOTION FROM HAMILTON'S PRINCIPLE
		1'2 1'2
		1" ,) alp
		1'211 11 1'2 au a(8u)
		1" 1,) alp
		image1
		image2
page594
		1'2 1'211 [8(8U) 8(881)
		8ndt = NllB-- + MllB--
		8A 8A AB
		+ Nll-8v + Mll-Mh + Nll-8w
		8f3 8f3 Rex
		8(8v) 8 (88z) 8B
		+ NzzA-- + MzzA-- + Nzz-8u
		8f3 8f3 8a
		8B AB
		+ Mzz-881 + Nzz-8w
		8(ou) 8A 8(Mh)
		+ NlzA-- - N1z-8u + M1zA--
		8f3 8f3 8f3
		- Mlz-cS81 + NIZB-- - Nlz-cSv
		.. 8f3 8a 8a
		8(88z) 8B
		+ M1zB-- - Mlz-;-88z + QZ3ABo8z
		8a va
		AB 8(cSw)
		- QZ3 R{3 ov + QZ3AM + QJ3AB081
		- QI3-R oU + Q13B-- da df3 dt
		1'211 8(8u)
		NllB-- dadf3dr
		= 1'2 [-1 r ~(NllB)Ou da df3 + r NIIB8u df3] dt
		1" 1'211 [ 0 0
		- ondt = -- (NllB)cSu - -(MllB)881
		8A 8A AB
		+ NII-cSv + Mil-of), + Nll-8w
		of3 of3 - Rex
		1'2 8T dt = -ph 1'21 r (u 8u + v 8v + w 8w)AB da df3 dt (15.165)
		image1
page595
		image1
		image2
page596
		oa Ra
		image1
		image2
		image3
page597
		___________________________________ .... -----------""'!!,.d
		image1
page598
		- [ ou 1 0 ] - }
		image1
		image2
page599
		f £ (Nil - NII)8u + [(N12 + ~:2) - (N12 + ~:2)] 8,
		+ Q13 + B ----a{3 - QI3 + B ----a{3 . 8w
		Rp
		Ra
		+ (M u - Mu) 8ellB dfJ dt = 0
		or u =u }
		image1
		image2
page600
		B=R
		f3 = 0,
		A=l,
		v =0,
		v =0,
		u =0,
		15.8 CIRCULAR CYLINDRICAL SHELLS
		image1
		image2
		image3
		image4
		table1
		table2
page601
page602
		a2u 1 :... va2u v aw 1 + v a2v (1 - v2)p ,j2u
		ax2 + 2R2 ae2 + R a; + 2R ax ae = E at2
		1 - v a2v 1 a2v 1 aw 1 + v a2u (1 - v2)p a2v
		-2- ax2 + R2 ae2 + R2 7iB + 2R ax ae = E at2
		(V au 1 av w )
		- R ax + R2 ae + R2
		_ h2 (a4w 2- a4w 2- a4w) _ (1- v2)p a2w
		12 ax4 + R2 ax2 ae2 + R4 ae4 - E at2
		v(O, e, t) = 0
		(au v av v)
		Nxx(O,e,t)=c ax+Rae+ RW (O,e,t)=o
		( a2w v av v a2w)
		Mxx(O, e, t) = D - ax2 + R2 ae - R2 ae2 (0, e, t) = 0
		v(l,e,t)=0
		w(l,e,t)=o
		(au v av v)
		Nxx(l,e,t) = C - + -- + -w (l,e,t) = 0
		ax R ae R
		(a2w v av v a2w)
		Mxx(l, e, t) = D - ax2 + R2 ae - R2 ae2 (l, e, t) = 0
		image1
		image2
page603
		, /
		, /
		'--'"
		, /
		~--
		~~ mrrx
		~ ~ . mrr x. II
		~ ~ mrrx
		w(x, 0) = L.-, L.-, Cmn sin -/- cosnO coswt
		h2
		image1
		image2
		image3
		image4
		image5
page604
		l+v
		VA
		VA
		image1
page605
		c(J2U 1 - v a2u ~ aw 1 + v J2v ) _ hu
		. ax2 + 2R2 ae2 + R ax + 2R ax ae - P
		-2- ax2 + R2 ae2 + R2 ao + '2R ax ao
		D (1 - v a2v 1 a2v 1 a3w a3w)
		+ R2 -2- ax2 + R2 ao2 - R2 ao3 - ax2 ao = phv
		image1
		image2
		image3
		image4
page606
		(au + ~ av + ~w) (0 e) = 0
		ax R ae R '
		1 a4w 1 a3v)
		image1
		image2
page607
		(mrr)2 D n2 2]
		mrrx
		. mrrx
		. mrrx
		+ C2 (c I + v mrr n) + C3 (c ~ mrr) = 0
		Cl (c 1 + v ~ !:)
		[ 1 - v (mrr)2 n2 D 1 - v
		+ C2 -C -2- -t- - C R2 - R2 2
		+ C3 [-c !:. _ E... n3.~ E... (mrr)2 n] = 0
		Cl (C .!:. mrr) + C2 [-C !:. _ E... (~)2 n _ E... ~]
		+ C3 [_.£ _ D (mrr)4 _ E... (2) (mrr)2 n2
		_ E... n4 + PhW2] = 0
		du = C (m1rrf +C 1; v (~f
		v mrr
		1 - v (mrr)2 (n )2 D 1 - v
		(mrr)2 E... (~)2
		image1
		image2
page608
		Cn Dn (mrr)2 Dn (n)2
		C (mrr)4 (mrr)2 (n)2
		cv + PI CV + P2CV + P3 =
		=0
		1
		image1
		image2
page609
		15.9 EQUATIONS OF MOTION OF CONICAL AND SPHERICAL
