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Answers to Odd-Numbered Homework Exercises

Mechanical and Structural Vibrations Jerry H. Ginsberg, John Wiley and Sons, Inc. 2001 Chapter 1 1.1 1.3 1.5 1.7 1.9

1.11

1.13 1.15 1.17

1.19 1.21

k1 (k2 + k3 ) + k4 k1 + k2 + k3 3 (EI/L3 ) k (2k + 3EI/L3 ) keq = k 2 + 9k (EI/L3 ) + 9 (EI/L3 )2 8 3 m¨ y + µyú + ky = 0 2 µ 3 ¶ 1 1 2¨ mL θ + k − mgL θ = 0 3 2 ¶ µ 1 2 9 1 2ú 1 2ú 9 3 2¨ mL θ1 + cL θ1 − cL θ2 + mgL + kL θ1 − kL2 θ2 = 0 64 16 8 32 4 8 ¶ µ 1 2ú 1 2ú 3 2 1 1 9 2¨ 2 mL θ2 − cL θ1 + cL θ2 − kL θ1 + mgL + kL θ2 = 0 3 8 4 8 2 16 keq =

(a) m1 x¨1 + (k1 + k2 + k4 ) x1 − k2 x2 = F,

m2 x¨2 − k2 x1 + (k2 + k3 ) x2 = 0

k2 k3 (b) keq = k1 + k4 + k2 + k3 ¶ ¶ µ µ R32 R32 R12 R12 (b) M11 = I1 2 + I2 2 + I3 (a) M11 = I1 + 2 I2 + 2 I3 , R2 R3 R1 R2 25 M11 = mL2 48 µ ¶ 1 c1 + c2 L2 , C11 = C22 = C12 = −c2 L2 4   1 2 1 1 2 − kL  kL + mgL  4 2 4  [K] =    1 2 1 1 2 − kL kL + mgL 4 4 2 K11 = 3.28k

1.23

1 Q1 = F L 4

1.25

Q1 = 0,

Q1 = F r sin (β)

2

    yG

   

1.27

{q} =

1.29

 kY [K] =   −kY `         y   1       {q} = y2  ,            y3   

1.31

1.33

    θ   

1.37



0   m  [M] =    2 0 mrG          L −kY `  , {Q} =     2   −L (` + s)  kY ` + kT   

 5  1 0 0          [C] = c  [M] = m  0 2 0  ,  −2       0 0 0 3          1    5 −2 0              {Q} = F [K] = k  −2 3 −1  , 2                  3  0 −1 1          yG     1 0  , {q} = [K] = k  , [M] = m       3 2   θ (ccw)   0 b  4        −1    2 −b    , {Q} = F [C] = c    5 2    −1b   −b b 2 2 q1 = x1 (cart), q2 = s2 (block parallel to incline)     cos (θ)   3 0    , [K] = k    0 1 cos (θ) 1         θ1    1 0   {q} = , [M] = I0         θ2  0 1           0 2 −1   2  {Q} = [K] = kR  ,      F r sin (β)  −1 1 ¶ µ 1 1 2¨ mL θ + k − mgL θ = 0 3 2  [M] = m  

1.35

,



3

−2



0    3 −1     −1 1

2



b   5 2  b b 2

3

1.39

y = vertical displacement, m¨ y + 0.6580ky = 0

1.41

(a) `0 = 1.632L,

1.43

24mr2 ¨θ + 4mgrθ = 0

1.45 1.47

1.49

1.51 1.53

1.55

5 (b) mL2 ¨θ + 2.402mgLθ = 0 3

25 1 mL2 ¨θ + kL2 θ = F L 48 4        θ (bar, ccw)  , {q} =      y (block)  



 1 2 0   m1 L 3  [M] =    0 m2      − 8 FL 0.4k2 L  15 , {Q} =     0 k2  

 (0.64k1 + k2 ) L2 [K] =   0.4k2 L         x  m 0 0   G         , [M] =  {q} = y  0 m 0 G               θ   0 0 IG     [K] = k    

3.5

0

(4L − 7b)

0

0.5

0

      

       

(3.5b2 + 2L2 − 4bL) 3 mg q = y − L, m¨ q + 3.253 q=0 4 L mg (a) `0 = 0.866L + 1.7321 , (b) K11 = 2mgL + 0.25kL2 k kL > 592 (c) mg (4L − 7b)

0

q = θAB − 65o 2.512mL2 q¨ + 1.4665cL2 qú + 1.4665kL2 q = −0.2707F L

1.57

¨ + 1.5831kR2 ψ = 1.2582F R, ψ = θ − π/2 4.614mR2 ψ

      

4

Chapter 2 2.1 2.3

F = 500 cos (5πt ± 1.2611) N x (0) = 17.324 mm,

xú (0) = −10.465 m/s,

max(¨ x) = 2.193 (104 ) m/s2 @ t = 2.5 ms

max (x) ú = 20.94 m/s @ t = 4 ms, 2.5

min (x) @ t = 2.5 ms

(b) Vú = Re {471 exp [i (ωt + 0.982)]} volt/s

(a) V = Re {1.2 exp [i (ωt − 0.589)]} volt,

(c)max Vú = 471 volt/s when t = 0.0135, 0.215, ... sec 2.7

(a) x = 14.25 sin (Ωt + 2.449) ,

(b) t = 0.693/Ω for x = 0,

2.9

(a) t = 0.010472 sec for F = 0,

(b) F = 200 cos (250t) + 346 sin (250t)

2.11

B = 10

20 q t , 10 n

(c) t = 2.26/Ω for xú = 0

0

20

0

1

2

3

4

5

6

7

4

5

6

7

4

5

6

7

t n

B=6

20 q t ,6 n

0

20

0

1

2

3 t n

B=5

10 q t ,5 n

0

10

0

1

2

3 t n

2.13

p1 = 0.005 cos (878πt − φ1 ) ,

2.15

u = −0.01 cos (50T + 0.6) − 0.05 cos (10T + 0.6) + 0.04952

2.17

(a) q = 0.10146 m @ t = 2.73 ms,

2.19

k = 392.3 N/m, m = 0.1956 kg ¸1/2 · 4kR − 2mg ωnat = , Unstable if m2 g > 2kR (3m1 + 8m2 ) R

2.21

p2 = 0.005 cos (882πt − φ1 − 0.20π) Pa

(b) qú = 51 m/s @ t = 12.10 ms

5

2.23 2.25

Keq = 1.583 (108 ) N/m,

Ceq = 2.315 (104 ) N-s/m

(a) ω nat = 200 rad/s, ζ = 0.20 (b) q = 0.012 exp (−40t) [cos (195.96t) + 0.2041 sin (195.96t)] m (c) min q = −0.00632 m @ t = 0.01603 sec (c) q = 0 @ t = 0.00904 sec

2.27 2.29

C = 10.208 N-s/m, max (q) = 0.695 mm @ t = 16.57 ms (a) δ = 0.297, ω nat = 62.90 rad/s, ζ = 0.04723 (b) t > 2.507 sec, (c) t > 1.257 sec, (d) qú0 = −1.904 m/s

2.31

π , ζ = cos (πta /tb ) , tb sin (πta /tb ) ¶¸ · µ ta πqmax ta exp π cot π v0 = tb sin (πta /tb ) tb tb

ωnat =

2.33

x = 2.207 m @ t = 0.10 sec

2.35

EI = 88.83 (106 ) N/m,

2.37

(a) cT = 21.21 N-m-s/rad,

2.39

t = ta + 4tb

c = 9.818 (105 ) N-s/m (b) t = 0.4664 sec,

(a) L = 69.87 mm, c = 80.42 N-s/m,

(c) t = 0.4965 sec

(b) t = 0.09224 sec

(c) θ = 2.400 (10−6 ) [56.60 exp (−3.72t) − 3.72 exp (−56.60t) rad] 2.41

t = 8.625 sec

2.43

(a) µk = 0.02516,

2.45

0 q val n

(b) x < 7.8 mm for dry friction, x < 7.801 mm for viscous friction

5

10

0

2 τ

4

n

maximum positive q @ ω nat t = 0.556

6

2.47

q = A sin (ωt) + B cos (ωt) + C1 exp (−0.6417ωnat t) + C2 exp (−1.5583ω nat t) ω 2nat − ω2 2ζωω nat F0 F0 h i, i B=− h 2 m (ω 2 − ω2 ) + 4ζ 2 ω2 ω 2 m (ω 2 − ω 2 )2 + 4ζ 2 ω2 ω 2 nat nat nat nat ¶ ¶ µ µ v0 − ωA v0 − ωA C1 = −1.7002B + 1.0911 , C2 = 0.7002B − 1.0911 ω nat ωnat A=

2.49

q=

F0 cos [(ω 2 − ω 1 ) t] − cos (ω nat t) F0 cos [(ω 2 + ω 1 ) t] − cos (ω nat t) ¡ 2 ¡ 2 ¢ − 2¢ 2m 2m ωnat − (ω 2 − ω 1 ) ω nat − (ω 2 + ω 1 )2 2

2.51

1 q t n 0 1

0

20

40

60

80

t n

2.53

q = βr (t) − 2βr (t − τ )

2.55

q = F0 u (t) −

2.57

q = 104 r (t) − 104 r (t − 0.02) − 200u (t − 0.02)

