HOMEWORK 01C

Eigenvalues

Problem 1:

Problem 2:

Problem 3:

Problem 4:

Lecture 1

Problem 5:

Problem 6:

The solution of Problem 1 in Homework 01B gives the equation motion below,

Problem 1:

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fkx2xc2x2m5

Find the eigenvalues of the system for the values of m=2 kg, k=3240 N/m ve c=380 Ns/m. Is the system stable? Write the form of free vibration response. Determine the values of Δt ve t∞ for the system’s response to be analyzed.

Here, f(t) is the input, x(t) is the generalized coordinate.

Eigenvalue equation: 2.5ms2+2cs+2k=0

System is stable, form of the response: t9.1422

t1.91 eAeA)t(x

Δt=0.0022, t∞=0.7

TkL9

cL

9

mL 222

The solution of Problem 2 in Homework 01B gives the equation motion below,

Problem 2:

)t53.376cos(Ae)t( t135

f0=63.6943 Hz, ξ=0.3375, Δt=0.7854 x 10-3 s, t∞=0.0465 s

Find the eigenvalues of the system for the values of m=1.8 kg, L=0.42m, k=32000 N/m, c=486 Ns/m. Is the system stable? Find undamped natural frequency and damping ratio. Determine the values of Δt ve t∞ for the system’s response to be analyzed.

Here, T(t) is the input, θ(t) is the generalized coordinate.

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Problem 3:

t65.773

t79.2821

t62.171A eAeA)t73.20cos(eA)t(x

f0=4.33 Hz, ξ=0.648, Δt=0.0041 s, t∞=0.36

A2

A2

A2

x

9/kL113/kL

3/kLk5.0x

9/cL113/cL

3/cLc5.0x

3/mL40

0m

111

11

xmL2x)3/cL4(x)3/kL4(Lf

xc5.0kx5.0

The solution of Problem 3 in Homework 01B gives the equation motion below. Here, f(t) ve x1(t) are the inputs, xA(t) ve θ(t) are the generalized coordinates.

Find the eigenvalues of the system for the values of m=20 kg, L=0.6 m, k=42000 N/m, c=2000 Ns/m. Is the system stable? Write the form of free vibration response. Find undamped natural frequency in terms of Hz and damping ratio. Determine the values of Δt ve t∞ for the system’s response to be analyzed.

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Gy2y1

k c k c

L1 L2

m,IyA yB

Problem 4:m=1050 kg, I=670 kg-m2 k=35300 N/m, c=2000 Ns/m L1=1.7 m, L2=1.4 m

For the system shown in the figure, yA are yB the inputs, y1 and y2 are the generalized coordinates. Find undamped natural frequencies in terms of Hz and damping ratio of the system. Determine the values of Δt ve t∞ for the system’s response to be analyzed.

f1=1.2968 Hz, f2=2.5483 Hz s1,2=1.8809±7.9283i (ξ=0.2308), s3,4=-7.2627±14.2698i (ξ=0.4536) Δt=0.0196 s, t∞=3.34 s

Solution:

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Gy2y1

k c k c

L1 L2

m,IyA yB

m=1050 kg, I=670 kg-m2 k=35300 N/m, c=2000 Ns/m L1=1.7 m, L2=1.4 m

Solution:

21

11

G yL

Ly

L

L1y

)LLL( 21

L

yy 12 )1(

2

12

2

21

11

1 L

yyI2

1y

L

Ly

L

L1m

2

1E

2

B22

A12 )yy(k2

1)yy(k

2

1E

2B21A1 y)yy(cy)yy(cW

BB

AA

2

1

2

1

2

1

2

21

211

211

2

21

ycky

ycky

y

y

k0

0k

y

y

c0

0c

y

y

L

Im

L

L

L

Im

L

L1

L

LL

Im

L

L1

L

L

L

Im

L

L1

48.38532.190

32.19087.283M

Problem 5: m1=5.8 kg, m2=3.2 kg, k1=4325 N/m, k2=3850 N/m, k3=3500 N/m, c1=37.2 Ns/m, c2=33.5 Ns/m, c3=32 Ns/m.

m2

f2 x2

k3 c3

m1

f1 x1

k2 c2

k1 c1

f1=4.6549 Hz, f2=8.4979 Hz s1,2=-3.76±29.01i (ξ=0.129), s3,4=-12.57±51.89i (ξ=0.235) Δt=0.0059 s, t∞=1.67 s

222

2111 xm2

1xm

2

1E 2

232

1222112 xk

2

1xxk

2

1xk

2

1E

232122221112211 xxc)xx(xxcxxcxfxfW

2

1

2

1

322

221

2

1

322

221

2

1

2

1

f

f

x

x

kkk

kkk

x

x

ccc

ccc

x

x

m0

0m

In the system shown in the figure, f1 and f2 are the inputs, x1 and x2 are the generalized coordinates. Find undamped natural frequencies in terms of Hz and damping ratio of the sytem. Determine the values of Δt ve t∞ for the system’s response to be analyzed.

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Solution:

Problem 6: m1=250 kg, m2=350 kg, k=37000 N/m, c=1500 Ns/m, L1=1.2 mm1, L1

θ

yA

m2

k c

f=1.4692 Hz, s1,2=-1.7308 ± 9.0768i (ξ=0.1872).

2122

211

21

11 Lm2

1

12

Lm

2

1

2

Lm2

1E

2A12 yLk

2

1E

)yL()yL(cW A1A1

A1A121

21

212

211 ykLycLkLcLLm3

Lm

Solution:

In the system shown in the figure, yA is the input, θ(t) is the generalized coordinate. Find undamped natural frequencies in terms of Hz and damping ratio of the system.

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