Modeling and simulation of a vibratory machine mechanical system with elastic-damped supporting elements.

Tataru, Mircea Bogdan ; Rus, Alexandru ; Abrudan, Gheorghe 等

1. INTRODUCTION

Mathematical modeling for different kind of mechanical systems is already well-known. This paper presents a specific way of using theory, mathematical calculus, systems theory, numerical calculus and soft techniques for a vibratory machine modeling

2. GENERAL MODEL

Calculus model for this machine is obtained using the schematic showed in the in Fig. 1 (see Munteanu, M., 1986, 1974)).

In order to determine the mathematical model of this model of this systems many authors (Buzdugan, Gh. et. al., 1975), (Munteanu, M., 1986,) are using the well-known fundamental equation of dynamic equilibrium between inertia forces on one part and resultant between applied forces and elastic-damped supports reaction forces on the other side. It may be obtained,

[FIGURE 1 OMITTED]

in this way, the following system of three equations on x,y, and z:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

and: m--is the inertia mass of all the system, [k.sub.x1,2,3,4], [k.sub.y1,2,3,4], [k.sub.zi,2,3,4],--are the rigidity constants of the four supports and, [k.sub.x] , [k.sub.y], [k.sub.z]--are their equivalent rigidity constants for each direction, [c.sub.x1,2,3,4], [c.sub.y1,2,3,4], [c.sub.z1,2,3,4],--are the damping constants of the four supports and, [c.sub.x], [c.sub.y], [c.sub.z]--are their equivalent damping constants for each direction, [m.sub.es], [m.sub.ed]--is the non-equilibrated mass of the generators left and right side respectively, [e.sub.s], [e.sub.d]--is the distance between weight centre and rotation axis of the non--equilibrated masses left and right side respectively, [[omega].sub.s], [[omega].sub.d]--is the angular velocity of the generators, left-side and right-side, respectively, [[phi].sub.s], [[phi].sub.d]--is the initial angular position of the weight centre of the non-equilibrated masses, left-side and right-side respectively.

It may be obtained the differential system of equations for this model based on (1), (2) and expressed it in matrix forms (see (Buzdugan, Gh. et.al. 1975) and (Ionescu, V, 1986)):

{r"} = [[H.sub.i]]x{{[F.sub.s]}+{[F.sub.d]} -[[H.sub.a]].{r'}-[[H.sub.e].{r}) (3)

where:

{[F.sub.s]} = [H[[alpha].sub.s]].[[H.sub.[pi]].[h.sub.mes].[H.sub.acfs] [[H.sub.[phi]s]].{[u.sub.gs]};

{[F.sub.d]} = [H[[alpha].sub.d]].[[H.sub.[pi]].[h.sub.med].[H.sub.acfd] [[H.sub.[phi]d]].{[u.sub.gd]}; (4)

in which: [{r"}={x" y" z"}.sup.t] - -is the accelerations vector; {r'}={x' y' z'}t--is the speeds vector; {r}=[{x y z}.sup.t]--is the displacements vector; {[u.sub.gs]}, {[u.sub.gd]}--are motion unit module vectors for the left and right generator; [[H.sub.i]]--is the inertia transfer matrix; [[H.sub.e]]--is the rigidity transfer matrix; [[H.sub.as]], [[H.sub.ad]]--is the damping transfer matrix; [[H.sub.[pi]]], [[H.sub.[phi]s]], [[H.sub.[phi]d]]--and geometrical transfer matrices; [H.sub.mes] = [m.sub.es], [H.sub.med] = [m.sub.ed]--inertia transfer matrices for the left and right generators; [H.sub.acfs] = [e.sub.s]. [[omega].sub.s.sup.2], [H.sub.acfd]=[e.sub.d]. [[omega].sub.d.sup.2]--centrifugal acceleration transfer matrices

[FIGURE 2 OMITTED]

In the end, based on the equations written above it may be obtained a structural scheme for the analyzed system (Fig. 2).

3. SIMULATION MODELS.

All the models provide the same pattern, because are following the same or same like steps (Bernhard P. & Algis Dziugys, 2002), (*** Matlab with Simulink[R]R13, 2003). Blocks presented in the simulation scheme in the next figures are the correspondent of the general one in the Fig. 2. Diagrams of the models provide trajectory information shown in Fig. 4 and kinematic parameters like displacements and their time derivative up to acceleration. A more detailed model for these mechanical systems is given in Fig. 3. This provides cinematic detailed aspects concerning measurements points and detailed manner of evaluating motion in the inertia center of the working element.

It may also be mentioned that modeling elastic-damped supports is similarly to the model presented in Fig. 2 (the same feedback connection aspect), as it may be seen in Fig. 4. In modeling elastic-damped mechanical systems a six degree of freedom joint with elastic and damping properties was used to model.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Displacements, rotations and trajectory of the working element are given in Fig. 5:

Accelerations for distinctive different points of the working element of the vibratory machine are given in Fig. 6.

This paper does not treat these cases. Studies for these malfunctioning cases make the object of more detailed working papers. Here is only the "perfect" example of the system, with an accent on modeling method and ways of realization.

4. CONCLUSIONS

As one may observe this work has developed such an understanding that creates a possibility to directly model and simulate in virtual form mechanical motion of vibratory machines.

5. REFERENCES

Bernhard Peters, Algis Dziugys, (2002), Numerical simulation of the motion of granular material using object-oriented techniques, Computer methods in applied mechanics and engineering, pg.1984 Elsevier Press Release

Buzdugan, Gh. et. al., (1975), Vibratiile sistemelor mecanice (Vibrations of Mechanical Systems), Editura Academiei Romane, Bucuresti (Romanian edition)

Ionescu, V., (1986), Teoria sistemelor (Systems Theory), vol.1, Editura Didactica si Pedagogica, Bucuresti (Romanian edition)

Munteanu, M., (1986), Introducere in dinamica masinilor vibratoare (Introduction to vibratory Machines Dynamics), Editura Academiei R.S. Romania, Bucuresti (Romanian edition)

Munteanu, M., (1974), Dinamica masinilor vibratoare suspendatepe elemente elastic (Dynamics of vibratory machines, elastic elements supported), Constructia de masini no 25, (Romanian edition)

***, (2003) Matlab with Simulink[R]R13, Help Documentation, SimMechanics Toolbox