Theoretical and experimental models for the dynamic response analysis of the mechanisms with deformable elements applied to automobiles.

Dumitru, Nicolae ; Rosca, Adrian Sorin ; Dumitru, Sorin 等

1. INTRODUCTION

Finite element formulation has demonstrated that it is an effective method not only for deformable body structures, but also for the linear and non-linear kinematics problems for rigid bodies (positions, velocities, accelerations and shocks). They are found in the works of Aviles (Aviles & Ajuria, 2000; Aviles & Hernandez, 1996; Aviles & Garcia, 1985), Aggirebeitia, Fernandez-Bustos, (Aggirebeitia & Fernandez-Bustos, 2003), and Dumitru (Dumitru & Cherciu, 2007), where modelling is performed with bars for mechanisms with joints of revolution and prismatic, for troubleshooting kinematics.

2. THE MOTION EQUATIONS IN THE NEWTON--EULER FORMALISM, WITH THE CONSIDERATION OF THE DEFORMABLE KINEMATIC ELEMENTS

It is proposed a dynamical analysis of a mobile mechanic system by the overlap of the solid rigid motion with the one of solid deformable. The mobile system configuration (multibody) leads to an equations system with the form of the relation (1), meaning:

[phi](q, t ) = 0; (1)

q = [[q.sub.r], [q.sub.f]]; (2)

where: [phi] = [{([[phi].sub.1], [[phi].sub.2], ... [[phi].sub.nc]}.sup.T]--it is a vectorial constraint equation, t--time, q--generalized coordinates vector; [q.sub.r] = [[[r.sup.T], [phi]].sup.T]--the position vector of the kinematic element, [q.sub.f]--the elastic or flexible coordinates vector.

Generalized elastic coordinates vector can be introduced using finite element method. The movement equation in the Newton-Euler formalism completed with the method of Lagrange multipliers, can be written as:

M [??] + Kq + J[q.sup.T] x [lambda] = [Q.sup.a] + [Q.sup.n]; (3)

where: M--mass matrix, K--rigidity matrix, [J.sub.q]--jacobian matrix, [lambda]--Lagrange multipliers vector, [Q.sup.a]--applied exterior generalized vector, [Q.sup.n]--square vector of the velocities which contain the gyroscopic and Coriolis components obtained by the kinetic energy difference in relation to the time and in relation with the generalized coordinates of the mechanism. Taking into account the generalized coordinates vector q from the relation (2), the motion equation (3) can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

3. ELASTODYNAMIC MODELLING OF THE CONNECTING ROD FROM A WINDSHIELD WIPER MECHANISM

Based on mathematical models presented above, we intend to analyse the dynamic behavior of the connecting rod of a windshield wiper mechanism, whose kinematics scheme is shown in figure 1. Known (the data entry program developed in Maple): lengths of kinematic elements of the mechanism in mm, kinematic elements masses [kg], and the mechanical moments of inertia, law of variation of generalized coordinate established by the experimental question, respectively:

[[phi].sub.1] = 122 x t+15 x sin(2 x t) (5)

[FIGURE 1 OMITTED]

4. NUMERICAL PROCESSING FOR A MOVEMENT SINUSOIDAL TO THE ENGINES INTERLOCK LAW

As a numerical processing for these mechanisms, we obtain the results for the plane and spatial connecting rods (figure 2, 3). The results of the modal dynamic analyse for the same input data with the ADAMS program were presented in figure 4.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

5. EXPERIMENTAL MODEL

It was made a scale model of natural casings Matiz, with the windshield and windshield wiper system. The model is shown in Figure 5. The following parameters were determined: the electric actuator shaft's torque M(Nm), T1 connecting rod's force F_T1(N), T2 connecting rod's force F_T2(N), wiper windshield no:1 force F_S1(N), ), wiper windshield no:2 force F_S2(N), displacement, Crs(grd), transversal acceleration in connecting rod no: 1, AccT1(m/s2 transversal acceleration in connecting rod no: 2, AccT2(m/s2). There are two types of analysis: analysis of the time and analysis of the frequency.

In the paper was preferred representation of the characteristics depending on the type parameters. In all representations it was maintained the degree course of the connecting rod, which is reported to the Y1 axes, from the left of the graphics. Subsequently other characteristics are reported for Y4 axes. In the analysis of the frequency's case, the spectral decomposition of the recording it is performed by the application techniques using Fourier transformer FFT (Fast Fourier Transform). In the paper, spectral decomposition was performed as a continuation of the analysis of the time, for each temporary representation corresponding a representation in frequency. Excluding acceleration, all spectral parameters are consistent in 0 ... 5 Hz, and the representation was limited to this domain (figure 6).

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

6. CONCLUSIONS

It was presented a Lagrange formulation of finite element for deformable elements connected in multibody systems.

For each finite element were considered 4 systems of reference. After numerical processing of mathematical models there are obtained the laws of variation in time for the resulting elastic dispacements, resulting elastic strains and tensions for both connecting rod of the mechanism considered flexible elements.

Model built with the ADAMS software enable the dynamic modal analysis for each kinematic element of the mechanism and also for the entire assembly when kinematic elements are considered as deformable.

The results of this analysis are confirmed also by experimental research and are materialized through diagrams, 2D and 3D graphics and simulations of the functionality of the mechanism in three-dimensional space.

7. REFERENCES

Aggirebeitia, J.; Aviles, R.; De Bustos, I.F.; Ajuria, G. (2003). Inverse Position Problem in Highly Redundant Multibody Systems in Environments with Obstacles, Mechanisms & Machine Theory, pp.1215-1235, vol.38

Aggirebeitia, J.; Aviles, R.; De Bustos, I.F.; Ajuria, G. (2002). A Method for the Study of Position in Highly Redundant Multibody Systems in Environments with Obstacles, IEEL Transactions and Robotics and Automation, pp.257-263, vol.18, no.2

Amirouche, F. (1992). Computational Methods in Multibody Dynamics, Prentice-Hall

Aviles, R.; Ajuria, G.; Amezua, E.; Gomez Garraz, V. (2000). A Finite Element Approach to the Position Problem in Open Loop Variable Geometry Trusses, Finite Elements in Analyis and Design, pp.233-255, vol.34

Aviles, R.; Ajuria, M.B.; Garcia de Jalon, J. (1985). A Fairly General Method for the Optimum Synthesys of Mechanisms, Mechanism and Mach. Theory, vol.20

Aviles, R.; Ajuria, M.B.; Hormaza, M.V.; Herandez, A. (1996). A Procedure Based on Finite Elements for the Solution of Nonlinear Problems in the Kinematic Analysis of Mechanisms, Finite Elements in Analyis and Design, pp.304-328, vol.22

Dumitru N., Cherciu M., Althalabi Z., (2007). Theoretical and Experimental Modelling of the Dynamic Response of the Mechanisms With Deformable Kinematics Elements, Proceedings of IFToMM, Besancon, France

Fernandez-Bustos, I.; Aggirebeitia, J.; Ajuria, G.; Angulo, C.A. (2003). A New Finite Element to Represent Prismatic Joint Constraints in Mechanisms, Finite Elements in Analysis and Design, article in press