Driven Harmonic Oscillator

Adding an external sinusoidal driving force to a simple harmonic oscillator model gives rise to the Driven Harmonic Oscillator model.  If we include a viscous drag force (friction) the net force on the mass m is now the sum of three forces.

Fnet = Fspring + Fdrag + Fexternal

where Fspring = -k x, Fdrag = -b v, and Fexternal = FD sin ( ωD t ).  The spring constant k, the damping constant b, the drive amplitude FD, and the drive frequency ωD along with the initial conditions x(t=0) and v(t=0) are inputs into the model. 

Substituting the net force into Newton's Second Law gives the dynamical equation for the sinusoidally driven oscillator

d2x / dt2 = - (k/m) x - (b/m) dx / dt + (FD /m) sin ( ωD t ) .

Note that the drive frequency ωD is not the natural frequency ω0 = (k/m)1/2  of the undriven simple harmonic motion.  As shown in the activities, the oscillator's displacement x is a maximum when it is driven at it natural frequency  ωD = ω0The analytic solution the differential equation is well know and consists of two parts.  The first part, known as the homogeneous or transient solution, is the solution without a driving force.  The solution is a damped sinusoidal oscillation if  the damping constant b < 4 m k is

xH(t) = A0 e-bt/2m cos( ω0t + φ) .

Note that the homogenous solution will not be sinusoidal if b 4 m k.  One modeling activity is to set the drive amplitude FD to zero and to vary the damping constant b in order to observe sinusoidal and non-sinusoidal  homogeneous solutions.

The second part of the solution is known as the inhomogeneous or steady state solution.  The inhomogeneous solution is a constant-amplitude sinusoidal function with frequency equal to the drive frequency ωD

xI(t) = A(ωD) cos( ωDt + γ) .

Although the amplitude is constant, its value A(ωD) depends on the drive frequency ωD  and other parameters.  The phase shift  γ between the external force and the steady state solution is also function of ωD .  The complete solution to the dynamical equations is the sum of the homogenous and inhomogeneous solutions.

x(t) = xH(t)  + xI(t)