F0 F0 r (t) + r (t − τ ) τ τ

0.002 0.001

q j

0 0.001 0.002

0

0.02

0.04

0.06

0.08

t j

2.59

¶ µ π q = F0 c (t) + F0 s t − 2ω d

2.61

(a) c = 489.9 N-s/m,

2.63

q = P g (t) + P g (t − τ ) + P g (t − 2τ ) + · · ·, ωnat τ = 2π gives maximum q ½ ¾ 2 [1 − cos (ω nat t)] h (t) q = −αc sin (ω nat t) − ωnat τ

2.65

(b) x = 0.7053 [exp (−2.0568t) − exp (−72.944t)] m

7 15

2.67 acc mag j max acc min acc

10

j

j

5

0

0

1

2 Ωj

3

4

8

Chapter 3 3.1

ζ = 0 : Q = −840 cos (110t − 1.5) N,

3.3

ζ = 0 : |F | < 395.6 N,

3.5

ζ = 0.4 : Q = 3619 cos (110t + 0.3051) N

ζ = 0.05 : |F | < 1260.2

ω = 950 Hz : q = 3.117 (10−5 ) sin (1900πt − 0.01234) m ω = 1050 Hz : q = 2.965 (10−5 ) sin (2100πt − 3.129) m

3.7

(a) ω nat = 80π rad/s,

(b) ζ = 0.06290

(c) M = 0.6442 kg, K = 4.069 (104 ) N/m, C = 26.37 N-s/m (d) |q| = 0.00588 m, 3.9

(e) qú = 19.63 cos (80πt) m/s

|F | = 104.88 N & relative arg (F ) = 3.132 rad @ ω = 75 rad/s |F | = 152.63N & relative arg (F ) = 3.135 rad @ ω = 85 rad/s

3.11

(a) φ = 134.3o @ 105 Hz,

3.13

(a) γ = 0.05098 & |q| = 4.84 mm,

3.15

(b) |q| = 1.687 mm & φ = 152.4o @ 110 Hz (b) |q| = 4.83 mm

(a) k = 4000 N/m, c = 2000/πω N-s/m (b) x = Re [(−14.85 + 18.60i) exp (i50t)] mm

3.17

3.19 3.21

3.23

8 8 βω 2 (a) Ceq = βωX, γ eq = X 3π K · 3π µ ¶¸ 8 β 2 (b) kX 1 − r2 + i r X =F 3π M

(a) ζ = 0.11467,

(b) εm = 0.19568 kg-m,

ωnat = 408 rad/s,

∆ω = 28 rad/s,

εm = 2.35 kg-m,

min (|q|) = 4.7 mm

(a) ω nat = 30π rad/s,

(c) min (|Y |) = 2.448 mm

ζ = 0.035

(b) c = 1.109 (104 ) N-s/m

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

9

3.25

3.27

¶ µ 1 5 1 2 ¨ I1 + mL θ + cL2 θú + kL2 θ = −mεLω 2 sin (ωt) 9 9 9 mεL r2 |θ| = £ ¤1/2 5 2 2 I1 + mL2 (1 − r2 ) + 4ζ r2 9 (a) |χ| = 0.0541 rad, arg (χ) = −0.823 rad, (b) µ > 7.73 n-s/m

3.29

Rc = 0.665 mm @ r = 0.5,

3.31

R = 20 mm, ε < 0.1704 mm

3.33

ω/ωnat = 0.843 and ζ = 0.356, or ω/ωnat = 1.352 and ζ = 0.265 ¡ ¢ 2 2 1/2 2 r r 1 + 4ζ ω |Ftr | = εmω 2nat £ , r= ¤ 1/2 ωnat (1 − r2 )2 + 4ζ 2 r2 k = 3.03 (105 ) N/m, c = 1.937 (103 ) N-s/m

3.35 3.37 3.39 3.41 3.43 3.45

Rc = 20 mm @ r = 1,

Rc = 2.67 mm @ r = 2

iF F0 = F, Fn = − if n 6= 0 πn · µ ¶¸ ∞ P 1 − exp (−λ) 2πt Q=F exp in λ + 2iπn T · n=−∞ ¸ ∞ P 2F F F + sin (Ωt) + cos (2nΩt) q= 2 2 2 πK 2 (K − MΩ2 ) n=1 π (1 − 4n ) (K − 4n MΩ ) P X1 = (0.02412 + 0.0405i) k 0.007

3.47

0.006 θ jj 0.005 0.004

0

0.5

1 t jj T

1.5

2

10

3.49

2

1

0

1

0

0.5

1

1.5

2

wT=0.2pi wT=2pi wT=20pi

3.51

(a) 33% error in amplitude and 4o error in phase for Þrst harmonic, 0.2% error in amplitude and 0.3o error in phase for tenth harmonic (b) 125% error inamplitude and 10o error in phase for Þrst 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 Þrst harmonic, , 0.3%error in amplitude and 4o error in phase for tenth harmonic

3.53

1.5

For λ = 1 : 2 ω nat .y j, 1 z'' j, 1

1

0.5

0

0

0.2

0.4

0.6

0.8

1

t j

3.55 3.57

|Yn /An | = 1.127 (10−12 ) and arg (Yn /An ) = 180o for n ≤ 32 P [ω 2 (τ 2 − t2 ) + 2 − (ω2nat τ 2 + 2) cos (ωnat τ )] if t < τ Mω 4nat τ 2 nat P q= {2 cos [ωnat (t − τ )] − 2ωnat τ sin [ωnat (t − τ )] Mω 4nat τ 2

q=

− (ω 2nat τ 2 + 2) cos (ω nat τ )} if t > τ

11 40

3.59

20

q n QQ

0 n 20 40

0

0.5

1

1.5

2

2.5

3

3.5

t n

3.61

4 2 0 2 4

0

0.5

1

1.5

2

Displacement (nondim.) Force (nondim.)

3.63

2.5

3

3.5

4

4.5

Time (sec)

ζ = 0.2: ω nat ≈ 2960 rad/s, ∆ω ≈ 1380 rad/s ζ = 0.002: ωnat ≈ 3140 rad/s, ∆ω ≈ 17 rad/s

3.65 3.67

ωnat ≈ 24.1 rad/s, ζ ≈ 0.0249 80 60

40 20

0 0.5

3.69

1

1.5

Actual D Direct DFT Hanning 0

For β = 6 : Im G j, 4

0.05

0.1 0.05

0 Re G j, 4

0.05

5

5.5

4

12 0.0015

3.71

0.001 G n

5 .10

4

0

0

50

100

150 ω n

200

250

300

13

Chapter 4 4.1

4.3

4.5

ω 1 = 6.021 rad/s, {φ1 } = [1 − 0.275]T ω 2 = 20.341 rad/s {φ2 } = [1 7.275]T µ ¶1/2 k ω 1 = 0.3660 , {φ1 }T = [1 − 1.2529]T m µ ¶1/2 k ω 2 = 2.326 {φ2 } = [1 0.9128]T m Case (a): ω 1 = 7.404 rad/s, {φ1 } = [1 0.5909]T ω 2 = 60.37 rad/s {φ2 } = [1 − 6.34]T Case (a): ω 1 = 14.028 rad/s, {φ1 }T = [1 0.2668]T

4.7

4.9

4.11

4.13

ω 2 = 100.81 rad/s {φ2 } = [1 − 16.339]T ³ g ´1/2 β = 4 : ω1 = 0.794 , {φ1 }T = [1 1.312]T L ³ g ´1/2 ω2 = 7.422 {φ2 } = [1 − 2.122]T L ³ g ´1/2 α = 2 : ω1 = 1. 2247 , {φ1 }T = [1 1 1]T L ³ g ´1/2 ω2 = 2. 7386 , {φ2 } = [1 0 − 1]T L ³ g ´1/2 ω3 = 4. 4159 , {φ3 } = [1 − 2 1]T L ω1 = 233.5 rad/s, ω 2 = 316.2 rad/s    0.2697 0.4472   [Φ] =    0.4045 −0.4472

ω2 = 165.8 rad/s, K22 = 75000 N/m, K12 = K21 = 15000 N/m   3 1 √   √  21 14  [Φ] =    1 2  √ −√ 21 14

14

4.15

ω1 = 6.02 rad/s, ω 2 = 20.34 rad/s  

 0.4908 0.0954   [Φ] =    −0.1349 0.6941

4.17

µ

mL EA

2

¶1/2

ω = 1.564, 4.54, 7.07, 8.91, 9.87

1

0

1

2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Position (x/L)

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

4.19

4.21

4.23

ω = 8.152, 9.94, 19.162 rad/s 

0.924 0.162  2.502   [Φ] = (10−2 )  6.669  −0.352 −0.217   0.773 −2.088 −0.027 ω = 1.5939, 3.761, 5.326, 8.055 rad/s 

        

1.566 −1.560 3.851   0.534      1.356 3.069 −1.947 −2.225     [Φ] = (10−3 )     2.916 1.467 3.752 0.518        3.977 −3.025 −1.896 −0.089

kB = 20.95 kN/m, ω 1 = ω 2 = 7.74 rad/s    1 0   [φ] =    0 1

15

4.25 4.27

4.29

k ω1 = 0, ω 2 = 1.581 m

¶1/2

µ

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  [φ] =    −1.039 1.295 −0.910 −1.675 −0.577 0.577     3.578 −1.901 0.035 0  0.401 −0.242   −1.385 0.577 0.577 0.057 1.175 −1.155             0.126    0.874  k sin (ωt) cos (ωt) + {q} =     mg      −0.831/L   0.831/L 

                  

6

Displacement (m)

