Quick guide

Sliders — masses, springs, geometry, gravity (display), initial conditions
Play / Stop / Reset — simulation control
Scroll to zoom, drag to pan on plots
Double-click main canvas to reset view
Right panel — time histories and state projections with energy level sets

Designed for laptop/desktop.

Equations

Dynamics (matrix form)

With $\mathbf{q}=\begin{bmatrix} x_1 & x_2 \end{bmatrix}^{\!T}$, $\mathbf{M}\ddot{\mathbf{q}}+\mathbf{K}\mathbf{q}=\mathbf{0}$:

\[ \begin{bmatrix} m_1+\dfrac{J_1}{r^2} & 0 \\[6pt] 0 & m_2 \end{bmatrix} \begin{bmatrix} \ddot{x}_1 \\[4pt] \ddot{x}_2 \end{bmatrix} \;+\; \begin{bmatrix} 4k_1+k_2 & -k_2 \\[4pt] -k_2 & k_2 \end{bmatrix} \begin{bmatrix} x_1 \\[4pt] x_2 \end{bmatrix} \;=\; \begin{bmatrix} 0 \\[4pt] 0 \end{bmatrix} \]

Energy

\[ T = \tfrac12\left(m_1+\frac{J_1}{r^2}\right)\dot{x}_1^2 + \tfrac12 m_2\,\dot{x}_2^2 \] \[ U = \tfrac12 k_1 (2x_1)^2 + \tfrac12 k_2 (x_2-x_1)^2 \]

Vibrazioni a 2 DOF

Simulatore dinamico

Prof. Marco Gabiccini & Martino Gulisano
Meccanica Applicata alle Macchine
Tutti i diritti riservati

Mass and geometry

$m_1$ (kg): 5.0

$r$ (m): 0.80

$m_2$ (kg): 4.0

$L_1$ (m): 0.10

$L_2$ (m): 0.10

$N_{\mathrm{nastro}}$ (m): 6.0

Molle e gravità

$k_1$ (N/m): 40

$k_2$ (N/m): 35

$g$ (m/s²): 9.81

Condizioni iniziali

$x_1(0)$ (m): 0.120

$\dot{x}_1(0)$ (m/s): 0.00

$x_2(0)$ (m): 0.000

$\dot{x}_2(0)$ (m/s): 0.00

Equazioni

Simulazione

$T_{\max}$ (s):

Stato corrente

Set initial conditions and run.

Legge oraria $x_1(t)$ ▼

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

Legge oraria $x_2(t)$ ▼

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

STATE SPACE: projection onto $(x_1,x_2)$ plane ▼

Horizontal: $x_1$ · Vertical: $x_2$ · Iso-$U$ · scroll: zoom

STATE SPACE: projection onto $(\dot{x}_1,\dot{x}_2)$ plane ▼

Horizontal: $\dot{x}_1$ · Vertical: $\dot{x}_2$ · Iso-$T$ · scroll: zoom

STATE SPACE: projection onto $(x_1,\dot{x}_1)$ plane ▼

Plane projection of the $\mathbb{R}^4$ state — trajectory only (not iso-$E$ or iso-$U$)

STATE SPACE: projection onto $(x_2,\dot{x}_2)$ plane ▼

Plane projection of the $\mathbb{R}^4$ state — trajectory only (not iso-$E$ or iso-$U$)