A diving board is modelled as a single-degree-of-freedom spring-mass-damper system:…

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 diving board is modelled as a single-degree-of-freedom spring-mass-damper system: an effective end mass

m = 50\,\mathrm{kg}

, effective stiffness

k = 2.0\times10^{4}\,\mathrm{N\,m^{-1}}

and effective damping constant

c = 200\,\mathrm{N\,s\,m^{-1}}

, so the tip deflection

y(t)

(positive downwards) obeys

m\ddot{y}+c\dot{y}+ky = F(t).

The board's own mass and any air resistance beyond the stated damping are ignored. The board is undisturbed,

y(0)=0

\dot{y}(0)=0

, when a diver steps on at

t=0

and applies a constant downward force

F(t)=F_0=1.0\times10^{3}\,\mathrm{N}

, maintained for

\tau_p = 0.50\,\mathrm{s}

, after which the diver leaves and

F

returns to zero. ((a)) Find the damped angular frequency

\omega_d

and natural period

T

of the unforced board, and confirm

\tau_p > T

. ((b)) Using the convolution solution built from the causal Green function of this operator, find the maximum downward deflection reached while the diver's force is still applied, and the time (measured from

t=0

) at which it occurs. ((c)) Express this peak as a percentage overshoot above the static deflection

F_0/k

the board would settle to under a steady load.

Related theory

Ordinary differential equations

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