4.31

¶1/2 k , ω 3 = 2.739 m µ ¶1/2 µ ¶1/2 k k ω1 = ω2 = ω3 = 0, ω4 = ω5 = 1.25 , ω6 = 1.732 m m  µ

4

2

0

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Time (sec)

Floor 1 Floor 2 Floor 3 Floor 4

4.33 4.35

4.37

q1 =10.97 mm, q2 = 17.07 mm @ t = 2 sec.     0.7071 −1   −0.2 cos (3t − π/4) + 0.1414 cos(2t) + 4.926 sin(2t)   q=   0.15972 cos (3t − π/4) − 0.0714 cos(4t) − 0.7261 sin(4t) 1.4042 1  ηj = Φ2j [400u (t, ω j ) − 200r (t, ω j ) + 200r (t − 2, ω j )]         0.1169 0.6974   η 1 (t)    q (t) =     η 2 (t)   0.0986 −0.0165 

      

m

16

4.39

P = mv

1

x 1,n 0

x 2,n

1

0

5

10

15

20

25

30

35

t n

4.41

4.43

ζ 1 = 0.0600, ζ 2 = 0.10182, (ω d )1 = 49.94, (ωd )1 = 96.15 rad/s ( " #) ¡ ¢ 1 ζωj ηj = 2 1 − exp −ζ j ωj t cos ((ωd )j t) + sin ((ω d )j t) h (t) ωj (ω d )j          0.1452 −0.2808   η1   q=     η2   0.1986 0.1028  0.005

q

0

1,p

0.005

0

20

40

60

80

100

120

140

t p

4.45

0.01 q 1,n q 2,n q 3,n

0

0.01

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

t n

4.47

Y1 = −0.2036 + 0.0070i, Y2 = −0.2707 + 0.0176i m

4.49

√ √ a = c = 0.5547/ m, b = 0.8321/ m, θ = −33.69o 10

4.51 Re Y 1,n Im Y 1,n

0

10

0

1

2 Ω

n

3

4

17

4.53

mgL mgL mgL |θ1 | = 1.624, |θ2 | = 4.898, |θ3 | = 8.254 Γ0 Γ0 Γ0

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 @ ω = 1.182 rad/s

4.59

|y| = [0.1114 0.0529 0.0045 0.0013]T m @ ω = 1.306 rad/s " # ∞ ∞ X X exp (i2πnt/T ) {Φj } (Φ1j P1 + Φ2j P2 ) {q} = T ω 2 T 2 − 4π 2 n2 n=−∞ j j=1

18

Chapter 5 5.1

q1 = 6.71 sin (20t − 2.29) , q2 = 17.53 sin(20t − 2.292) mm 10

5.3

5 Re X Im X

1 ,n 0

1 ,n 5

10

0

0.5

1

1.5

2

2.5

3

Ωn 0.1

Amplitude (m)

5.5

0.01

1 .10

3

1 .10

4 0

5

10

15

20

Speed (m/s)

Front, c = 5 kN-s/m Back, c = 5 kN-s/m Front, c = 0.5 kN-s/m Back, c = 0.5 kN-s/m

5.7

For ψ = 20o : 0.1

0.1 θx n

y G 0.05 n

0

θy n 0

10 v n

0.05

20 0

0

10

20

v n

5.9

γ = 0.004 : {Y } = [0.807 exp (1.234i) 0.417 exp (−2.018i) 0.3372 exp (−1.903i)] m

19 0.1

5.11 Amplitude (m)

0.01 1 .10

3

1 .10

4

1 .10

5

1 .10

6 0

2

4

6

8

10

12

Frequency (rad/s)

Floor 4 Floor 3 Floor 2 Floor 1

5.13

For γ = 0.0001 : 4 1 .10 . 3 Z 6 , k 1 10 Z 7,k

100

Z 8,k Z 9,k Z 10 , k

10

1

0.1 0.8

0.9

1

1.1

1.2 Ω

1.3

1.4

1.5

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 | = 0.1944εm, |y2 | = 0.250εm

5.19

(a) k2 = 266.7 N/m (b) θ1 = 2.30 (10−3 ) sin (20t − 1.905) , θ2 = 7.32 (10−3 ) sin (20t − 2.247) rad Case (a)

5.21 5 qa 1,p qa 2,p

Case (b)

5 qb 1,p qb 2,p

0

5 0

0.5 t p 2 .π

1

0

5

0

0.5 t p 2 .π

1

20

5.23

(a) M11 = m1 + m3 , M22 = m1 R2 + m3 (`2 + κ23 ) M33 = m2 , M44 = m2 κ22 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 = εmω2 cos (ωt) , Q2 = −εm`ω2 sin (ωt) k2 (b) m2 = 2 , 2ω (c)

100 10 Y 1,n

1

Y 2,n

0.1

0.01 Y 3,n . 3 1 10 Y 4 , n 1 .10 4 5 1 .10 6 1 .10

0

50

100 ω n

5.25

0.003

0.002 x j

x j0.001

0

0

0.1

0.2 t j

0.3

150

200

21 0.02

5.27 q j,1

0.01

q j,2 q j,3

0

0.01

0

0.2

0.4

0.6

0.8

1

t j

5.29

ω1 = 5.734, ω 2 = 18.478 rad/s, ζ 1 = 0.0019, ζ 2 = 0.0020  

 0.0124 −0.298   [Φ] =    0.0514 0.213

22

Chapter 6 

6.1





 2.036 1.527 1.221   3.403 2.202 1.801       8   [M] =   1.527 1.221 1.018  kg, [K] = 10  2.202 1.801 1.601       1.221 1.018 0.872 1.801 1.601 1.481 Cjn = 4000 N-s/m, {Q} = F [0.50 0.25 0.125]T 

6.3

6.5

ψ j = (x/L)j−1 ,

¶ jπx ψ j = sin , L µ



    N/m   

       1  0.5 0.333     1             [M] = ρAL  0.5 0.333 0.25  , {Q} = F 1                  1  0.333 0.25 0.2     0  0 0  EA   0 1 [K] = 1 L    0 1 1.333 



 0.5 0.167 0.056          + k  0.167 0.056 0.019           0.056 0.019 0.006 

−0.09 0 −0.007   0.75       −0.09 0.75 −0.097 0     [M] = ρA0 L    0 −0.097 0.75 −0.099        −0.007 0 −0.099 0.75   0 −0.015   0.75 −0.113       −0.113 0.75 −0.105 0  EA0    [K] =   L   0 −0.105 0.75 −0.103       −0.015 0 −0.103 0.75

23

6.7

6.9

6.11

6.13

6.15

¸ · ¸ (n − 1) π (j − 1) π cos Mjn = 9ρJL cos 2 2        1 if j = n = 1          +ρJL 0.5 if j = n > 1               0 otherwise     1   2 GJ  2 π (j − 1) (n − 1) + 2 if j = n > 1  Kjn =  L     2 otherwise  ¸ · (j − 1) π Qj = Γ cos 2 ³ x ´j GJ 1 jn ρJL + If , Kjn = ψj = , Mjn = L j"+ n + 1 j+n−1 L µ ¶j+n µ ¶j+n # 1 2 , Qj = −Γ + Cjn = χ 3 3 ³ x ´j+1 1 (j 2 + j) (n2 + n) EI k ρAL, Kjn = ψj = , Mjn = + L j+n+3 j +n−1 L3 2j+n+2 c Cjn = j+n+2 , Qj = F 2 ¶ µ µ ¶ ³ nπ ´ jπx 1 jπ ψ j = sin , Mjn = ρALδ jn + m sin sin L 2 4 4 4 2 2 π j n EI δ jn Kjn = 2 L3 x´ x³ 1− , q4 = y of the block ψj = L L   ·

0 −0.0072 0  0.0244    0 0.0171 0 0  [M] = ρAL    −0.0072 0 0.0168 0    0 0 0 0.25

          

24

0 −10.08 −15   15.99       0 66.20 0 0   EI   , {Q} = F [K] = 3     L     −0.25  0 224.83 15      −10.08                  0  −15 0 15 60 3 .10

4

6 .10

7

2 .10

4

4 .10

7

1 .10

4

2 .10

7

Rotation (rad)

Rotation (rad)

6.17

        0.25                0 





ω = 0.95ω1 ω = 1.05ω1

0

0

0.5

ω = 0.95ω2

ω = 1.05ω2

0

1

0

Position (x/L)

6.19

0.5

0.01 X 1,k X 2,k X 3,k

1 .10

3

1 .10

4

1 .10

5

1 .10

6

1 .10

7 0

500

1000

1500

2000

2500

3000

3500

ω k

6.21

1/2

ω = [0.700 3.490 7.7641]T (E/ρL2 ) 4 Ψ Ψ Ψ

p,1

2

p,2 p,3

0

2

0

0.2

0.4

0.6

0.8

1

x p

6.23

1

Position (x/L)

1/2

{ω} = [0.860 3.426 6.664]T (E/ρL2 )

1.2

4000

25

2 1 0 1 2

6.25

0

0.2

0.4

Mode 1 Mode 2 Mode 3

0.6

0.8

1

1/2

N = 4 : {ω} = [5.355 17.072 51.094 196.706]T (EI/ρAL4 ) 4

Ψ Ψ Ψ Ψ

p,1

2

p,2 p,3 0

p,4

2

0

0.2

0.4

0.6

0.8

1

x p

6.27

Case 1: k = 20EI/L3 , First three modes: 1/2

{ω} = [4.56 22.95 61.72]T (EI/ρAL4 ) 2

0.5

1 Ψ Ψ Ψ

p,1

0

0

p,2 p,3

1 2 3

0

0.2

0.4

0.6

0.8

1

x p

6.29

For m = 2ρAL : 1/2

{ω} = [9.019 61.825 94.222 202.384]T (EI/ρAL4 )

