Welcome to the Mechanical Model of the Pendulum

It consists of three interacting objects:
• Support,
• Connecting rod, and
• Massive particle.

The support restricts the motion of the connected end of the rod by allowing only the rotation by angular velocity ω around point O, the center of the rotation. The x and y components of the velocity of this point are zero. There are also force components in the x and y direction due to the support reaction (see Figure down below).

The effects of the connecting rod are transformations of the angular velocity ω into linear velocity of the other end of the rod relative to the center of rotation v_OP, and of the force at the other end into the torque T around the center of rotation. This velocity is orthogonal to the axis of the rod (see Figure up above). Its orthogonal components are given by expressions:
v_OPx = L∙ω∙cos(φ)=-y_P∙ω
v_OPy = L∙ω∙sin(φ)=x_P∙ω
Note that φ is an angle of the rod's axis with y axis, and
x_P=L∙sin(φ) and y_P=-L∙cos(φ)
are coordinates of the other end of the rod. Note that due to motion of the rod and the particle in the coordinate plane x-y, the following simple relation holds
ω=dφ/dt,
and hence the rod's angle can be evaluated simply by integrating the angular velocity. For 3-dimensional motions these relationships are much more complex.
Finally, by applying simple kinematics to the Figure up above the velocity of end point P is given by the relationships:
v_Px=v_Ox-y_P ∙ω
v_Py=v_Oy+x_P∙ω

At the end, we consider the dynamics of massive particle, which is connected at the other end of the rod. Its main property is the inertia, which is governed by its mass m. It is driven by forces F_Px and F_Py of the connected rod and gravity of Earth's attraction of the particle G_y=-m∙g, where g is the gravity (9.81 m/s²). By applying the Newton's 2^nd law we obtain
m∙dv_Px/dt=F_x, and
m∙dv_Py/dt=F_y+G_y.

Now we can start with Bond Graph modeling. For a better understanding of Bond Graph models and their correlation to mechanical model of pendulum described up above, we overwrite the corresponding relations over the bond graphs, the efforts above or to left of the bonds, and the flows below or to the right of them. If we want we may first return back by clicking the button below.