		--+-+'--------- =
		-- + - ox + . -- - Qo = 0
		a2u
		a2v
		image1
page610
		B2w
		aq, ae
		15.10 EFFECT OF ROTARY INERTIA AND SHEAR DEFORMATION
		image1
page611
		image1
		table1
		table2
page612
		1 - V
		of u, v, and w.
		image1
		image2
		image3
		image4
		image5
		image6
		table1
page613
		( U 1 aw)
		(vI aw)
		112 112
		~ ~ ah
		112 11211 [ a a
		aB AB a
		act Rf3 a{3
		a aB a
		aB AB
		a
		AB a ]
		image1
page614
		}'1 a
		}'1 p
		AB .
		o 8
		image1
		image2
		image3
		image4
page615
		RaNxx + aNex R h·· Rf
		-- -.-= pu- x
		ax ae
		aNxe a Nee ..
		R-- + -- + Qez = Rphu - Rie
		ax ae
		aQxz aQez ..
		R-- + -- - Nee = Rphw - Riz
		ax ae
		RaMxx ... aMex _ RQ = RPh3 .i~
		ax + .. ae xz 12 'I' x
		RaMxe aMee _ RQ = RPh3 .i~
		ax + ae ez 12 'l'e
		[au ( 1 au w)]
		N =C -+v --+-
		x ax R ae R
		( 1 au w au)
		Ne = C - - + - + v-
		R ae R ax
		1 - v ( 1 au au)
		Nxe = Nex = -2-C Rae + ax
		(a1/1"x va1/1"e)
		Mx = D ax + R ae
		(1 a1/1"e a1/1"x)
		Me=D --+v-
		R ae ax
		1 - v (1 a1/1"x a1/1"e)
		Mxe=Mex=-2-D RM+ ax
		Qx = kGh (~: + 1/1"x)
		Qe = kGh [~ ~; - (~ - 1/1"e ) ]
		image1
		table1
page616
		a2w 1 a2w
		v2w = ax2 + R2 ae2
		v(O,e,t)=o
		(a1/1x va1/16)
		Mxx(O,e,t)=D ax + Rae (O,e,t)=o
		[au (laV w)]
		Nxx(O,e,t)=c ax+v Rae+ R (O,e,t)=o
		1/16(0, e, t) = 0
		a2u 1 - v 1 a2u 1 + v 1 a2v v aw p(1 - v2) a2u
		ax2 + -2- R2 af)2 + -2- R ax ae + Ii~ = E at2 (15.341)
		1 a2v 1 - v a2v 1 + v 1 a2u 1 aw k ( v 1 aw)
		R2 ae2 + -2- ax2 + -2- R ax ae + R2 ae + Ii 1/16 - R + Rae
		= E at2
		_( 2 a1/1x 1 a1/16 1 av) 1 (1 av . w au)
		k v w + ax + R ae - R2 ae - R R ae + R + v ax
		=
		E at2
		(a21/1x 1 - va21/1x 1 + v a21/16) _ 12k (aw ,/,)
		ax2 + 2R2 ae2 + 2R ax ae h2 ax + 'l'X
		p(1 - v2) a21/1x
		=
		E at2
		(2- a21/16 1 - v a21/16 1 + v a21/1x ) _ 12k (.!. aw ,Ir _~)
		R2 ae2 + 2 ax2 + 2R ax ae h2 R ae + '1'6 R
		p(1 - v2) a21/16
		=
		E at2
		image1
		image2
		image3
		image4
		image5
page617
		[ ~~ + v (~ ~~ + ;)] (0. e) = 0
		[~~ + v (~ ~~ + ;)] (l, e) = 0
		image1
page618
		1 + v n m7C
		2 R Z
		... R Z
		k
		d33=k {(m;f +(~f}+ ~2
		12k 1 - v (n )2(nt7C)2
		d45 = d54 = - -- - -
		2 R Z
		12k 1 - v (mz7C)2 + (_Rn)2
		image1
		table1
page619
		. ·(m7r)Z
		dll = -/-
		v m7r
		(m7r)Z k
		/ + RZ
		k
		_ ( m7r ) Z 1
		-m7r
		12k (m7r)Z
		image1
		image2
		image3
		image4
page620
		{ i~ } = { ~} (l5.MM)
		Notes
		image1
		image2
page621
		15.11 RECENT CONTRIBUTIONS
		.,
		j
		image1
page622
		REFERENCES
		image1
page623
		PROBLEMS
		image1
		image2
page624
		0= --+v--- -+W
		image1
page625
		16
		Elastic Wave Propagation
		16.1 INTRODUCTION
		16.2 ONE-DIMENSIONAL WAVE EQUATION
		c-=-
		image1
		image2
page626
		c=.JP/p
		a~ = 1 a~ _ -c aYJ - 1
		a~ ' at -, ax - ,
		aw aw a~ aw aT/ aw aw
		-=--+--=-+-
		ax a~ ax aYJ ax a~ aYJ
		a2w a2w a~ a2w aYJ a2w a~ a2w aYJ a2w a2w a2w
		_ = __ + + + __ = -+2--+-
		ax2 a~2 ax a~ oYJ ax a~ aYJ ax aYJ2 ax a~2 a~ aYJ aT/2
		16.3 TRAVELING-WAVE SOLUTION
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page627
		-=--+--=-e-+e- (16.7)
		aw
		8f = h(~) (16.10)
		I
		image1
		image2
page628
		1 x
