A mass of 1 kgkg is attached to the end of a spring whose restoring force is 180 NmNm. The mass is in a medium that exerts a viscous resistance of 156 NN when the mass has a velocity of 6 msms. The viscous resistance is proportional to the speed of the object.Suppose the spring is stretched 0.07 mm beyond the its natural position and released. Let positive displacements indicate a stretched spring, and suppose that external vibrations act on the mass with a force of 7sin(3t)7sin(3t) NN at time tt seconds.Find an function to express the steady-state component of the object's displacement from the spring's natural position, in mm after tt seconds. (Note: This spring-mass system is not "hanging", so there is no gravitational force included in the model.) ### Spring-Mass System with Damping and External Vibrations#### Problem DescriptionA mass of \( 1 \, \text{kg} \) is attached to the end of a spring whose restoring force is \( 180 \, \frac{N}{m} \). The mass is in a medium that exerts a viscous resistance of \( 156 \, N \) when the mass has a velocity of \( 6 \, \frac{m}{s} \). The viscous resistance is proportional to the speed of the object.Suppose the spring is stretched \( 0.07 \, \text{m} \) beyond its natural position and released. Let positive displacements indicate a stretched spring, and suppose that external vibrations act on the mass with a force of \( 7 \sin(3t) \, N \) at time \( t \) seconds.#### ObjectiveFind a function to express the *steady-state component* of the object's displacement from the spring's natural position, in meters, after \( t \) seconds.**Note:** This spring-mass system is not "hanging," so there is no gravitational force included in the model.\[ u(t) = \boxed{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \]

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A mass of 1 kgkg is attached to the end of a spring whose restoring force is 180 NmNm. The mass is in a medium that exerts a viscous resistance of 156 NN when the mass has a velocity of 6 msms. The viscous resistance is proportional to the speed of the object. Suppose the spring is stretched 0.07 mm beyond the its natural position and released. Let positive displacements indicate a stretched spring, and suppose that external vibrations act on the mass with a force of 7sin(3t)7sin(3t) NN at time tt seconds. Find an function to express the steady-state component of the object's displacement from the spring's natural position, in mm after tt seconds. (Note: This spring-mass system is not "hanging", so there is no gravitational force included in the model.)