Quick guide

Sliders — parameters, geometry, spring constants, initial conditions.
Play / Stop / Reset — simulation control.
Scroll to zoom, drag to pan on all views.
Double-click to reset any view.

Designed for laptop/desktop — not tested on touch devices.

Equations of motion (small oscillations)

\[ (J_1 + m_1 a^2 + m_2 b^2)\,\ddot\theta + \bigl[(k_2+k_3)b^2 + 2k_1 c^2\bigr]\,\theta + (k_2-k_3)\,b\,r\,\varphi = 0 \] \[ J_2\,\ddot\varphi + (k_2+k_3)\,r^2\,\varphi + (k_2-k_3)\,b\,r\,\theta = 0 \]

Matrix form

\[ \mathbf{M}\,\ddot{\mathbf{q}} + \mathbf{K}\,\mathbf{q} = \mathbf{0} \] \[ \begin{bmatrix} J_1+m_1 a^2+m_2 b^2 & 0 \\ 0 & J_2 \end{bmatrix} \begin{bmatrix}\ddot\theta \\ \ddot\varphi\end{bmatrix} + \begin{bmatrix} (k_2+k_3)b^2+2k_1 c^2 & (k_2-k_3)\,b\,r \\ (k_2-k_3)\,b\,r & (k_2+k_3)\,r^2 \end{bmatrix} \begin{bmatrix}\theta \\ \varphi\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix} \]

The equations are decoupled when k₂ = k₃ (off-diagonal terms vanish).

2-DOF: T-shape + disk

Small oscillations simulator

Day mode

Mechanical parameters ▼

$m_1$ (kg):

$J_1$ (kg m²):

$m_2$ (kg):

$J_2$ (kg m²):

$r$ (m):

Geometry ▼

$a$ (m):

$b$ (m):

$c$ (m):

Spring constants ▼

$k_1$ (N/m):

$k_2$ (N/m):

$k_3$ (N/m):

Initial conditions ▼

$\theta(0)$ (rad):

$\dot\theta(0)$ (rad/s):

$\varphi(0)$ (rad):

$\dot\varphi(0)$ (rad/s):

Simulation

$T_{\max}$ (s):

Show geometric points

Current state

Set values and press Play.

$\theta(t)$ — T-shape rotation ▼

scroll: zoom · drag: pan · dbl-click: reset

$\varphi(t)$ — disk rotation ▼

scroll: zoom · drag: pan · dbl-click: reset