Distinct interactions between electric drive and non-uniformly geared transmission
The four-unit crank gear is a basic kinematic structure that forms the basis of the kinematic structure of numerous mechanisms. Crank gears are used, for example, as a guide gear to guide a tool on a certain path or as a transmission gear to create a fixed connection between the drive position and the output position.
The latter task is also performed, for example, by the series connection of two four-unit crank gears in the drive train of the screen sorter in Fig. 1.
At IGMR, the test bench shown in Fig. 2 was developed for the investigation of a four-unit crank gear unit. The crank gear is driven by a synchronous servo motor. If the controller parameters of the servo amplifier are set in accordance with the manufacturer's instructions, there will be significant fluctuations in the course of the drive speed, which are due to the uneven load on the drive. Fig. 3 shows a measured actual speed curve for a ramp-shaped or constant target speed curve.
Simulation model of the mechatronic system
Figure 4: Architecture of the crank gear test bench
For the overall system of the test bench, a model was created in order to investigate the dynamic behaviour of this system in a purely virtual environment, Figure 4. In this step, it has already been determined which physical effects are to be taken into account and which properties can be described with the model. It should be noted that the model parameters introduced for the model description can be determined on the real system with sufficient accuracy, e. g. by calculation, measurement or tests.
Multi-body model of the mechanical subsystem
Figure 5: Multi-body model of the crank gear and calculated drive torque curve
Simulation of mechanical systems is possible with multi-body simulation programs such as ADAMS. Figure 5 shows the multi-body model of the crank gear. Such a model can already be used in the design phase for dimensioning the gear components and the drive.
Servo motor and servo amplifier model
Figure 6: Simulation of the vibration behaviour with consideration of the interactions
In order to take into account the interaction between the drive, the control system and the gear unit during the simulations, the dynamic behaviour of the motor and the servo amplifier must also be recorded by the simulation model.
The servo amplifier provides speed control with downstream current control in the form of cascade control. This was implemented for the virtual environment in MATLAB/Simulink. In the present case, only the parameters of an equivalent direct current motor were given in the motor manufacturer's data sheets, even though it is a three-phase motor. For the purposes of the virtual test bench environment in Fig. 3, however, such a modeling of the synchronous servomotor was sufficient. The engine model was also created in MATLAB/Simulink and is illustrated at the top right of Figure 6.
Numerical analysis of time behaviour
A co-simulation of MATLAB/Simulink and ADAMS provides time curves of the state variables such as the motor current curve shown in Fig. 6. It is possible to investigate the effects of individual error sources (e. g. support elasticity) or disturbance variables (e. g. friction) on the dynamic system behavior by modifying the model. The model opens up the possibility to analyse the effects relevant for the dynamic behaviour of the design phase by parameter variation. However, it is not possible to make a reliable statement about cases that have not been investigated. This possibility is only possible with the help of vibration models.
Model-based analysis of the dynamic system
Fig. 7: Multi-body model and vibration model of the crank gear
Ultimately, the task of modelling is to design the model as simple as possible and only as complex and extensive as necessary. Often, multi-body models can be simplified and abstracted even further by using equivalent replacement models for parts of the multi-body model reduced to rack axles. These strongly abstracted models are called vibration models.
Due to their high degree of abstraction, vibration models are particularly well suited for mathematical description and analysis by equations in symbolic form. The use of symbolic equations offers, for example, the possibility of analytically determining the relevance of individual parameters for the motion or stability behavior.
For the crank gear, for example, a vibration model with three degrees of freedom can be created, Fig. 7. In it, the play in the planetary gear is taken into account by the spring/damper element c1/k1 and the elasticity of the swing arm is represented by the spring/damper element c2/k2 in the vibration model, while all masses and mass moments of inertia of the output model are reduced to the mass moments of inertia J1 to J4 and JK in the vibration model. The symbolic equations of motion of this system were used in the design of vibration reduction measures to analyze the system in terms of vibration causes and countermeasures.