Explanation Given: The value of mass is 2 kg and the spring constant k is 32 N/m . Calculation: First convert the unit of spring constant ( k ) in kg/sec 2 as follows, 1 N/m = kg/sec 2 32 N/m = 32 kg/sec 2 The force acting on spring is expressed by a function as follows: f ( t ) = 68 e − 2 t cos 4 t The differential equation for the spring mass system is expressed as, d 2 x d t 2 + k m x = f ( t ) (1) Substitute k = 32 and m = 2 in the equation (1) as follows: d 2 x d t 2 + 32 2 x = 68 2 e − 2 t cos 4 t d 2 x d t 2 + 16 x = 34 e − 2 t cos 4 t Therefore, the equation of motion is, d 2 x d t 2 + 16 x = 34 e − 2 t cos 4 t (2) As there is no force acting on the system the value of x ( 0 ) = 0 and at equilibrium, it is x ′ ( 0 ) = 0 . The homogeneous differential equation of equation (2) is, d 2 x d t 2 + 16 x = 0 (3) The auxiliary equation of equation (3) is, m 2 + 16 = 0 m 2 = − 16 m = ± 4 i Therefore, the root m 1 is − 4 i and m 2 is 4 i . The roots are complex and hence the solution is, x = e α t [ c 1 cos ( β t ) + c 2 sin ( β t ) ] Here, the value of α = 0 , β = 4 . Substitute α = 0 , β = 4 and find the complementary solution as, x c = c 1 cos ( 4 t ) + c 2 sin ( 4 t ) (4) The particular solution is x p = e − 2 t ( A cos 4 t + B sin 4 t ) . Differentiate x p with respect to t . x ′ p = e − 2 t ( − 2 A cos 4 t − 2 B sin 4 t − 4 A sin 4 t + 4 B cos 4 t ) Now, differentiate x ′ p with respect to t . x ″ p ( t ) = − 2 e − 2 t ( − 2 A cos 4 t − 2 B sin 4 t − 4 A sin 4 t + 4 B cos 4 t ) + e − 2 t ( 8 A sin 4 t − 8 B cos 4 t − 16 A cos 4 t − 16 B sin 4 t ) = e − 2 t ( − 12 A cos 4 t − 12 B sin 4 t + 16 A sin 4 t − 16 B cos 4 t ) Substitute the values x p ( t ) , x ′ p ( t ) , x ″ p ( t ) in the given differential equation

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 3.8, Problem 37E

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To find: the equation of motion in the absence of damping.

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Chapter 3.8, Problem 36E

Chapter 3.8, Problem 38E

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the equation of motion in the absence of damping.

Solution Summary: The author explains the equation of motion in the absence of damping. The unit of spring constant k is 32N/m.

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Chapter 3 Solutions