Program Explanation Given information: The stiffness matrix for the given mass and spring system is written as, K = [ − ( k 1 + k 2 ) k 2 k 2 − ( k 2 + k 3 ) ] The value of each mass and spring constant is written below. m 1 = 1 , m 2 = 2 k 1 = 1 , k 2 = 2 , k 3 = 2 The walls are not exist for the given spring mass system. Initial conditions for the system is written as, x 1 ( 0 ) = 0 , x 2 ( 0 ) = 0 x ′ 1 ( 0 ) = 0 , x ′ 2 ( 0 ) = 0 The given external forces F 1 ( t ) and F 2 ( t ) acting on the masses m 1 and m 2 respectively as, F 1 ( t ) = 0 F 2 ( t ) = 120 cos 3 t Explanation: The matrix differential equation for the given system is written as, M x ″ = K x + F Here the mass matrix M is written as, M = [ m 1 0 0 m 2 ] M = [ 1 0 0 2 ] Similarly the stiffness matrix K is written as, K = [ − ( k 1 + k 2 ) k 2 k 2 − ( k 2 + k 3 ) ] K = [ − ( 1 + 2 ) 2 2 − ( 2 + 2 ) ] K = [ − 3 2 2 − 4 ] Similarly the external force vector F is written as, F = [ F 1 ( t ) F 2 ( t ) ] F = [ 0 120 cos 3 t ] F = [ 0 120 ] cos 3 t Therefore, matrix differential equation is written as, [ 1 0 0 2 ] x ″ = [ − 3 2 2 − 4 ] x + [ 0 120 ] cos 3 t Reduce the above differential equation in simplest form as, x ″ = [ 1 0 0 2 ] − 1 [ − 3 2 2 − 4 ] x + [ 1 0 0 2 ] − 1 [ 0 120 ] cos 3 t x ″ = [ 1 0 0 0.5 ] [ − 3 2 2 − 4 ] x + [ 1 0 0 0.5 ] [ 0 120 ] cos 3 t x ″ = [ − 3 2 1 − 2 ] x + [ 0 60 ] cos 3 t Consider the periodic particular solution as x p ( t ) = c cos 3 t and so derivatives are written as, x p ( t ) = c cos 3 t x ′ p ( t ) = − 3 c sin 3 t x ″ p ( t ) = − 9 c cos 3 t Substitute, the above periodic particular solution and its derivative in the matrix differential equation and solve for the vector c as, − 9 c cos 3 t = [ − 3 2 1 − 2 ] c cos 3 t + [ 0 60 ] cos 3 t − 9 c = [ − 3 2 1 − 2 ] c + [ 0 60 ] − [ 0 60 ] = [ − 3 2 1 − 2 ] c + 9 c − [ 0 60 ] = [ 6 2 1 7 ] [ c 1 c 2 ] Further, solve the above equation as, 6 c 1 + 2 c 2 = 0 c 1 + 7 c 2 = − 60 Solve the system of equations by substitution method and obtain the value of c 1 and c 2 as, c 1 = 3 c 2 = − 9 Therefore, the period solution x p ( t ) = c cos 3 t of the equation is written as, x p ( t ) = c cos 3 t = [ c 1 c 2 ] cos 3 t = [ 3 − 9 ] cos 3 t Solve the homogeneous part of the non-homogeneous equation as follows: The matrix A for the homogeneous part is written below

MyLab Math with Pearson eText -- 24-Month Standalone Access Card -- For Differential Equations and Boundary Value Problems: Computing and Modeling Tech Update

5th Edition

ISBN: 9780134872971

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

Publisher: PEARSON

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Chapter 5.4, Problem 9P

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Program Description: Purpose of the problem is to find the motion described by mass and spring system as a superposition of oscillations at three different frequencies.

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Chapter 5.4, Problem 8P

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MyLab Math with Pearson eText -- 24-Month Standalone Access Card -- For Differential Equations and Boundary Value Problems: Computing and Modeling Tech Update

Ch. 5.1 - Let A=[2347] and B=[3451]. Find (a) 2A+3B; (b)...Ch. 5.1 - Prob. 2P Ch. 5.1 - Find AB and BA given A=[203415] and B=[137032].Ch. 5.1 - Prob. 4P Ch. 5.1 - Prob. 5P Ch. 5.1 - Prob. 6P Ch. 5.1 - Prob. 7P Ch. 5.1 - Prob. 8P Ch. 5.1 - Prob. 9P Ch. 5.1 - Prob. 10P

