Upload
puhumight
View
284
Download
2
Embed Size (px)
Citation preview
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(")
8/10/2019 Ginsberg Answers
2/67
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
8/10/2019 Ginsberg Answers
3/67
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, #= ! ! $/2
8/10/2019 Ginsberg Answers
4/67
4
Chapter 2
2.1 F= 500 cos (5$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
8/10/2019 Ginsberg Answers
5/67
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
8/10/2019 Ginsberg Answers
6/67
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)
8/10/2019 Ginsberg Answers
7/67
7
2.67
0 1 2 3 40
5
10
15
acc magj
maxaccj
minaccj
#j
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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, ,
8/10/2019 Ginsberg Answers
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 )
8/10/2019 Ginsberg Answers
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!
8/10/2019 Ginsberg Answers
12/67
12
3.71
0 50 100 150 200 250 3000
5 &104
0.001
0.0015
Gn
%n
8/10/2019 Ginsberg Answers
13/67
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
$%%%&
8/10/2019 Ginsberg Answers
14/67
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
$%%&
8/10/2019 Ginsberg Answers
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)
+((,((-
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
#
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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'&
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
$%%&
8/10/2019 Ginsberg Answers
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
$%%%%%%%%%%&
8/10/2019 Ginsberg Answers
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
$%%%%%%%%%%&
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
+
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
$%%%%%%&
, {%}= [5.01 19.36] rad/s
1 2 30
0.01
0.02
0.03
Exact
Ritz
First mode
Generalized coordinate
1 2 30.05
0
0.05
0.1
Exact
Ritz
Second mode
Generalized coordinate
8/10/2019 Ginsberg Answers
31/67
31
6.57 %1= [7.769 4.302 3.620]T (EI /1AL4)
1/2
0 2 4 6 80.5
0
0.5
1
,'j 1!
, fundj 1!
, fundj 2!
, fundj 3!
j
8/10/2019 Ginsberg Answers
32/67
32
Chapter 7
7.1 Axial force:E A2u
2x=ku
Torsion moment:GJ2!
2x=0!
Bending moment:EI22w
2x2 =0
2w
2x
Shear force: ! 22x
EI
22w
2x2
= kw
7.3 EI24w
2x4 +1A w= 0
w=2w
2x = 0 @ x = 0
EI22w
2x2
+b
2
EI23w
2x3
+b2
6
m2
..w
2x
= 0 @ x = L
EI23w
2x3!m
w+
b
2
2..w
2x
= 0@ x =L
7.5 %1= 0, #1= C11, %j =
E
1L2
1/2j$, #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
8/10/2019 Ginsberg Answers
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)