1/2

, ω rb = 9.798 (EI/ρAL4 )

26 2 Ψ Ψ Ψ Ψ

p,1

1

p,2 0 p,3 p,4

1

2

0

0.2

0.4

0.6

0.8

1

x p

6.31

1 .10

3

1 .10

4

Y p , 1 1 .10 5 Y 6 p , 2 1 .10 7 Y p , 3 1 .10 1 .10

8

1 .10

9 0

0.2

0.4

0.6

0.8

1

x p

0.02

6.33

Disp 0.01 p 0

6.35

0

0.1

Results for N = 4 :

0.3

0.4

0.5

t p

Initial displacement

1 ua p,1

0.2

0.5

u0 x p 0 0.5

0

0.2

0.4

0.6 x p

0.8

1

27

0

0.2

0.4

0

0.2

0.4

1/4 period 1/2 period 1 period

0.6

0.8

1

0.001

6.37

5 .10

4

5 .10

4

w 1,p

0

0.001

0

1

2

3

4

5

t p

0.447 0.364 0.328 0.328

6.39

0

ω=

0.324

0.324

0.323

0.323

3.379

3.378

3.329

3.329

4.539

4.537

4.345

4.345

13.697 13.577 9.773

9.748

9.606

4.251 3.926 3.497

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

6.388 6.195

18.469 18.361 11.461 11.451

2.415 2.409 2.406 2.405

6.41

0 ω=

5.744 5.527 5.522

0

0

0

0

0

0

0

0

0

0

0

0

0

9.939 8.666

27.836 27.75

35.293 35.213 0

2.405

2.405

5.521

5.521

8.657

8.655

15.312 11.946 11.794 21.974 15.546 0

16.795

29.944

46.975

28 First mode

2

Fourth mode

20

15

1.5

10 1 5 0.5

0

0

0

0.5

5

1

0

0.5

Position (x/L)

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

6.43

ω=

6.45

1

Position (x/L)

N=4 N=5 N=6

1

2

3

4

5

6

7

8

9

10

1

6.979

6.866

6.855

6.855

6.853

6.852

6.851

6.851

6.851

6.851

2

0

28.376

28.023

28.023

27.981

27.941

27.93

27.93

27.926

27.921

3

0

0

80.579

80.579

80.398

80.229

80.184

80.184

80.167

80.146

4

0

0

0

157.914

157.914

157.914

157.914

157.914

157.914

157.914

5

0

0

0

0

226.132

220.716

219.38

219.38

218.943

218.372

6

0

0

0

0

0

316.433

313.472

313.472

312.703

311.735

7

0

0

0

0

0

0

463.531

463.531

462.703

461.71

8

0

0

0

0

0

0

0

631.655

631.655

631.655

9

0

0

0

0

0

0

0

0

763.438

751.843

10

0

0

0

0

0

0

0

0

0

922.521

First mode

2 1.5 1 0.5 0

0

0.1

N=4 N=6 N=8

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

29 Second mode

2 1 0 1 2

0

0.1

N=4 N=6 N=8

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Third mode

2 1 0 1 2

0

0.1

N=4 N=6 N=8

6.49

0.4

0.5

0.6

0.7

0.8

6.149

6.149

6.149

25.212 25.212 25.117

25.117

25.116

25.116

25.11

25.11

62.135 62.135

61.825

61.825

61.74

61.74

61.708

122.645 122.645 121.811 121.811 121.554 121.554

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

202.384 202.384 200.988 200.988 200.453 302.761 302.761 300.686 300.686 422.755 422.755 420.024

100

u p

1

6.149

6.16

At x = L/2, |w| = 2.718θ0 meter

6.51

0.9

6.153

0 ω =

0.3

6.153

6.16

6.47

0.2

50

0

0

0.2

0.4

0.6 x p

0.8

1

562.961 562.961 0

722.876

30 7 1 .10

6.53 Force (N)

6 1 .10 5 1 .10 4 1 .10 3 1 .10 100 0

5000



6.55

4 4 4 1.5 .10 2 .10 2.5 .10 Frequency (rad/s)

First mode

4 4 .10

    , {ω} = [5.01 19.36] rad/s   

0.01

0

2 Generalized coordinate

Exact Ritz

Second mode

0.1

0.05

1

4 3.5 .10



0.02

0

4 3 .10

Method (a) Method (b)

 1 1   For [Ψ] =   1 0   1 −1 0.03

4 1 .10

3

0.05

1

2 Generalized coordinate

Exact Ritz

3

31

6.57

1/2

ω1 = [7.769 4.302 3.620]T (EI/ρAL4 ) 1 φ'

j,1

φ fund j, 1

0.5

φ fund j, 2 φ fund j, 3

0

0.5

0

2

4 j

6

8

32

Chapter 7 7.1

7.3

7.5 7.7 7.9

∂u = ku ∂x ∂θ Torsion moment: GJ = κθ ∂x ∂ 2w ∂w Bending moment: EI 2 = κ µ ∂x 2 ¶ ∂x ∂ ∂ w Shear force: − EI 2 = kw ∂x ∂x 4 ∂ w EI 4 + ρA w ¨=0 ∂x ∂w w= =0@x=0 ∂x .. ∂ 2w b ∂ 3 w b2 ∂ w EI 2 + EI 3 + m =0@x=L ∂x 2 µ ∂x 6 ¶ ∂x .. ∂ 3w b ∂w EI 3 − m w =0@x=L ¨+ ∂x 2 ∂x ¶1/2 ¶ µ µ E jπx ω 1 = 0, ψ 1 = C11 , ωj = if j ≥ 2 jπ, ψ j = C1j cos ρL2 L ³α x´ GJρL j tan (α) = , ψ = C sin 2j j Ip Gα2 − κρL2 L Axial force: EA

1/2

{ω} = [22.37 61.67 120.90]T (EI/ρAL4 ) 2 1 0 1 2

7.11

0

0.2

Mode 1 Mode 2 Mode 3

0.4

0.6

0.8

1

2α2 tan (α) − tanh (α) + tan (α) tanh (α) = 0 κ · ³ ³ α x ´¸ αj x ´ sin (αj ) j ψ j = C1j sin − sinh L sinh (αj ) L Asymptotic: αj → jπ, 1% low for j = 3

7.13

2 sin (α) sinh (α) − µα3 [sin (α) cosh (α) − sinh (α) cos (α)] = 0 · ³ ³ x ´¸ x´ sin (αj ) ψ j = C1j sin αj − sinh αj L sinh (αj ) L

33

7.15 7.17

C1j ψj → 2i

½ · ¸ · ¸¾ (2j − 1) πx (2j − 1) πx exp i − exp −i 2L 2L

tan (α) + tanh (α) = 0 if α 6= 0 ½ ³ ³ x ´ cos (α ) + cos (α ) x´ j j ψ j = C1j sin αj + sinh αj + L L sin (αj ) − sinh (αj ) h ³ x´ ³ x ´io × cos αj + cosh αj L L ³ x´ ³ h ³ x´ x ´i → C2j − cos αj + sin αj − exp −αj L L L

Note: x axis is reversed in the following graphs.

Mode #2

2

Mode #8

2

0 0 2 4

1

0.5

Exact Asymptotic

2

0

1

0.5

0

Mode #16

2

0

2

7.19

1

0 h ³ x´ ³ n ³0.5 x ´ ³ x´ x ´io ψ j → C1j sin αj − cos αj − exp −αj + (−1)j exp −αj 1 − L L L L

Mode #5

2

Mode #10

2

0 0 2 4

0

0.5 Position (x/L)

Exact Asymptotic

1

2

0

0.5 Position (x/L)

1

34

Mode #15

2

0

2

0

0.5

1

position (x/L)

7.21 7.23

7.25

n ³ ³ ³ x´ h x´ x ´io 1 + (−1 − i) exp iαj − exp −αj 1 − ψ j → C1j (−1 + i) exp −iαj 2 L L L  ³α´ ε    tan − =0 2 α Symmetric: ´ ³    ψ j = C2j cos αj x L  ³α´ ε    cot + =0 2 α Antisymmetric: ´ ³    ψ j = C1j sin αj x L Axial Symmetric: µ ¶1/2 ³ x´ E (2j − 1) π, ψ j = C1j sin 2αj , ω j = 2αj αj = 2 L ρL2 Flexural Symmetric:

¶1/2 EI tan (α) + tanh (α) = 0, ω j = ρAL4 · ³ ¸ ´ ³ x x´ cos (αj ) ψ j = C2j cos 2αj + cosh 2αj L cosh (αj ) L 4α2j

µ

Axial Antisymmetric:

µ ¶1/2 ³ x´ E αj = (j − 1) π, ψ j = C2j cos 2αj , j = 2, 3, ...; ωj = 2αj L ρL2

Flexural Antisymmetric:

¶1/2 EI tan (α) − tanh (α) = 0, ω j = ρAL4 · ³ ¸ ´ ³ x x´ sin (αj ) + sinh 2αj ψ j = C1j sin 2αj L sinh (αj ) L 4α2j

7.27

µ

Symmetric: ³α´ ³ α ´ 2 ρAL cot + coth + =0 2 · 2 α m ³ x´ ³ x ´¸ sin (αj /2) ψ j = C2j cos αj + cosh αj L sinh (αj /2) L

