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 = 1 k 1 = 1 , k 2 = 4 , k 3 = 1 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 ) = 96 cos 5 t F 2 ( t ) = 0 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 1 ] Similarly the stiffness matrix K is written as, K = [ − ( k 1 + k 2 ) k 2 k 2 − ( k 2 + k 3 ) ] K = [ − ( 1 + 4 ) 4 4 − ( 4 + 1 ) ] K = [ − 5 4 4 − 5 ] Similarly the external force vector F is written as, F = [ F 1 ( t ) F 2 ( t ) ] F = [ 96 cos 5 t 0 ] F = [ 96 0 ] cos 5 t Therefore, matrix differential equation is written as, [ 1 0 0 1 ] x ″ = [ − 5 4 4 − 5 ] x + [ 96 0 ] cos 5 t Reduce the above differential equation in simplest form as, x ″ = [ 1 0 0 1 ] − 1 [ − 5 4 4 − 5 ] x + [ 1 0 0 1 ] − 1 [ 96 0 ] cos 5 t x ″ = [ 1 0 0 1 ] [ − 5 4 4 − 5 ] x + [ 1 0 0 1 ] [ 96 0 ] cos 5 t x ″ = [ − 5 4 4 − 5 ] x + [ 96 0 ] cos 5 t Consider the periodic particular solution as x p ( t ) = c cos 5 t and so derivatives are written as, x p ( t ) = c cos 5 t x ′ p ( t ) = − 5 c sin 5 t x ″ p ( t ) = − 25 c cos 5 t Substitute, the above periodic particular solution and its derivative in the matrix differential equation and solve for the vector c as, − 25 c cos 5 t = [ − 5 4 4 − 5 ] c cos 5 t + [ 96 0 ] cos 5 t − 25 c = [ − 5 4 4 − 5 ] c + [ 96 0 ] − [ 96 0 ] = [ − 5 4 4 − 5 ] c + 25 c − [ 96 0 ] = [ 20 4 4 20 ] [ c 1 c 2 ] Further, solve the above equation as, 20 c 1 + 4 c 2 = − 96 4 c 1 + 20 c 2 = 0 Solve the system of equations by substitution method and obtain the value of c 1 and c 2 as, c 1 = − 5 c 2 = 1 Therefore, the period solution x p ( t ) = c cos 5 t of the equation is written as, x p ( t ) = c cos 5 t = [ c 1 c 2 ] cos 5 t = [ − 5 1 ] cos 5 t Solve the homogeneous part of the non-homogeneous equation as follows: The matrix A for the homogeneous part is written below. A = [ − 5 4 4 − 5 ] So, obtain the characteristic equation of matrix A to obtain the eigenvalues as, | − 5 − λ 4 4 − 5 − λ | = 0 ( − 5 − λ ) 2 − 4 2 = 0 ( − 5 − λ − 4 ) ( − 5 − λ + 4 ) = 0 ( − λ − 9 ) ( − λ − 1 ) = 0 Further, solve the above equation as, ( λ + 1 ) ( λ + 9 ) = 0 λ 1 = − 1 or λ 2 = − 9 The relation between natural frequency and eigenvalue in the mass spring system is defined as ω = − λ

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 8P

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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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PROBLEM 24 - 0586: A free harmonic oscillator (FHO) is a system whose behavior can be described by a second- order, linear differential equation of the form: = -Ay(t) (1) where A is a positive constant. Two FHO systems are a spring-mass system and an LC electric circuit: Y=dy/dt y(t) Spring- x- FHO V displacement =velocit q-charge in coulombs mass LC i circuit =curren in ampere Given the initial conditions y(0) = %3D Yo and y(0) = Zo , write a FORTRAN program that uses the modified Euler method to simulate this system from t = 0 to t %3D = tf . LINEAR SPRING OF STIFFNESS K C INDUCTOR CAPACITOR (HENRYS) 1 IM MASS (FARADS)

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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

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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