Explanation Given: The weight of the object is 4 pound, stretching length of the spring is 2 feet and damping force is equal to the 7 8 times the instantaneous velocity. Definition used: “(1) The Laplace transformation of a linear differential equation with constant coefficient is expressed as follows: a n L { d n y d t n } = a n [ s n Y ( s ) − s ( n − 1 ) y ( 0 ) − ⋯ − y ( n − 1 ) ( 0 ) ] (2) The linearity property of the Laplace transformation is given below. L { a f ( t ) + b g ( t ) } = a L { f ( t ) } + b L { g ( t ) } where a , b be any constant and f , g be any function of t ,. (3) The linear differential equation for the damping system is as follows: m x ″ + β x ′ + k x = 0 where, m is the mass, β is the damping coefficient and k is spring constant.” Calculation: The value of m is calculated as follows: m = W g = 4 32 = 1 8 The value of k is calculated as follows: W = k x 4 = k ⋅ 2 k = 2 The damping coefficient β of the system is given as 7 8 . Substitute the value of m , β and k in the equation of damping system m x ″ + β x ′ + k x = 0 . 1 8 x ″ + 7 8 x ′ + 2 x = 0 x ″ + 7 x ′ + 16 x = 0 Use the definition (1) to write the Laplace transform of the above linear differential equation. s 2 L { x } − s x ( 0 ) − x ′ ( 0 ) + 7 [ s L { x } − x ( 0 ) ] + 16 L { x } = 0 ( 1 ) The initial values are calculated as follows: Since the weight is released 18 inches above from the equilibrium position then the initial displacement is as follows: x ( 0 ) = − 18 inch = − 3 2 feet Initial velocity of the spring is represented as follows: x ′ ( 0 ) = 0 Substitute the value x ( 0 ) = − 3 2 and x ′ ( 0 ) = 0 in equation (1). ( s 2 + 7 s + 16 ) L { x } + 3 2 s + 21 2 = 0 ( s 2 + 7 s + 16 ) L { x } = − 3 s + 21 2 L { x } = − 3 s + 21 2 ( s 2 + 7 s + 16 ) The obtained equation is as follows: L { x } = − 3 s + 21 2 ( s 2 + 7 s + 16 ) ( 2 ) Equation (2) is simplified as follows using perfect square method

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 4.3, Problem 33E

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Chapter 4.3, Problem 32E

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