Ginsberg Answers

8/10/2019 Ginsberg Answers

1/67

Answers to Odd-Numbered Homework Exercises

Mechanical and Structural Vibrations

Jerry H. Ginsberg, John Wiley and Sons, Inc. 2001

Chapter 1

1.1 keq = k1(k2+ k3)

k1+ k2+ k3+ k4

1.3 keq = 3 (EI/L3) k (2k+ 3EI/L3)

k2 + 9k (EI/L3) + 9 (EI/L3)2

1.5 my+3

2 y+

8

3ky = 0

1.7

1

3mL2!+

k!

1

2mgL

!= 0

1.9 9

64mL2!1+

1

16cL2 !1 !

1

8cL2!2+

9

32mgL +

1

4kL2

!1 !

3

8kL2!2= 0

1

3mL2!2 !

1

8cL2!1+

1

4cL2!2 !

3

8kL2!1+

1

2mgL +

9

16kL2

!2= 0

1.11 (a) m1x1+ (k1+ k2+ k4) x1 ! k2x2= F, m2x2 ! k2x1+ (k2+ k3) x2= 0

(b)keq =k1+ k4+ k2k3k2+ k3

1.13 (a)M11=

I1+

R21

R22

I2+

R21

R23

I3

, (b)M11=

I1R2

3

R21

+ I2R2

3

R22

+ I3

1.15 M11=25

48mL2

1.17 C11= C22=

1

4c1+ c2

L2, C12= !c2L

2

1.19 [K] =

!""#

1

4kL2 +

1

2mgL !

1

4kL2

!

1

4kL2

1

4kL2 +

1

2mgL

$%%&

1.21 K11= 3.28k

1.23 Q1=1

4F L

1.25 Q1= 0, Q1= F r sin(")


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2

1.27 {q}=

'(()((*

yG

!

+((,((-

, [M] =

!""#

m 0

0 mr2G

$%%&

[K] =

!""# kY !kY`!kY` kY`2 + kT

$%%& , {Q}=

'(()((*

L

!L (` + s)

+((,((-

1.29 {q}=

'(((((()((((((*

y1

y2

y3

+((((((,((((((-

, [M] =m

!""""""#

1 0 0

0 2 0

0 0 3

$%%%%%%&

, [C] =c

!""""""#

5 !2 0

!2 3 !1

0 !1 1

$%%%%%%&

[K] =k

!

""""""#

5

!2 0

!2 3 !1

0 !1 1

$

%%%%%%&, {Q}= F

'

(((((()((((((*

1

2

3

+

((((((,((((((-

1.31 {q}=

'(()((*

yG

!(ccw)

+((,((-

, [M] =m

!""#

1 0

0 3

4b2

$%%& , [K] =k

!""#

2 b

b 5

2b2

$%%&

[C] =c

!

""#2 !b

!b 52

b2

$

%%& , {Q}= F

'(()((*

!1

!12

b

+((,((-

1.33 q1= x1 (cart), q2= s2 (block parallel to incline)

[M] =m

!""#

3 cos (!)

cos(!) 1

$%%& , [K] =k

!""#

3 0

0 1

$%%&

1.35 {q}=

'(()

((*

!1

!2

+((,

((-, [M] =I0

!""#

1 0

0 1

$%%&

[K] =kR2

!""#

2 !1

!1 1

$%%& , {Q}=

'(()((*

0

F r sin(")

+((,((-

1.37 1

3mL2!+

k ! 1

2mgL

!= 0


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3

1.39 y= vertical displacement,my+ 0.6580ky = 0

1.41 (a)`0= 1.632L, (b) 5

3mL2!+ 2.402mgL!= 0

1.43 24mr2!+ 4mgr!= 0

1.45 25

48mL2!+ kL2!=

1

4F L

1.47 {q}=

'(()((*

! (bar, ccw)

y (block)

+((,((-

, [M] =

!""#

1

3m1L

2 0

0 m2

$%%&

[K] =

!""#

(0.64k1+ k2) L2 0.4k2L

0.4k2L k2

$%%& , {Q}=

'(()((*

! 815

F L

0

+((,((-

1.49 {q}=

'(((((()((((((*

xG

yG

!

+((((((,((((((-

, [M] =

!""""""#

m 0 0

0 m 0

0 0 IG

$%%%%%%&

[K] =k

!""""""#

3.5 0 (4L! 7b)

0 0.5 0

(4L! 7b) 0 (3.5b2 + 2L2 ! 4bL)

$%%%%%%&

1.51 q= y ! 34

L, mq+ 3.253mg

L q= 0

1.53 (a) `0= 0.866L + 1.7321mg

k , (b)K11= 2mgL + 0.25kL

2

(c) kL

mg >592

1.55 q= !AB ! 65o

2.512mL2q+ 1.4665cL2 q+ 1.4665kL2q=!

0.2707F L

1.57 4.614mR2#+ 1.5831kR2#= 1.2582F R, #= ! ! $/2


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4

Chapter 2

2.1 F= 500 cos (5$t 1.2611)N

2.3 x (0) = 17.324mm, x (0) = !10.465m/s, min(x) @t = 2.5msmax( x) = 20.94m/s@ t = 4ms, max(x) = 2.193 (104) m/s2 @t = 2.5ms

2.5 (a)V = Re {1.2exp[i (%t! 0.589)]} volt, (b) V = Re {471exp[i (%t + 0.982)]} volt/s

(c)max V= 471 volt/s whent = 0.0135, 0.215,...sec

2.7 (a) x = 14.25sin(!t + 2.449) , (b)t = 0.693/!for x = 0, (c)t = 2.26/!for x= 0

2.9 (a) t = 0.010472 sec forF = 0, (b)F= 200 cos (250t) + 346 sin (250t)

2.11

0 1 2 3 4 5 6 720

0

20B = 10

q tn

10!

tn

0 1 2 3 4 5 6 720

0

20B = 6

q tn

6!

tn

0 1 2 3 4 5 6 710

0

10B = 5

q tn

5!

tn

2.13 p1= 0.005cos (878$t! &1) , p2= 0.005 cos (882$t! &1 ! 0.20$)Pa

2.15 u= !0.01cos(50T+ 0.6)! 0.05cos(10T+ 0.6) + 0.049522.17 (a)q= 0.10146 m @t = 2.73 ms, (b) q= 51 m/s @t = 12.10 ms

2.19 k= 392.3 N/m, m= 0.1956 kg

2.21 %nat =

4kR ! 2mg

(3m1+ 8m2) R

1/2, Unstable ifm2g >2kR


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5

2.23 Keq = 1.583 (108)N/m, Ceq = 2.315 (10

4)N-s/m

2.25 (a) %nat = 200 rad/s, '= 0.20

(b)q= 0.012exp(

!40t)[cos(195.96t) + 0.2041 sin (195.96t)] m

(c) minq= !0.00632 m @t = 0.01603 sec

(c)q= 0 @ t = 0.00904 sec

2.27 C= 10.208 N-s/m,max (q) = 0.695 mm @t = 16.57 ms

2.29 (a) (= 0.297, %nat = 62.90rad/s, '= 0.04723

(b)t >2.507sec, (c)t >1.257sec, (d) q0= !1.904 m/s

2.31 %nat = $tbsin ($ta/tb), '= cos($ta/tb) , t= ta+ 4tb

v0= $qmax

tbsin ($ta/tb)exp

$

tatb

cot

$

tatb

2.33 x= 2.207 m @t = 0.10 sec

2.35 EI= 88.83(106)N/m, c= 9.818 (105) N-s/m

2.37 (a)cT = 21.21 N-m-s/rad, (b) t = 0.4664 sec, (c) t = 0.4965 sec

2.39 (a) L = 69.87 mm, c= 80.42 N-s/m, (b)t = 0.09224 sec

(c)! = 2.400 (10!6)[56.60exp(!3.72t)! 3.72exp(!56.60t)rad]2.41 t= 8.625 sec

2.43 (a)k= 0.02516, (b)x


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6

2.47 q= A sin(%t) + B cos(%t) + C1exp (!0.6417%nat t) + C2exp (!1.5583%nat t)

A=F0m

%2nat ! %2

h(%2nat ! %2)2 + 4'2%2%2nati

, B= !F0m

2'%%nat

h(%2nat ! %2)2 + 4'2%2%2nati

C1= !1.7002B+ 1.0911

v0 ! %A%nat

, C2= 0.7002B ! 1.0911

v0 ! %A%nat

2.49 q=

F02m

cos [(%2 ! %1) t]! cos(%nat t)%2nat ! (%2 ! %1)2

! F02m

cos[(%2+%1) t]! cos(%nat t)%2nat ! (%2+%1)2

2.51

0 20 40 60 801

0

1

2

q tn

tn

2.53 q= "r (t)! 2"r (t! ))

2.55 q= F0u (t)!F0)

r (t) +F0)

r (t! ))

2.57 q= 104r (t)! 104r (t! 0.02)! 200u (t! 0.02)

0 0.02 0.04 0.06 0.080.002

0.001

0

0.001

0.002

qj

tj

2.59 q= F0c (t) + F0s

t! $

2%d

2.61 (a)c = 489.9 N-s/m, (b) x = 0.7053[exp(!2.0568t)! exp(!72.944t)]m

2.63 q= P g (t) + P g (t!)) + P g (t

!2)) + , %nat)= 2$ gives maximumq

2.65 q= !*c

sin(%nat t)!2

%nat) [1! cos(%nat t)]

h (t)