		_ {A sin w (t - ~)
		fix - C1)
		t
		I
		-10 0 0
		w(x, 1) = f (t - ~) + g (1 + ~)
		1 x
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page629
		w
		r=-=­
		f=;="i.
		16.4 WAVE MOTION IN STRINGS
		I
		.,;
		image1
page630
		image1
		image2
		image3
		image4
		image5
		image6
page631
		o
		. ~ "
		- ; .... ~._-><_./.
		-\
		11X
		image1
		image2
page632
		1 1 lx
		1 1 lx
		1 1 lx-ct 1 1 lx+ct
		lx-ct lx+ct lx+ct
		image1
page633
		w(x, t) = -[-R(x - ct) + R(x + ct)] = - V(y)dy
		2c 2c x-ct
		image1
		image2
		image3
		table1
page634
		-1a -20
		ow
		7it(x,O) = Va == qo
		image1
		image2
		image3
		image4
page635
		•
		•
		16.5 REFLECTION OF WAVES IN ONE-DIMENSIONAL
		16.5.1 Reflection at a Fixed or Rigid Boundary
		u(O, t) = °
		image1
		image2
		table1
page636
		AU
		image1
		image2
		image3
page637
		c-.--
		Fixed 1/: f"'-......
		/ o-·_-·---·_-·--~·_-·_-· ..... _-·_-·+~ . ...;.
		_'~"7'::'='::'='7"-=--:=:--=--:::,=,+_/:_,-,-,----,-C • ~---' .. ~_+...,;:
		, ,
		'"
		" . ~
		" ' .
		... .
		~------'-----_ .•.. ..;--
		-. 4_:...-=-..:.==·==..:.~·==.:.._=_..: '
		o
		•
		.+.~
		16.6
		REFLECTION AND TRANSMISSION OF WAVES
		AT THE INTERFACE OF TWO ELASTIC MATERIALS
		image1
		image2
		image3
		image4
page638
		x
		x
		_. E -: :::::.::.:.:::.: ::-.-
		-----~c
		-)
		-0
		-. L_~--=-"':=-:"-==-;"~'==~-=--:_ .
		~-----
		image1
		image2
		image3
		image4
		image5
		image6
page639
		-
		I
		- -'\AL
		(A. + 2tth -(0, t) = (A. + 2tth-(0, t)
		ax ax
		(16.53)
		image1
		image2
		image3
		image4
		table1
page640
		- = ---- + ---- = ---- + ---
		ox - OS2 oX - C2 d~2
		---+--=-a--
		(x) I-a ( x)
		r t + - = --p t + -
		s (t _ ~) = _2 p (t _ ~)
		image1
		image2
page641
		COMPRESSIONAL AND SHEAR WAVES
		v=O
		image1
		image2
		image3
page642
		o
		o
		o
		ou
		image1
		image2
		image3
page643
		o
		).. + 2IJ-
		image1
		image2
		table1
page644
		w=O
		f3= ­
		image1
		table1
page645
		o
		table1
		table2
page646
		av
		[au av]
		ay ax
		av _
		+- =0
		ax4 c2 at2
		16.8 FLEXURAL WAVES IN BEAMS
		image1
		image2
		image3
		image4
page647
		W = 2rrf = kv
		A A pA
		Wi =w+Aw
		image1
page648
		image1
		image2
		image3
page649
		. t:.w dw d(kv)
		g Ak ••.• O t:.k dk dk
		dk
		16.9 WAVE PROPAGATION IN AN INFINITE ELASTIC MEDIUM
		132 t:.2 = cfV'2 t:.
		CI = (A:2/LY/2
		image1
		image2
page650
		2 ( au OW) a2 (ou OW)
		/-L"V --+- =p- --+-
		. OZ oy at2 az ay
		at
		C2 = (~Y/2
		image1
		image2
		image3
page651
		Wx = wy = Wz = 0:
		aw av au aw av au 0
		ay - az = az - ax = ax - ay =
		at/J
		ax
		at/J
		at/J
		w=-
		az
		w _ ~ (aw _ av) _ ~ (a2t/J _ a2t/J) = 0
		x - 2 ay az - 2 ayaz azay
		~ = au + av + aw = V2t/J
		. ax ay az
		a~ = ~(V2t/J) = V2 (at/J) = v2u
		ax ax ax
		a~ = v2w
		a2u
		a2v
		at
		a2w
		image1
page652
		ou
		ox
		(A + 2/.L) ox2 = p ot2 (16.129)
		/.L ox2 = Pai2 (16.130)
		/.L ox2 = P ot2 (16.131)
		C ox2 = ot2 (16.133)
		image1
page653
		- = -- + --:-=----'. (16.137)
		ar2 r ar
		or
		16.10 RAYLEIGH OR SURFACE WAVES
		image1
page654
		I
		y 2 az ax 2
		a A a2u
		a2v
		J1,\l2v = p- (16.144)
		at2
		aA a2w
		az at
		au aw
		ax az
		a2 a2
		ax2 az2
		a¢J a1/f
