# Damping of Oscillations of a Rotary Pendulum System
## 3 Problem Formulation and System Description  
A dynamic system can be determined by modeling the system using the equations of motion and then solving for the Laplace or Fourier transform of the output in terms of the input. The transfer function of a rotary pendulum system is a mathematical representation of the system's behavior. It describes the relationship between the input (angular acceleration of the carousel) and the output (angular position of the load). A damped oscillatory system is mainly expressed by a second-order linear transfer function, which consists of estimated values of the natural frequency, \(\omega\), and damping, \(b\). In this case, we consider the linearization of the pendulum model for small deflection values [47; 48] and a rope that is solid, inelastic, and not subject to any twisting, friction, or other non-ideal properties. On the contrary, we do not assume any other oscillating modes given by the flexibilities of the arm, column, etc. The dynamics of a gantry crane usually consist of two interconnected parts that represent the oscillation of the load in each axis. The dynamics of the motor can affect the dynamic properties, but if the dynamics of the motor (response) are faster than the dynamics of the load, there is no significant influence on the position of zeros in the transfer function of the system. Then, the motor should be able to effectively control the load without significantly changing the dynamic characteristics of the overall system. Its contribution to the overall system's response can only be in the rounding of the waveforms, which means that the response is smoother, and thus the motor is unlikely to be a limiting factor in controlling the system's response. If the bandwidth of the motor servo is much higher than the natural frequency of the load, it means that the motor responds quickly to changes in the control signal, which will ensure fast and accurate control of the load. The dynamics of the actuators (also current loop, etc.) can be defined by a first-order transfer function with a low time constant, but it is omitted because the electrical time constant of the actuator is negligible compared to the mechanical time constant of the entire system. The subject of the analysis in this article is a rotating carousel system, which consists of a fixed structure and arms that rotate around the axis of the vertical structure. Both arms have oscillating rope-suspended loads that exhibit low damping properties. A similar mechanism is the rotary pendulum, which has several basic similarities with the described system. The chain carousel rotates at a prescribed angular velocity about a vertical axis, while the loads move in several degrees of freedom that are coupled. The dynamics of these systems are usually characterized by significant feedforward nonlinearities, nonholonomic constraints, a non-minimum phase, and underactuated dynamics. A chain carousel model with a diameter of 62 cm and a rope length of 22 cm in the Cartesian plane is shown in Figure 4.  
Figure 4: Block diagram of the carousel model in Cartesian coordinates. This case is interesting because two different accelerations controlled by one motor affect the deflection of the load relative to the stable position. One acceleration is the tangential acceleration proportional to the angular acceleration, which we directly influence by controlling the torque of the motor driving the mechanism. \[a_{tang}=r\times\epsilon=r\times\frac{d\omega}{dt}. \tag{1}\]  
The second acceleration is the centrifugal acceleration, which is proportional to the square of the angular velocity of the load. \[a_{rad}=r\times\omega^{2}. \tag{2}\]  
The resulting acceleration amplitude acting on the hinge is  
\[\sqrt{ax^{2}+ay^{2}}=\sqrt{a_{rad}^{2}+a_{tang}^{2}}. \tag{3}\]  
This acceleration turns with the motion of the arm. The resulting block diagram in Figure 5 represents an intuitive principle model from the distribution point of view, split into tangential and radial components.