Program Explanation Given Information: A mass system with m = 3 , c = 30 , k = 63 , x 0 = 2 , v 0 = 2 is performing free damped motion. Explanation: For a mass performing harmonic motion with m = 3 , c = 30 , k = 63 the circular frequency is given by, ω 0 = k m ........ (1) Substitute m = 3 and k = 63 in equation (1) as, ω 0 = 63 3 = 21 = 21 rad / s Now, the system of mass with m = 3 , c = 30 , k = 63 , x 0 = 2 , v 0 = 2 is performing free damped motion, the differential equation takes the form of x " + 2 p x ' + ω 0 2 x = 0 such that p = c 2 m ........ (2) Substitute m = 3 and c = 30 in equation (2) as, p = 30 2 ⋅ ( 3 ) = 5 Now, for m = 3 , c = 30 , k = 63 the value of c 2 − 4 k m is as below, c 2 − 4 k m = 30 2 − 4 ⋅ 63 ⋅ ( 3 ) = 900 − 756 = 144 > 0 As c 2 − 4 k m > 0 , the motion is over damped motion and the position function is given by, x ( t ) = c 1 e r 1 t + c 2 e r 2 t ........ (3) such that r 1 = − p + ( p 2 − ω 0 2 ) 1 2 and r 2 = − p − ( p 2 − ω 0 2 ) 1 2 ........ (4) Substitute p = 5 and ω 0 = 21 in equation (4) to get r 1 and r 2 as, r 1 = − p + ( p 2 − ω 0 2 ) 1 2 = − 5 + ( 5 2 − ( 21 ) 2 ) 1 2 = − 5 + ( 25 − 21 ) 1 2 r 1 = − 3 And, r 2 = − p − ( p 2 − ω 0 2 ) 1 2 = − 5 − ( 5 2 − ( 21 ) 2 ) 1 2 = − 5 − ( 4 ) 1 2 r 2 = − 7 Substitute p = 5 , r 1 = − 3 and r 2 = − 7 in equation (3) to get the position function x ( t ) as, x ( t ) = c 1 e − 3 t + c 2 e − 7 t ........ (5) Differentiate equation (5) with respect to t and obtainthe velocity function as, d d t [ x ( t ) ] = d d t [ c 1 e − 3 t + c 2 e − 7 t ] v ( t ) = − 3 c 1 e − 3 t − 7 c 2 e − 7 t Hence the velocity function is, v ( t ) = − 3 c 1 e − 3 t − 7 c 2 e − 7 t ........ (6) Now, to find c 1 and c 2 use the initial conditions x 0 = 2 and v 0 = 2 . For x 0 = 2 , put t = 0 and x ( t ) = 0 in equation (5) as, x ( t ) = c 1 e − 3 t + c 2 e − 7 t 2 = c 1 e 0 + c 2 e 0 c 1 + c 2 = 2 Hence, c 1 + c 2 = 2 ........ (7) For v 0 = 2 , put t = 0 and v ( t ) = 0 in equation (6) as, v ( t ) = − 3 c 1 e − 3 t − 7 c 2 e − 7 t 2 = − 3 c 1 e 0 − 7 c 2 e 0 − 3 c 1 − 7 c 2 = 2 Hence, − 3 c 1 − 7 c 2 = 2 ........ (8) Solve equation (7) and (8) to get the value of c 1 and c 2 , Thus, c 1 = 4 and c 2 = − 2 Substitute c 1 = 4 and c 2 = − 2 in equation (5) to get the position function of given mass system as, x ( t ) = 4 e − 3 t − 2 e − 7 t Hence the given system of mass is overdamped and has position function as x ( t ) = 4 e − 3 t − 2 e − 7 t

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

5th Edition

ISBN: 9780321816252

Author: C. Henry Edwards, David E. Penney, David Calvis

Publisher: PEARSON

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Chapter 3.4, Problem 16P

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Program Description: Purpose of the problem is to obtain the position function x(t) for the mass system with m=3 , c=30 , k=63 , x0=2 , v0=2 , discuss whether the motion is over-damped, critically damped or underdamped, find undamped position function and construct a figure that illustrates the effect of damping.

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Chapter 3.4, Problem 15P

Chapter 3.4, Problem 17P

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Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

Ch. 3.1 - In Problems 1 through 16, a homogeneous...Ch. 3.1 - Prob. 2P Ch. 3.1 - Prob. 3P Ch. 3.1 - Prob. 4P Ch. 3.1 - Prob. 5P Ch. 3.1 - Prob. 6P Ch. 3.1 - Prob. 7P Ch. 3.1 - Prob. 8P Ch. 3.1 - Prob. 9P Ch. 3.1 - Prob. 10P

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Solution Summary: The author explains that the problem is to obtain the position function x(t) for a mass performing harmonic motion. They then discuss whether the motion is over-damped, critically damped or under

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