vibration problems 2004

8/6/2019 Vibration Problems 2004

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Mechanical Engineering Ph.D. Preliminary Qualifying Examination

Vibration

January 26, 2004

This is one of five problems. You are required to work four of the five problems. Note thatProblem 5 is mandatory. Clearly indicate which problems you are choosing. Show all work on

the exam sheets provided and write your student personal identification (PID) number on each

sheet. Do not write your name on any sheet.

Your PID number:____________________________

Problem 1. A mass M hanging from a spring with stiffness N/m is in equilibrium state. A

second mass drops through a height and sticks with

k

m h M without rebound as shown.

a. Draw free-body diagram of the system.b. Write down the equation of motion with respect to the static equilibrium position.c. Determine the subsequent free vibration motion.

M

m

h

k

M

m

h

k


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Vibration

January 26, 2004

This is one of five problems. You are required to work four of the five problems. Note that

Problem 5 is mandatory. Clearly indicate which problems you are choosing. Show all work onthe exam sheets provided and write your student personal identification (PID) number on each


Your PID number:____________________________

Problem 2. Consider a double pendulum system as shown. Each bar has an equal mass m , mass

moment of inertia with respect to the center of mass of the bar , and length l . Assume that

these two bars are under forced vibration motion through harmonic excitations and the

amplitudes of motion are small.

cJ

tM sin

1

2

l, m,Jc

l, m,Jc

Msint

1

2

l, m,Jc

l, m,Jc

Msint

Figure 2. Schematic of a double pendulum system under forced vibration motion.

1. Derive equations of motion for the system shown in Figure 2.2. Determine the forced vibration motion responses.3. Under what condition will the amplitude of the top bar vanish?


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Vibration

January 26, 2004




Your PID number:____________________________

Problem 3

The fundamental natural frequency of a string (length l, mass m, tension force T = Po) is

ml

Pf o

2

1= . Elongation of the string is proportional to the tension force, ( oo PPkll = ) . How

the natural frequency of the string is changing with increasing tension depending on the design of

the tensioning device.

In Fig. 1, the distance between two supports is constant ( lo = const), the tension is applied by

turning the tension adjuster, like in a string instrument (guitar). Thus, the strings stretch is

changing.

In Fig. 2, the tension is applied as shown, by moving one end of the string relative to the other, so

that the strings length is changing.

a. Derive expressions forfas function ofTfor both cases.b. In what casefis changing more?

Figure 1

Figure 2.


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Vibration

January 26, 2004




Your PID number:____________________________

Problem 4. Massive container 2 (massM) is suspended on spring 1 (stiffness k1). To limit lateral

displacements of container 2 and for cushioning lateral impacts, light guides 3 are pressed to

sides of container 2 by four springs 4, each having transverse (perpendicular to its axis) stiffness

k2. There is viscous (sliding) friction between guides 3 and container 2 with coefficient of

viscous friction ..

a. Draw the free body diagram.b. Derive equations of motion for vertical vibrations of container and write expressions for its

natural frequencies (Neglect masses of guides).

Figure


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Vibration

January 26, 2004


Problem 5 is mandatory. Clearly indicate which problems you are choosing. Show all work on

the exam sheets provided and write your student personal identification (PID) number on each


Your PID number:____________________________

Problem 5. Block (mass m) is connected by spring with stiffness k1 with the rigid wall and

move without friction as shown. It is also attached to belt which is going without slip around

pulley (moment of inertiaJ) and is attached by its other end to the rigid wall spring via spring

with stiffness k2 . Whenx = 0 and = 0, springs k1 and k2 are extended by the amounts x10 and

x20, respectively, from their unstretched (free) configurations. Assume that springs remain in

tension throughout the motion of the system.

a. Using free body diagram, derive the differential equations of motion in terms ofresponse variablex. What is the equivalent mass and what is the equivalent stiffness of

the system?

b. Verify the result in part (a) using the energy method.c. Determine the natural frequency.d. Express the equation of motion in coordinate ; determine the natural frequency;

equivalent moment of inertia; and equivalent torsional stiffness

e. How many degrees of freedom has this system?

vibration problems 2004

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