The animation compares the oscillation behavior of two spring-mass systems: one without damping (harmonic oscillation) and one with damping (damped oscillation). Amplitude, frequency, and damping constant are adjustable. The motions are shown both mechanically and in a time diagram.
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Description of the Animation
The animation visualizes two fundamental types of mechanical oscillations side by side:
Undamped oscillation (blue): The mass oscillates with constant amplitude around its equilibrium position. The motion follows the equation:
\[ y(t) = -A \cdot \cos(\omega t) \]
Damped oscillation (red): Due to frictional forces, the amplitude decreases exponentially. The motion is described by:
\[ y(t) = -A \cdot e^{-\delta t} \cdot \cos(\omega t) \]
Here, \( A \) is the initial amplitude, \( \omega = 2\pi f \) is the angular frequency, and \( \delta \) is the damping constant.
Note: The animation uses the same angular frequency \( \omega \) for both oscillations. In reality, the frequency of the damped oscillation is slightly lower: \( \omega_d = \sqrt{\omega_0^2 – \delta^2} \). For weak damping, this difference is negligible.
The left area of the animation shows the two spring-mass systems. The springs are drawn dynamically and adapt to their respective displacement. The right area displays a coordinate system showing the time evolution of both oscillations. The dashed envelope \( \pm A \cdot e^{-\delta t} \) illustrates the exponential amplitude decay of the damped oscillation. Connecting lines between masses and curve markers allow direct correlation to the diagram.
Interactive Controls
The following parameters are adjustable:
- Amplitude A: Determines the maximum displacement from the equilibrium position (20 to 120 pixels). Both oscillations start with this amplitude.
- Frequency f: Controls the oscillation frequency from 0.1 to 2.0 Hz. The period \( T = 1/f \) is shown in the current values display.
- Damping δ: Controls the strength of damping from 0 to 2. At δ = 0, the red oscillation also behaves undamped; at high values, it decays quickly.
- Start/Pause: Starts or pauses the animation. The simulation runs over a time span of 10 seconds.
- Reset: Resets the time to zero and restarts the display.
The lower left area shows the current values: time \( t \), the instantaneous displacements \( y_1 \) and \( y_2 \), and the period \( T \).
Overview
| Title | Damped mass-spring system |
| Target Audience | Teachers and Lecturers |
| Features | Full-screen mode Lossless scaling Large screens and projectors supported |
| License | Open Source – CC BY 4.0 |
| Attribution Notice | Created with AI assistance |