		ax az
		a¢J a1/f
		az ax
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page655
		W = ~ (OW _ ov) = 0
		x 2 oy oz
		Wz = ~ (:: - :~) = 0
		c = - == w)..
		1 ox2 OZ2 1 ot2 [
		image1
		image2
		image3
		image4
		table1
page656
		a = (n2 _ ;;Y/2
		~ = (n2 _ ;;Y/2
		au
		(Tzz = A (a2¢ + a2t/1 + a2¢ _ a2t/1) + 2J1- (a2¢ _ a2t/1)
		ax2 axaz az2 axaz az2 axaz
		image1
		image2
		image3
		image4
page657
		(au aw)
		(aw av)
		[ 2 ]2 ( 2)2 (2) ( 2)
		image1
		image2
		image3
		image4
		image5
		image6
page658
		or
		image1
		image2
		image3
		image4
		image5
		image6
		image7
		image8
		image9
page659
		2 1 - 2v 1
		P -8p --p --=0
		2 + v'3'
		n ",3
		image1
page660
		- = 0.91940C2
		image1
		image2
		image3
		image4
page661
		: c2 =.!. : :
		----I------T------
		o
		------r------ ------r------
		------1- ----I------T------
		16.11 RECENT CONTRIBUTIONS
		image1
		image2
		image3
		image4
page662
		REFERENCES
		image1
page663
		PROBLEMS
		{ 2, -I < x < 1
		image1
		image2
page664
		p (aw)2
		2 at
		P (8W)2
		2 ax
		,
		I'''''''''';'''~'n e:u-.~""""""*''''''''''''''''~'''''''".,eti'''''''''''''''''_i'''''''''''~_~ ~_~,_"""","" •.•.••••• """,,,-_,, •. _ ••.•.• ~""'""'_.~.~,~~
page665
		Approximate Analytical Methods
		17.1 INTRODUCTION
		image1
		image2
		image3
page666
		17.2 RAYLEIGH'S QUOTIENT
		. 2 0 dx
		image1
		image2
		image3
page667
		image1
		image2
		image3
		image4
page668
		17.3
		RAYLEIGH'S METHOD
		I
		1~~"''''''''''''''","","--"~~4_'''''''';'''"''-'''''''''''''''''"'''~''~~''''''''~'''~'' __ •• "'"' ..••.•.• _-~~~~~l:; •••••.•
page669
		it' = ~EI t [o2W(X, t)]2 dx
		2 10 ox2
		1 l' [OW(X, t)]2 d
		T=-pA --- x
		2 0 ot
		I
		----I
		image1
		image2
page670
		" " "
		image1
		image2
		image3
		image4
		image5
		image6
		image7
		image8
		image9
		image10
		image11
		image12
		image13
		image14
		image15
		image16
		image17
		image18
		image19
		image20
page671
		-
		N
		I
		,..---,
		~l'"
		-I ••.
		;:1'"
		'" •..
		'""'
		I
		....---
		~I'"
		-I ••.
		+
		JIM ...
		'-'
		0,:
		~
		l::
		­
		N
		I
		....---
		~IM
		M q:,
		_I"' •..
		+
		-I ••.
		;:1'"
		'""'
		....---
		JI~
		+
		-I ••.
		+
		JIM ...
		'-'
		-
		0,:
		~
		'"
		~
		5
		5
		653
		image1
		image2
		image3
		image4
		image5
		image6
		image7
		image8
		image9
		image10
		image11
		image12
		image13
		image14
		image15
		image16
		image17
		image18
		image19
		image20
page672
		image1
		image2
		image3
		image4
		image5
		table1
page673
		2 10
		R(X (x» = w = =------''-'-1---'---
		R = w2 = 2 = ----
		(El
		image1
		image2
page674
		IpA (...LC219) pAl4
		f1I.1
		pAL
		I~
		f1I.1 f1I.1 f1I.1
		pAL pAL pAL
		image1
		image2
		image3
		image4
		image5
		image6
page675
		T=-pA --- x+-m ~~-
		I'll 1
		-dx-2- = 6C(1 - x)
		image1
		image2
		image3
page676
		R =It} = 2 = _
		/£.1
		/£.1 ffI.1
		image1
		image2
page677
		2 Jo at
		1 1/ ( X) [dX(X)]2, 1 2
		rrmax = - E Ao 1 - - -- dx + -kX (I)
		* 1 t ( X) 2
		dX Crr rrx
		1/ ( X) 2 2 rrx 21 (3 1 )
		I
		[ 1 ] 1/2
		image1
		image2
page678
		W (r) = c (1 _ ::) 2
		V=7rD -+-- -2(l-v)-_- rdr
		image1
		image2
		image3
		image4
page679
