Spring-Mass-Damper System with External Forcing

Simulate the forced harmonic oscillator \(m\ddot{y}+c\dot{y}+ky=F(t)\) — observe resonance, beats, impulse response, and transient behaviour in real time.

Governing Equation

The simulation models a mass \(m\) attached to a spring with natural (rest) length \(L_0\) [m] and stiffness \(k\), subject to viscous damping \(c\) and an external force \(F(t)\). Let \(x\) denote the total extension from the fixed support. Newton's second law gives

\[m\ddot{x}+c\dot{x}+k(x-L_0)=mg+F(t).\]

The static equilibrium satisfies \(k(x_{\rm eq}-L_0)=mg\), so \(x_{\rm eq}=L_0+mg/k\). Defining displacement from static equilibrium \(y=x-x_{\rm eq}\) eliminates gravity and reduces the equation to

\[m\ddot{y}+c\dot{y}+ky=F(t),\qquad y(0)=y_0,\quad\dot{y}(0)=v_0.\]

The rest length \(L_0\) sets the visual scale of the animation. Larger \(L_0\) provides more canvas space for large-amplitude oscillations. The natural frequency is \(\omega_0=\sqrt{k/m}\) and the damping ratio is \(\zeta=c/(2\sqrt{km})\).

Classification by Damping Ratio

Underdamped \((\zeta<1)\): oscillations with exponentially decaying amplitude. Discriminant \(c^2-4mk<0\).
Critically damped \((\zeta=1)\): fastest return to equilibrium without oscillation.
Overdamped \((\zeta>1)\): slow exponential return, no oscillation.

Observable Phenomena

Resonance (undamped): When \(\omega_\text{forcing}=\omega_0\) and \(c=0\), amplitude grows without bound as \(t\sin(\omega_0 t)\).
Resonance (damped): Amplitude peaks near \(\omega_0\) but remains bounded; the steady-state amplitude is \(F_0/(c\omega_0)\) at exact resonance.
Beats: When the forcing frequency is close but not equal to \(\omega_0\), the solution exhibits slow oscillation of amplitude — a beating pattern at frequency \(|\omega-\omega_0|/2\).
Transient and steady state: The general solution is the sum of the homogeneous (transient) and particular (steady-state) solutions. For \(c>0\), the transient decays and only the steady-state remains.

Special Forcing Functions

Dirac delta \(\delta(t-c)\): an instantaneous impulse at \(t=c\). Implemented by adding \(\Delta v=A/m\) to the velocity when \(|t-c|<\Delta t/2\).

Heaviside step \(u(t-c)\): equals 0 for \(t<c\) and 1 for \(t\ge c\), enabling delayed forcing activation.

Expression examples: 5*cos(2*t), 3*delta(t-10), u(t-5)*cos(2*(t-5)), 2*delta(t-10)+u(t-35)*cos(2*(t-35)).

Numerical Integration

Velocity Verlet (Störmer method): Symplectic integrator used for non-stiff systems (\(c^2-4mk\le5\)). Fixed step \(\Delta t=0.01\,\text{s}\).
Backward Euler / BDF-2: A-stable implicit methods for stiff (heavily overdamped) systems.
Impulse handling: The delta function impulse is applied as a direct velocity jump.

How to Use

Adjust \(m,c,k,L_0\) and initial conditions using sliders or value boxes. The equation updates automatically.
Increase \(L_0\) to lengthen the spring and allow larger initial displacements \(y_0\).
Enter any expression for \(F(t)\) in the forcing function field. Use t as the time variable.
Select an example from the dropdown to load preset parameters.
Click Reset to restart. Click Pause/Resume to freeze and continue.

\(m\ddot{y}+c\dot{y}+ky=F(t)\)

Parameters & Initial Conditions

Mass \(m\) [kg]

Damping \(c\) [N·s/m]

Spring \(k\) [N/m]

Rest length \(L_0\) [m]

Initial displacement \(y_0\) [m]

Initial velocity \(v_0\) [m/s]

End time \(T\) [s]

Forcing function \(F(t)\)

Load example

Displacement \(y(t)\) vs time

Spring-mass animation