1

DYNAMIC ANALYSIS

Homework 2

INSTRUCTIONS

There is no specific format for this homework assignment although answers should be neatly set out and legible if handwritten. Not only the final results but also important intermediate results should be included so that the train of thought can be followed easily. The homework should be uploaded to Moodle using the icon Upload Homework 2. The assignment is due by midnight (11:59 pm) on the day specified in the Module Guide i.e. on November 8th.

1.

Determine the natural frequency of the clamped beam shown in Figure 1 using a generalized SDOF system.

Figure 1 Clamped beam

The displacement , is approximated by the product of the following interpolation function

1.25 0.25

and the displacement at the right end . Use a value of for the generalized mass.

x

L

E, A, I,

2

2.The truss element shown in Figure 2 has four nodes.

Figure 2 Cubic truss element

The displacement , is approximated by the following expression

,

916 3

3

916 3

3

2716 3

2716 3

Calculate the consistent mass matrix of the element.

Calculate the lumped mass matrix of the element.

Calculate the consistent load vector of the element for a constant distributed load .

E, A,

2a/3

u1 u4 u2

2a/3x1 3 2

u34

a/3 a/3

3

3.

Determine the mass, damping and stiffness matrices of the system shown in Figure 3 using Lagrange Equations. Discretize the cantilever using only one frame element.

Figure 3 Cantilever connected to two springs and a damper

The translational generalized coordinates shall coincide with the global (x,y) axes.

Figure 4 Frame element

The stiffness matrix of the frame element shown in Figure 4 is defined by

0 0

0 0

0 126 0

12

6

0 64 0

6

2

0 0 0 0

0 12 6 0

12

6

0 62 0

6

4

,

L

E, A, I, k

c

L = 2 mE = 28 GPaA = 0.025 m2I = 20500 cm4 = 2500 kg/m3k = 120 kN/mc = 125 Ns/m = 30 = 20

y

x

k