		17.4 RAYLEIGH-RITZ METHOD
		X(x) = L:>i4>i(X)
		image1
		image2
		image3
page680
		R-w ----_
		image1
		image2
page681
		* 1 L L LT-
		. /
		ki' = EI----dx
		image1
		image2
		image3
		image4
		image5
page682
		aN -T
		ac = c [k]
		aD -T
		ac = c [m]
		or
		[[k] -). (n)[m]] C = 0
		I[k] - ). (n)[m]1 = 0
		image1
		image2
		image3
		image4
		image5
		image6
		image7
page683
		o
		I~
		A(x) =Ao (1- ft)
		.1
		image1
		image2
		image3
		image4
page684
		8Ci r;;ix
		~=T+f2+[3
		image1
		image2
		image3
		image4
page685
		~ I
		a Jrmax EAo (3 4 5)
		a Jrmax EAo (5 4 9)
		-- = -- -C2 + -CI + -:;CJ
		aJrmax EAo (21 5 9)
		-- = -- -C3 + -CI + -c,
		~ ] {CI}
		I !]
		:s 10
		~: ~~]
		image1
		image2
		image3
page686
		1-0.2034J
		R=--= I
		1-0.871OJ
		/ 2/
		image1
		image2
		image3
page687
		aT;;ax pAol [(3 1) ( 1 ) ( 1 )]
		----
		16+4
		---
		=
		image1
		image2
		image3
		image4
		image5
		image6
page688
		"16+4
		4
		1
		---
		"16+4
		[k]2 = A[m]2
		[k] =
		[m] =
		17.5 ASSUMED MODES METHOD
		image1
		image2
		image3
page689
		1 t [au ]2
		1 n n [t ] 1 n n
		image1
		image2
page690
		image1
		image2
		image3
		image4
		image5
page691
		w2[m]X = [k]X (17,61)
		17.6 WEIGHTED RESIDUAL METHODS
		inD
		Ej W = 0,
		R«$, x) = A<P - AB-;P
		17.7 GALERKIN'S METHOD
		image1
		image2
page692
		IC(i)}
		c(i) = c{ ... '
		image1
		image2
page693
		[ )4 ] [()4 ]
		lx=o
		lx=o cos L - 1 Cl T - {3 cos L + ClfJ
		[ -- 4] ]
		+ c2 (T) :- fJ cos L + C2fJ dx = 0
		lx=o cos L - 1 CI T - fJ cos L + clfJ
		[( )4] ]
		c, ( H en' - p'] - p'] - c,p' = 0
		H er)' - p'] - p' -p'
		=0
		image1
page694
		For cvz:
		image1
		image2
		image3
		image4
page695
		image1
		image2
		image3
		image4
		image5
page696
		c = [X]p
		[k]c = A[m]c
		IA2[m]+A[d]+[k]1 =0
		[xf [m] [X] = [I]
		A 2[m][X]p + A(a[m] + .8[k])[X]p + [k][X]p = 6
		image1
		image2
		image3
		image4
page697
		a[1] + fJ[A] = [y] == [2S1Wl 2S2W2 ... 0]
		AC = AC
		A2C = -A[mr1[d]c - [mr1[k]c
		{ C } [[0] [1]] { C }
		A AC = -[mr1[k] -[m]-l[d] AC
		or
		AY = [B]y
		image1
		image2
		image3
		image4
page698
		17.8 COLLOCATION METHOD
		[l] ]
		i = 1,2, ... , n
		[[OJ
		1,
		image1
		image2
		image3
page699
		image1
		image2
page700
		I
		I
		d4 [( 2JTX) ( 4JTX)]
		[ ( 2JTX) ( 4JTX)]
		(2JT)4 2JTX (4JT)4 4JTX
		, I I ,
		[ ( 2JTX) ( 4JTX)]
		image1
		image2
		image3
		image4
		image5
page701
		C, [- (~)' Co'i - «I-CO' ~)] +C2 [- (4n' co,,, -<o-CO'''>] = 0
		c, [- (~)' co,,, - <O-CO'''>] + C2 [ - (4;)' CO, 2" - «I - CO, 2,,>] ~ 0
		C,(-<)+C2 [en' -2<] =0
		c, [(~)' - 2<] +02 [_ (4;)'] =0
		(2 )4
		=0
		(El
		(El (El
		image1
		image2
		image3
		image4
page702
		(¥ / - 2A
		17.9 SUBDOMAIN METHOD
		1; R(¢(x» dx = 0,
		image1
page703
		o R dx = 10 . -Cl -t- cos -t- - C2 -t- cos -t-
		I I
		I t
		image1
		image2
		image3
		image4
		image5
page704
		- en3 - Ai + A~ -Ai
		16,,4
		C2 = A (1/4) c)
		1 i.l0 LEAST SQUARES METHOD
		image1
page705
		8 (1/ ) 1/ 8 R
		i = 1,2, ... , n
		tn {d[ d4J'(X)] }
		image1
page706
		- -
		image1
		image2
page707
		- ( 21rX) ( 41rX)
		'11 - oR-
		R(W(x»-(W(x»dx = 0,
		_ d4W _ (21r)4 21rx (4Jl')4 41rx