35

7.29

7.31

7.33

Antisymmetric: ³α´ ³ α ´ 2 ρAL tan + tanh + =0 2 · 2 α m ³ x ´¸ ³ x´ cos (αj /2) ψ j = C1j sin αj − sinh αj L cosh (αj /2) L Z L Example 7.5: ρA Ψk Ψj dx = δ jk 0 Z L d2 Ψk d2 Ψj EI EI dx + κ 3 Ψk (L) Ψj (L) = ω 2j δ jk 2 2 dx dx L Z0 L Exercise 7.12: ρA Ψk Ψj dx + µρAΨk (L) Ψj (L) = δ jk 0 Z L d2 Ψk d2 Ψj EI dx = ω2j δ jk 2 2 dx dx 0 · Z L 5 dΨk dΨj ρA Ψk Ψj dx + m Ψk Ψj + b2 12 dx dx 0 µ ¶¸¯ dΨj dΨk ¯¯ b − = δjk + Ψj Ψk ¯ 2 dx dx x=0 ¸¯ · Z L d2 Ψk d2 Ψj dΨk dΨj ¯¯ EI dx + k Ψ Ψ + k = ω 2jk δ jk S k j T ¯ 2 2 dx dx dx dx x=L 0 ( ) µ ¶ ∞ 0 L Γ 1 0 0 2 X C2j C (t ) + [1 − cos (αj t0 )] h (t0 ) At either end: θ = Ψj 2 πR4 E 2 21 α 2 j j=2 Total rotation

40

θ tot p

0.2

θ tor p

20

Rotation due to deformation

0.1

0

0

0

2

4 t p

7.35

0.1

0

2

4 t p

¸ ∞ ³ x´· 1 2F L3 X µ 2 2 2 sin jπ sin (µjπ τ ) − sin (j π τ ) w= EI j=1 j 2 π 4 (j 2 − µ2 ) L j ¶1/2 ¶1/2 µ µ EI EI v , (v)cr = π, τ = t where µ = 2 (v)cr ρAL ρAL4

36

7.37

Z ∞ £¡ ¢ ¤ ρAL4 F X 1 1 w (x, t) = vj Ψj (x) cos (ω j t) , vj = [Ψj (Ly)] 1 − y 3 + 3y − 1 dy EI j=1 6 0 0.5

w end 1,p

0

0.5

0

2

4

6

8

t p

7.39

· ¸ · ¸ ∞ 1 8vL X (2j − 1) π x (2j − 1) π cbar t u (x, t) = − 2 sin sin π cbar j=1 (2j − 1)2 2 L 2 L ∂u at x = 0 is a square wave whose amplitude is −v/cbar , ∂x and whose period is 2L/cbar.

7.41

f0 L4 (0.512) , 180o out-of-phase from a sine EI f0 L4 Structural damping: |w| = (0.458) , 153.54o lag relative to a sine EI

No damping: |w| =

7.43 w 1,p w 2,p w 3,p

0.8

0

0.6

w def 1 , p 0.001

w def 0.002 3,p

0.2 0

7.45

w def 2,p

0.4

0

0.5

1

1.5

0.003

t p

0

0.5

1 t p

1.5

µ ¶ ¸ ∞ X x 1 1 2 0 0 u= C1j ν j1 − sin (ωt) + B ν 1 + EA/kL L 1 + EA/kL j2 j=1 ³ x´ ω [ω sin (ωt) − ω j sin (ω j t)] sin αj × 2 ωj − ω2 L µ ¶ ∞ X B 1 B 1 2 0 0 F = − sin (ωt) + ν C αj ν j1 − L 1 + EA/kL L j=1 1j 1 + EA/kL j2 · B 1−

ω [ω sin (ωt) − ω j sin (ω j t)] − ω2 hR i−1/2 1 where C1j = 0 sin (αj y)2 dy ×

ω 2j

37

7.49 7.51

¶ µ ¨θ 1¨ 2 x x2 x3 (a) wbc = θt + [1 − cos (Ωt)] −3 2 + 3 2 L 2Ω2 L L Fbottom = EA Re [ik (B1 exp (ikL) − B2 exp (−ikL)) exp (iωt)] · ¸ 2k utop = Re u1 exp (iωt) ∆ u1 u1 where B1 = (iEAk + K + iωc − mω 2 ) , B2 = (iEAk − K − iωc + mω 2 ) i∆ i∆ ∆ = EAk cos (kL) + (K + iωc − mω2 ) sin (kL)

7.53

Midpoint displacement

Amplitude

1 0.1 0.01 3 1 .10 4 1 .10 5 1 .10

0

500

1000

1500

2000

Frequency (rad/s)

7.55

Midpoint displacement

Amplitude

0.004

0.002

0

0

50

100

150

Frequency (Hz)

7.57

7.59

7.63

∂χ ∂w = 0 and −χ=0 @ x=L ∂x ∂x µ ¶ ∂χ ∂w (b) w = = 0 @ x = 0, χ = 0 and κGA − χ + mw ¨=0@x=L ∂x ∂x ³ x ´j ¡ ¢ (a) (ψ w )j = ψ χ j = L ³ x ´j ¡ ¢ ³ x ´j (b) (ψ w )j = , ψχ j = 1 − L L (a) w = χ = 0 @ x = 0,

100

kL Tim n kL cl n

10 1 0.1 0.01

2

4

6 n

8

10

38 1

7.65

0.1 0.01 3 . 1 10 w classic kL 1 .10 4 p 5 1 .10 6 1 .10 7 1 .10 w kL p

0

2

4 kL p

6

8

39

Chapter 8 x x , ψ u2 = 1 − L L 

8.1

ψ u1 =

8.3

 2 1   1 −1  1  , [K e ] = EA   [M e ] = ρAL     6 L  1 2 −1 1 x x ψ1 = , ψ2 = 1 − L   L  

8.5





 2 1   1 −1  1  , [K e ] = GJ   [M e ] = ρIL     6 L  1 2 −1 1

{qe } = [u1 w1 θ1 u2 w2 θ2 u3 w3 ]T

x2 x2 x2 x x x + 2 2 , ψ u2 = − + 2 2 , ψ u3 = 4 − 4 2 L L L L µ L L ¶ x2 x2 x3 x4 x x3 x4 −4 2 +5 3 −2 4 ψ w1 = 1 − 11 2 + 18 3 − 8 4 , ψ w2 = L L L L L L L L ¶ µ 2 x2 x3 x4 x x3 x4 −3 3 +2 4 ψ w3 = −5 2 + 14 3 − 8 4 , ψ w4 = L L L L L2 L L x2 x3 x4 ψ w5 = 16 2 − 32 3 + 16 4 L L L      [R] [0] [0]    cos (β) sin (β)     e  , [R ] =  [0] [R] [0]  [R] =        − sin (β) cos (β)   [0] [0] [R] ψ u1 = 1 − 3

8.9

8.11

{q e } = [wg1 θg1 wg2 θg2 wg3 θg3 ]T

1 2 1 2 Sjj = 1, Sj(j+3) = 1 for j = 1, ..., 4, Sjn = Sjn = 0 otherwise 

       h i   ˆ = M         

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



8.13

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 4.167     4.167 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 DeÞne X axis horizontal, number mesh points from the left.        h i   ˆ = K         

{ˆ q} = [ˆ ug1 w ˆg1 θg1 uˆg2 w ˆg2 θ ˆg3 w ˆg3 θg3 uˆg4 w ˆg4 θg4 ]T g2 u   1 2  Sjj = Sj(j+3) =1     3 3 3 3 3 3 For j = 1, ..., 6, n = 1, ..., 12 : S1,10 = S2,11 = S3,12 = S4,7 = S5,8 = S6,9 =1       k  Sjn = 0 otherwise  

1  2 1   for e = 1, 2, 3 [γ e ] = L   6  1 2 ¸T ¸T · · f f 1 2 {f } = 0 0 0 0 0 , {f } = 0 0 0 f 0 2 2

{f 3 } = [0 0 0 0 0 0]T , {F } = [H1 V1 0 0 0 0 − F 0 0 H4 V4 M4 ]T 3 X {Q} = {F } + [S e ]T [Re ]T [γ e ] {f e } with β 1 = β 2 = 30o , β 3 = 90o e=1

8.15

{ˆ q } = [ˆ ug1 w ˆg1 θg1 uˆg2 w ˆg2 θg2 uˆg3 w ˆg3 θg3 uˆg4 w ˆg4 θg4 ]T {qc } = [ˆ ug1 uˆg4 w ˆg4 θg4 ]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

DeÞne X axis horizontal, number mesh points from the bottom left. {ˆ q} = [ˆ ug1 w ˆg1 θg1 uˆg2 w ˆg2 θg2 uˆg3 w ˆg3 θg3 uˆg4 w ˆg4 θg4 ]T

41

{qf } = [ˆ ug2 w ˆg2 θg2 uˆg3 w ˆg3 θg3 ]T {ω} = [796

Φ =

8.19

3116

0 0 0 1 0 2 0 3 0.7418 4 0.0016 5 -1.1906 6 0.7418 7 -0.0016 8 -1.1906 9 0 10 0 11 0

7283

1

2 0 0 0 0 0 0 0.0008 0.2915 0.0286 -0.0616 8.5135 -16.2526 -0.0008 0.2915 0.0286 0.0616 -8.5135 -16.2526 0 0 0 0 0 0

