Advanced structural systems - mechanical engineering

Problem Set 2 – KB6005 – Free vibration of undamped two-degree-of-freedom systems

1. Find the natural frequencies of the system shown below assuming m1 = m, m2 = 2m, k1 = k,

k2 = 2k.

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

1

2

0 27 3 ,

0 3 3

m M K

m

        

   . The first vibration mode natural frequency and mode shape are

given by (1)

1

1 2,

1 X

     

  . Determine the masses m1 and m2 and the second natural

frequency and second mode shape of the system.

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

0 4 ,

0 4 4 12

m k M K

        

    . Obtain the relation between m and k which results in

1

1 X

    

 

as one of the system’s mode shapes.

Problem Set 2 – KB6005 – Free vibration of undamped two-degree-of-freedom systems

4. For the system shown below:

a) Derive the equations of motion.

b) Calculate the natural frequencies and mode shapes.