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
$%%%%%%%%%%%%%%%%%%&
8/10/2019 Ginsberg Answers
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
$%%& fore = 1, 2, 3
{f1}= 0 0 0 0 f2
0T
, {f2}= 0 f2
0 0 f 0T
{f3}= [0 0 0 0 0 0]T , {F}= [H1 V1 0 0 0 0 ! F 0 0 H4 V4 M4]T
{Q}= {F}+3X
e=1
[Se]T [Re]T [+e] {fe} with "1= "2= 30o, "3= 90
o
8.15 {q}= [ug1 wg1 !g1 ug2 wg2 !g2 ug3 wg3 !g3 ug4 wg4 !g4]T
{qc}= [ug1 ug4 wg4 !g4]T = [0 0 0 0]T
A1,9= A2,1= A3,2= A4,3= A5,4= A6,5= 1
A7,6= A8,7= A9,8= A10,10= A11,11= A12,12= 1
Aj,n= 0 otherwise
8.17 DeneXaxis horizontal, number mesh points from the bottom left.
{q}= [ug1 wg1 !g1 ug2 wg2 !g2 ug3 wg3 !g3 ug4 wg4 !g4]T
8/10/2019 Ginsberg Answers
41/67
41
{qf}= [ug2 wg2 !g2 ug3 wg3 !g3]T
{%}= [796 3116 7283 16325 17399 24628]T rad/s
-
0
12
3
4
5
6
7
8
9
10
11
0 1 2 3 4 5
0 0 0 0 0 0
0 0 0 0 0 00 0 0 0 0 0
0.7418 0.0008 0.2915 0.3041 0.168 1.0042
0.0016 0.0286 -0.0616 -0.832 0.9752 0.5718
-1.1906 8.5135 -16.2526 5.7351 -8.2001 -7.6307
0.7418 -0.0008 0.2915 -0.3041 0.168 -1.0042
-0.0016 0.0286 0.0616 -0.832 -0.9752 0.5718
-1.1906 -8.5135 -16.2526 -5.7351 -8.2001 7.6307
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
+
8.19 {q}= [uA wA !A uB wB !B uC wC !C uD wD !D]T
{%}= [0 849.8 3674 4070 5657 9049 15273 17802 21365 31862]T rad/s
-
1
2
3
4
5
67
8
9
10
11
12
13
1 2 3 4 5 6 7 8 9 10
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0.517 1.187 4.14 3.973 0.955 9.587 1.247 9.782 0.401 0.732
-0.129 -0.103 -0.079 0.131 0.13 0.04 -0.038 -0.005 0.016 0.396
0.224 0.176 0.077 -0.089 0.306 -0.251 -0.098 0.001 -0.379 0.062
0.517 -1.019 -2.554 -3.527 1.421 3.659 0.725 10.79 -2.402 -0.318-0.299 -0.144 -3.051 2.403 -1.944 1.157 8.56 0.906 9.316 0.509
-0.259 0.431 -0.162 -0.238 0.486 0.334 -0.417 -0.428 0.78 -0.317
0.448 -0.75 0.216 0.562 -0.218 -0.86 -0.221 1.098 -0.099 -0.309
0.517 -2.568 1.787 4.794 -4.466 -12.683 0.382 23.831 -10.07 -2.473
-0.129 -0.104 -0.077 0.136 0.161 0.046 0.111 -0.076 -0.219 -0.439
0.075 0.06 0.044 -0.079 -0.093 -0.027 -0.064 0.044 0.126 0.254
-0.299 -0.314 2.95 -2.374 1.269 -2.042 6.862 1.373 6.729 3.702
+
8/10/2019 Ginsberg Answers
42/67
42
Chapter 9
9.1 (a)#j = x
Lsin
j$
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
$%%%%%%%%%%%%%%&
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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}
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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]
$%%&
8/10/2019 Ginsberg Answers
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
X displacementY displacement
Second mode
0 5 100.05
0
0.05
0.1
X displacementY displacement
Third mode
0 5 100.1
0.05
0
0.05
0.1
X displacementY displacement
Fourth mode
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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+
8/10/2019 Ginsberg Answers
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
+((((((((((((((((((,((((((((((((((((((-
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
8/10/2019 Ginsberg Answers
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
$%%%%%%&
8/10/2019 Ginsberg Answers
57/67
57
[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
$%%%%%%%%%%%%%%%%%%&
8/10/2019 Ginsberg Answers
58/67
58
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
+((((((((((((((((((,((((((((((((((((((-
8/10/2019 Ginsberg Answers
59/67
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
+((((((((((((((((((,((((((((((((((((((-
8/10/2019 Ginsberg Answers
60/67
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 < !
8/10/2019 Ginsberg Answers
61/67
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
8/10/2019 Ginsberg Answers
62/67
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
Time (nondimensional)
0 5 10 15 20 25 30 350.01
0
0.01Relative rotation
Time (nondimensional)
11.29
0 1 2 3 4 54
2
0
2
4Real part of eigenvalues
Rotation rate (nondimensional)
8/10/2019 Ginsberg Answers
63/67
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
8/10/2019 Ginsberg Answers
64/67
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
8/10/2019 Ginsberg Answers
65/67
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
8/10/2019 Ginsberg Answers
66/67
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
8/10/2019 Ginsberg Answers
67/67
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!