Mechanical Resonant System

This simulation shows how the resonant frequency of a mass–stiffness–damping system changes with adjustments of these parameters.

First plot – Impedance vs Frequency: Shows how the magnitude of the total impedance (the system’s “resistance to motion”) changes with frequency.

• Mass reactance increases with frequency — it resists high-frequency motion (inertia).
• Stiffness reactance decreases with frequency — it resists low-frequency motion (spring restoring force).
• Resistance is flat — it’s the damping term that’s constant across frequency.
• Total impedance shows a dip at resonance: that’s where the spring and mass effects cancel, leaving only damping. At this dip, the system moves most easily (maximum amplitude for a given force).

Second plot – Phasor diagram: A vector (phasor) diagram of the impedance components at the selected frequency.

• The horizontal axis = real part (resistance).
• The vertical axis = imaginary part (reactance).
• The mass and stiffness phasors point in opposite vertical directions — they are 90° out of phase and opposite in sign.
• The resistance phasor lies along the real axis.
• The total impedance is their vector sum — it shows both the total magnitude and phase of the system at that frequency.

Third plot – Admittance vs Frequency: Shows how the system’s admittance (ease of motion) changes across frequency. The resonant peak occurs where the total impedance in the first plot is minimum — when mass and stiffness reactances cancel out, leaving only resistance to limit motion.

Fourth plot – Force & Response Waveform: Shows a short segment of the input sine-wave force (black, dotted) and the system response (red) at the selected frequency. The response is scaled so that its peak matches the input amplitude at resonance, allowing both peaks to be visible. Notice how the phase of the responses changes as the driving frequency changes. For frequencies below the resonant frequency, the response leads the driving force (by up to 90°). At the resonant frequency the displacement and the driving force are in phase. For frequencies above the resonant frequency, the displacement lags behind the driving force (by up to 90°).