Keywords: washing machine, dynamic of rigid bodies, modified Lagrange method, Rodriguez method for symbolical calculation, simulation, PID controller.

The modeling of non-linear dynamics has been attracting increasing attention in recent years, but nevertheless three vital phases in the design of the dynamic system of a washing machine must be considered: modeling, experimental validation and parameter optimization [1]. Moreover, a very important issue is to form such a mechanical model, which would enable designer parametric solution for different combinations of washing machines regarding their functionality demands. Also, most of the research work concerning the modeling of washing machines is done by the appliance industry itself and the number of publication on this topic is therefore very limited [2,3].

In this paper, an alternative robotic approach for modeling of multi-body dynamics of rigid bodies overcomes the difficulties of classical approaches, which refer to washing machines as rigid multi-body systems (MBS). Modified Langrangian equations of motion in covariant form will be introduced instead of regular Newton-Euler and Langrange method for the MBS description. The proposed approach is "robotic" based on Rodrigues' method [4] which utilized a basic metric tensor, Christoffel symbols of the first kind and generalized forces which act on the given system.

Here,a mechanical model of the washing machine can be considered as a tree structured MBS consisting of (n+1) rigid bodies interconnected by n one-degree-of-freedom (DOF) joints. In our case, since many parts of the washing machine are rigidly connected through screws or welding points, the whole system can be modelled using four different rigid bodies. Moreover, one can obtain symbolic expressions using modified Langrange's equations of motion of given system (left side) which are suitable for automatic setting of differential equations. Also, the analytical form of the generalized (gravitational forces, spring forces, viscous damping forces) will be obtained in a symbolic way which is also suitable for automatic setting. Now, one can obtain a symbolic model developed for the dynamic simulation for washing machine within the symbolic toolbox of the MATLAB environment.

Also,using a symbolic model, it is obvious that modifying only one parameter of the mechanical system, we could minimize efforts to re-evaluate the equations of the mechanical system. Simulation work has been undertaken to verify the theoretical approach. The simulation results for the proposed two-dimensional model are conducted for the determination of the following quantities: response function for impulse excitation, response function at steady rotation and the solution flowchart.

One may conclude that the symbolic model enables analysis, synthesis, testing and validation of control systems.

1

Wagner F., "Dynamics Of Washing Machines", Fort. Berichte VDI Reihe 11, Nr. 287, VDI Verlag, 2000.

2

Conrad D.C., "The fundamentals of automatic washing machine design based upon dynamic constraints", Ph.D. Thesis, Purdue University, 1994.

3

Türkay O.S., "Formulation and implementation of parametric optimization of a washing machine suspension system", Mechanical Systems and Signal Processing, 1995.

4

A. Shabana, "Computational dynamics", 3rd edition, John Wiley and Sons, 2001.

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