Damped Oscillator

anypinn create my-project --template damped-oscillator

Harmonic oscillator with damping. Recovers damping ratio \(\zeta\) from displacement observations.

Background

The damped harmonic oscillator is a fundamental model in mechanics, electrical engineering, and control theory. It describes a system subject to a restoring force proportional to displacement and a dissipative force proportional to velocity. The damping ratio \(\zeta\) classifies the response into three regimes: underdamped (\(\zeta < 1\), oscillatory decay), critically damped (\(\zeta = 1\), fastest non-oscillatory return), and overdamped (\(\zeta > 1\), sluggish return). The template uses an underdamped configuration (\(\zeta = 0.15\)).

Governing Equations

Second-order ODE with learnable damping ratio \(\zeta\):

\[ \ddot{x} + 2\zeta\omega_0\,\dot{x} + \omega_0^2\,x = 0 \]

Equivalently, as a first-order system with velocity \(v = \dot{x}\):

\[ \begin{cases} \dfrac{dx}{dt} = v \\[8pt] \dfrac{dv}{dt} = -2\zeta\omega_0\,v - \omega_0^2\,x \end{cases} \]

where:

• \(x(t)\): displacement
• \(v(t)\): velocity
• \(\omega_0\): natural frequency (known)
• \(\zeta\): damping ratio (to recover)

Default Configuration

The generated template uses the following values.

Parameters to recover:

Symbol	Code constant	True value
\(\zeta\)	TRUE_ZETA	\(0.15\)

Known constants:

Symbol	Code constant	Value
\(\omega_0\)	TRUE_OMEGA0	\(2\pi \approx 6.283\)

Initial conditions: \(x(0) = 1.0, \quad \dot{x}(0) = 0.0\)

Domain: \(t \in [0, 5]\) s

Features Demonstrated

• Second-order ODE support via ODEProperties.order
• Native higher-order initial condition enforcement (ODEProperties.dy0)
• Scalar Parameter recovery

Results