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 = ω0 . The 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)