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7

2.67

0 1 2 3 40

5

10

15

acc magj

maxaccj

minaccj

#j


8/67

8

Chapter 3

3.1 '= 0 : Q= !840 cos (110t! 1.5)N, '= 0.4 : Q= 3619 cos (110t + 0.3051) N

3.3 '= 0 : |F|< 395.6 N, '= 0.05 : |F|< 1260.2

3.5 %= 950Hz: q= 3.117 (10!5)sin(1900$t! 0.01234) m

%= 1050 Hz : q= 2.965 (10!5) sin (2100$t! 3.129) m

3.7 (a) %nat = 80$ rad/s, (b)'= 0.06290

(c)M= 0.6442 kg, K= 4.069(104) N/m,C= 26.37N-s/m

(d) |q|= 0.00588 m, (e) q= 19.63cos(80$t) m/s

3.9 |F|= 104.88 N & relative arg(F) = 3.132 rad @ % = 75 rad/s

|F|= 152.63N & relative arg(F) = 3.135 rad @ % = 85 rad/s

3.11 (a) & = 134.3o @ 105 Hz, (b) |q|= 1.687 mm & & = 152.4o @ 110 Hz

3.13 (a) += 0.05098 & |q|= 4.84 mm, (b) |q|= 4.83 mm

3.15 (a) k = 4000 N/m, c= 2000/$% N-s/m

(b)x = Re [(!14.85 + 18.60i)exp(i50t)] mm

3.17 (a) Ceq = 8

3$"%X, +eq =

8

3$

"%2

K X

(b)kX

1! r2 + i

8

3$

"

Mr2X

=F

3.19 (a) '= 0.11467, (b),m= 0.19568 kg-m, (c) min (|Y|) = 2.448 mm

3.21 %nat = 408 rad/s, "%= 28 rad/s, '= 0.035

,m= 2.35kg-m, min(|q|) = 4.7 mm

3.23 (a) %nat = 30$ rad/s, (b)c = 1.109 (104) N-s/m

(c) |y|= 8.90 mm, 109.7o above or ! 70.3o below horizontal


9/67

9

3.25

I1+

5

9mL2

!+

1

9cL2!+

1

9kL2!= !m,L%2 sin(%t)

|!|= m,L

I1+5

9mL2

r2

(1! r2)2 + 4'2r2

1/2

3.27 (a)|-|= 0.0541 rad,arg(-) = !0.823 rad, (b) >7.73 n-s/m

3.29 Rc= 0.665mm @r = 0.5, Rc = 20 mm @r = 1, Rc= 2.67 mm @r = 2

3.31 R= 20 mm, ,


10/67

10

3.49

0 0.5 1 1.5 21

0

1

2

wT=0.2piwT=2piwT=20pi

3.51 (a) 33% error in amplitude and 4o error in phase for rst harmonic,

0.2% error in amplitude and 0.3oerror in phase for tenth harmonic

(b) 125% error inamplitude and 10o error in phase for rst harmonic,

0.6% error in amplitude and 0.4o error in phase for tenthharmonic

(c) 15% error in amplitude and 60o error in phase for rst harmonic,

0.3%error in amplitude and 4o error in phase for tenth harmonic

,

3.53 For . = 1 :

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

% nat

2

yj 1!&

z''j 1!

tj

3.55 |Yn/An|= 1.127(10!12)and arg(Yn/An) = 180

o forn " 32

3.57 q= P

M%4nat)2

[%2nat()2 ! t2) + 2! (%2nat)2 + 2) cos (%nat))] ift )


11/67

11

3.59

0 0.5 1 1.5 2 2.5 3 3.5 440

20

0

20

40

qn

QQn

tn

3.61

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.54

2

0

2

4

Displacement (nondim.)Force (nondim.)

Time (sec)

3.63 '= 0.2:%nat# 2960 rad/s,"

% # 1380 rad/s'= 0.002: %nat# 3140 rad/s, "% # 17 rad/s

3.65 %nat# 24.1 rad/s, '# 0.0249

3.67

0.5 1 1.50

20

40

60

80

Actual D

Direct DFTHanning

3.69 For "= 6 :

0.05 0 0.05

0.1

0.05

0

Im Gj 4!

Re Gj 4!


12/67

12

3.71

0 50 100 150 200 250 3000

5 &104

0.001

0.0015

Gn

%n


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13

Chapter 4

4.1 %1= 6.021 rad/s, {&1}= [1 ! 0.275]T

%2= 20.341 rad/s {&2}= [1 7.275]T

4.3 %1= 0.3660

k

m

1/2, {&1}

T = [1 ! 1.2529]T

%2= 2.326

k

m

1/2{&2}= [1 0.9128]

T

4.5 Case (a): %1= 7.404 rad/s, {&1}= [1 0.5909]T

%2= 60.37rad/s {&2}= [1 ! 6.34]T

Case (a): %1= 14.028 rad/s, {&1}T

= [1 0.2668]T

%2= 100.81 rad/s {&2}= [1 ! 16.339]T

4.7 "= 4 : %1= 0.794 g

L

1/2, {&1}

T = [1 1.312]T

%2= 7.422 g

L

1/2{&2}= [1 ! 2.122]T

4.9 *= 2 : %1= 1. 2247 g

L

1/2, {&1}

T = [1 1 1]T

%2= 2. 7386 gL1/2 , {&

2

}= [1 0 !

1]T

%3= 4. 4159 g

L

1/2, {&3}= [1 ! 2 1]T

4.11 %1= 233.5 rad/s, %2= 316.2 rad/s

[#] =

!""#

0.2697 0.4472

0.4045 !0.4472

$%%&

4.13 %2= 165.8rad/s, K22= 75000 N/m, K12= K21= 15000 N/m

[#] =

!"""#

3$21

1$14

! 1$21

2$14

$%%%&


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14

4.15 %1= 6.02 rad/s, %2= 20.34 rad/s

[#] =

!""#

0.4908 0.0954

!0.1349 0.6941

$%%&

4.17

mL

EA

1/2%= 1.564, 4.54, 7.07, 8.91, 9.87

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92

1

0

1

2

Mode 1Mode 2Mode 3Mode 4Mode 5

Position (x/L)

4.19 %= 8.152, 9.94, 19.162 rad/s

[#] = (10!2)

!""""""#

2.502 0.924 0.162

!0.352

!0.217 6.669

0.773 !2.088 !0.027

$%%%%%%&

4.21 %= 1.5939, 3.761, 5.326, 8.055 rad/s

[#] = (10!3)

!""""""""""#

0.534 1.566 !1.560 3.851

1.356 3.069 !1.947 !2.225

2.916 1.467 3.752 0.518

3.977 !3.025 !1.896 !0.089

$%%%%%%%%%%&

4.23 kB = 20.95 kN/m, %1= %2= 7.74rad/s

[&] =

!""#

1 0

0 1

$%%&


15/67

15

4.25 %1= 0, %2= 1.581

k

m

1/2, %3= 2.739

k

m

1/2

4.27 %1= %2= %3= 0, %4= %5= 1.25

k

m

1/2, %6= 1.732

k

m

1/2

[&] =

!""""""""""""""""""#

1 1 1 1 1 1

!1.732 !0.140 2.067 1.618 !0.618 0.577

1 1 1 0.901 !1.035 !1

!1.039 1.295 !0.910 !1.675 !0.577 0.577

0.401 !0.242 3.578 !1.901 0.035 0

!1.385 0.577 0.577 0.057 1.175

!1.155

$%%%%%%%%%%%%%%%%%%&

4.29 k

mg{q}=

'(()((*

0.874

0.831/L

+((,((-

cos(%t) +

'(()((*

0.126

!0.831/L

+((,((-

sin(%t)

4.31

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.22

0

2

4

6

Floor 1Floor 2Floor 3Floor 4

Time (sec)

Displacement(m)

4.33 q1= 10.97 mm, q2= 17.07 mm @t = 2sec.

4.35 q=

!""#

0.7071 !1

1.4042 1

$%%&

'(()((*

!0.2cos(3t! $/4) + 0.1414cos(2t) + 4.926 sin(2t)

0.15972 cos (3t! $/4)! 0.0714 cos(4t)! 0.7261 sin(4t)

+((,((-

m

4.37 /j = #2j[400u (t,%j)! 200r (t,%j) + 200r (t! 2,%j)]

q(t) =

!""#

0.1169 0.6974

0.0986 !0.0165

$%%&

'(()((*

/1(t)

/2(t)

+((,((-


16/67

16

4.39 P =mv

0 5 10 15 20 25 30 351

0

1

x1 n!

x2 n!

tn

4.41 '1= 0.0600, '2= 0.10182, (%d)1= 49.94, (%d)1= 96.15 rad/s

/j = 1

%2j

(1! exp

!'j%jt

"cos((%d)jt) +

'%j

(%d)jsin((%d)jt)

#)h (t)

q=

!

""#0.1452 !0.2808

0.1986 0.1028

$

%%&

'(()((*

/1

/2

+((,((-

4.43

0 20 40 60 80 100 120 1400.005

0

0.005

q1 p!

tp

4.45

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.01

0

0.01

q

1 n!

q2 n!

q3 n!

tn

4.47 Y1= !0.2036 + 0.0070i, Y2= !0.2707 + 0.0176i m

4.49 a=c = 0.5547/$

m, b= 0.8321/$

m, != !33.69o

4.51

0 1 2 3 410

0

10

Re Y1 n!

Im Y1 n!

