Can you derive the mathematical model describing the flexible joint experiment?
The Rotatary Flexible Joint experiment can be used to investigate resonance phenomena and vibrations. This labatory experiment consits of 4 major parts:
These parts are illustrated in the figure below:
The head is attached to the base, and can be rotated along the \(z\) axis via a motor (not shown in the figure). The rotation of the head relative to the \(x\) axis is denoted by \(\varphi_M\). Attached to the head is the arm, which can rotate freely along the \(z\) axis. Let the arm angle relative to the \(x\) axis be denoted \(\varphi_A\), while the angle relative to the head be denoted \(\alpha\). As shown in the figure, springs are attached to the head and the arm via screws, creating a connection between the arm and the head.
The goal of this experiment is to control the arm angle \(\varphi_A\) by manipulating the input voltage \(V_M\) of the motor.
As mentioned earlier, the head is connected to a motor. The circuit diagram for the electrical parts is given below:
The motor is driven by the input voltage \(V_M\) and current \(I_M\). The resistance of the windings in the motor is given by \(R_M\), and the back-emf voltage is given by \(u_M\). All friction losses are combined into the viscous friction coefficient \(B_M\), which includes the friction losses in the bearings, the gearbox, and the brushes. The rotation angle of the motor shaft is denoted by \(\varphi_M\) (=rotation angle of the head), while its torque is denoted by \(M_M\). Finally, the torque applied to the head is given by \(T\).
Let us look at the moving parts of the experiment.
The equivalent moment of inertia \(J_{eq}\) represents the combined moment of inertia of the motor, the gearbox, and the head with respect to the \(z\) axis. Likewise, the moment of inertia along the \(z\) axis for the arm is given by \(J_A\). The damping coefficient in the pivot between the arm and the head is denoted by \(c\), and the equvalent stiffness of the springs is denoted by \(K_S\).
The back-emf voltage can be computed via \[u_M = K_M \dot{\varphi}_M,\] while the torque produced by the motor can be computed via \[M_M = K_A I_M.\] Both \(K_M\) and \(K_A\) are constant motor parameters. The combined friction losses are given via \[M_B = B_M \dot{\varphi}_M. \]
The viscious damping in the pivot point between the head and the arm can be found via \[M_c = c \dot{\alpha}. \]
The torque generated by the sprigns is approximated by a linear torsion spring with a stiffness \( K_S \). Following this, the potential energy of the springs can be found via \[V = \frac{1}{2} K_S \alpha^2.\]
The following tasks should be solved:
| Variable | Description | Value | Unit |
|---|---|---|---|
| \(R_M\) | Motor resistance | 2.604 | \(\Omega\) |
| \(K_M\) | Motor constant | 0.558 | \(\mathrm{Vs}/\mathrm{rad}\) |
| \(K_A\) | Motor constant | 0.558 | \(\mathrm{Nm}/\mathrm{A}\) |
| \(B_M\) | Motor friction | 0.0696 | \(\mathrm{Nms}/\mathrm{rad}\) |
| \(J_{eq}\) | Equivalent moment of inertia | 0.0036229 | \(\mathrm{kg}\mathrm{m}^2\) |
| \(J_{A}\) | Arm moment of inertia | 0.0039716 | \(\mathrm{kg}\mathrm{m}^2\) |
| \(K_{S}\) | Spring stiffness | 0.2015 | \(\mathrm{Nm}/\mathrm{rad}\) |
| \(c\) | Pivot damping coefficient | 0.003217 | \(\mathrm{Nms}/\mathrm{rad}\) |