UNIVERSITY OF TORONTO

FACULTY OF APPLIED SCIENCE AND ENGINEERING

MIE 402S

Vibrations

FINAL EXAMINATION

Friday April 28, 2017

Examiner: J.K. Mills

Question Mark

1 /20

2 /20

3 /20

4 /20

5 /20

/100

Last name First Name Student Number

Instructions to Candidates:

Type X Exam (Open book - all written materials permitted) All calculators permitted Exam Duration: 2.5 Hours Attempt All Questions Answer Questions in Space Provided (Use the left hand page if necessary) Maximum Value of Exam: 50% of Final Grade This exam paper has 11 pages

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(20%) 1) A schematic diagram of a lathe is shown below. As shown, the lathe is supported at opposite ends, and this support is modelled with two springs. The mass of the lathe is m=200 kg. The stiffness of the springs is k1 = k2 = k = 20,000 N/rn. Ignore the mass of the springs. The inertia about the centre of mass, indicated by C. G., is J0

A B

v(t) .v(t)

1J*n t'

"I " F 12

la) Using the coordinates, y(1) and 0(t), write the differential equations which describes the dynamics of the lathe in symbolic form, i.e. use symbols, rn, k, J0, for the mass, stiffness, inertia.

ib) Assuming a harmonic solutiony(t) = Ycos ((ot + ) and 0(1) = ecos(wt + ) determine the two natural vibration frequencies of the lathe, in terms of the lathe dynamic parameters and lengths. Answer the problem using methods and techniques taught in the MIE402S course lectures.

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Question 1 (Con't)

ic) Determine rj and r2 , the ratios of the magnitude of the mode shapes of the lathe. Answer the problem using methods and techniques taught in the MIE402S course lectures.

1 d) Determine the first and second mode of vibration of the lathe, in terms of the vibration amplitude magnitude, 1' and 6. and also in terms of Y and r and r2. Answer the problem using methods and techniques taught in the MIE402S course lectures.

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(20%) 2) A torsional system is shown below. A cylinder with inertia J0 is connected to a torsional shaft. The torsional shaft is fixed to a solid base. The torsional stiffness of the shaft is given as k1 N-rn/radian. A sinusoidal torque given by T= sin (wt) is applied to the cylinder. The frequency w is close to, or equal to the resonant frequency of the cylinder and torsional shaft.

T sin (cot)

e1

al

\\\\\\ \\

You are to design a vibration absorber such that the motion of the cylinder, ® , is close to or zero.

Illustrate (sketch) the vibration damper in the figure above.

Write the dynamic equations of motion of the system as you have shown in tyeh figure above, introducing additional variable where needed.

Question 2 (Con't)

2b) Derive the magnitude of the relevant mode amplitudes form the dynamic equations derived in part 2b).

2c) Determine the dynamic parameters of the vibration absorber and calculate the dynamic parameters of the vibration absorber such that the motion of the cylinder is zero, i.e.e(t) =0.

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(20%) 3) A dynamic system is shown below. The force of gravity does not act on the system. The applied force is given by F(r) = F, cos(vt). The system initial conditions are

given as x(0) = x0 and (0) =

F(t) X(t)

3a) Draw a free body diagram of the above system. Write the differential equation describing the motion of the spring mass damper systems.

3b) Write the homogeneous solution x(t), for the above system. Verify that the homogeneous solution is a solution of the differential equation of part 3(a). Answer the problem using methods and techniques taught in the MIE402S course lectures.

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Question 3 (Con't)

3c) Derive the particular solution for the above system. Answer the problem using methods and techniques taught in the MTE402S course lectures.

3d) Consider the case where the excitation frequency of the applied force F(t) is equal to the natural vibration frequency of the system. Derive the complete solution to the system response using the initial conditions. Answer the problem using methods and techniqLes taught in the MIE402S course lectures.

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(20%) 4) A mass-spring-damper system is shown below. The system dynamic parameters are k= 1000 N/rn, c=400 N-s/m., rn = 10 kg. An impulse input, denoted byf(t) and shown below, is applied to the system. The system is initially at rest at t = , i.e. x(0) = x = 0 and (0) = = 0. Gravity does not act on the system

At)

ft

i t Time (s)

4a) Calculate the initial conditions of the system after the application of the first pulse at t = 0. Answer the problem using methods and techniques taught in the MIE402S course lectures.

9

Question 4 (Con't)

4b) Derive the response due to the application of the two pulses as shown in the above plot of f(t) vs. t. Answer the problem using methods and techniques taught in the MIE402S course lectures.

LD

(20%) 5) A system, which moves in the horizontal plane, is shown below. moment of inertia of the pulley is J=1 kg-m2 .

The polar

xT

3000 N

0 C

IY

)O N/rn

5a) Replace the system of springs by a single spring of equivalent stiffness where x is the displacement of the block of 2 kg. and x is used as the coordinate. Answer the problem using methods and techniques taught in the MIE402S course lectures.

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Question 5 (Con't)

Sb) Replace the system of springs by a single spring of equivalent torsional stiffness when the clockwise angular rotation of the disk, 0, is used as the coordinate. Answer the problem using methods and techniques taught in the MIE402S course lectures.

5e) Determine the natural frequency of vibration of the system. Answer the problem using methods and techniques taught in the MIE402S course lectures.