Dynamics & Control · Lab

Vibrations Lab — Mass–Spring–Damper

m·x¨ + c·x˙ + k·x = F(t). Set the physical mass, damping and stiffness, then read free decay, forced and harmonic response, resonance, transmissibility and isolation.

Physical parameters

Mass m [kg]2

Damping c [N·s/m]3

Stiffness k [N/m]50

Excitation

x(0) [m]

v(0) [m/s]

Derived parameters

Natural frequency ωₙ	5 rad/s
Natural frequency fₙ	0.7958 Hz
Critical damping c_c	20 N·s/m
Damping ratio ζ	0.15
Damping regime	underdamped
Damped frequency ω_d	4.943 rad/s
Damped period T_d	1.271 s
Logarithmic decrement δ	0.9533
Quality factor Q	3.333
Static compliance 1/k	0.02 m/N

Presets

Displacement x(t) [m]

Time view

Window [s]

output solid · variant dashed amber · horizon 6.7 s

Zero-input response from the stated initial displacement and velocity; the envelope is drawn only for the underdamped case.

Energy

Initial total energy	0.0625 J
Final total energy	2.712e-6 J
Energy dissipated by damping	0.0625 J

Kinetic ½mv² plus spring potential ½kx². With c = 0 the total is constant; with c > 0 the difference is dissipated by the damper, never stored.

Pole–zero map of X(s)/F(s)

poles = × · zeros = ○ · dashed line = imaginary axis

Numeric poles and zeros
Root	Re(s)	Im(s)	Multiplicity
Pole ×	-0.75	4.9434	1
Pole ×	-0.75	-4.9434	1

The poles are the roots of m·s² + c·s + k, the same quantities the damping regime and damped natural frequency above are read from.

Continue with this system

The current model travels with the link; nothing is re-entered and nothing opens on its own.

Transfer functionSame physical system as X(s)/F(s) = 1 / (m s² + c s + k), with m, c and k substituted · force-input channel only, static gain = compliance 1/k [m/N]. Base excitation is not carried over.Second orderSame system normalised: ωn = 5 rad/s, ζ = 0.15. The normalised model has unit static gain, so its step amplitude differs from the physical compliance 1/k by the factor k.

What this Lab models

The single-degree-of-freedom system m·x¨ + c·x˙ + k·x = F(t): free decay, forced and harmonic response, resonance, and — for base excitation — displacement transmissibility and isolation. Relevant both as a control-loop plant and as a mechanical vibration model.