Spring-Mass-Damper System with External Forcing

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

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

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

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 \(\Delta t=0.005\,\text{s}\), advanced by real elapsed time so playback is frame-rate independent.
• Backward Euler / BDF-2: A-stable implicit methods for stiff (heavily overdamped) systems.
• Gauss–Legendre (2-stage): A 4th-order, symplectic, A-stable implicit Runge–Kutta method, selectable from the Numerical method menu. The system is linear in the state, so the 2-stage equations are a fixed \(4\times4\) solve (precomputed when \(m,c,k\) change); each step is then one \(4\times4\) matrix product plus two evaluations of \(F(t)\) — as cheap as BDF-2 but far more accurate for smooth forcing.
• 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 the 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.