#n


17/67

17

4.53 mgL

$0|!1|= 1.624,

mgL

$0|!2|= 4.898,

mgL

$0|!3|= 8.254

4.55 |y1|= 0.1538 m, |y2|= 0.0926 m, vcr = 1.98,2.96 m/s

4.57 |y|= [0.1084 0.0704 0.0080 0.0005]T m@ %= 1.182 rad/s

|y|= [0.1114 0.0529 0.0045 0.0013]T m@ %= 1.306 rad/s

4.59 {q}= T"X

j=1

"{#j} (#1jP1+ #2jP2)

"Xn=!"

exp(i2$nt/T)

%2j T2 ! 4$2n2

#


18/67

18

Chapter 5

5.1 q1= 6.71sin(20t! 2.29) , q2= 17.53 sin(20t! 2.292) mm

5.3

0 0.5 1 1.5 2 2.5 310

5

0

5

10

Re X1 n!

Im X1 n!

#n

5.5

0 5 10 15 201 &10

4

1 &103

0.01

0.1

Front, c = 5 kN-s/mBack, c = 5 kN-s/m

Front, c = 0.5 kN-s/m

Back, c = 0.5 kN-s/m

Speed (m/s)

Amplitude(m)

5.7 For # = 20o :

0 10 200

0.05

0.1

$ xn

$ yn

vn

0 10 200

0.05

0.1

y Gn

vn

5.9 += 0.004 :

{Y}= [0.807exp(1.234i) 0.417exp(!2.018i) 0.3372exp(!1.903i)]m


19/67

19

5.11

0 2 4 6 8 10 121 &10

6

1 &105

1 &104

1 &103

0.01

0.1

Floor 4Floor 3Floor 2Floor 1

Frequency (rad/s)

Amplitude(m)

5.13 For += 0.0001 :

0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.1

1

10

100

1 &103

1 &104

Z6 k!

Z7 k!

Z8 k!

Z9 k!

Z10 k!

#k

5.15 m >4.46 kg and k (N/m) = 400m(kg)

5.17 (a)k2= 63.2 kN/m, (b)k2= 35.5 kN/m, (c)|y1|= 0.1944,m, |y2|= 0.250,m

5.19 (a) k2= 266.7 N/m

(b)!1= 2.30(10!3)sin(20t! 1.905) , !2= 7.32(10!3)sin(20t! 2.247) rad

5.21

0 0.5 1

5

0

5

Case (a)

q a1 p!

q a2 p!

tp

2'&

0 0.5 15

0

5Case (b)

qb1 p!

qb2 p!

tp

2'&


20/67

20

5.23 (a) M11= m1+ m3, M22= m1R2 + m3(`

2 +023)

M33= m2, M44= m2022

K11= 2 (k1+ k2) , K22= 0.0648k1+ 0.0128k2

K33= 2k2, K44= 0.0128k2

K13= K31= !2k2, K24= K42= !0.0128k2

C11= C33= c, C13= !c

Q1= ,m%2 cos(%t) , Q2= !,m`%2 sin(%t)

(b)m2= k22%2

,

(c)

0 50 100 150 2001 &10

6

1 &105

1 &104

1 &103

0.01

0.1

1

10

100

Y1 n!

Y2 n!

Y3 n!

Y4 n!

%n

5.25

0 0.1 0.2 0.30

0.001

0.002

0.003

x1( )

j

x2( )

j

tj


21/67

21

5.27

0 0.2 0.4 0.6 0.8 10.01

0

0.01

0.02

qj 1!

qj 2!

qj 3!

tj

5.29 %1= 5.734, %2= 18.478rad/s, '1= 0.0019, '2= 0.0020

[#] =

!""#

0.0124 !0.298

0.0514 0.213

$%%&


22/67

22

Chapter 6

6.1 [M] =

!""""""#

2.036 1.527 1.221

1.527 1.221 1.018

1.221 1.018 0.872

$%%%%%%&

kg, [K] = 108

!""""""#

3.403 2.202 1.801

2.202 1.801 1.601

1.801 1.601 1.481

$%%%%%%&

N/m

Cjn = 4000 N-s/m, {Q}= F[0.50 0.25 0.125]T

6.3 #j = (x/L)j!1 , [M] =1AL

!""""""#

1 0.5 0.333

0.5 0.333 0.25

0.333 0.25 0.2

$%%%%%%&

, {Q}= F

'(((((()((((((*

1

1

1

+((((((,((((((-

[K] =EA

L

!""""""#

0 0 0

0 1 1

0 1 1.333

$%%%%%%&

+ k

!""""""#

0.5 0.167 0.056

0.167 0.056 0.019

0.056 0.019 0.006

$%%%%%%&

6.5 #j = sinj$x

L

, [M] =1A0L

!""""""""""#

0.75 !0.09 0 !0.007

!0.09 0.75

!0.097 0

0 !0.097 0.75 !0.099

!0.007 0 !0.099 0.75

$%%%%%%%%%%&

[K] =EA0

L

!""""""""""#

0.75 !0.113 0 !0.015

!0.113 0.75 !0.105 0

0 !0.105 0.75 !0.103

!0.015 0 !0.103 0.75

$%%%%%%%%%%&


23/67

23

6.7 Mjn = 91JL cos

(j ! 1) $

2

cos

(n! 1)$

2

+1JL

'(((((()((((((*

1ifj = n = 1

0.5ifj = n >1

0 otherwise

+((((((,((((((-

Kjn =GJ

L

'(()((*

1

2$2 (j ! 1) (n! 1) + 2ifj =n >1

2 otherwise

+((,((-

Qj = $ cos

(j ! 1)$

2

6.9 #j =

xLj

, Mjn = 1j+ n + 11JL + If, Kjn = jnj+ n! 1 GJL

Cjn = -

"1

3

j+n+

2

3

j+n#, Qj = !$

6.11 #j =x

L

j+1

, Mjn = 1

j+ n + 31AL, Kjn =

(j2 +j) (n2 + n)

j+ n! 1EI

L3 +

k

2j+n+2

Cjn = c

2j+n+2, Qj =F

6.13 #j = sinj$x

L , Mjn =1

21AL(jn + m sin

j$

4 sinn$

4 Kjn =

$4j2n2

2EIL3

(jn

6.15 #j = x

L

1! x

L

, q4= y of the block

[M] =1AL

!""""""""""#

0.0244 0 !0.0072 0

0 0.0171 0 0

!0.0072 0 0.0168 0

0 0 0 0.25

$%%%%%%%%%%&


24/67

24

[K] =EI

L3

!""""""""""#

15.99 0 !10.08 !15

0 66.20 0 0

!10.08 0 224.83 15!15 0 15 60

$%%%%%%%%%%&

, {Q}=F

'(((((((((()

((((((((((*

0.25

0

!0.250

+((((((((((,

((((((((((-6.17

0 0.5 10

1 &104

2 &104

3 &104

Position (x/L)

Rotation(rad)

0 0.5 10

2 &107

4 &107

6 &107

Position (x/L)

Rotation(rad)

%= 0.95%2

%= 0.95%1

%= 1.05%1%= 1.05%2

6.19

0 500 1000 1500 2000 2500 3000 3500 40001 &10

7

1 &106

1 &105

1 &104

1 &103

0.01

X1 k!

X2 k!

X3 k!

%k

6.21 %= [0.700 3.490 7.7641]T (E/1L2)1/2

0 0.2 0.4 0.6 0.8 1 1.22

0

2

4

*p 1!

*p 2!

*p 3!

xp

6.23 {%}= [0.860 3.426 6.664]T (E/1L2)1/2


25/67

25

0 0.2 0.4 0.6 0.8 12

1

0

1

2

Mode 1

Mode 2Mode 3

6.25 N= 4 :{%}= [5.355 17.072 51.094 196.706]T (EI /1AL4)1/2

0 0.2 0.4 0.6 0.8 12

0

2

4

*p 1!

*p 2!

*p 3!

*p 4!

x

6.27 Case 1: k= 20EI/L3

,First three modes:

{%}= [4.56 22.95 61.72]T (EI /1AL4)1/2

0 0.2 0.4 0.6 0.8 13

2

1

0

1

2

0*

p 1!

*p 2!

*p 3!

0.5

x

p

6.29 Form = 21AL:

{%}= [9.019 61.825 94.222 202.384]T (EI /1AL4)1/2

, %rb = 9.798(EI /1AL4)

1/2


26/67

26

0 0.2 0.4 0.6 0.8 12

1

0

1

2

*p 1!

*p 2!

*p 3!

*p 4!

xp

6.31

0 0.2 0.4 0.6 0.8 11 &10

9

1 &108

1 &107

1 &106

1 &105

1 &104

1 &103

Yp 1!

Yp 2!

