Coefficients

Damping b

Stiffness k

Driving frequency ω

Trajectories 1 / 12

Selected initial condition (x, x′) at t = 0

Position x(0)Velocity x′(0)

Click the phase portrait to add a trajectory, or add a copy and edit its initial values. Coefficient changes apply to every trajectory.

Time range

Start timeEnd time

Choose any interval within −500…500. Initial conditions remain at t = 0.

Display

Show forcing cos(ωt) Compare periodic response Show phase vector field

DRIVEN MOTION

Damped, forced oscillator

Underdamped

Motion

Following trajectory 1

x 0.000x′ 0.000F 1.000

Spring and damper schematic · displacement scaled to the plotted range.

Phase portrait

(x, x′)

Click to add (x(0), x′(0)). Bright trails and filled dots animate for every trajectory. Hollow circles stay at t = 0. The vector field follows the current forcing phase.

t = 0.00

Displacement over time

Trajectories x(t)cos(ωt)Periodic response

Frequency response

Amplitude · log scale

Same b and k, varying drive Ω. Amplitudes above 10³ are clipped.

What to look for

Natural frequency √k
Periodic amplitude
Phase lag

R = 1 / √((k − ω²)² + b²ω²)