Quick guide

Sliders — geometry, parameters, ICs (see textbook for symbols)
Play / Stop / Reset — simulation control
Scroll to zoom, drag to pan on time plots
Double-click main canvas to reset view
Right panel — time histories $\alpha(t)$, $\beta(t)$, $\gamma(t)$ (angles in rad)

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

Equations

Dynamics (matrix form, fully expanded)

$\mathbf{q}=\begin{bmatrix} \alpha & \beta & \gamma \end{bmatrix}^{\!T}$ (rad); $a,b$ (m); $g$ (m/s²); $k_1,k_2$ (N/m); masses (kg); inertias (kg·m²).

\[ \begin{bmatrix} J_1+m_1\,a^2+J_3+m_3\,(a+b)^2 & 0 & 0 \\[8pt] 0 & J_2+m_2\,b^2+m_4\,(a+b)^2 & 0 \\[8pt] 0 & 0 & J_4 \end{bmatrix} \begin{bmatrix} \ddot{\alpha} \\[4pt] \ddot{\beta} \\[4pt] \ddot{\gamma} \end{bmatrix} \;+\; \begin{bmatrix} (k_1+k_2)\,a^2-m_1\,g\,a-m_3\,g\,(a+b) & -k_2\,a\,b & 0 \\[8pt] -k_2\,a\,b & k_2\,b^2+m_2\,g\,b+m_4\,g\,(a+b) & 0 \\[8pt] 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \alpha \\[4pt] \beta \\[4pt] \gamma \end{bmatrix} \;=\; \begin{bmatrix} 0 \\[4pt] 0 \\[4pt] 0 \end{bmatrix} \]

Scalar rows (same model)

\[ \begin{aligned} \bigl(J_1+m_1 a^2+J_3+m_3(a+b)^2\bigr)\ddot{\alpha} &= -k_1 a^2\alpha -k_2(\alpha a-\beta b)a + m_1 g a\,\alpha + m_3 g(a+b)\alpha,\\ \bigl(J_2+m_2 b^2+m_4(a+b)^2\bigr)\ddot{\beta} &= -k_2(\beta b-\alpha a)b - m_2 g b\,\beta - m_4 g(a+b)\beta,\\ J_4\,\ddot{\gamma} &= 0. \end{aligned} \]

Sistema linearizzato a 3 DOF

Simulatore dinamico

Prof. Marco Gabiccini & Martino Gulisano
Meccanica Applicata alle Macchine
All rights reserved

Geometry

$a$: 0.200 m

$b$: 0.200 m

Inertia

$m_1$: kg

$m_2$: kg

$m_3$: kg

$m_4$: kg

$J_1$: kg·m²

$J_2$: kg·m²

$J_3$: kg·m²

$J_4$: kg·m²

Springs / $g$

$k_1$: 80 N/m

$k_2$: 80 N/m

$g$: 9.81 m/s²

Initial state

$\alpha_0$: 2.0 °

$\dot{\alpha}_0$: 0.00 rad/s

$\beta_0$: 0.0 °

$\dot{\beta}_0$: 0.00 rad/s

$\gamma_0$: 30 °

$\dot{\gamma}_0$: 0.50 rad/s

Equations

Simulation

$T_{\max}$:

State

Set initial conditions and run.