16325

24628]T rad/s

17399

3

4

5

0 0 0 0.3041 -0.832 5.7351 -0.3041 -0.832 -5.7351 0 0 0

0 0 0 0.168 0.9752 -8.2001 0.168 -0.9752 -8.2001 0 0 0

0 0 0 1.0042 0.5718 -7.6307 -1.0042 0.5718 7.6307 0 0 0

{ˆ q } = [ˆ uA w ˆA θA uˆB w ˆB θB uˆC w ˆC θC uˆD w ˆD θD ]T {ω} = [0 849.8 3674 4070 5657 9049 15273 17802 21365 31862]T rad/s

Φ=

1 2 3 4 5 6 7 8 9 10 11 12 13

1

2

3

4

0 0 0.517 -0.129 0.224 0.517 -0.299 -0.259 0.448 0.517 -0.129 0.075 -0.299

0 0 1.187 -0.103 0.176 -1.019 -0.144 0.431 -0.75 -2.568 -0.104 0.06 -0.314

0 0 4.14 -0.079 0.077 -2.554 -3.051 -0.162 0.216 1.787 -0.077 0.044 2.95

0 0 3.973 0.131 -0.089 -3.527 2.403 -0.238 0.562 4.794 0.136 -0.079 -2.374

5

6 0 0 0 0 0.955 9.587 0.13 0.04 0.306 -0.251 1.421 3.659 -1.944 1.157 0.486 0.334 -0.218 -0.86 -4.466 -12.683 0.161 0.046 -0.093 -0.027 1.269 -2.042

7

8 0 0 0 0 1.247 9.782 -0.038 -0.005 -0.098 0.001 0.725 10.79 8.56 0.906 -0.417 -0.428 -0.221 1.098 0.382 23.831 0.111 -0.076 -0.064 0.044 6.862 1.373

9

10

0 0 0.401 0.016 -0.379 -2.402 9.316 0.78 -0.099 -10.07 -0.219 0.126 6.729

0 0 0.732 0.396 0.062 -0.318 0.509 -0.317 -0.309 -2.473 -0.439 0.254 3.702

42

Chapter 9 9.1

9.3

³ x´ x sin jπ , a1j = 0.5 sin (0.5jπ) ; j = 1, ..., N L L ¶ µ ¶j+1 µ 2x 2jπx 1 2 (b) Left segment: ψ wj = ; Right segment: ψ wj = sin L L 2 (j + 1) 2jπ For j = 1, ..., N : a1j = 1, a1(j+N) = 0, a2j = , a2(j+N) = − cos (jπ) L ³ x ´j ³ x ´j+1 ³ x ´j L 1 1 2 Bar 1 is horizontal, ψ 1uj = , ψ 1wj = , ψ 2uj = ψ 2wj = L L L (a) ψ j =

For j = 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

9.5

ψ `wj

=

³ x ´j+1 `

L

,

j+1 j a3(j+N) = , a3(j+3N) = , anj = 0 otherwise L ³ x ´j L ` ` ψ θj = ; ` = 1, 2 L

For j = 1, ..., N : a1j = 1, a1(j+2N ) = −1

j+1 , a2(j+3N ) = 1 L1 j+1 j+1 , a3(j+2N) = − , anj = 0 otherwise a3(j+N) = L L2 a2j =

9.7

ψj =

³ x´ x sin jπ L L

1 1 1 1 [M 1 ] {¨ q1 } + [K  ] {q } = {Q } + [a ] {λ} , where



0.0901 0.019 −0.0072 0.0035   0.1413      −0.0901 0.1603 −0.0973 0.0225 −0.0092        1  [M ] = ρAL  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



      EI  1 [K ] = 3  L       

43.4

−74.6

75.9

−74.6

368.3

−459.5

75.9

−459.5

−90.2

298.3

107.3 −286.9

107.3

298.3

−286.9

1559.3

−1629.1

816.6

−1629.1

4590.4

−4293.9

816.6

[a1 ] = [0.5 0 − 0.50 0 0.5]

9.9

−90.2

−4293.9

10825.3

{Q1 } = [0.5303 − 0.75 0.5303 0 − 0.5303]T µ ¶j µ ¶j+1 µ ¶j−1 µ ¶j−1 x1 x1 x2 x2 1 1 2 2 ψ uj = , ψ wj = , ψ uj = , ψ wj = L L L2 L2 µ 1 ¶j µ 1 ¶j+1 x3 x3 ψ 3uj = , ψ 3wj = , where L1 = 2 m, L2 = L3 = 4 m L3 L3 h iT T T T T T T {q} = {qu1 } {qw1 } {qu2 } {qw2 } {qu3 } {qw3 }

               

ρAL1 ρAL1 ρAL2 ρAL2 , (Mw1 )jn = , (Mu2 )jn = , (Mw2 )jn = j+n+1 j +n+3 j+n−1 j+n−1 ρAL ρAL 3 3 (Mu3 )jn = , (Mw3 )jn = j+n+1 j +n+3 EA EI (j + 1) j (n + 1) n jn (Ku1 )jn = , (Kw1 )jn = 3 L1 j + n − 1 L1 j +n−1     0 if j or n = 1 2 (Ku )jn =  EA (j − 1) (n − 1)   otherwise j+n−3  L2    0 if j or n = 1 or 2 2 (Kw )jn =  EA (j − 1) (j − 2) (n − 1) (n − 2)   otherwise L2 j+n−5 jn EA EI (j + 1) j (n + 1) n , (Kw3 )jn = 3 (Ku3 )jn = L3 j + n − 1 L3 j +n−1 (Mu1 )jn =

(Q1u )j = (Q1w )j = (Q1u )j = (Q1w )j = (Q1u )j = 0, (Q3w )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 j+1 , a5(j+5N) = − L2 L3

44

                   

9.11

[Mu1 ] [0] [0] [0] [0] [0]

[0]

[0]    [Mw1 ] [0] [0] [0] [0]     [0] [Mu2 ] [0] [0] [0]    {¨  q} 2 [0] [0] [Mw ] [0] [0]      [0] [0] [0] [Mu3 ] [0]    3 [0] [0] [0] [0] [Mw ]           +         

[0]

[Ku1 ] [0] [0] [0] [0] [0]

[0]



[0]

[0]

[0]

[0]

[0]



[0]    [Kw1 ] [0] [0] [0] [0]     [0] [Ku2 ] [0] [0] [0]    {q} = {Q} + [a]T {λ}  2 [0] [0] [Kw ] [0] [0]      3 [0] [0] [0] [Ku ] [0]    [0] [0] [0] [0] [Kw3 ]

[a] {q} = {0} µ ¶j−1 µ ¶ x1 ³ x2 x1 ´ 2jπx1 1 2 ψ wj = 1− sin , ψ θj = L L   L3  L   q1}     EI [K 1 ] [0] [0]  ρAL [M 1 ]   {¨  L3   +     GJ  {¨  [0] ρJL [M 2 ]  q2}  [0] [K 2 ] L       {0}   = + [a]T {λ}     {Q2 }   [a] {q} = {0}

     {q1 }     {q2 }

      

45



−3

−4

−5



0.017 −7.604 (10 ) −4.511 (10 ) −8.274 (10 )        −7.604 (10−3 ) 0.017 −7.687 (10−3 ) −4.753 (10−4 )     [M 1 ] =     −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   0.5 0.333 0.25   1       0.5 0.333 0.25 0.2   2   [M ] =     0.333 0.25 0.2 0.167       0.25 0.2 0.167 0.143 



 66.20 −47.41 −6.33 −2.06       −47.41 574.3 −431.3 −47.41     [K 1 ] =     −6.33 −431.3 2460 −1727        −2.06 −47.41 −1727 7282  

9.13

0 0   0 0       0 1 1 1   2 ,  [K ] =    0 1 1.333 1.5        0 1 1.5 1.8

{Q2 } = [−R 0 0 0]T

2

Ψ Ψ Ψ Ψ

p,1

1

p,2 p,3 p,4

0

1

2

0

0.2

0.4

0.6 x p

0.8

1

46

9.15

10 1 0.1 0.01 3 1 .10 4 1 .10 5 1 .10 6 1 .10 7 1 .10 8 1 .10 9 1 .1010 1 .10 11 1 .10

Displacement at left force

0

50

100

150

200

250

300

350

400

Frequency (Hz)

Vertical Horizontal 4 1 .10 5 1 .10 6 1 .10 7 1 .10 8 . 1 10 9 . 1 10 10 . 1 10 11 1 .10

Displacement at right force

0

50

100

150

200

250

300

350

400

Frequency (Hz)

Vertical Horizontal

6 1 .10 Displacement (nondim)

9.17

5 1 .10 4 1 .10 3 1 .10 100

0

0.2

0.4

0.6

0.8

1

Frequency (nondim)

Horizontal Vertical

9.19

1 .10

4

1 .10

5

1 .10

6

1 .10

7

1 .10

8 0

500

Displacement Rotation 1 Rotation 2

1000

1500

2000

2500

47

9.21

Fixed interface modes: Left: Clamped-clamped normal modes, Right: Hinged-clamped normal modes " µ ¶2 µ ¶3 # 2x L 2x ` ` , ` = 1, 2 Constraint modes: ψ C` − + 1 = 2 L L £© ª © ª ¤T {q} = qwF 1 q1C1 qwF 2 q2C2

9.23

a1(N+1) = a1(2N +2) = 1, a1j = 0 otherwise h© ª © ª © F 2 ªT © F 2 ªT T T C1 C1 C1 {q} = qwF 1 qw1 qw2 qθ1 · ·· qθF 1 qw qθ C2 C2 C2 C2 C2 C2 · · · qw1 qw2 qθ1 qw3 qw4 qθ2

