Design of mechatronic systems consisting of drive, control and non-uniformly geared transmissions

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Non-uniform transmissions in combination with controlled drives are used, for example, for motion tasks in which the temporal course of the output motion along the path curve is changed. In the case of fast-running transmissions with uneven transmissions and, in particular, those designed in a lightweight design to reduce drive power, the flexibility of individual components results in undesired vibration movement.

A typical example of uneven gear transmission is the simple four-unit crank gear shown in Fig. 1. It has a very flexible swing arm, so that the elasticity leads to a vibrational movement superimposed on the set motion, Fig. 2.

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Figure 2: Calculated and measured vibration angle progression for a run-up process

If high demands are placed on the accuracy of the movements, the output system must be extended by components for vibration reduction. These components contain, for example, model-based controls in which the motion errors due to the compliant components or the dynamic behaviour of the drives are taken into account. These are complex dynamic systems that have to be considered and designed as a whole. Within the framework of synthesis, the overall system is to be designed in such a way that the desired dynamic behaviour is achieved. This requires four steps:

  • the synthesis of the output system consisting of an uneven transmission and drive motor,
  • the analysis of the initial system and
  • the synthesis of the regulator system
  • the implementation of the controller system


Synthesis of the initial system

For the non-uniform transmission, a suitable transmission structure and the corresponding kinematic dimensions are determined within the framework of a structural and dimensional synthesis. Knowledge stores and synthesis programs are used here for example. Subsequently, the dimensioning of the components and the bearings is carried out. During the design phase, the freedom of play and stiffness of the components required for synthesis may not be guaranteed.

The synthesis of the drive is usually limited to the selection of a suitable motor, which fulfils the required speed and torque curves and also allows sufficient leeway for position intervention by the control system.


Analysis of the initial system

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Figure 3: Architecture of the crank mechanism test bench

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Figure 4: Multi-body model and vibration model of the crank mechanism

At the IGMR, a test bench for the crank gear was created from Fig. 1 in order to initially design vibration reduction measures in a purely virtual environment and then to be able to test them in practice on the real transmission, Fig. 3.

A good knowledge of the physical system as a whole in the form of physical and mathematical models is necessary for the targeted design of vibration reduction measures. The motion differential equation of the corresponding vibration model in Fig. 4 can be used for the analysis of the initial system. For example, it provides information about

  • the mechanisms of formation of oscillations.
  • the feasibility of different vibration reduction measures.
  • the necessary measured variables and actuators.
  • the requirements placed on the actuators used to reduce vibration.

In the present example, the linearized motion equation shows excitation torques that are independent of external process loads and drive torques. These cause the system to vibrate and can be interpreted as a known disturbance curve, suggesting the creation of a feedforward control.

In the differential equation, technically feasible measures for passive vibration reduction lead on the one hand to the desired proportions with a dampening effect and on the other hand to stronger proportions with a stimulating effect. Therefore, the use of semi-active or active actuators is more promising.

When designing a (semi)active feedforward control, knowledge of the known disturbance curve can be used to estimate, for example, the required power requirements of the actuators.


Synthesis of the regulator system

In order to improve the dynamic behaviour of the output system, i. e. to minimise unwanted vibrations, the output system must be extended by the additional components (actuators and controls) for vibration reduction.

As a rule, non-uniform transmissions are used. non-linear time-invariant systems. When the equation of motion is linearized with respect to a nominal position, a linear but time-variant system is created. For such systems, the classical methods of control engineering designed for linear time-invariant systems cannot be directly applied.

However, a wide variety of structures are available for suitable, mostly condition-based control systems. For the control of non-linear time-invariant systems, for example, the harmonic balance method can be used for stationary operation and the method of compensation and final coupling for transient operation. Adaptive control systems often have to be designed to control linear time variants. All methods have in common that they require a mathematical model of the route.

After the controller system structure has been selected, the control algorithms and, if necessary, the control algorithms are designed. of the status monitor. Numerous design methods for control algorithms are known for the design of the control system and, if necessary, of a pre-filter. This includes, for example, the minimization of quality functionalities with optimum controls (Ricatti controller) or the pole specification.


Implementation and testing of mitigation measures

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Fig. 5: Conversion of a semi-active feedforward and a P-controller

Programs such as Matlab/Simulink can be used to implement the control concepts. In this case, the control system can first be created on a design computer and tested in a virtual environment, Fig. 3, and then the control system can be ported directly to the hardware of a control computer.

For example, the previously mentioned semi-active pre-control of magnetorheological brakes has been designed and tested for the crank mechanism under consideration. It leads to a clearly measurable reduction in vibrations, Fig. 5.

 

CONTACT


Institute of Mechanism Theory, Machine Dynamics and Robotics

RWTH Aachen University

Eilfschornsteinstraße 18

52062 Aachen

Germany

 

Phone: +49 241 80 95546

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