Damped Driven Oscillation
Explore m*xddot+b*xdot+k*x=F0*cos(omega*t), distinguishing a transient trajectory from steady-state amplitude and damping-dependent displacement resonance.
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Damped Driven Oscillation
Concept Overview
A damped driven oscillator extends simple harmonic motion by adding a resistive damping force proportional to velocity and a periodic external driving force. The equation of motion ism · x″ + b · x′ + k · x = F₀ · cos(ω · t). For positive damping, the free transient decays and the motion approaches a forced response at the driving frequency. With zero damping the free motion persists; at exact resonance the driven motion grows without a bounded steady state.
Steady-State Amplitude
The steady-state solution is of the form x(t) = A · cos(ω · t - φ). The amplitude A is given by:
Let's calculate the steady amplitude for a specific set of parameters. Given parameters: m = 1 kg, k = 4 N/m, b = 1 kg/s, F₀ = 1 N, ω = 2 rad/s.
- Calculate the spring and mass term: (k - m · ω²)² = (4 - 1 · 2²)² = (4 - 4)² = 0.
- Calculate the damping term: (b · ω)² = (1 · 2)² = 4.
- Substitute into the amplitude equation to find the final expected result: A = 1 / √(0 + 4) = 1 / 2 = 0.5 m.
At exact resonance without damping (b = 0), there is no finite steady-state amplitude. The displacement has a growing envelope. With F₀ = 0, the forced response is zero even at resonance.
At zero frequency the drive is constant: the static displacement is F₀/k = 1/4 = 0.25 m for the worked parameters. For positive damping, minimizing the squared denominator gives the displacement-response peak at ω = √max(0, k/m − b²/(2m²)). Here it is √3.5 ≈ 1.8708 rad/s, below the natural frequency √(k/m) = 2. The peak is at zero when the expression inside the maximum is nonpositive. With no drive there is no driven peak.
Interactive Experiment
You can reproduce this behavior using the Damped Driven Oscillation module:
- Set Mass (m) to 1.0 kg and Stiffness (k) to 4.0 N/m.
- Set Damping (b) to 1.0 kg/s.
- Set Drive Force (F₀) to 1.0 N and Drive Freq (ω) to 2.0 rad/s.
- Click Step for fixed 0.1-second updates or Play for continuous motion. Observe the initial transient motion which eventually decays.
- Notice that the steady-state oscillation amplitude stabilizes near 0.5 m. The view fits the displacement automatically; pixel length is not a fixed physical scale.
Assumptions and Limitations
This model assumes linear damping (viscous friction proportional to velocity) and a perfectly linear spring (Hooke's Law holds for all extensions). In reality, real springs become non-linear at large displacements, and damping can be non-linear (e.g., aerodynamic drag proportional to velocity squared). Motion starts at rest and uses fourth-order Runge–Kutta integration with substeps no larger than 0.005 s; it remains a numerical approximation. Parameter changes restart the simulation and each run is limited to 120 s. The frequency-response curve is sampled every 0.02 rad/s; it cannot represent an infinite pole as a finite peak.
References and Further Reading
To dive deeper into the mathematical derivation and physical intuition of driven harmonic oscillators, you can consult these standard physics textbooks and resources which provide excellent foundational coverage:
- OpenStax University Physics Volume 1: 15.5 Damped Oscillations - Covers the foundational theory of damping in harmonic motion.
- OpenStax University Physics Volume 1: 15.6 Forced Oscillations - Explains driven systems, steady-state response, and resonance curves.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Damped Driven Oscillation module.
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