Mass-Spring-Damper

Explore free and forced vibration through numerical integration, synchronized animation, response plots, phase trajectories, and energy flow.

Model and Simulation

Set the physical system, initial state, forcing, and integration method.

Physical parameters

Mass, m (kg) Damping, c (N·s/m) Stiffness, k (N/m)

Initial state and time

x₀ (m) v₀ (m/s) Duration (s) Δt (s)

Excitation

Force type Amplitude, F₀ (N) Frequency, f (Hz) End frequency (Hz)

Numerics

Integrator Playback speed

Ready to simulate.

Advanced Spreadsheet

Edit parameter cells and the formulas used to produce every time step. Row 2 is the initial state; formulas shown for row 3 fill downward automatically using relative cell references.

Parameters

Name	Unit	Description
m	kg	Mass
c	N·s/m	Viscous damping
k	N/m	Linear stiffness
dt	s	Time increment
duration	s	Simulation duration
x0	m	Initial displacement
v0	m/s	Initial velocity
A	N	Force amplitude available to formulas
f	Hz	Frequency available to formulas

Time History

A3fx

	A · n	B · t	C · Force	D · x	E · v	F · a
2	0	0	0	x0	v0	=(C2-cE2-kD2)/m
3+

Supported: + − * / ^, parentheses, cell references, named parameters, PI, and SIN COS TAN EXP SQRT ABS MIN MAX SIGN. Example harmonic force: =A*SIN(2*PI*f*B3). Calculations are limited to 50,000 rows, five seconds, and 500 characters per formula.

Edit cells, then calculate the sheet.

Calculated ValuesFirst 100 rows shown

	A · n	B · t	C · Force	D · x	E · v	F · a

Mechanical Animation

The plot cursor and mechanism share the same simulation time.

t = 0.000 s

Simulation time

Natural frequency—

Damping ratio—

Response class—

Max |x|—

Max |a|—

Steps / runtime—

Response Figures

Drag the time-history plots to inspect a time; scroll over them to zoom horizontally.

Selected solverAnalytical

Displacement x(t)

Velocity, acceleration, and force

Phase portrait: v(x)

Energy

Numerical Diagnostics

Free vibration includes an analytical displacement reference. Large Euler energy growth indicates numerical instability.

Quantity	Value	Interpretation

Model Reference

Equation of motionm ẍ + c ẋ + kx = F(t)

Natural frequencyωₙ = √(k/m)

Damping ratioζ = c / (2√km)

Damped frequencyωd = ωₙ√(1 − ζ²)

Mechanical energyE = ½mv² + ½kx²

Dissipated energyEd = ∫ cv² dt

RK4 evaluates four slopes per step and is generally much more accurate than Forward Euler at the same Δt. Euler is included to illustrate time-step sensitivity and numerical instability.