Yp 3!

xp

6.33

0 0.1 0.2 0.3 0.4 0.50

0.01

0.02

Dispp

t

6.35 Results forN= 4 :

0 0.2 0.4 0.6 0.8 10.5

0

0.5

1Initial displacement

u ap 1!

u 0 xp

xp


27/67

27

0 0.2 0.4 0.6 0.8 10.4

0.2

0

1/4 period1/2 period

1 period

6.37

0 1 2 3 4 50.001

5 &104

0

5 &104

0.001

w1 p!

tp

6.39

%

0.447

0

0

0

0

0

0

0

0.364

4.251

0

0

0

0

0

0

0.328

3.926

6.388

0

0

0

0

0

0.328

3.497

6.195

13.697

0

0

0

0

0.324

3.379

4.539

13.577

18.469

0

0

0

0.324

3.378

4.537

9.773

18.361

27.836

0

0

0.323

3.329

4.345

9.748

11.461

27.75

35.293

0

0.323

3.329

4.345

9.606

11.451

16.795

35.213

46.975

+

6.41

%

2.415

0

0

0

0

0

2.409

5.744

0

0

0

0

2.406

5.527

9.939

0

0

0

2.405

5.522

8.666

15.312

0

0

2.405

5.521

8.657

11.946

21.974

0

2.405

5.521

8.655

11.794

15.546

29.944

+


28/67

28

0 0.5 10

0.5

1

1.5

2

N = 1N = 2N = 3N = 4N = 5N = 6

First mode

Position (x/L)

0 0.5 15

0

5

10

15

20

N = 4N = 5N = 6

Fourth mode

Position (x/L)

6.43

%

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 10

6.979 6.866 6.855 6.855 6.853 6.852 6.851 6.851 6.851 6.851

0 28.376 28.023 28.023 27.981 27.941 27.93 27.93 27.926 27.921

0 0 80.579 80.579 80.398 80.229 80.184 80.184 80.167 80.146

0 0 0 157.914 157.914 157.914 157.914 157.914 157.914 157.914

0 0 0 0 226.132 220.716 219.38 219.38 218.943 218.372

0 0 0 0 0 316.433 313.472 313.472 312.703 311.735

0 0 0 0 0 0 463.531 463.531 462.703 461.71

0 0 0 0 0 0 0 631.655 631.655 631.655

0 0 0 0 0 0 0 0 763.438 751.843

0 0 0 0 0 0 0 0 0 922.521

+

6.45

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

N = 4

N = 6N = 8

First mode


29/67

29

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

1

0

1

2

N = 4

N = 6N = 8

Second mode

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

1

0

1

2

N = 4

N = 6N = 8

Third mode

6.47

%

6.16

25.212

0

0

00

0

0

0

6.16

25.212

62.135

0

00

0

0

0

6.153

25.117

62.135

122.645

00

0

0

0

6.153

25.117

61.825

122.645

202.3840

0

0

0

6.149

25.116

61.825

121.811

202.384302.761

0

0

0

6.149

25.116

61.74

121.811

200.988302.761

422.755

0

0

6.149

25.11

61.74

121.554

200.988300.686

422.755

562.961

0

6.149

25.11

61.708

121.554

200.453300.686

420.024

562.961

722.876

+

6.49 At x = L/2, |w|= 2.718!0 meter

6.51

0 0.2 0.4 0.6 0.8 10

50

100

up

xp


30/67

30

6.53

0 5000 1 &104

1.5 &104

2 &104

2.5 &104

3 &104

3.5 &104

4 &104

100

1 &10

3

1 &104

1 &105

1 &106

1 &107

Method (a)Method (b)

Frequency (rad/s)

Force(N)

6.55 For[%] =

!""""""#

1 1

1 0

1 !1

$%%%%%%&

, {%}= [5.01 19.36] rad/s

1 2 30

0.01

0.02

0.03

Exact

Ritz

First mode

Generalized coordinate

1 2 30.05

0

0.05

0.1

Exact

Ritz

Second mode

Generalized coordinate


31/67

31

6.57 %1= [7.769 4.302 3.620]T (EI /1AL4)

1/2

0 2 4 6 80.5

0

0.5

1

,'j 1!

, fundj 1!

, fundj 2!

, fundj 3!

j


32/67

32

Chapter 7

7.1 Axial force:E A2u

2x=ku

Torsion moment:GJ2!

2x=0!

Bending moment:EI22w

2x2 =0

2w

2x

Shear force: ! 22x

EI

22w

2x2

= kw

7.3 EI24w

2x4 +1A w= 0

w=2w

2x = 0 @ x = 0

EI22w

2x2

+b

2

EI23w

2x3

+b2

6

m2

..w

2x

= 0 @ x = L

EI23w

2x3!m

w+

b

2

2..w

2x

= 0@ x =L

7.5 %1= 0, #1= C11, %j =

E

1L2

1/2j$, #j =C1jcos

j$x

L

ifj% 2

7.7 tan(*) = GJ1L

IpG*2 ! 01L2, #j =C2jsin

*jxL

7.9 {%}= [22.37 61.67 120.90]T (EI /1AL4)

1/2

0 0.2 0.4 0.6 0.8 12

1

0

1

2

Mode 1Mode 2Mode 3

7.11 tan(*)! tanh(*) +2*2

0 tan(*)tanh(*) = 0

#j =C1j

sin*jx

L

! sin(*j)

sinh(*j)sinh

*jx

L

Asymptotic: *j& j$, 1%low for j = 3

7.13 2sin(*)sinh(*)! *3 [sin(*)cosh(*)! sinh(*)cos(*)] = 0

#j =C1j

sin

*j

x

L

! sin(*j)

sinh(*j)sinh

*j

x

L


33/67

33

7.15 #j&C1j2i

exp

i

(2j ! 1) $x2L

! exp

!i (2j ! 1)$x

2L

7.17 tan(*) + tanh (*) = 0if*6= 0

#j =C1j sin*jx

L + sinh*j

x

L +

cos(*j) + cos (*j)

sin(*j)! sinh(*j)h

cos*j

x

L

+ cosh

*j

x

L

io& C2j

h! cos

*j

x

L

+ sin

*j

x

L

! exp

!*j

x

L

iNote: x axis is reversed in the following graphs.

00.514

2

0

2

ExactAsymptotic

Mode #2

00.512

0

2Mode #8

00.512

0

2Mode #16

7.19 #j& C1jn

sin*j

x

L

! cos

*j

x

L

! exp

!*j

x

L

+ (!1)j exp

h!*j

1! x

L

io

0 0.5 14

2

0

2

ExactAsymptotic

Mode #5

Position (x/L)

0 0.5 12

0

2Mode #10

Position (x/L)


34/67

34

0 0.5 12

0

2Mode #15

position (x/L)

7.21 #j&1

2C1j

n(!1 + i)exp

!i*j

x

L

+ (!1! i)exp

i*j

x

L

! exp

h!*j

1! x

L

io

7.23 Symmetric:

'(()((*

tan*

2

! ,

*= 0

#j =C2jcos*j

x

L

Antisymmetric:

'(()((*

cot*

2

+ ,

* = 0

#j =C1jsin*j

x

L

7.25 Axial Symmetric:

*j =(2j ! 1)

2 $, #j =C1jsin

2*j

x

L

, %j = 2*j

E

1L2

1/2Flexural Symmetric:

tan(*) + tanh (*) = 0, %j = 4*2

j EI

1AL41/2

#j =C2j

cos

2*j

xL

+ cos(*

j)cosh(*j)

cosh

2*jxL

Axial Antisymmetric:

*j = (j ! 1) $, #j =C2jcos

2*jx

L

, j= 2, 3,...; %j = 2*j

E

1L2

1/2Flexural Antisymmetric:

tan(*)! tanh(*) = 0, %j = 4*2j

EI

1AL4

1/2

#j =C1j

sin

2*j xL

+ sin(*j)sinh(*j)sinh

2*j xL

7.27 Symmetric:

cot*

2

+ coth

*2

+

2

*

1AL

m = 0

#j =C2j

cos

*j

x

L

+

sin(*j/2)

sinh(*j/2)cosh

*j

x

L


35/67

35

Antisymmetric:

tan*

2

+ tanh

*2

+

2

*

1AL

m = 0

#j =C1j sin*jx

L!

cos(*j/2)

cosh(*

j/2)

sinh*jx

L

7.29 Example 7.5:

Z L0

1A%k%jdx= (jkZ L0

EId2%kdx2

d2%jdx2

dx +0EI

L3%k(L)%j(L) =%

2j(jk

Exercise 7.12:

Z L0

1A%k%jdx + 1A%k(L)%j(L) =(jkZ L0

EId2%kdx2

d2%jdx2

dx= %2j(jk

7.31 Z L

0

1A%k%jdx + m%k%j+ 512

b2d%k

dx

d%j

dx

! b2

%k

d%jdx

+%jd%kdx

x=0

=(jkZ L0

EId2%kdx2

d2%jdx2

dx +

kS%k%j+ kT

d%kdx

d%jdx

x=L

=%2jk(jk

7.33 At either end: != $

$R4E

(1

2C021(t

0)2 +"X

j=2

C02j*2j

%j

L

2

[1! cos(*jt0)] h (t0)

)

0 2 40

20

40Total rotation

$ totp

tp

0 2 40.1

0

0.1

0.2Rotation due to deformation

$ torp

tp

7.35 w=2F L3

EI

"Xj=1

1

j2$4 (j2 ! 2)sinj$

x

L

sin(j$2))!

jsin(j2$2))

where = v

(v)cr, (v)cr=

EI1AL2

1/2$, )=

EI1AL4

1/2t


36/67

36

7.37 w (x, t) =1AL4F

EI

"Xj=1

vj%j(x)cos(%jt) , vj =1

6

Z 10

[%j(Ly)]

1! y3

+ 3y ! 1

dy

0 2 4 6 80.5

0

0.5

w end1 p!

tp

7.39 u (x, t) = ! 8vL$2cbar

"Xj=1

1

(2j ! 1)2sin

(2j ! 1)$

2

x

L

sin

(2j ! 1)$

2

cbar t

L

2u

2x atx = 0 is a square wave whose amplitude is!v/cbar ,

and whose period is 2L/cbar.

7.41 No damping: |w|=f0L

4

EI (0.512) , 180o out-of-phase from a sine

Structural damping: |w|=f0L

4

EI (0.458) , 153.54o lag relative to a sine

7.43

0 0.5 1 1.50

0.2

0.4

0.6

0.8

w1 p!

w2 p!

w3 p!

tp

0 0.5 1 1.50.003

0.002

0.001

0

w def1 p!

w def2 p!

w def3 p!

tp

7.45 u= B

1! 1

1 + EA/kL

x

L

sin(%t) + B

"Xj=1

C21j

30j1 !

