A mass-spring-damper obeys m x +b x +kx=F(t) with , ,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Simple, damped and driven oscillations

Conditions of Validity, Ch Linear ordinary differential equations: existence, the Wronskian, and a kernel in disguise

Problem

A mass-spring-damper obeys

m\ddot{x}+b\dot{x}+kx=F(t)

with

m=0.010\,\mathrm{kg}

b=0.10\,\mathrm{N\,s\,m^{-1}}

k=100\,\mathrm{N\,m^{-1}}

x

the displacement in metres and

F(t)

the applied force in newtons; the coefficients are constant and hence continuous for all

t

, so the existence and uniqueness theorem applies for all time. The system starts at rest. ((a)) Show the motion is underdamped and find the damped angular frequency

\omega_d

. ((b)) Construct the initial-value Green function

G(t,\xi)

giving the displacement produced by a unit impulsive force at

t=\xi

, and state the jump it must satisfy at

t=\xi

. ((c)) Evaluate

G(t,\xi)

for

t-\xi=0.010\,\mathrm{s}

Related theory

Ordinary differential equations

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