Advanced structural systems - mechanical engineering

Problem Set 3 – KB6005 – Free vibration of undamped three-degree-of-freedom systems

1. For the system shown below:

(a) Derive the equations of motion.

(b) Write the equations of motion in matrix form.

(c) Find the natural frequencies of the system.

(d) Obtain the mode shapes of the system.

(e) Repeat parts (c) and (d) using MATLAB and assuming m =1 kg and k = 100 N/m.

Compare numerical results to analytical ones.

2. The mass and stiffness matrices of a three-degree-of-freedom system are given by:

0 0 2 0

0 0 , 2

0 0 0

m k k

M m K k k k

m k k

       

              

. Obtain the natural frequencies and mode shapes of the

system.

3. The mass and stiffness matrices of a three-degree-of-freedom system are given by

1

2

3

0 0 72 40 0

0 0 , 40 120 80

0 0 0 80 144

m

M m K

m

       

              

. If one natural frequency of the system and the

corresponding mode shape are given by (1)

1

1.0

=3.6703, 1.1265

1.0

X

   

      

, obtain the values of m1,

m2, and m2. Then calculate the other two natural frequencies and mode shapes.

Problem Set 3 – KB6005 – Free vibration of undamped three-degree-of-freedom systems

4. For the system shown below:

(a) Draw the free-body-diagram for each mass.

(b) Derive the equations of motion and write them in matrix form.

(c) Calculate the natural frequencies and mode shapes.

5. For the systems shown below, draw the free-body-diagrams for each mass.

(a)

(b)