ME 370.3. Spring 2018 Name(s)

Homework 11 Due: In class, Friday April 13

1. [44 pts] The math model of the system in Prob. 2 of hw10 is given by

�𝑚𝑚1 + 𝑚𝑚2 𝑚𝑚2𝐿𝐿𝑚𝑚2𝐿𝐿 𝐽𝐽2 + 𝐿𝐿2𝑚𝑚2

� ��̈�𝑥1�̈�𝜃� + �𝑐𝑐 0

0 0� ��̇�𝑥1�̇�𝜃� + �𝑘𝑘 0

0 𝑚𝑚2𝑔𝑔𝐿𝐿� �𝑥𝑥1𝜃𝜃 � = �00�

Let 4.0=L m, 200=k N/m, 02 =J , 0=c , and 0.221 == mm kg.

(1) (2 pts) The 1st natural frequency of the system in rad/s is

a) 37.11 =nω b) 56.21 =nω

c) 34.41 =nω d) 83.71 =nω

(2) (2 pts) The 2nd natural frequency of the system in rad/s is

a) 14.61 =nω b) 21.81 =nω

c) 11.101 =nω d) 41.111 =nω

(3) (4 pts) The 1st mode shape of the system is

a)

721.01

b)

− 997.0

1

c)

993.0120.0

d)

− 913.0

409.0

(4) (4 pts) The 2nd mode shape of the system is

a)

− 451.0

1 b)

607.01

c)

977.0212.0

d)

− 951.0

309.0

(5) (3 pts) Write down the general expression for the free vibration of the undamped system with four undetermined constants: )1(X , 1φ , )2(X and 2φ .

(6) (3 pts) Suppose you displace the cart and the pendulum together to the right by an amount of 0.1 m. You then release the whole thing with �̇�𝜃(0) = −1.0 rad/s to initiate a free vibration of the system. Write down the initial conditions for the problem.

�𝑥𝑥1𝜃𝜃 � = ��̇�𝑥1�̇�𝜃� =

(7) (4 pts) The two constants for the 1st mode in the free-vibration expression yield the following:

a) 226.0sin 1)1( =φX b) 321.0sin 1

)1( =φX

c) 478.0sin 1)1( =φX d) 611.0sin 1

)1( =φX

(8) (4 pts) The two constants for the 2nd mode in the free-vibration expression yield the following:

a) 151.0sin 2)2( =φX b) 236.0sin 2

)2( =φX

c) 327.0sin 2)2( =φX d) 542.0sin 2

)2( =φX

(9) (8 pts) The four constants for the free vibration expression of Part (5) are:

a) X(1) = 0.282, X(2) = 0.237, 1φ = 2.213, 2φ = 1.465

b) X(1) = 0.380, X(2) = 0.282, 1φ = 2.213, 2φ = 1.465

c) X(1) = 0.768, X(2) = 0.370, 1φ = 2.843, 2φ = 0.691

d) X(1) = 0.590, X(2) = 0.137, 1φ = 2.843, 2φ = .0691 (10) (10 pts) Program in matlab the expression for )(tθ you obtain in steps (5) and (8) to

plot the free vibration of the pendulum swing for a time period of 𝑡𝑡 = 2𝜋𝜋/𝜔𝜔𝑛𝑛1. Use sufficient data points to generate a smooth )(tθ curve with plotting grids, making sure ICs are satisfied to be correct. Examine the plot to envision how the pendulum swings with two modes of contributions. Submit your program along with the plot.

2. [10 pts] Consider a system with the following governing equations:

�10 5 05 50 100 10 25

� ��̈�𝑦1�̈�𝑦2�̈�𝑦3� + �

1000 −120 −270−120 1500 90−270 90 400

� ��̇�𝑦1�̇�𝑦2�̇�𝑦3� + �

2 × 105 −105 0−105 2 × 105 0

0 0 5 × 105� �𝑦𝑦1𝑦𝑦2𝑦𝑦3� = �

000�

Use Matlab to determine the natural frequencies and the mode shapes of the system. Include your Matlab program along with the results.

Problem 2