1

1 + EA/kL30j2

%

%2j! %2[% sin(%t)! %jsin (%jt)] sin

*j

x

LF = !B

L1

1 + EA/kLsin (%t) + B

L

"Xj=1

C21j*j30j1 ! 11 + EA/kL3

0

j2

%

%2j! %2[% sin(%t)! %jsin (%jt)]

whereC1j =hR1

0 sin(*jy)

2 dyi!1/2


37/67

37

7.49 (a)wbc=1

2!t2

x

L+

!

2!2[1 ! cos(!t)]

!3 x

2

L2+

x3

L3

7.51 Fbottom=EA Re [ik (B1exp (ikL)!B2exp (!ikL))exp(i%t)]

utop = Re2k" u1exp (i%t)

whereB1=

u1i"

(iEAk+ K+ i%c!m%2) , B2= u1i"

(iEAk !K! i%c + m%2)

" =EAk cos(kL) + (K+ i%c!m%2)sin(kL)

7.53

0 500 1000 1500 20001 &105

1 &104

1 &103

0.01

0.1

1Midpoint displacement

Frequency (rad/s)

Amplitude

7.55

0 50 100 1500

0.002

0.004Midpoint displacement

Frequency (Hz)

Amplitude

7.57 (a) w = - = 0@ x = 0, 2-

2x = 0 and

2w

2x! -= 0 @ x =L

(b)w =2-

2x = 0 @ x = 0, -= 0and0GA

2w

2x! -

+ mw= 0@ x = L

7.59 (a) (#w)j =#!

j=

xL

j

(b) (#w)j =x

L

j

,#!

j=

1! x

L

j

7.63

2 4 6 8 100.01

0.1

1

10

100

kLTimn

kL cln

n


38/67

38

7.65

0 2 4 6 81 &10

71 &106

1 &105

1 &104

1 &103

0.01

0.1

1

w kLp

w classickLp

kLp


39/67

39

Chapter 8

8.1 #u1= x

L, #u2= 1!

x

L

[Me] =161AL

!""# 2 11 2

$%%& , [Ke] = EAL!""# 1 !1!1 1

$%%&

8.3 #1= x

L, #2= 1!

x

L

[Me] =1

61IL

!""#

2 1

1 2

$%%& , [Ke] =

GJ

L

!""#

1 !1

!1 1

$%%&

8.5 {qe}= [u1 w1 !1 u2 w2 !2 u3 w3]T

#u1= 1! 3 xL+ 2 x2

L2, #u2= !xL+ 2 x

2

L2, #u3= 4 xL

! 4 x2

L2

#w1= 1! 11x2

L2+ 18

x3

L3! 8 x

4

L4, #w2= L

x

L! 4 x

2

L2+ 5

x3

L3! 2 x

4

L4

#w3= !5x2

L2+ 14

x3

L3! 8 x

4

L4, #w4= L

x2

L2! 3 x

3

L3+ 2

x4

L4

#w5= 16x2

L2! 32 x

3

L3+ 16

x4

L4

8.9 [R] =

!

""#cos(") sin (")

! sin(") cos (")

$

%%& , [Re

] =

!""""""#

[R] [0] [0]

[0] [R] [0]

[0] [0] [R]

$%%%%%%&

8.11 {qe}= [wg1 !g1 wg2 !g2 wg3 !g3]T

S1jj = 1, S2

j(j+3)= 1forj = 1,..., 4, S1

jn = S2

jn = 0otherwise

hMi

=

!""""""""""""""""""#

0.4457 0.0754 0.1543 !0.0446 0 0

0.0754 0.0165 !0.0446 !0.0123 0 0

0.1543 !0.0446 0.8914 0 0.1543 !0.0446

!0.0446 !0.0123 0 0.0329 !0.0446 !0.0123

0 0 0.1543 !0.0446 0.4457 !0.0754

0 0 !0.0446 !0.0123 !0.0754 0.0165

$%%%%%%%%%%%%%%%%%%&


40/67

40

hKi

=

!""""""""""""""""""#

6.944 4.167 !6.944 4.167 0 0

4.167 3.333 !4.167 1.667 0 0

!6.944 !4.167 13.889 0 !6.944 4.1674.167 1.6670 0 6.667 !4.167 1.667

0 0 !6.944 !4.167 6.944 !4.167

0 0 4.167 1.667 !4.167 3.333

$%%%%%%%%%%%%%%%%%%&

8.13 DeneXaxis horizontal, number mesh points from the left.

{q}= [ug1 wg1 !g1 ug2 wg2 !g2 ug3 wg3 !g3 ug4 wg4 !g4]T

Forj = 1,..., 6, n= 1,..., 12 :

'(((((()((((((*

S1jj =S2j(j+3)= 1

S31,10= S32,11= S

33,12= S

34,7= S

35,8= S

36,9= 1

Skjn = 0otherwise

[+e] =1

6L

!""#

2 1

1 2

$%%& fore = 1, 2, 3

{f1}= 0 0 0 0 f2

0T

, {f2}= 0 f2

0 0 f 0T

{f3}= [0 0 0 0 0 0]T , {F}= [H1 V1 0 0 0 0 ! F 0 0 H4 V4 M4]T

{Q}= {F}+3X

e=1

[Se]T [Re]T [+e] {fe} with "1= "2= 30o, "3= 90

o

8.15 {q}= [ug1 wg1 !g1 ug2 wg2 !g2 ug3 wg3 !g3 ug4 wg4 !g4]T

{qc}= [ug1 ug4 wg4 !g4]T = [0 0 0 0]T

A1,9= A2,1= A3,2= A4,3= A5,4= A6,5= 1

A7,6= A8,7= A9,8= A10,10= A11,11= A12,12= 1

Aj,n= 0 otherwise

8.17 DeneXaxis horizontal, number mesh points from the bottom left.

{q}= [ug1 wg1 !g1 ug2 wg2 !g2 ug3 wg3 !g3 ug4 wg4 !g4]T


41/67

41

{qf}= [ug2 wg2 !g2 ug3 wg3 !g3]T

{%}= [796 3116 7283 16325 17399 24628]T rad/s

-

0

12

3

4

5

6

7

8

9

10

11

0 1 2 3 4 5

0 0 0 0 0 0

0 0 0 0 0 00 0 0 0 0 0

0.7418 0.0008 0.2915 0.3041 0.168 1.0042

0.0016 0.0286 -0.0616 -0.832 0.9752 0.5718

-1.1906 8.5135 -16.2526 5.7351 -8.2001 -7.6307

0.7418 -0.0008 0.2915 -0.3041 0.168 -1.0042

-0.0016 0.0286 0.0616 -0.832 -0.9752 0.5718

-1.1906 -8.5135 -16.2526 -5.7351 -8.2001 7.6307

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

+

8.19 {q}= [uA wA !A uB wB !B uC wC !C uD wD !D]T

{%}= [0 849.8 3674 4070 5657 9049 15273 17802 21365 31862]T rad/s

-

1

2

3

4

5

67

8

9

10

11

12

13

1 2 3 4 5 6 7 8 9 10

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0.517 1.187 4.14 3.973 0.955 9.587 1.247 9.782 0.401 0.732

-0.129 -0.103 -0.079 0.131 0.13 0.04 -0.038 -0.005 0.016 0.396

0.224 0.176 0.077 -0.089 0.306 -0.251 -0.098 0.001 -0.379 0.062

0.517 -1.019 -2.554 -3.527 1.421 3.659 0.725 10.79 -2.402 -0.318-0.299 -0.144 -3.051 2.403 -1.944 1.157 8.56 0.906 9.316 0.509

-0.259 0.431 -0.162 -0.238 0.486 0.334 -0.417 -0.428 0.78 -0.317

0.448 -0.75 0.216 0.562 -0.218 -0.86 -0.221 1.098 -0.099 -0.309

0.517 -2.568 1.787 4.794 -4.466 -12.683 0.382 23.831 -10.07 -2.473

-0.129 -0.104 -0.077 0.136 0.161 0.046 0.111 -0.076 -0.219 -0.439

0.075 0.06 0.044 -0.079 -0.093 -0.027 -0.064 0.044 0.126 0.254

-0.299 -0.314 2.95 -2.374 1.269 -2.042 6.862 1.373 6.729 3.702

+


42/67

42

Chapter 9

9.1 (a)#j = x

Lsin

j$

x

L

, a1j = 0.5sin(0.5j$) ; j = 1,...,N

(b) Left segment: #1wj =

2xLj+1

; Right segment: #2wj = sin

2j$xL

Forj = 1,...,N : a1j = 1, a1(j+N)= 0, a2j =

2 (j+ 1)

L , a2(j+N)= !

2j$

L cos(j$)

9.3 Bar 1 is horizontal, #1uj=x1

L

j

, #1wj =x1

L

j+1

, #2uj =#2wj =

x2L

j

Forj = 1,...,N : a1j = 1, a1(j+2N)= 0.5, a1(j+3N)= 0.866

a2(j+N)= 1, a2(j+2N)= !0.866, a2(j+3N)= 0.5

a3(j+N)= j+ 1

L

, a3(j+3N)= j

L

, anj = 0 otherwise

9.5 #`wj =x`

L

j+1

, #`"j =x`

L

j

; `= 1, 2

Forj = 1,...,N : a1j = 1, a1(j+2N)= !1

a2j =j+ 1

L1, a2(j+3N)= 1

a3(j+N)= j+ 1

L , a3(j+2N)= !

j+ 1

L2, anj= 0 otherwise

9.7 #j = x

Lsin

j$

x

L[M1] {q1}+ [K1] {q1}= {Q1}+ [a1] {.} ,where

[M1] =1AL

!""""""""""""""#

0.1413 0.0901 0.019 !0.0072 0.0035

!0.0901 0.1603 !0.0973 0.0225 !0.0092

0.019 !0.0973 0.1639 !0.0993 0.0237

!0.0072 0.0225 !0.0993 0.1651 !0.1001

0.0035 !

