Non-Dimensionalization of the Damped Harmonic Oscillator Equation
We start with the damped harmonic oscillator equation (no external forcing):
$$ m\frac{d^2x}{dt^2} + \alpha\frac{dx}{dt} + kx = 0 $$Here:
- \(m\) is the mass
- \(\alpha\) is the damping coefficient
- \(k\) is the spring constant
To find the natural timescale, we ask ourselves 'What sets the timescale of motion if only inertia and the restoring spring force are present?'. Ignoring damping, balance inertia and the spring force:
$$ m\frac{x}{t^2} \sim kx $$This estimates the timescale over which inertia and the restoring force balance.
We obtain
$$ \frac{m}{t^2} \sim k $$So the natural time scale is
$$ t_0 = \sqrt{\frac{m}{k}} $$This is the characteristic oscillation timescale of the undamped system.
We introduce nondimensional variables
$$l = \frac{x}{x_0}, \quad \tau = \frac{t}{t_0}$$where:
- \(l\) is a dimensionless length
- \(\tau\) is a dimensionless time
The variable \( l \) therefore measures displacement relative to the characteristic scale \( x_0 \). Similarly, the variable \( \tau \) measures time relative to the characteristic scale \( t_0 \).
We apply the chain rule to compute derivatives with respect to \(t\).
First derivative
We start with:
$$ \frac{dx}{dt} = \frac{dx}{d\tau} \frac{d\tau}{dt} $$Since \(x = x_0 \, l(\tau)\), we have
$$ \frac{dx}{d\tau} = x_0 \frac{dl}{d\tau} $$And since \( \displaystyle \tau = \frac{t}{t_0} \), we get
$$ \frac{d\tau}{dt} = \frac{1}{t_0} $$Therefore
$$ \frac{dx}{dt} = \frac{x_0}{t_0} \frac{dl}{d\tau} $$Second derivative
Now differentiate again with respect to \(t\)
$$ \frac{d^2x}{dt^2} = \frac{d}{dt} \left( \frac{x_0}{t_0} \frac{dl}{d\tau} \right) $$ $$ \frac{d^2x}{dt^2} = \frac{x_0}{t_0} \frac{d}{dt} \left( \frac{dl}{d\tau} \right) $$Applying the chain rule again
$$ \frac{d}{dt} \left( \frac{dl}{d\tau} \right) = \frac{d}{d\tau} \left( \frac{dl}{d\tau} \right) \frac{d\tau}{dt} $$This gives
$$ \frac{d}{dt} \left( \frac{dl}{d\tau} \right) = \frac{1}{t_0} \frac{d^2l}{d\tau^2} $$ $$\therefore \hspace{3mm} \frac{d^2x}{dt^2} = \frac{x_0}{t_0^2} \frac{d^2l}{d\tau^2} $$Substituting into:
$$ m\ddot{x} + \alpha\dot{x} + kx = 0 $$gives
$$ m \left(\frac{x_0}{t_0^2} \frac{d^2l}{d\tau^2}\right) + \alpha \left(\frac{x_0}{t_0} \frac{dl}{d\tau}\right) + k(x_0 l) = 0 $$Dividing by \( k x_0 \)
$$ \frac{m}{k t_0^2} \frac{d^2l}{d\tau^2} + \frac{\alpha}{k t_0} \frac{dl}{d\tau} + l = 0 $$Using the natural time scale \( \, \displaystyle t_0 = \sqrt{\frac{m}{k}} \, \) we simplify the coefficients and the equation becomes
$$ \frac{d^2l}{d\tau^2} + \frac{\alpha}{\sqrt{mk}} \frac{dl}{d\tau} + l = 0 $$We define the dimensionless damping parameter
$$ \gamma = \frac{\alpha}{\sqrt{mk}} $$The system reduces to
$$ \frac{d^2l}{d\tau^2} + \gamma \frac{dl}{d\tau} + l = 0 $$All damped harmonic oscillators collapse to this universal form after non-dimensionalization. The only parameter that controls the dynamics is the dimensionless damping ratio \(\gamma\), which compares damping strength to the natural oscillation scale of the system.
The value of \( \gamma \) determines the qualitative behavior of the oscillator. For \( \gamma < 2 \), the system is underdamped and oscillates while gradually losing energy. At \( \gamma = 2 \), the system is critically damped, corresponding to the threshold between oscillatory and non-oscillatory motion. This is the fastest return to equilibrium without overshooting. For \( \gamma > 2 \), the system is overdamped and returns to equilibrium without oscillating.
The critical value \( \gamma = 2 \) comes from the characteristic equation
$$ r^2 + \gamma r + 1 = 0 $$whose discriminant \( \gamma^2 - 4 \) changes sign at \( \gamma = 2 \), marking the transition between oscillatory and non-oscillatory motion.