¤T

C C1 C C1 C ψ C1 w1 = ψ w (x1 /L1 ) , ψ w2 = ψ χ (x1 /L1 ) , ψ θ1 = ψ θ (x1 /L1 ) C C2 C C2 C ψ C2 w1 = ψ w (x2 /L2 ) , ψ w2 = ψ χ (x2 /L2 ) , ψ θ1 = ψ θ (x2 /L2 ) C C2 C C2 C ψ C2 w3 = ψ w (1 − x2 /L2 ) , ψ 24 = ψ χ (1 − x2 /L2 ) , ψ θ2 = ψ θ (1 − x2 /L2 )

a1(2N +1) = 1, a1(4N+4) = −1, a2(2N+2) = a2(4N +6) = 1, af 3(2N+3) = 1, a3(4N +5) = −1 9.25

See Answer  9.21 for basis function deÞnitions and [a]  n¡ ¢ o F C 1/2 [I]4×4 (ρAL3 ) M`  £ `¤    M = n¡ ¢ oT  F C 1/2 (ρAL3 ) M` 0.001190ρAL3 oT n FC = [−0.03161 0.01147 − 0.00585 0.00354] (M 1 ) n oT 2 FC (M ) = [−0.04551 0.01620 − 0.00827 0.00500]  n¡ ¢ o h¡ ¢ i FC ` FF 3 1/2 K` (ρAL ) K  £ `¤ EI  4×4 K = n¡ ¢ oT ρAL4  FC 1/2 (ρAL3 ) K` 8 (ρAL3 ) FF

FF

FF

FF

FF

   

FF

(K 1 )1,1 = 8009, (K 1 )2,2 = 60857, (K 1 )3,3 = 233882, (K 1 )4,4, = 639101 FF

FF

(K 2 )1,1 = 3804, (K 2 )2,2 = 3994, (K 2 )3,3 = 173881, (K 2 )4,4 = 508582 n oT oT n FC FC (K 1 ) = [0 0 0 0] , (K 2 ) = [64.6 113.0 163.4 213.6]      [K 1 ] [0]   [M 1 ] {0}      [M] =   , [K] =   T 2 2 {0} [M ] [0] [K ]

48

9.27

9.29

1 .10

4

1 .10

5

1 .10

6

1 .10

7

1 .10

8 0

500

Displacement Rotation of bar 1 Rotation of bar 2

{ω} = [57.88

138.54

1000

1500

202.23

2000

2500

425.12]T rad/s

DeÞne global XY coordinate system with X to the right and Y upward. First mode

0.06

Second mode

0.1

0.04

0.05

0.02 0

0 0.02

0

0.1

5

10

X displacement Y displacement Third mode

0.05

0

0.1

5

10

5

10

X displacement Y displacement Fourth mode

0.05

0.05

0 0

0.05

0.05 0

5

10

X displacement Y displacement

0.1

0

X displacement Y displacement

49

Chapter 10 10.1

{λ} = [−3.061 + 10.303i − 3.061 − 10.303i − 5.189 + 12.530i ... −5.189 − 12.530i]T



0.074 + 0.001i 0.010 + 0.062i 0.010 − 0.062i  0.074 − 0.001i    0.055 + 0.011i 0.055 − 0.011i −0.016 − 0.036i −0.016 + 0.036i  [ψ] =    −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

0  −1500 480    480 −240 0  [S] =    0 0 0.8533    0 0 0

           



0 0 −1500 480             0 0 480 −240 0      , [R] =       48 52 12  0   −1500        480 −240 12 −4 0.1067 0

{λ} = [−5.928 + 20.088i − 5.928 − 20.09i − 43.29 + 37.04i − 43.29 − 37.04i]T rad/s  

10.5

 −0.015 + 0.009i    −0.042 + 0.013i  {ψ} =    −0.089 − 0.352i    −0.015 − 0.930i λ 1 = 0.502

0.867i

−0.015 − 0.009i 0.002 + 0.008i

   −0.042 − 0.013i −0.010 − 0.012i −0.010 + 0.012i     −0.089 + 0.352i −0.356 − 0.284i −0.356 + 0.284i     −0.015 + 0.930i 0.871 + 0.182i 0.871 − 0.182i λ 3 = 0.516

Mode 1

1

0

0

0.5

5

6

Real Imag

7

8

9

10

1

0.859i

Mode 2

0.5

0.5

0.5

0.002 − 0.008i

0

2

4

6

8

10

50

λ5 = 0.5 0.869i Mode 3

1

1

0.5

0.5

0

0

0.5

0

2

4

6

8

10

λ9 = 0.498 0.87i Mode 5

1

0.5

0.5

0

0

0

2

4

6

8

10

λ13 = 0.489 0.875i Mode 7

1

0.5

0.5

0

0

0

2

4

6

8

10

λ17 = 0.491 0.874i Mode 9

1

0.5

0.5

0

0

0.5

0

2

4

6

8

0

10

0.5

4

6

8

10

2

4

6

8

10

λ15 = 0.487 0.877i Mode 8

0

2

4

6

8

10

λ19 = 0.497 0.871i Mode 10

1

0.5

2

λ11 = 0.508 0.864i Mode 6

1

0.5

0.5

0

1

0.5

0.5

λ7 = 0.512 0.862i Mode 4

0

2

4

6

8

10

51

10.7

{λ} =  [−1.323 + 6.700i − 1.323 − 6.700i − 2.064 + 7.949i − 2.064 − 7.949i]T 0.028 − 0.043i −0.017 + 0.043i −0.017 − 0.043i   0.028 + 0.043i      −0.014 + 0.029i −0.014 − 0.029i −0.017 − 0.026i −0.017 + 0.026i     [ψ] =     −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 (ωnat )1 = 6.829, ζ 1 = 0.194, (ω nat )2 = 8.212, ζ 2 = 0.251

10.9

Case (a): {λ} =

³ g ´1/2 L

{Ψ1 } = {Ψ∗2 } =

{Ψ5 } = {Ψ∗6 } =

[1.225i − 1.225i − 0.0038 + 1.259i − 0.0038 − 1.259i ...

                                                          

−0.011 + 1.324i − 0.011 − 1.324i]T            0.577i  −0.001 − 0.688i                         0.577i  0                       0.577i 0.001 + 0.688i  ∗ , {Ψ3 } = {Ψ4 } =         −0.707  0.866 + 0.001i                        −0.707  0                       −0.866 − 0.001i  −0.707     −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

Case (b): ³ g ´1/2 [−0.431 1.225i − 1.225i − 0.75 + 1.011i ... {λ} = L

10.11

−0.75 − 1.011i − 4.069]T                 −0.565i 0.577i                                 1.130i 0.577i                              −0.565i   0.577i  ∗ , {Ψ2 } = {Ψ3 } = {Ψ1 } =            0.243i  −0.707                                 −0.487i −0.707                              0.243i   −0.707                  0.241 + 0.729i 0.184                                 0 −0.368                              −0.241 − 0.729i   0.184  ∗ {Ψ4 } = {Ψ5 } = , {Ψ6 } =            −0.918 − 0.303i  −0.748                                 0 1.496                              0.918 + 0.303i   −0.748  0.1 x 1,p x 2,p 0

0

0.2

0.4

0.6

0.8 t p

1

1.2

1.4

53

10.13 0.05

0

0

1

Floor 4 Floor 3 Floor 2 Floor 1

2

3

4

0.1

0.15

5

6

7

8

9

10

0.02

10.15 x 1,p

0.01

x 2,p x 3,p

0

0.01

0

0.05

0.2

0.25

0.3

0.35

0.4

t p 1

10.17

0.5 X 1,p X 2,p

0

0.5

1

0

0.5

1

1.5 t p

2

2.5

54

10.19

0.6 X 1,p

0.4

X 2,p

0.2

X 3,p

0

X 4,p

0.2 0.4

0

1

2

3

4

5

6

7

8

t p 0.005

10.21 x 1,p

0

0.005

0

20

40

60

80

100

120

140

80

100

120

140

t p 0.01

x 2,p

0

0.01

0

20

40

60 t p

0.01

x 3,p

0

0.01

0

20

40

60

80 t p

100

120

140

9

10

55 0.01

10.23

0.005 x 1,p 0 0.005

0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

t p 5

x'' 1,p

0

5

0

0.2

0.4 t p

0.1

10.25 X 1,p X 2,p

0.01

1 .10

3 0

5

10 v p

15

20

56

Chapter 11 11.1 11.3

11.5

m¨ x + (k − mω2 ) x = − (k − mω 2 ) R0 + kL0 , ω < (k/m)1/2 for stability ¶ ¸ ·µ 4¨ 3w ¨ 1¨ 3 g g θ1 + θ2 + 2β − − θ1 − β θ2 = 0 3 2 2 L 2L L ·µ ¶ ¸ 1¨ ¨ 1 g 1 g 1w θ1 + ¨θ2 − β θ1 + β − − θ2 = 0 2 3 L 2 L 2L m [¨ x − 2ωyú − ω 2 (R + x)] + kx x = 0 m [¨ y + 2ωxú − ω 2 y] + ky y = 0

11.7

m¨ z + kz z = 0     0    1 0        0 1 0          2 0 0 L /12  

11.9

        xú C 0 −2ω 0 x¨C                0 0  y¨C  +    yúC  2ω           ¨θ   θú  0 0 0 

              