0.0092 0.0237 !

0.1001 0.1657

$%%%%%%%%%%%%%%&


43/67

43

[K1] =EI

L3

!""""""""""""""#

43.4 !74.6 75.9 !90.2 107.3

!74.6 368.3 !459.5 298.3 !286.9

75.9 !459.5 1559.3 !1629.1 816.6!90.2 298.3 !1629.1 4590.4 !4293.9

107.3 !286.9 816.6 !4293.9 10825.3

$%%%%%%%%%%%%%%&

[a1] = [0.5 0 ! 0.5 0 0 0.5]

{Q1}= [0.5303 ! 0.75 0.5303 0 ! 0.5303]T

9.9 #1uj=

x1L1

j, #1wj =

x1L1

j+1, #2uj =

x2L2

j!1, #2wj =

x2L2

j!1

#3uj =

x3L3

j, #3wj =

x3L3

j+1, whereL1= 2 m, L2= L3= 4m

{q}=h

{q1u}T

{q1w}T

{q2u}T

{q2w}T

{q3u}T

{q3w}TiT

(M1u)jn = 1AL1

j+ n + 1, (M1w)jn =

1AL1j+ n + 3

, (M2u)jn = 1AL2

j+ n! 1 , (M2w)jn =

1AL2j+ n! 1

(M3u)jn = 1AL3

j+ n + 1, (M3w)jn =

1AL3j+ n + 3

(K1u)jn =EA

L1

jn

j+ n! 1 , (K1w)jn =

EI

L31

(j+ 1)j (n + 1) n

j+ n! 1

(K2u)jn =

'(()((*

0ifj orn = 1

EA

L2

(j ! 1) (n! 1)j+ n! 3 otherwise

(K2w)jn =

'(()((*

0ifj orn = 1or 2

EA

L2

(j ! 1) (j ! 2) (n! 1) (n! 2)j+ n! 5 otherwise

(K3u)jn =EA

L3

jn

j+ n! 1 , (K3w)jn =

EI

L33

(j+ 1)j (n + 1) n

j+ n! 1

(Q1

u)j = (Q1

w)j = (Q1

u)j = (Q1

w)j = (Q1

u)j = 0, (Q3

w)j = 1

a1,j= 1, a1(3N+1)= !1, a2(j+N)= 1, a2(2N+1)= !1

a3(j+2N)= a3(j+5N)= 1, a4(j+3N)= 1, a4(j+4N)= !1

a5(j+3N)=j ! 1

L2, a5(j+5N)= !

j+ 1

L3


44/67

44

!""""""""""""""""""#

[M1u] [0] [0] [0] [0] [0]

[0] [M1w] [0] [0] [0] [0]

[0] [0] [M

2

u ] [0] [0] [0]

[0] [0] [0] [M2w] [0] [0]

[0] [0] [0] [0] [M3u] [0]

[0] [0] [0] [0] [0] [M3w]

$%%%%%%%%%%%%%%%%%%&

{q}

+

!""""""""""""""""""#

[K1u] [0] [0] [0] [0] [0]

[0] [K1w] [0] [0] [0] [0]

[0] [0] [K2u] [0] [0] [0]

[0] [0] [0] [K2w] [0] [0]

[0] [0] [0] [0] [K3u] [0]

[0] [0] [0] [0] [0] [K3w]

$%%%%%%%%%%%%%%%%%%&

{q}= {Q}+ [a]T {.}

[a] {q}={0}

9.11 #1wj = x1

L

1! x1

L

sin

2j$x1

L

, #2"j =

x2L3

j!1!""#

1AL [M1] [0]

[0] 1JL [M2]

$%%&

'(()((*

{q1}

{q2}

+((,((-

+

!""#

EI

L3 [K1] [0]

[0] GJ

L [K2]

$%%&

'(()((*

{q1}

{q2}

+((,((-

=

'(()((*

{0}

{Q2}

+((,((-

+ [a]T {.}

[a] {q}={0}


45/67

45

[M1] =

!""""""""""#

0.017 !7.604 (10!3) !4.511 (10!4) !8.274 (10!5)

!7.604 (10!3) 0.017 !7.687 (10!3) !4.753 (10!4)

!4.511 (10!4

) !7.687 (10!3

) 0.017 !7.696 (10!3

)

!8.274 (10!5) !4.753 (10!4) !7.696 (10!3) 0.017

$%%%%%%%%%%&

[M2] =

!""""""""""#

1 0.5 0.333 0.25

0.5 0.333 0.25 0.2

0.333 0.25 0.2 0.167

0.25 0.2 0.167 0.143

$%%%%%%%%%%&

[K1] =

!""""""""""#

66.20 !47.41 !6.33 !2.06

!47.41 574.3 !431.3 !47.41

!6.33 !431.3 2460 !1727

!2.06 !47.41 !1727 7282

$%%%%%%%%%%&

[K2] =

!""""""""""#

0 0 0 0

0 1 1 1

0 1 1.333 1.5

0 1 1.5 1.8

$%%%%%%%%%%&

, {Q2}= [!R 0 0 0 ]T

9.13

0 0.2 0.4 0.6 0.8 12

1

0

1

2

*p 1!

*p 2!

*p 3!

*p 4!

xp


46/67

46

9.15

0 50 100 150 200 250 300 350 401 &10

111&10

101&1091&

10

81&10

71&10

61&10

51&10

41&10

30.010.1

110

Vertical

Horizontal

Displacement at left force

Frequency (Hz)

0 50 100 150 200 250 300 350 401 &10

111&

10

101 &10

91 &10

8

1 &107

1 &106

1 &105

1 &104

Vertical

Horizontal

Displacement at right force

Frequency (Hz)

9.17

0 0.2 0.4 0.6 0.8 1100

1 &103

1 &104

1 &105

1 &10

Horizontal

Vertical

Frequency (nondim)

Displacement(nondim)

9.19

0 500 1000 1500 2000 25001 &10

8

1 &107

1 &106

1 &105

1 &104

DisplacementRotation 1Rotation 2


47/67

47

9.21 Fixed interface modes:

Left: Clamped-clamped normal modes, Right: Hinged-clamped normal modes

Constraint modes: #C`1 =L

2"!

2x`

L2

+ 2x`

L3

# , `= 1, 2{q}=

qF1w

qC11

qF2w

qC22

Ta1(N+1)= a1(2N+2)= 1, a1j = 0 otherwise

9.23 {q}=h

qF1wT

qF1"T

qC1w1 qC1w2 q

C1"1

qF2w

TqF2"

T

qC2w1 qC2w2 q

C2"1 q

C2w3 q

C2w4 q

C2"2

T#C1w1=#

Cw(x1/L1) , #

C1w2=#

C!(x1/L1) , #

C1"1 =#

C" (x1/L1)

#C2

w1=#C

w(x2/L2) , #C2

w2=#C!(x2/L2) , #

C2"1 =#

C" (x2/L2)

#C2w3=#Cw(1! x2/L2) , #C224 =#C!(1 ! x2/L2) , #C2"2 =#C" (1! x2/L2)

a1(2N+1)= 1, a1(4N+4)= !1, a2(2N+2)= a2(4N+6)= 1, af3(2N+3)= 1, a3(4N+5)= !1

9.25 See Answer 9.21 for basis function denitions and[a]

M`

=

!""#

[I]44 (1AL3)

1/2n

M`F Co

(1AL3)1/2

nM`F C

oT

0.0011901AL3

$%%&

n(M1)

F Co

T= [!0.03161 0.01147 ! 0.00585 0.00354]n

(M2)F CoT

= [!0.04551 0.01620 ! 0.00827 0.00500]

K`

= EI

1AL4

!""#

hK`F Fi

44(1AL3)

1/2n

K`F Co

(1AL3)1/2n

K`F CoT

8 (1AL3)

$%%&

(K1)F F1,1 = 8009, (K

1)F F2,2 = 60857, (K

1)F F3,3 = 233882, (K

1)F F4,4,= 639101

(K

2

)

F F

1,1 = 3804, (K

2

)

F F

2,2 = 3994, (K

2

)

F F

3,3 = 173881, (K

2

)

F F

4,4 = 508582n(K1)

F CoT

= [0 0 0 0] ,n

(K2)F CoT

= [64.6 113.0 163.4 213.6]

[M] =

!""#

[M1] {0}

{0}T [M2]

$%%& , [K] =

!""#

[K1] [0]

[0] [K2]

$%%&


48/67

48

9.27

0 500 1000 1500 2000 25001 &10

8

1 &107

1 &106

1 &105

1 &104

Displacement

Rotation of bar 1Rotation of bar 2

9.29 {%}= [57.88 138.54 202.23 425.12]T rad/s

Dene globalX Y coordinate system with Xto the right and Y upward.

0 5 100.02

0

0.02

0.04

0.06

X displacementY displacement

First mode

0 5 100.05

0

0.05

0.1


Second mode

0 5 100.05

0

0.05

0.1


Third mode

0 5 100.1

0.05

0

0.05

0.1


Fourth mode


49/67

49

Chapter 10

10.1 {.}= [!3.061 + 10.303i ! 3.061! 10.303i ! 5.189 + 12.530i ...