Ch. 5.1 - Prob. 11P Ch. 5.1 - Prob. 12P Ch. 5.1 - Prob. 13P Ch. 5.1 - Prob. 14P Ch. 5.1 - Prob. 15P Ch. 5.1 - Prob. 16P Ch. 5.1 - Prob. 17P Ch. 5.1 - Prob. 18P Ch. 5.1 - Prob. 19P Ch. 5.1 - Prob. 20P Ch. 5.1 - Prob. 21P Ch. 5.1 - Prob. 22P Ch. 5.1 - Prob. 23P Ch. 5.1 - Prob. 24P Ch. 5.1 - Prob. 25P Ch. 5.1 - Prob. 26P Ch. 5.1 - Prob. 27P Ch. 5.1 - Prob. 28P Ch. 5.1 - Prob. 29P Ch. 5.1 - Prob. 30P Ch. 5.1 - Prob. 31P Ch. 5.1 - Prob. 32P Ch. 5.1 - Prob. 33P Ch. 5.1 - Prob. 34P Ch. 5.1 - Prob. 35P Ch. 5.1 - Prob. 36P Ch. 5.1 - Prob. 37P Ch. 5.1 - Prob. 38P Ch. 5.1 - Prob. 39P Ch. 5.1 - Prob. 40P Ch. 5.1 - Prob. 41P Ch. 5.1 - Prob. 42P Ch. 5.1 - Prob. 43P Ch. 5.1 - Prob. 44P Ch. 5.1 - Prob. 45P Ch. 5.2 - Prob. 1P Ch. 5.2 - Prob. 2P Ch. 5.2 - Prob. 3P Ch. 5.2 - Prob. 4P Ch. 5.2 - Prob. 5P Ch. 5.2 - Prob. 6P Ch. 5.2 - Prob. 7P Ch. 5.2 - Prob. 8P Ch. 5.2 - Prob. 9P Ch. 5.2 - Prob. 10P Ch. 5.2 - Prob. 11P Ch. 5.2 - Prob. 12P Ch. 5.2 - Prob. 13P Ch. 5.2 - Prob. 14P Ch. 5.2 - Prob. 15P Ch. 5.2 - Prob. 16P Ch. 5.2 - Prob. 17P Ch. 5.2 - Prob. 18P Ch. 5.2 - Prob. 19P Ch. 5.2 - Prob. 20P Ch. 5.2 - Prob. 21P Ch. 5.2 - Prob. 22P Ch. 5.2 - Prob. 23P Ch. 5.2 - Prob. 24P Ch. 5.2 - Prob. 25P Ch. 5.2 - Prob. 26P Ch. 5.2 - Prob. 27P Ch. 5.2 - Prob. 28P Ch. 5.2 - Prob. 29P Ch. 5.2 - Prob. 30P Ch. 5.2 - Prob. 31P Ch. 5.2 - Prob. 32P Ch. 5.2 - Prob. 33P Ch. 5.2 - Prob. 34P Ch. 5.2 - Prob. 35P Ch. 5.2 - Prob. 36P Ch. 5.2 - Prob. 37P Ch. 5.2 - Prob. 38P Ch. 5.2 - Prob. 39P Ch. 5.2 - Prob. 40P Ch. 5.2 - Prob. 41P Ch. 5.2 - Prob. 42P Ch. 5.2 - Prob. 43P Ch. 5.2 - Prob. 44P Ch. 5.2 - Prob. 45P Ch. 5.2 - Prob. 46P Ch. 5.2 - Prob. 47P Ch. 5.2 - Prob. 48P Ch. 5.2 - Prob. 49P Ch. 5.2 - Prob. 50P Ch. 5.3 - Prob. 1P Ch. 5.3 - Prob. 2P Ch. 5.3 - Prob. 3P Ch. 5.3 - Prob. 4P Ch. 5.3 - Prob. 5P Ch. 5.3 - Prob. 6P Ch. 5.3 - Prob. 7P Ch. 5.3 - Prob. 8P Ch. 5.3 - Prob. 9P Ch. 5.3 - Prob. 10P Ch. 5.3 - Prob. 11P Ch. 5.3 - Prob. 12P Ch. 5.3 - Prob. 13P Ch. 5.3 - Prob. 14P Ch. 5.3 - Prob. 15P Ch. 5.3 - Prob. 16P Ch. 5.3 - Prob. 17P Ch. 5.3 - Prob. 18P Ch. 5.3 - Prob. 19P Ch. 5.3 - Prob. 20P Ch. 5.3 - Prob. 21P Ch. 5.3 - Prob. 22P Ch. 5.3 - Prob. 23P Ch. 5.3 - Prob. 24P Ch. 5.3 - Prob. 25P Ch. 5.3 - Prob. 26P Ch. 5.3 - Prob. 27P Ch. 5.3 - Prob. 28P Ch. 5.3 - Prob. 29P Ch. 5.3 - Prob. 30P Ch. 5.3 - Prob. 31P Ch. 5.3 - Prob. 32P Ch. 5.3 - Prob. 33P Ch. 5.3 - Verify Eq. 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Program Description: Purpose of the problem is to find the motion described by mass and spring system as a superposition of oscillations at three different frequencies.

Solution Summary: The author explains that the problem is to find the motion described by mass and spring system as a superposition of oscillations at three different frequencies.

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Program Description: Purpose of the problem is to find the motion described by mass and spring system as a superposition of oscillations at three different frequencies.

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Chapter 5 Solutions