7/18/2019 Problem Set

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CE 137  –  STRUCTURAL DYNAMICS AND EARTHQUAKE ENGINEERING

1ST PROBLEM SET

(NOT TO BE SUBMITTED)

1.  Determine the natural period of the system shown in Figure 1. Assume that the beam and

the springs supporting the weight are massless. Neglect damping.

Figure 1

 Answer:     

2.  The weight of the wooden block shown in Figure 2 is 10 lb and the spring stiffness is 100 lb/in.

A bullet weighing 0.5 lb is fired at a speed of 60 ft/s into the block and becomes

embedded in the block. Determine the natural frequency, maximum displacement and

minimum and maximum acceleration of the block in g’s. 

Figure 2

 Answers:  , , ,

3.  Derive the response for an overdamped free vibration. That is, solve for the response

displacement of a structure subject to initial boundary conditions of u=u(0) and

=

(0)

having an equation of motion given by

 

 Answer:  [  ] where  ( √  )

 

  ( √  )

 

√   

4.  A simply supported beam of negligible mass has a span of six meters. This beam is used to

carry a 70 kg air-conditioner, which is located two meters from the support operating at a

frequency of 200 rpm. The beam is made up of a material having a modulus of elasticity of

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20,700 MPa and whose cross section is 5” by 8”. Using the bandwidth method to evaluate

damping ratio, the angular frequencies corresponding to half-power points are 10 rad/s

and 5 rad/s respectively. The motor produces an unbalanced force of 2 kN.

a.  Calculate the maximum static deformation of the beam due to the unbalanced force.

 Answer: 4.212 mm

b.  Assuming negligible damping, what is the peak deformation in the beam? Answer: 4.5032

mm 

c.  Assuming damped conditions, what is the magnitude of force transmitted to the supports?

 Answer: 2138.24 N 

d.  Assuming damped conditions, what is the peak deformation in the beam? Answer: 4.5026

mm 

5.  Consider the water tank shown in Figure 3 which is subjected to ground motion produced

by a passing train in the vicinity of the tower. The ground motion is idealized as a harmonic

acceleration of the foundation of the tower with amplitude of 0.1g at a frequency of 10 Hz.

Determine the motion of the tower relative to the motion of its foundation. Assume aneffective damping coefficient of 10% of the critical damping in the system.

Figure 3

 Answer: 0.013 in

6.  A machine is supported on four steel springs for which damping can be neglected. The

natural frequency of vertical vibration of the machine-spring system is 200 cycles/min. The

machine generates a vertical force p(t) = po sin ωt. The amplitude of the resulting steady-

state vertical displacement of the machine is 0.2 in. when the machine is running at 20

rev/min. Calculate the amplitude of the vertical motion of the machine if the steel springsare replaced by four rubber isolators which provide the same stiffness, but introduce

damping equivalent to ζ = 25% for the system.

 Answer: 0.1997 in

7.  An SDOF system is subjected to a harmonic excitation P(t) = Po sin ωt where ω/ωn = 0.80

and the damping ratio of the system ζ = 5%. The amplitude of steady-state vibration is u0 =

0.54 inches. Suppose the stiffness of the system is doubled. Compute the new amplitude u0 

of steady-state vibration for the same harmonic excitation.

 Answer: 0.147 inches