How Is the Amplitude Calculated in a Damped Spring-Mass System?

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SUMMARY

The amplitude of a damped spring-mass system is calculated using the formula R2 = 1/((1-w²)² + w²y²), where w represents the angular frequency and y is the damping constant. This formula derives from the general solution to the second-order differential equation governing the system's motion. The initial conditions provided are u(0) = 0 and u'(0) = 0, which are essential for solving the initial value problem (IVP) associated with the system.

PREREQUISITES
  • Understanding of second-order differential equations
  • Familiarity with the concepts of damping in mechanical systems
  • Knowledge of angular frequency (w) and its role in oscillatory motion
  • Basic principles of spring-mass systems and Hooke's Law
NEXT STEPS
  • Study the derivation of the general solution for second-order linear differential equations
  • Learn about the effects of damping on oscillatory systems
  • Explore the relationship between angular frequency and amplitude in forced oscillations
  • Investigate the application of initial value problems in mechanical systems
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Students studying physics or engineering, particularly those focusing on dynamics, mechanical vibrations, and differential equations.

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Homework Statement


A mass of 4 kg is stretches a spring by 1 m. An external force of cos (!t)
N acts on the mass. Assume that the damping constant is nonzero and gravity is 10 ms^-2.
Consider a spring mass system described by the following IVP.

u''+yu'+u = cos(wt)
u(0) = 0
u'(0) = 0

1) Find the amplitude of the spring mass system in terms of w and y.

The Attempt at a Solution



The solution says

R2 = 1/((1-w2)2+w2y2)

But how is that the solution, I can't find how to compute the amplitude in my textbook, it just says R2=A2+B2.
 
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It comes from the general solution to the differential equation.
 

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