Dynamic answer modeling of a mechanism from harvester machines considering the elements deformable.

Ungureanu, Alin ; Geonea, Ionut ; Dumitru, Nicolae 等

1. INTRODUCTION

It is know the fact that in the mechanisms functioning appear vibrations, acoustic radiations and joints wear. From that reason it is necessary to perform an elastodynamic analysis of mechanisms operating at high velocities, much more than the dynamic analysis considering the elements as rigid ones.

The finite element formulation has demonstrated to be an efficient method not only for structures with deformable body but also for linear and nonlinear kinematic problems for rigid bodies (positions, velocities, accelerations and impacts). In order to solve the kinematic problems, these types of problems can be founded in scientific paper published by (Aggirebeitia & Aviles, 2003), (Fernandez-Bustos, 2003). In research paper of (Dumitru, 2007), it was realised the dynamic analysis of a wiper mechanism with deformable elements, and (Hernandez, 2003), where a beam modeling for mechanisms with prismatic and revolutions joints was performed.

2. DYNAMIC ANSWER MODELING OF THE OSCILLATORY WASHER MECHANISM

2.1 Kinematical analysis

The mechanisms with oscillatory washer are mechanisms with three kinematics joints. In figure 1 we present the 3D model, in Adams, of the mechanism with the kinematic joints.

[FIGURE 1 OMITTED]

We know as input dates for the kinematic analysis:

[FIGURE 2 OMITTED]

The mechanism from figure 2 can have the angle of the motor shaft, [alpha] between 10 ... 30[degrees], the elements lengths being: [l.sub.3] = 369.61 mm, [l.sup.4] = 403.64 mm. The angular velocity of the motor element, which is: 620 rot/min. For the kinematic analysis we use the kinematic scheme represented in figure 2.

The space covered by the knife is give by the relation:

x = l sin [alpha](1- cos[omega]t/cos [alpha][square root of 1 + [tg.sup.2][alpha] x [cos.sup.2][omega]t]) (1)

The knife linear velocity is determined with the relation:

[V.sub.x] = dx/dt = l x sin[alpha][omega] x sin [omega]t x [mu] (2)

Where:

[mu] = 1/cos [alpha][(1 + [tg.sup.2][cos.sup.2][omega]).sup.3/2] = [cos.sup.3][xi]/cos[alpha] (3)

The knife acceleration is determined with the relation:

[a.sub.x] = d[V.sub.x]/dt = l x sin [alpha][[omega].sup.2] * cos [omega]t x v (4)

Where:

[upsilon] = 1 + [3tg.sup.2][alpha] - [2tg.sup.2][alpha][cos.sup.2] [omega]t/cos [alpha][(1 + [2tg.sup.2][alpha] [cos.sup.2] [omega]t).sup.5/2] (5)

Upon the computerised processing of the kinematics model of the mechanism, with the Maple computer program, we present in figure 3, the graphics of variation of the knife mass centre position, velocity and acceleration, upon the time and the angle of the motor shaft [alpha].

From the figure 3, we observe that the motor shaft angle varies between 10 ... 30 degrees, corresponding to the existing oscillatory washer mechanisms. From figure 3, a) we observe that at measure the angle of the shaft increase also the knife covered space increase. It is observed that the mechanism can have the space realised by the knife from 50 mm to 160 mm.

[FIGURE 3 OMITTED]

With MSC.Adams we performed the dynamic analysis of the mechanism. In figure 4 we present the time variation laws of the connecting forces from D joint (see figure 1).

[FIGURE 4 OMITTED]

If we analyse the way that the components varies we see that the great value has the z axis component, from 3750N to -1000 N, other two components having smaller values.

3. MODAL DYNAMIC ANALYSIS WITH ADAMS

The equations that describe a flexible body motion are (Craig & Bampton, 1998):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

F = 0

where: L is the Lagrange operator, defined as: L = T--V, T and V represent the kinetic energy, and the potential energy; F is the dissipation function; F are the constraints equations; [lambda] --is the Lagrange multipliers vector for constraints; [xi]--generalised coordinates; Q--generalised forces (projected on [xi]).

3.1 Graphical results

We have performed the modal dynamic analysis of the mechanism considering the elements as deformable. We use the MSC.Adams software, considering the element 4 as flexible. We present in the following, the deformed shape for the element 4, considered deformable, for two modes of vibration.

[FIGURE 5 OMITTED]

With MSC.Adams we could determinate the mechanism elements mass centre elastic displacements.In figures 6 and 7, we present the transversal elastic displacement and velocity, for the node placed in the mass centre of the element, reported to the global coordinate system.

[FIGURE 6 OMITTED]

4. CONCLUSIONS

It is well known that the elastic deformation has a significant effect on the dynamic behaviour of high speed mechanisms. In addition to the inertia force caused by the rigid body motion of a flexible mechanism, the inertia force due to the elastic vibration of flexible links plays an important role in the dynamics of flexible mechanisms and so should be eliminated.

The future research concern the problem of dynamic balancing of the flexible mechanism, that is more complicated that of the rigid body one.

5. REFERENCES

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

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

Craig, R.R. & Bampton, C.C. (1998). Coupling of Substructures for Dynamics Ana/yes, AIAA Journal, pp 1313-1319, 1998

Dumitru N. & Cherciu M. (2007). Theoretical and Experimental Modeling of the Dynamic Response of the Mechanisms with Deformable Kinematics Elements, IFToMM, 2007, Besancon, France

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

Hernandez, A., Altuzarra, O., Aviles, R.. & Petuya, V. (2003). Kinematic Ana/ysis of Mechanism via a Velocity Equation on a Geometric Matrix, Mechanisms & Machine Theory, vol.38, pp 1413-14295, 2003