Program Explanation Explanation: For a mass-spring-dashpot system the differential equation of motion is x " + 2 p x ' + ω 0 2 x = 0 ...... (1) Such that p = c 2 m and ω 0 = k m It is given that the system is under damped and the solution of the differential equation for the under damped case is, x ( t ) = e − p t ( A cos ω 1 t + B sin ω 1 t ) Such that ω 1 = ω 0 2 − p 2 Thus the general solution of the differential equation in equation (1) is, x ( t ) = e − p t ( A cos ω 1 t + B sin ω 1 t ) ...... (2) Differentiate equation (2) with respect to t as, d d t [ x ( t ) ] = d d t [ e − p t ( A cos ω 1 t + B sin ω 1 t ) ] x ' ( t ) = − p e − p t ( A cos ω 1 t + B sin ω 1 t ) + e − p t ( − A ω 1 sin ω 1 t + B ω 1 cos ω 1 t ) x ' ( t ) = A [ e − p t ( − p cos ω 1 t − ω 1 sin ω 1 t ) ] + B [ e − p t ( − p sin ω 1 t + ω 1 cos ω 1 t ) ] Hence we get x ' ( t ) = A [ e − p t ( − p cos ω 1 t − ω 1 sin ω 1 t ) ] + B [ e − p t ( − p sin ω 1 t + ω 1 cos ω 1 t ) ] ...... (3) Now, to find A and B use the initial conditions x 0 = x ( 0 ) and v 0 = x ' ( 0 ) For x 0 = x ( 0 ) , put t = 0 and x ( t ) = x 0 in equation (2) as, x ( t ) = e − p t ( A cos ω 1 t + B sin ω 1 t ) x 0 = e 0 ( A cos 0 + B sin 0 ) x 0 = ( A + 0 ) A = x 0 Hence, A = x 0

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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A naïve solution for the above-stated problem is to evaluate all of the possible combinations of different pieces of varying size to find the maximum revenue-generating combination. Although this method seems the easiest and quick to practice, it has exp…

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Chapter 3.4, Problem 30P

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Program Description: Purpose of the problem is to show that x(t)=e−pt(x0cosω1t+v0+px0ω1sinω1t) for the under damped case of a mass-spring-dashpot system with initial conditions x0=x(0) and v0=x'(0)

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Chapter 3.4, Problem 29P

Chapter 3.4, Problem 31P

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Chapter 3 Solutions

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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Program Description: Purpose of the problem is to show that x ( t ) = e − p t ( x 0 cos ω 1 t + v 0 + p x 0 ω 1 sin ω 1 t ) for the under damped case of a mass-spring-dashpot system with initial conditions x 0 = x ( 0 ) and v 0 = x ' ( 0 )

Solution Summary: The author explains that the differential equation of motion for a mass-spring-dashpot system is x"+2px'+omega.

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Program Description: Purpose of the problem is to show that x(t)=e−pt(x0cosω1t+v0+px0ω1sinω1t) for the under damped case of a mass-spring-dashpot system with initial conditions x0=x(0) and v0=x'(0)

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Chapter 3 Solutions