		R(W(X»=---AW=-Cl - COS--C2 - cos-
		dx4 t t t t
		4 [( 2Jl'X) ( 41rX)]
		t - oR(W(x» t [(21r)4 21rx (4Jl')4 41rx
		o R(W(x» OCl dx = 10 Cl -t- cos -t- + C2 -t- cos -t-
		4 ( 21rX) 4 ( 41rX)]
		2Jl' 21r X 4 21r x
		image1
		image2
		image3
page708
		~(EJ)I/2
		image1
		image2
		image3
		image4
		image5
		image6
page709
		( EI )1/2
		[ t - ). + fP ).2 ] {CI} {oJ
		P i28 -:- 16X + fX2 C2 = °
		image1
		image2
page710
		.fo oc;
page711
		fo
		-=24
		aR
		17.11 RECENT CONTRIBUTIONS
		image1
		image2
page712
		image1
page713
		REFERENCES
		image1
page714
		PROBLEMS
		.-.-._.-.J._._._._._._._._._._.
		j.-y~
		image1
		image2
		image3
page715
		d [ dU(X)]
		21 21
		image1
		image2
		image3
		image4
page716
		~ I
		2/ - 2/
		W(x) = c) (1- 7)4 +cz7 (1- 7)4
		image1
		image2
		image3
		image4
		image5
page717
		W(x) = CI (1- Tf +c2T (1- Tf
		W(x) =c(I-COS~)
		. 2L
		W(r) = CI (1 _ ~:) 2 + C2 (1 - ~: y
		image1
		image2
		image3
		image4
page718
		A
		Basic Equations of Elasticity
		A.I STRESS
		A.2 STRAIN-DISPLACEMENT RELATIONS
		image1
		image2
		image3
		image4
page719
		z
		/"
		I au
		t
		au""
		- •. -------
		image1
		image2
		image3
		image4
page720
		= {dy + [v + (ovjoy)dy] - v} - dy _ ov
		dy oy
		[v + (ovjox)dx] - v [u + (oujoy)dy] - u (A.4)
		~=~+~~ +
		dx + [u + (oujox)dx] - u dy + [v + (ovjoy)dy]- v
		ou ov
		ow ov
		Eyz = oy + oz
		ou ow
		oz ox
		ov ou
		- ox - oy
		Wz = ~ (:: - :;) (A. 10)
		Wx = ~ (~; - :~) (All)
		W = ~ (ou _ ow). (AI2)
		y 2 oz ox
		w. = ~ (OV _ OU) (AI3)
		- 2 ox oy
		A.3 ROTATIONS
		image1
		image2
		image3
		image4
page721
		ctuu---uu-------- A;u" B T
		c' _u_u_: E
		: ~
		, au +-.J._. _.~,
		I- dx -I
		A.4 STRESS-STRAIN RELATIONS
		image1
		table1
page722
		A.5 EQUATIONS OF MOTION IN TERMS OF STRESSES
		( au.x)
		image1
		image2
		image3
		image4
		image5
		table1
page723
		a2u
		aaxx oaxy aazx a2u
		ax ay az at2
		aaxy aayy aayZ a2v
		-+-+-=p-
		ax ay az at2
		aazx aayZ aazz a2w
		-+-+-=p-
		ax ay az at2
		A.6 EQUATIONS OF MOTION IN TERMS OF DISPLACEMENTS
		a a a a2u
		a ( au) a [ (av au)] a [ (aw au)] a2u
		- AD.+2J.L- +- J.L -+- +- J.L -+- =p_.
		ax ax ay ax ay az ax az at2
		a2 a2 a2
		V2=-+-+­
		image1
		image2
page724
		oy t
		image1
		image2
		image3
page725
		B
		Laplace and Fourier Transforms
		707
		."
		image1
		image2
		table1
page726
		s
		e-as
		2as
		F(s - a)
		'Vvvv .
		:~.
		26
		27
		image1
		image2
		image3
		image4
		image5
		image6
		image7
		table1
page727
		Table B.l (continued)
		28
		o
		29
		image1
		image2
		image3
		image4
		image5
		table1
page728
		I{x)
		Ixl <a
		{I,
		image1
		image2
		image3
		image4
		table1
page729
		711
		· .. ·1
		image1
		image2
		table1
		table2
page730
		image1
page731
		Index
		713
		'J
page732
		E
		D
		image1
		image2
		image3
		image4
		image5
page733
		H
		image1
		image2
page734
		K
		L
		image1
page735
		N
		o
		image1
		image2
		image3
page736
		R
		image1
		image2
		image3
		image4
page737
		-- ··1
		image1
page738
		V
		r:/:'~~nJ:~~:··:··J(.'~~~"~)":~' ::~:,-~~~ee-Of-freedom system, 33
		::\;. !,-, tj 169
		\~:~·~·~:~._;f?'5'J··:~~·~~4; w
		image1
                        
