Assignment 1 1

Assigned Problem Set 1Covers roughly 2 weeks of lecture material

1. (20 points) Perform the following tasks for the system shown in Figure 1:

(a) Write the equation of motion using the total displacement coordinate (static and dynamic)assuming that all the loads are applied simultaneously. Assume the bar is rigid with a totalmass µL = m. What role does the weight of the bar play in the equation of motion?

O

a

b

L

d

P (t)

k c

m

Figure 1: Problem 1

(b) Modify the equation on motion in terms of the dynamic displacement coordinate only. Whathappens to the effect of the weight of the bar?

(c) Assume that the left support, O, is rotated clockwise by 90 ◦ resulting in the system assuminga vertical configuration supported at the top at O. The spring, damper, and point load stillremain perpendicular to the rigid bar. Show all the relevant loads including the effects ofself-weight and formulate the equations of motion in terms of the total coordinates.

(d) Rewrite the equations of motion in terms of the dynamic coordinate only. What is the effectof the weight of the bar? What does this imply about the effective stiffness of the system?

(e) Rotate the support clockwise by an angle α. The spring, damper, and point load still remainperpendicular to the rigid bar. Find the equation of motion in terms of both the total dis-placement coordinates and dynamic displacement coordinates. Comment on the role playedby the self weight of the bar in both cases.

(f) Using α = 0 ◦ and 90 ◦, verify the results from parts (a) through (d).

CIVE 405: Structural Dynamics Winter 2014

Assignment 1 2

2. (10 points) What is the natural frequency of the system shown in Figure 2. Neglect theself weight of the beam. Assume the beam has a flexural stiffness EI.

2k

EI

m

L/2 L/2

Figure 2: Problem 2

3. (10 points) Write the equation of motion for the system in Figure 3. The horizontal bar isrigid and massless, the vertical bar is rigid with a mass per unit length of µ.

O

LL

2L

P (t)

k

m

y

µ

Figure 3: Problem 3

CIVE 405: Structural Dynamics Winter 2014

Assignment 1 3

4. (10 points) In order to determine the dynamic characteristics of a single storey frame shownin Figure 4, a field test is performed. The frame is pulled by a lateral load P a distance 1 cm andinstantaneously released. If the force exerted to displace the structure is 180 kN, the maximumdisplacement in the return swing (after one full cycle) is 0.5 cm. The mass of the structure is 105kg. Determine the damped and undamped natural frequency and the number of cycles to reducethe amplitude of the response to 10% of the initial displacement.

P

c

k/2 k/2

m

x0

Figure 4: Problem 4

CIVE 405: Structural Dynamics Winter 2014