Damped Forced Oscillations -- Practical Resonance
by Mike Martin

Consider a damped spring-mass system with external sinusoidal forcing:

m x '' (t) + b x ' (t) + k x (t)= Fo cos(ω t)

Submit positive real values for the mass, m, the damping constant, b, the spring constant, k, and the forcing amplitude, Fo. For these parameter values, the transient solution dies out over time and we therefore focus on the steady-state solution. The steady-state solution is sinusoidal whose amplitude is a function of the forcing frequency, ω. By examining the graph of this amplitude we can readily identify the maximal amplitude and the forcing frequency that produces it. It can be shown that there is a single maximum value if

b2 < 2 k m

and is monotonically decreasing otherwise. The amplitude is plotted on the interval [0, W], where the final forcing frequency value, W, is input by the user. x (t) represents the displacement of the mass at time t, x ' (t) represents the velocity of the mass at time t, and x '' (t) represents the acceleration of the mass at time t.

Enter in the value of the parameters and select the plotting bounds, then press the "Plot" button.

m =  

b =  

k =  

Fo =  

W =  


MathWorld's Simple Harmonic Oscillator

Wikipedia's Harmonic Oscillator

Wikipedia's Simple Harmonic Motion

Wikipedia's Resonance