!5.189! 12.530i]T

[#] =

!""""""""""#

0.074! 0.001i 0.074 + 0.001i 0.010 + 0.062i 0.010! 0.062i

0.055 + 0.011i 0.055! 0.011i !0.016! 0.036i !0.016 + 0.036i

!0.219 + 0.760i !0.219! 0.760i !0.822! 0.198i !0.822 + 0.198i

!0.285 + 0.534i !0.285! 0.534i 0.529! 0.011i 0.529 + 0.011i

$%%%%%%%%%%&

10.3 [S] =

!""""""""""#

!1500 480 0 0480 !240 0 0

0 0 0.8533 0

0 0 0 0.1067

$%%%%%%%%%%&

, [R] =

!""""""""""#

0 0 !1500 4800 0 480 !240

!1500 48 52 12

480 !240 12 !4

$%%%%%%%%%%&

{.}= [!5.928 + 20.088i ! 5.928! 20.09i ! 43.29 + 37.04i ! 43.29! 37.04i]T rad/s

{#}=

!""""""""""#

!0.015 + 0.009i

!0.015

!0.009i 0.002 + 0.008i 0.002

!0.008i

!0.042 + 0.013i !0.042! 0.013i !0.010! 0.012i !0.010 + 0.012i

!0.089! 0.352i !0.089 + 0.352i !0.356! 0.284i !0.356 + 0.284i

!0.015! 0.930i !0.015 + 0.930i 0.871 + 0.182i 0.871! 0.182i

$%%%%%%%%%%&

10.5 . 1 0.502 0.867i+ .3 0.516 0.859i+

5 6 7 8 9 100.5

0

0.5

1

RealImag

Mode 1

0 2 4 6 8 101

0.5

0

0.5Mode 2


50/67

50

0 2 4 6 8 100.5

0

0.5

1Mode 3

.5 0.5 0.869i+

0 2 4 6 8 100.5

0

0.5

1Mode 4

.7 0.512 0.862i+

0 2 4 6 8 100.5

0

0.5

1Mode 5

.9 0.498 0.87i+

0 2 4 6 8 100.5

0

0.5

1Mode 6

.11 0.508 0.864i+

0 2 4 6 8 100.5

0

0.5

1Mode 7

.13 0.489 0.875i+

0 2 4 6 8 100.5

0

0.5

1Mode 8

.15 0.487 0.877i+

0 2 4 6 8 100.5

0

0.5

1Mode 9

.17 0.491 0.874i+

0 2 4 6 8 100.5

0

0.5

1Mode 10

.19 0.497 0.871i+


51/67

51

10.7 {.}= [!1.323 + 6.700i ! 1.323! 6.700i ! 2.064 + 7.949i ! 2.064! 7.949i]T

[#] =

!"""

"""""""#

0.028 + 0.043i 0.028! 0.043i !0.017 + 0.043i !0.017! 0.043i

!0.014 + 0.029i !0.014! 0.029i !0.017! 0.026i !0.017 + 0.026i

!0.326 + 0.133i !0.326! 0.133i !0.303! 0.226i !0.303 + 0.226i

!0.178! 0.130i !0.178 + 0.130i 0.239! 0.084i 0.239 + 0.084i

$%%%

%%%%%%%&

(%nat )1= 6.829, '1= 0.194, (%nat )2= 8.212, '2= 0.251

10.9 Case (a):

{.}= g

L

1/2[1.225i ! 1.225i ! 0.0038 + 1.259i ! 0.0038! 1.259i ...

!0.011 + 1.324i

!0.011

!1.324i]T

{%1}={%#

2}=

'(((((((((((((((((()((((((((((((((((((*

0.577i

0.577i

0.577i

!0.707

!0.707

!0.707

+((((((((((((((((((,((((((((((((((((((-

, {%3}= {%#

4}=

'(((((((((((((((((()((((((((((((((((((*

!0.001! 0.688i

0

0.001 + 0.688i

0.866 + 0.001i

0

!0.866! 0.001i

+((((((((((((((((((,((((((((((((((((((-

{%5}={%#

6}=

'(((((((((((((((((()((((((((((((((((((*

!0.002! 0.378i

0.003 + 0.755i

!0.002! 0.378i

0.500 + 0.002i

!1.000! 0.004i

0.500 + 0.002i

+((((((((((((((((((,((((((((((((((((((-


52/67

52

Case (b):

{.}= g

L

1/2[!0.431 1.225i ! 1.225i ! 0.75 + 1.011i ...

!0.75! 1.011i ! 4.069]T

{%1}=

'(((((((((((((((((()((((((((((((((((((*

!0.565i

1.130i

!0.565i

0.243i

!0.487i

0.243i

+((((((((((((((((((,((((((((((((((((((-

, {%2}={%#

3}=

'(((((((((((((((((()((((((((((((((((((*

0.577i

0.577i

0.577i

!0.707

!0.707

!0.707

+((((((((((((((((((,((((((((((((((((((-

{%4}={%#

5}=

'(((((((((((((((((()((((((((((((((((((*

0.241 + 0.729i

0

!0.241! 0.729i

!0.918! 0.303i

0

0.918 + 0.303i

+((((((((((((((((((,((((((((((((((((((-

, {%6}=

'(((((((((((((((((()((((((((((((((((((*

0.184

!0.368

0.184

!0.748

1.496

!0.748

+((((((((((((((((((,((((((((((((((((((-

10.11

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0

0.1

x1 p!

x2 p!

tp


53/67

53

10.13

0 1 2 3 4 5 6 7 8 9 10

0

0.05

Floor 4Floor 3Floor 2

Floor 1

10.15

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.01

0

0.01

0.02

x1 p!

x2 p!

x3 p!

tp

10.17

0 0.5 1 1.5 2 2.51

0.5

0

0.5

1

X1 p!

X2 p!

tp


54/67

54

10.19

0 1 2 3 4 5 6 7 8 9 100.4

0.2

0

0.2

0.4

0.6

X1 p!

X2 p!

X3 p!

X4 p!

tp

10.21

0 20 40 60 80 100 120 1400.005

0

0.005

x1 p!

tp

0 20 40 60 80 100 120 1400.01

0

0.01

x2 p!

tp

0 20 40 60 80 100 120 140.01

0

0.01

x3 p!

t


55/67

55

10.23

0 0.2 0.4 0.6 0.8 10.005

0

0.005

0.01

x1 p!

tp

0 0.2 0.4 0.6 0.8 15

0

5

x''1 p!

tp

10.25

0 5 10 15 201&

10

3

0.01

0.1

X1 p!

X2 p!

vp


56/67

56

Chapter 11

11.1 mx + (k !m%2) x= ! (k !m%2) R0+ kL0, %< (k/m)1/2 for stability

11.3

4

3 !1+

1

2 !2+

2"!3

2 g

L!3

2

w

L!1 ! "

g

L!2= 0

1

2!1+

1

3!2 ! "

g

L!1+

"! 1

2

g

L! 1

2

w

L

!2= 0

11.5 m [x! 2% y ! %2 (R + x)] + kxx= 0

m [y+ 2% x! %2y] + kyy = 0

mz+ kzz= 0

11.7

!""""""#

1 0 0

0 1 0

0 0 L2/12

$%%%%%%&

'(((((()((((((*

xC

yC

!

+((((((,((((((-

+

!""""""#

0 !2% 0

2% 0 0

0 0 0

$%%%%%%&

'(((((()((((((*

xC

yC

!

+((((((,((((((-

+

!""""""#

(k1+ k2)

m ! %2 0 0

0 (k3+ k4)

m ! %2 (k4 ! k3) L

2m

0 (k

4 !k3) L

2m

(k4+ k

3) L2

4m ! %2

$%%%%%%&

'(((((()

((((((*

xC

yC

!

+((((((,

((((((-

=

'(((((()((((((*

0

%2H

0

+((((((,((((((-

11.9 [M] {q}+ [[G] + [C]] { q}+ [[K]! [E]] {q}= {J}

where [M] =m

!""""""#

1 0 0

0 1 !L/2

0 !L/2 L2/3

$%%%%%%&

, [G] =m

!""""""#

0 !

2% L%

2% 0 0

!L% 0 0

$%%%%%%&


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[E] =m

!""""""#

%2 0 0

0 %2 !L%2/2

0 !L%2

/2 Lr%2

/2

$%%%%%%&

, [K] =

!""""""#

k1 0 0

0 k2 0

0 0 k3

$%%%%%%&

{J}=m

'(((((()((((((*

(r ! L/2)%2

0

0

+((((((,((((((-

11.11 {.}= [0.6i ! 0.6i 1i ! 1i 1.4i ! 1.4i]T

[#] =

!

""""""""""""""""""#

!0.606

!0.606 0 0 0.411 0.411

0.606i !0.606i 0 0 0.411i !0.411i

0 0 !0.707 !0.707 0 0

!0.364i 0.364i 0 0 0.575i !0.575i

!0.364 !0.364 0 0 !0.575 !0.575

0 0 !0.707i 0.707i 0 0

$

%%%%%%%%%%%%%%%%%%&

h#i

=

!""""""""""""""""""#

0.606 0.606 0 0 0.411 0.411

0.606i !0.606i 0 0 !0.411i 0.411i

0 0 !0.707 !0.707 0 0

0.364i !0.364i 0 0 0.575i !0.575i

!0.364 !0.364 0 0 0.575 0.575

0 0 !0.707i 0.707i 0 0

$%%%%%%%%%%%%%%%%%%&


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0.5 0 0.5

0.5

0

0.5

Mode pair #1

q 12 p!

q 11 p!