Document Text Contents
Page 1

Vibration of Continuous
Systents

Singiresu S. Rao
Professor and Chairman
Department of Mechanical and Aerospace Engineering
University of Miami
Coral Gables, Florida-

/

81CEHTllHNIAI.

~lll 8 0 7 ~z •

: G?WILEY :
z z
~ 2007;
"'--_--'r.IC.NTllMMI""
JOHN WILEY & SONS. INC.

Page 2

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Rao, S. S.
Vibration of Continuous Systems / Singiresu S. Rao.

p. cm.
Includes index.
ISBN-13 978-0-471-77171-5 (cloth)
ISBN-IO 0-471-77171-6 (cloth)'
I. Vibration-Textbooks. 2. Structural dynamics-Textbooks. -:. - 0,

1. Title. jiI " ••.,J __./.r; -;.

TA355.R378 2007 /~~(J
624.1'71-dc22 I 0~. ...,

2006008775 i~
Printed in the United States of America \ ~ ',1
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Page 369

,-
I 11.9 Response of'Bemns under Mu"'ingLoads ~351

p

1'---: .-x d~y
Figure 11.11 Concentrated load assumed to be uniformly distributed over a length 2Llx.

extended for all values of x and can be considered to be periodic with period L. The
Fourier series expansion of the extended periodic function converges to the function
i(x) in the original interval from Xo to Xo + L. As a specific case, if the function i(x)
is defined over the interval 0 to I, its Fourier series expansion in terms of only sine
terms is given by

00

'"' nJf X_ i(x) = ~ in sin -t-
'. n=l

where the coefficients in are giveri by
2 t . nrrx

in = 7 10 i(x) sm -t- dx

(11.157)

(11.158)

(11.159)

In the present case, the Fourier coefficients "in can be computed. using Eq. (11.156)
for i(x), as

in = ~[1d-aX (0) (sin n~x) dx + l:::x 2~x (sin n~x) dx
+ 1/ (0) (sin nrr

I
X) dX]

d+ax

P 1d+ax " nrrx d 2P. nrr d sin(nrr l:1.x/l)= - sm -- x = - sm -------
Il:1.x d-ax t I I nrrl:1.x/t

Since P is actually a concentrated load acting at x = d, we let l:1.x~ 0 in Eq.(11.159)
with

to obtain the coefficients

. sin(nrr l:1.x/l)
hm ----- = 1

ax--+o (llrr l:1.x/ t)

2P nrr din = -sin--
I t

(11.160)

(11.161)

I

i

1
i

i
i
I

J

Page 370

352 Transverse Vibration of Beams

Thus, the Fourier sine series expansion of the concentrated load acting at x = d can
be expressed as

Using d = vot in Eq. (11.162), the load distribution can be represented in terms of x
and t as

(11.162)

(11. 166)

2P 00 • nrrd mrx
f(x) = -1-LS10 -1- sin -1-

n=!

I )!/2aw(x, t) 2ds - dx = (dx)2 + [ ax dX] - dx

The problem of transverse vibration of beams subjected to axial force finds application
in the study of vibration of cables, guy wires, and turbine blades. Although the vibration
of a cable can be studied by modeling it as a taut string, many cables fail due to fatigue
caused by alternating flexure induced by vortex shedding in a light wind. In turbines,
blade failures are associated with combined transverse loads due to fluids flowing at
high velocities and axial loads due to centrifugal action.

The total response of the beam considering all components (or harmonics) of the load,
given by Eq.(11.163), can be expressed as

2PI3 ~ 1 1 nrrx ( 2rrvo 2rrvo )
w(x, t) = --4 ~""4 • 2sin - -1- sin --t - --I sinwnt

EIrr n=! n 1 - (2rrvo/lwn) I Wn
(11.165)

2P (. rrx . rrvot . 2rrx . 2rrvot . 3rrx . 3rrvot )
f(x t) = - S1O-S1O-- +S1O--sm--+sm--s1O __ + ' ..

