Forced Damped Harmonic Oscillator with Arbitrary Initial Conditions

Introduction

The motion of the subject harmonic oscillator is a result that reflects very many real-life mechanisms. Typically we expect a drag force that is proportional to the speed of the oscillator and a restoring force that is proportional to its displacement. The drag coefficient will be labeled b here and the restoring force coefficient will be labeled k. Of course, the coefficient of the acceleration will be the mass, m, of the oscillating object as expected from Newton's law of motion. The frequency of oscillation of the free oscillator will be complex since there is a drag force and this frequency will be labeled w. Since ω is complex, it will have a real part which I will name ω₀ and an imaginary part labeled α. For initial conditions, we may have arbitrary displacement, y(0), as well as arbitrary initial speed $\dot{y} (0)$ where 0 here indicates time=0, and time is labeled t here. These are the parameters of the free oscillator.

The parameters of the sinusoidal force are the peak force value, F, and the frequency of the force, ω_F.

Calculating the Motion of the Oscillator.

The values of ω₀ and α come from the solution to a quadratic equation of the damped and un-forced (free) harmonic oscillator:

$\begin{array}{l} m \ddot{y} + b \dot{y} + k y = 0 \\ L e t y = Re a l [Y \exp (i ω t)] \\ Y (- m ω^{2} + i ω b + k y) = 0 \\ ω = \frac{- i b \pm \sqrt{- b^{2} + 4 m k}}{2 m} \end{array}$

(1.1)

Assuming that 4mk>b² and separating ω into real and imaginary parts we have:

$\begin{array}{l} ω_{0} \equiv ω_{r e a l} = \sqrt{\frac{k}{m} - \frac{b^{2}}{4 m^{2}}} \\ α \equiv ω_{i m a g} = - \frac{b}{2 m} \end{array}$

(1.2)

For the sinusoidally forced harmonic oscillator we have the equation:

$\begin{array}{l} m \ddot{y} + b \dot{y} + k y = F \cos ω_{F} t \\ L e t y = Re a l [Y \exp (i ω_{F} t)] \\ Y (- m ω_{F}^{2} + i ω_{F} b + k y) = F \end{array}$

(1.3)

$Y = \frac{F}{(- m ω_{F}^{2} + i ω_{F} b + k)}$

(1.4)

$\begin{array}{l} a_{F} \equiv Y_{Re a l} = \frac{F (k - m ω_{F}^{2})}{{(k - m ω_{F}^{2})}^{2} + b^{2} ω_{F}^{2}} \\ b_{F} \equiv Y_{Im a g} = \frac{F b ω_{F}}{{(k - m ω_{F}^{2})}^{2} + b^{2} ω_{F}^{2}} \end{array}$

(1.5)

Combining these the general solution for both the transient and steady state is:

$y (t) = \exp (- α t) [c_{c} \cos (ω_{0} t) + c_{s} \sin (ω_{0} t)] + a_{F} \cos (ω_{F} t) + b_{F} s i n (ω_{F} t)$

(1.6)

The time derivative of the general solution is:

$\begin{array}{l} \dot{y} (t) = - α \exp (- α t) [c_{c} \cos (ω_{0} t) + c_{s} \sin (ω_{0} t)] + ω_{0} \exp (- α t) [- c_{c} s i n (ω_{0} t) + c_{s} c o s (ω_{0} t)] + \\ ω_{F} [- a_{F} s i n (ω_{F} t) + b_{F} \cos (ω_{F} t] \end{array}$ (1.7)

To match initial conditions, we use the values of both y and $\dot{y}$ at t=0.

$\begin{array}{l} y (0) = c_{c} + a_{F} \\ \dot{y} (0) = - α c_{c} + ω_{0} c_{s} + ω_{F} b_{F} \end{array}$

(1.8)

where both c_c and c_s have to be solved for.

For generality, I will assume that both y(0) and $\dot{y} (0)$ are non-zero and will now solve for the values of constants c_c and c_s. Setting up a matrix multiplier for these constants and using equation (1.8) we obtain:

$(\begin{matrix} 1 & 0 \\ - α & ω_{0} \end{matrix}) (\begin{matrix} c_{c} \\ c_{s} \end{matrix}) = (\begin{matrix} y_{0} - a_{F} \\ {\dot{y}}_{0} - ω_{F} b_{F} \end{matrix})$

(1.9)

The inverse of a 2x2 matrix is computed by swapping the diagonal elements and negating the off-diagonal elements and then dividing all elements by the determinant of the original matrix.

Multiplying equation (1.9) by the inverse of the matrix on its left we get:

$\frac{1}{ω_{0}} (\begin{matrix} ω_{0} & 0 \\ α & 1 \end{matrix}) (\begin{matrix} 1 & 0 \\ - α & ω_{0} \end{matrix}) (\begin{matrix} c_{c} \\ c_{s} \end{matrix}) = \frac{1}{ω_{0}} (\begin{matrix} ω_{0} & 0 \\ α & 1 \end{matrix}) (\begin{matrix} y_{0} - a_{F} \\ {\dot{y}}_{0} - ω_{F} b_{F} \end{matrix})$

(1.10)

$(\begin{matrix} c_{c} \\ c_{s} \end{matrix}) = \frac{1}{ω_{0}} (\begin{matrix} ω_{0} & 0 \\ α & 1 \end{matrix}) (\begin{matrix} y_{0} - a_{F} \\ {\dot{y}}_{0} - ω_{F} b_{F} \end{matrix}) = (\begin{matrix} y_{0} - a_{F} \\ \frac{α}{ω_{0}} (y_{0} - a_{F}) + \frac{{\dot{y}}_{0} - ω_{F} b_{F}}{ω_{0}} \end{matrix})$

(1.11)

We can also express the general solution as cosines only by applying the phase of the ω₀ and the ω_F terms separately.

$y (t) = \exp (- α t) c_{0} \cos (ω_{0} t + ϕ_{0}) + c_{F} \cos (ω_{F} t + ϕ_{F})$

(1.12)

where:

$c_{0} = \sqrt{c_{c}^{2} + c_{s}^{2}}$

(1.13)

$ϕ_{0} = \tan^{- 1} (\frac{c_{s}}{c_{c}})$

(1.14)

and

$c_{F} = \sqrt{a^{2} + b^{2}}$

(1.15)

$ϕ_{F} = \tan^{- 1} (\frac{b_{F}}{a_{F}})$

(1.16)