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0593_C13_fm Page 467 Monday, May 6, 2002 3:21 PM

Introduction to Vibrations 467

13.7. Wowk, V., Machinery Vibration, McGraw-Hill, New York, 1991.
13.8. Ayres, F., Theory and Problems of Differential Equations, Schaum’s Outline Series, McGraw-Hill,
New York, 1958.
13.9. Boyce, W. E., and DiPrima, R. C., Elementary Differential Equations, Wiley, New York, 1965.
13.10. Coddington, E. A., An Introduction to Ordinary Differential Equations, Prentice Hall, Englewood
Cliffs, NJ, 1961.
13.11. Golomb, M., and Shanks, M., Elements of Ordinary Differential Equations, McGraw-Hill, New
York, 1965.
13.12. Murphy, G. M., Ordinary Differential Equations and their Solutions, Van Nostrand, New York,
1960.
13.13. Rainville, E. D., Elementary Differential Equations, Macmillan, New York, 1964.
13.14. CRC Standard Mathematical Tables, CRC Press, Boca Raton, FL, 1972.
13.15. Schmidt, G., and Tondl, A., Non-Linear Vibrations, Cambridge University Press, New York, 1986.

Problems

Section 13.2 Solutions of Second-Order Ordinary Differential Equations
P13.2.1: Show that the solution of Eq. (13.2.1) may also be expressed in the form:

(
ˆ
x = Bsin ω t + φ ˆ )
ˆ
φ
where B ˆ and are constants. Compare this solution with the solutions given by Eqs.
(13.2.2) and (13.2.3).
P13.2.2: Find the solution to the equation:

˙˙ x + λ x = 0

if λ = 36 sec– , x(0) = 0, and (0) = 3 m/sec.
2
˙ x
2
P13.2.3: Solve Problem P13.2.2 if λ = –36 sec– .
P13.2.4: A particle P moves on a straight line as represented in Figure P13.2.4. Let x measure
the displacement of P away from a fixed point O as shown. Suppose x is given by the
expression:

x = 6cos 5 t + 8sin t 5

where x is measured in feet and t in seconds. Find the following:

a. Amplitude of the motion
b. Circular frequency
c. Frequency
d. Period
e. Phase