' I I I I I I I

(11.163)

The response of the beam under the nth component of the load represented by
Eq. (11.163) can be obtained using Eq. (El1.6.l2) as

2PI
3

nrrx ( 2rrvo 2rrvo )
w(x, t) = sin -- sin --t - -- sinwnt

EI(nrr)4 [1 - (2rrvO/lwn)2] I I wnl
(11.164)

11.10 TRANSVERSE VffiRATION OF BEAMS SUBJECTED TO
AXIAL FORCE

11.10.1 Derivation of Equations

Consider a beam undergoing transverse vibration under axial tensile force as shown in
Fig. 11.12(a). The forces acting on an element of the beam of length dx are shown in
Fig. 11.12(b). The change in the length of the beam element is given by

Page 737

strain energy, 571
stress-strain relations, 562
theory of surfaces. 541

Single-degree-of-freedom system, 33
critically damped. 36
damped harmonic response, 40
forced vibration, 36, 41
free vibration, 33
under general force, 41
under harmonic force. 36
overdamped, 36
underdamped, 35

Skew plate, 540
Solid mechanics. 104
Spectral diagram, 25
Spherical shell, 546, 561, 591
Spring element, 1
Standing wave, 612
State space, 54
State vector, 54
Static deflection, 39
Stiffness matrix, 43
Strain-displacement relations, 700
Strain energies of structural elements, 652
Strain energy, 104
Stress, 700
Stress-strain relations, 703
String

boundary conditions, 211
finite length, 194
forced vibration, 183
free vibration, 181
harmonic waves, 611
infinite, 210
transverse vibration, 205
traveling wave solution, 210
wave motion, 611

Sturm-Liouville problem, 154
classification, 155
periodic, 155
regular, 155
singular, 155

Subdomain method, 684
Support motion

longitudinal vibration of bars, 257
Surface waves, 635
S waves, 625
Synchronous motion, 23

T
Terminology, 21
Theory of surfaces. 541

Three-dimensional vibration of circular ring,
393

Timoshenko-Gere theory. 300
Torsional properties of shafts. 310
Torsional rigidity, 303
Torsional vibration of circular rings, 406
Torsional vibration of shafts. 115, 271

elementary theory, 271
forced vibration, 292
free vibration, 276, 289
noncircular shafts, 295, 299
Timoshenko-Gere theory, 300

Transformation of relations. 486
Transform method in beams, 385
Transform pair, 175
Transient motion, I, 4
Transients, 42
Transmission of waves, 619
Transverse vibration of plates, 457

boundary conditions, 465, 489
circular plates, 485
on elastic foundation, 521
equation of motion, 457
forced vibration, 479
free vibration, 471, 511
frequency equations. 480
Mindlin theory, 499
mode shapes, 475, 480
rotary inertia and shear deformation, 499
with variable thickness, 529

Transverse vibration of strings, 205
Transverse vibration of thin beams, 71,116,

185,317
under axial force, 352
coupled bending-torsional vibration, 380
on elastic foundation, 364
equation of motion, 317
Euler-Bernoulli theory, 317
flexural waves, 628
forced vibration, 344
frequencies and mode shapes, 326
infinite length, 385
with in-plane loads, 523
on many supports,359
under moving load, 350, 367
orthogonality of normal modes, 339
Rayleigh's theory, 369
response due to initial conditions, 341
rotating, 357
Timoshenko theory, 371
transformation methods, 385

-- ··1

Page 738

720 Index

Traveling wave solution, 210,
608

U
Uncoupled equations, 48
Undamped system, 47, 52
Undamped vibration, 15
Underdamped system, 35

V
Variational approach, 85

membrane, 423
plate, 458
shaft, 272
string, 235
thick plate, 505

Variational methods, 85
in solid mechanics, 104

Variation operator, 89
Vibration

analysis, 16
beams, 317, 369,371
beams on elastic foundation, 364
circular cylindrical shell, 582
circular rings, 393,406
concept, 1
continuous beam, 359
curved beams, 393, 408
developments, 5
forced, 52, 53
free, 47
history, 8
importance, 4
membranes, 420
multidegree-of-freedom system, 43
origins, 5
plates, 457, 485, 499
problems, 15
rotating beam, 357
shafts, 271

,/....• shells, 54]
..t' " ." .. , .••.:r:/:'~~nJ:~~:··:··J(.'~~~"~)":~'::~:,-~~~ee-Of-freedom system, 33/, V(J~'\thick beams, 371

I} . :~iscous damping coefficient, 170
l q . iscous]y damped system, 34, 54,

,". 1/ 1 .•
::\;. !,-, tj 169
\~:~·~·~:~._;f?'5'J··:~~·~~4; w

~" • -t ",...r l r .L:"."f'~~"';;.~~:>Warping function, 296

Wave equation
D'A]embert's solution, 608
membrane, 421
one-dimensional, 607
string, 207
traveling-wave solution, 608
two-dimensional,610

Wavelength, 612
Wave number, 612
Wave packet, 629
Wave propagation, 607

in infinite elastic medium, 631
traveling wave, 608

Waves
compressional, 623
dilatational, 631
distortional, 631
equivo]uminal, 634
flexural, 628
harmonic, 611
irrotationa], 634
]ongitudinal, 634
primary, 634
P, 623, 634
Rayleigh, 635
rotational, 634
shear, 623, 625
standing, 612
surface, 635
S,625
traveling, 210, 608

Wave solution
compression a] waves, 623
dilatational waves, 631
distortional waves, 632
flexural waves, 628
graphical interpretation, 614
group velocity, 629
interface of two materials, 619
membrane, 425
P waves, 623
Rayleigh waves, 635
reflection of waves, 617, 622
shear waves, 623, 625
string, 210, 611
surface waves, 635
S waves, 625
transmission of waves, 619
wave packet, 629

Weighted residual methods, 673

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