0.5 0 0.5

0.5

0

0.5

Mode pair #3

q 32 p!

q 31 p!

0 5 101

0.5

0

0.5

1

q 11 p!

q 12 p!

"

0 5 100.5

0

0.5

q 31 p!

q 32 p!

"

11.13

{.}=

'(((((((((((((((((()

((((((((((((((((((*

!3.75(10!5) + 0.274i

!3.75(10!5)! 0.274i

1.314 (10!3) + 19.868i

1.314 (10!3)!

19.868i

!1.332 (10!3) + 20.132i

!1.332(10!3)! 20.132i

+((((((((((((((((((,

((((((((((((((((((-

, {%1}=

'(((((((((((((((((()

((((((((((((((((((*

!1.531 (10!6)! 5.25(10!10) i

1.150 (10!8)! 5.594 (10!5) i

!0.122! 1.789 (103) i

2.01(10!10)!

4.195(10!7) i

1.532 (10!5) + 5.27(10!9) i

489.898 + 0.034i

+((((((((((((((((((,

((((((((((((((((((-

{%3}=

'(((((((((((((((((()((((((((((((((((((*

!1.532 (10!3)! 0.308i

0.308! 1.532 (10!3) i

4.689 (10!3) + 7.051 (10!5) i

6.127! 0.031i

0.031 + 6.127i

!1.395 (10!3) + 0.093i

+((((((((((((((((((,((((((((((((((((((-

, {%5}=

'(((((((((((((((((()((((((((((((((((((*

!1.542 (10!3)! 0.306i

0.306! 1.542(10!3) i

4.537 (10!3) + 6.79(10!5) i

6.168! 0.031i

0.031 + 6.168i

!1.373 (10!3) + 0.091i

+((((((((((((((((((,((((((((((((((((((-


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59

n%1o

=

'(((((((((((((((((()((((((((((((((((((*

1.532 (10!6) + 5.24(10!10) i

1.15(10!8)! 5.594 (10!5) i

!0.122! 1.789 (103

) i

!2.01(10!10) + 4.20(10!7) i

1.532(10!5) + 5.25(10!9) i

489.898 + 0.034i

+((((((((((((((((((,((((((((((((((((((-

,n%3o

=

'(((((((((((((((((()((((((((((((((((((*

1.532 (10!3) + 0.308i

0.308! 1.532 (10!3) i

4.689(10

!3

) + 7.05(10

!5

) i

!6.127 + 0.031i

0.031 + 6.127i

!1.395 (10!3) + 0.093i

+((((((((((((((((((,((((((((((((((((((-

n%5

o=

'(((((((((((((((((()((((((((((((((((((*

!1.542(10!6)! 0.306i

!0.306 + 1.542 (10!3) i

!4.537(10!3)! 6.791 (10!5) i

6.168! 0.031i

!0.031! 6.168i

1.373 (10!3)! 0.091i

+((((((((((((((((((,((((((((((((((((((-

11.15

{.}=

'(((((((((((((((((()((((((((((((((((((*

0.417i

!0.417i

1.252i

!1.252i

2.481i

!2.481i

+((((((((((((((((((,((((((((((((((((((-

, {%1}=

'(((((((((((((((((()((((((((((((((((((*

0.014i

2.794 (10!3)

!1.540i

!5.812 (10!3)

1.166 (10!3) i

0.643

+((((((((((((((((((,((((((((((((((((((-


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60

{%3}=

'(((((((((((((((((()((((((((((((((((((*

0.277

!0.377i

4.961 (10

!3

)

0.347i

0.472

6.209 (10!3) i

+((((((((((((((((((,((((((((((((((((((-

, {%5}=

'(((((((((((((((((()((((((((((((((((((*

0.248i

!0.212

1.035 (10

!3

) i

!0.616

!0.527i

!2.568 (10!3)

+((((((((((((((((((,((((((((((((((((((-

n%1

o=

'(((((((((((((((((()((((((((((((((((((*

0.014i

!2.794(10!3)

!1.540i

!5.812(10!3)

!1.166(10!3) i

0.643

+((((((((((((((((((,((((((((((((((((((-

,n%3

o=

'(((((((((((((((((()((((((((((((((((((*

!0.277

!0.377i

!4.961 (10!3)

!0.347i

0.472

!6.209 (10!3) i

+((((((((((((((((((,((((((((((((((((((-

n%5

o=

'(((((((((((((((((()((((((((((((((((((*

0.248i

0.212

1.035 (10!3) i

!0.616

0.527i

!2.568(10!3)

+((((((((((((((((((,((((((((((((((((((-

11.17 Unstable for1 < !


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61

11.23 (a) Amplitudes as a function of rotation rate:

0 5 10 15 20 25 30 35 40 45 500.01

0.1

1

10

100

1 &103

Lower y

Lower zUpper yUpper z

(b) Orbits at ! = 0.836:

0.5 0 0.51

0.5

0

0.5

1Lower mass

q2 p!

q1 p!

4 2 0 2 46

4

2

0

2

4

6Upper mass

q4 p!

q3 p!

11.25

0.006 0.004 0.002 0 0.002 0.004 0.0060.006

0.004

0.002

0

0.002

0.004

0.006Upper mass

y displacement

zdisplacement

0.0022 0 0.00220.0032

0

0.0032Lower mass

y displacement

zdisplac

ement


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62

11.27 Motion at % = 0.6 (k1/m)1/2 :

0 5 10 15 20 25 30 350.5

0

0.5

1Relative x displacement

Time (nondimensional)

0 5 10 15 20 25 30 35

0

Relative y displacement


0 5 10 15 20 25 30 350.01

0

0.01Relative rotation


11.29

0 1 2 3 4 54

2

0

2

4Real part of eigenvalues

Rotation rate (nondimensional)


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63

0 1 2 3 4 50

20

40

60

Imaginary part of eigenvalues

Rotation rate (nondimensional)

Divergence instability if% > 3.564(EI /1AL4)1/2

11.31 vcrit=$ (EI /1AL2)

1/2

0 0.5 10.3

0.2

0.1

0

Real

Imag

First mode

x/L

0 0.5 10.04

0.02

0

0.02

0.04

Real

Imag

Third mode

x/L

0 0.5 10.02

0.01

0

0.01

0.02

Real

Imag

Fifth mode

x/L 0 0.5 10.01

0.005

0

0.005

0.01

RealIma

Seventh mode

x/L


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64

Chapter 12

12.1 %nat = 40$ rad/s, 'E= 0.0222, 'I= 0.0111, ,= 0.445 mm

12.3

100 50 0 50 100

50

0

50

Y Cp

X Cp

12.5 Flutter instability at% = 5217 rad/s, Critical speeds are % = 632 and 941 rad/s

12.7

0.9 0.95 1 1.050

5

10

15

20

25

30

Case a: X

Case a: YCase b: XCase b: Y

Case c: XCase c: Y


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65

12.9

0 500 1000 1500 20001 &10

7

1 &106

1 &105

1 &104

1 &103

0.01

0.1

1

Displacement

Transverse rotation

12.11 Equations of motion are (12.4.10) with:

K11 = kY A+ kY B, K22= kZA+ kZB , K33= kZAb2 + kZB(L! b)2

K44 = kY Ab2 + kY B(L! b)2 , K14= K41= !bkY A+ (L! b) kY B

K23 = K32= bkZA ! (L! b) kZB

M11 = M22= m, M33= M44= Iyy, G34= !G43= 2%Ixx

0 100 200 300 400 500 6000

500

1000

1500

2000Campbell diagram

Rotation rate (rad/s)

Eige

nvalue,imaginarypart(rad/s)

Synchronous line

%crit= 219, 240,and424 rad/s


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66

First critical mode

0.005 0 0.0050.005

0

0.005

Z Ap

Y Ap

0.005 0 0.0050.005

0

0.005

Z Bp

Y Bp

Second critical mode

0.004 0 0.0040.004

0

0.004

Z Ap

Y Ap

0.004 0 0.0040.004

0

0.004

Z Bp

Y Bp

Third critical mode

0.002 0 0.0020.002

0

0.002

Z Ap

Y Ap

0.002 0 0.0020.002

0

0.002

Z Bp

Y Bp

12.13 Critical displacements for isotropic bearings:

%1= 220 rad/s, |YC|=|ZC|= 3.40(10!3)m, |"Z|= |"Y|= 4.55(10

!3)rad

%2= 424 rad/s, |YC|=|ZC|= 0.77(10!3)m, |"Z|= |"Y|= 5.28(10

!3)rad

Critical displacements for orthotropic bearings:

%1= 229rad/s |YC|= 3.79(10!3) , |ZC|= 0.20(10

!3) m

|"Z|= 1.23(10!3) , |"Y|= 4.12(10

!3) rad

%2= 317 rad/s |YC|= 0.52(10!3) , |ZC|= 2.82(10

!3) m

|"Z|= 2.21(10!3) , |"Y|= 0.76(10

!3) rad


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67

%3= 424 rad/s |YC|= 0.77(10!3) , |ZC|= 0.76(10

!3) m

|"Z|= 5.28(10!3) , |"Y|= 5.28(10

!3) rad

12.15

5 0 52.5

0

2.5Orthotropic bearings

Y displacement

Zdisplacemen

t

10 5 0 5 1010

5

0

5

10Orthotropic shaft

Y displacement

Zdisplacement

12.17 ! = 0.767 :

10 5 0 5 1010

5

0

5

10Center's Path Relative to Fixed XYZ

q fixed2 p!

q fixed1 p!

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