(k1 + k2 ) 0 0 − ω2  m   (k4 − k3 ) L (k3 + k4 ) + 0 − ω2  m 2m   (k4 + k3 ) L2 (k4 − k3 ) L 0 − ω2 2m 4m        0          2 = ω H             0  

[M] {¨ q } + [[G] + [C]] ú + [[K] − [E]]{q} = {J}   {q}

       x    C         y  C            θ  



0 0  −2ω Lω   1  0            where [M] = m  0 1 −L/2  , [G] = m  2ω 0 0           0 −L/2 L2 /3 −Lω 0 0

57



11.11

2

0  ω   [E] = m  ω2  0   0 −Lω 2 /2     (r − L/2) ω2     {J} = m 0        0





 k1 0 0       −Lω 2 /2  , [K] =   0 k2 0     0 0 k3 Lrω 2 /2         0

      

{λ} =[0.6i − 0.6i 1i − 1i 1.4i − 1.4i]T

         [ψ] =          

−0.606

−0.606

0

0

0.411

       h i   ˜ = ψ         

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   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

58 Mode pair #1

Mode pair #3

0.5

q1

2,p

0.5

q3

0

0.5

2,p

0

0.5

0.5

0

0.5

0.5

q1 1,p

1 q1 1,p

0.5

0.5

0.5

q3 1,p

0

q1 2,p

0 q3 1,p

q3 2,p

0

0.5 1

0

5

0.5

10

τp

11.13

    −3.75 (10−5 ) + 0.274i         −3.75 (10−5 ) − 0.274i         1.314 (10−3 ) + 19.868i

0

5

10

τp

                   

    −1.531 (10−6 ) − 5.25 (10−10 ) i         1.150 (10−8 ) − 5.594 (10−5 ) i         −0.122 − 1.789 (103 ) i

                   

, {Ψ1 } =        −3    1.314 (10 ) − 19.868i  2.01 (10−10 ) − 4.195 (10−7 ) i                              −3 −5 −9     −1.332 (10 1.532 (10 ) + 20.132i ) + 5.27 (10 ) i                             −3    −1.332 (10 ) − 20.132i  489.898 + 0.034i             −3 −3     −1.532 (10 −1.542 (10 ) − 0.308i ) − 0.306i                             −3 −3     0.308 − 1.532 (10 0.306 − 1.542 (10 ) i ) i                            4.689 (10−3 ) + 7.051 (10−5 ) i   4.537 (10−3 ) + 6.79 (10−5 ) i    {Ψ3 } = , {Ψ5 } =             6.127 − 0.031i 6.168 − 0.031i                                 0.031 + 6.127i 0.031 + 6.168i                             −3 −3    −1.395 (10 ) + 0.093i −1.373 (10 ) + 0.091i 

{λ} =

59

n o ˜1 = Ψ

n o ˜5 = Ψ

11.15

{λ} =

    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                −5 −9   1.532 (10 ) + 5.25 (10 ) i                 489.898 + 0.034i       −6   −1.542 (10 ) − 0.306i               −3   −0.306 + 1.542 (10 ) i              −3 −5  −4.537 (10 ) − 6.791 (10 ) i                     

6.168 − 0.031i

−0.031 − 6.168i 1.373 (10−3 ) − 0.091i

        0.417i                 −0.417i                1.252i       −1.252i                  2.481i                −2.481i 

, {Ψ1 } =

n o ˜3 = Ψ

                  

    0.014i         2.794 (10−3 )         −1.540i

   −5.812 (10−3 )         1.166 (10−3 ) i         0.643

                                      

    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

                                      

60

    0.277         −0.377i         4.961 (10−3 )

                   

    0.248i         −0.212         1.035 (10−3 ) i

                   

11.17

, {Ψ5 } =             0.347i −0.616                                 0.472 −0.527i                              6.209 (10−3 ) i   −2.568 (10−3 )              0.014i −0.277                      −3    −2.794 (10 −0.377i )                     n o    −4.961 (10−3 ) n o  −1.540i ˜ ˜ Ψ1 = , Ψ3 =       −3    −5.812 (10 −0.347i )                      −3    0.472 −1.166 (10 ) i                         −6.209 (10−3 ) i 0.643         0.248i                 0.212              −3  n o  1.035 (10 ) i  ˜5 = Ψ       −0.616                 0.527i               −2.568 (10−3 )  

11.19

Divergence instability at ω = 1

11.21

Divergence instability beyond ω = 0.732

{Ψ3 } =

Unstable for 1 < Ω < 1.4

                                      

61

11.23

(a) Amplitudes as a function of rotation rate: 3 1 .10 100 10 1 0.1 0.01

0

5

10

15

Lower y Lower z Upper y Upper z

20

25

30

35

40

45

50

(b) Orbits at Ω = 0.836: Lower mass

1

Upper mass

6 4

0.5

2 q

q

2, p

4, p

0

0 2 4

0.5

6 1

0.5

0 q

4

2

4

1, p

Upper mass

0.006

Lower mass

0.004

z displacement

z displacement

0 q 3,p

0.5

11.25 0.0032

2

0

0.002 0 0.002

0.0032 0.0022

0

0.0022

y displacement

0.004 0.006 0.006

0.004

0.002

0

0.002

y displacement

0.004

0.006

62

11.27

Motion at ω = 0.6 (k1 /m)1/2 : Relative x displacement

1 0.5 0 0.5

0

5

10

15

20

25

30

35

25

30

35

25

30

35

Time (nondimensional)

Relative y displacement

0

0

5

10

15

20

Time (nondimensional)

Relative rotation

0.01

0

0.01

0

5

10

15

20

Time (nondimensional)

11.29

Real part of eigenvalues

4

2

0

2

4

0

1

2

3

Rotation rate (nondimensional)

4

5

63

Imaginary part of eigenvalues 60

40

20

0

0

1

2

3

4

5

Rotation rate (nondimensional) 1/2

Divergence instability if ω > 3.564 (EI/ρAL4 ) 11.31

1/2

vcrit = π (EI/ρAL2 ) First mode 0

0.02

0.1

0

0.2

0.02

0.3

0

0.5

Third mode

0.04

1

0.04

0

x/L

Real Imag

Fifth mode

Seventh mode

0.01

0.01

0.005

0

0

0.01 0.02

1

x/L

Real Imag 0.02

0.5

0.005 0

0.5 x/L

Real Imag

1

0.01

0

0.5 x/L

Real Imag

1

64

Chapter 12 12.1

ωnat = 40π rad/s, ζ E = 0.0222, ζ I = 0.0111, ε = 0.445 mm

12.3 50

YC

p

0

50 100

50

0 XC

12.5 12.7

50

100

p

Flutter instability at ω = 5217 rad/s, Critical speeds are ω = 632 and 941 rad/s 30

25

20

15

10

5

0 0.9

0.95

Case a: X Case a: Y Case b: X Case b: Y Case c: X Case c: Y

1

1.05

65

1

12.9

0.1 0.01 3

1 .10

4

1 .10

5

1 .10

6

1 .10

7 0

500

1000

Displacement Transverse rotation

1500

2000

Equations of motion are (12.4.10) with: K11 = kY A + kY B , K22 = kZA + kZB , K33 = kZA b2 + kZB (L − b)2 K44 = kY A b2 + 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 = 2ωIxx Campbell diagram

2000 Eigenvalue, imaginary part (rad/s)

12.11

1 .10

1500

1000 Synchronous line 500

0

0

100

200

300

400

Rotation rate (rad/s)

ω crit = 219, 240, and 424 rad/s

500

600

66 First critical mode 0.005

ZA p

0.005

ZB p

0

0.005 0.005

0

0

0.005 0.005

0.005

YA p

0

0.005

YB p

Second critical mode 0.004

ZA

p

0.004

ZB

0

0.004 0.004

0 YA

p

0

0.004 0.004

0.004

0

0.004

YB p

p

Third critical mode 0.002

0.002

ZA p

ZB p

0

0.002 0.002

0

0.002

0

0.002 0.002

YA p

12.13

0

0.002

YB p

Critical displacements for isotropic bearings: ω 1 = 220 rad/s, |YC | = |ZC | = 3.40 (10−3 ) m, |β Z | = |β Y | = 4.55 (10−3 ) rad ω 2 = 424 rad/s, |YC | = |ZC | = 0.77 (10−3 ) m, |β Z | = |β Y | = 5.28 (10−3 ) rad Critical displacements for orthotropic bearings: ω 1 = 229 rad/s |YC | = 3.79 (10−3 ) , |ZC | = 0.20 (10−3 ) m |β Z | = 1.23 (10−3 ) , |β Y | = 4.12 (10−3 ) rad ω 2 = 317 rad/s |YC | = 0.52 (10−3 ) , |ZC | = 2.82 (10−3 ) m |β Z | = 2.21 (10−3 ) , |β Y | = 0.76 (10−3 ) rad

67

ω 3 = 424 rad/s

|YC | = 0.77 (10−3 ) , |ZC | = 0.76 (10−3 ) m |β Z | = 5.28 (10−3 ) , |β Y | = 5.28 (10−3 ) rad

Orthotropic bearings

2.5

0

2.5

5

0

Orthotropic shaft

10

5

5

Z displacement

Z displacement

12.15

0 5

Y displacement

10

10

5

0

Y displacement

12.17

Ω = 0.767 : Center's Path Relative to Fixed XYZ

10

5

q fixed 2,p

0

5

10

10

5

0 q fixed 1,p

5

10

5

10