On dynamics stochastic evaluation of embedded systems protection against vibration.

Nastac, Silviu ; Debeleac, Carmen ; Curtu, Ioan 等

1. INTRODUCTION

This research tries to achieve positive, conclusive and gainful partial solutions for the large problematics of the protection against undesirable and nocive vibration action.

According to the protective technical solutions analyzed by the authors, the major problems appear at evaluation of the isolation degree.

The entire ensemble of the theoretical approaches (Axinti, 2008; Bratu, 1990; Bratu, 2000; Harris et al., 2002) supposses a simple lumped computation models (sometimes, with nonlinear approximations), with one to few degrees of freedom, and with concise excitation signals--harmonical or periodical time functions.

The differences between the theoretical or computational results and the instrumental experimental tests can be explained through the impossibility to simulate the real excitations, besides the inclusion of the entire set of protective device characteristics into the transfer function used on numerical simulations.

Supposing the general heuristic character of the real device transfer functions, but the essential parameters that leads to the main evolution of system, it could be declare that a right spectral composition of the input dictate a corresponding spectral composition at the output.

At this moment it had to be said that these researches have a leading significance for a virtual analysis, based on computer simulations, of behaviour dynamics for supposed technical systems. These are useful both for the concurent engineering concept implementations, and when the practical instrumental tests are impossible to use or difficult to run.

2. BASIC COMPUTATIONAL MODEL

In Figure 1 is depicted the schematic diagram of the basic model intended for computational dynamics development. The structure (m, J) is linked to the ground by means of complex visco-elastic devices, which supply both horizontal (X), and vertical (Z) movement restrictions.

It was supposed a horizontal displacement excitation that had, in a first approximation, a harmonical expression.

[FIGURE 1 OMITTED]

The matrix form of the moving equations is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

denote the stiffness matrix (the stiffness of each isolator have the same value for the same direction);

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

denote the dampings matrix (the damping coefficient of each insulation device have the same value for the same direction);

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

denotes the vector of generalized coordinates;

denote the final form of the stimulus forces vector. Supposing the notations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

results the simplified matrix forms of stiffness and dampings

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It had to be noted that the addition operator from the above expressions allow the total number of the insulation devices (usually t = 4).

In such conditions, after Laplace transformation of Eq. (1) results the system matrix B(s)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

and the transfer matrix H (s), defined as the inverse of the system matrix, is

[H(s)] = [[B(s)].sup.-1] = [[B(s)].sup.A]/[DELTA](s) (3)

where [[B(s)].sup.A] denote adjunct matrix and A(s) denote characteristic polynom of B(s), hence it satisfies the equation

[Q(s)] = [H(s)][F(s)] (4)

Extended version of Eq. (4) in terms of adjunct matrix is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

in which the specific discrete components have the next expressions

[A.sub.11] = ([ms.sup.2] + [c.sub.x]s + [k.sub.x])([Js.sup.2] + [c.sub.[phi]]s + [k.sub.[phi]])--[[c.sub.[phi]s + [k.sub.[phi]x]).sup.2] (6.1)

[A.sub.12] = ([C.sub.[phi]x]s + [k.sub.[phi]x]) ([c.sub.[phi]z] s + [k.sub.[phi]z]) (6.2)

[A.sub.13] = -([C.sub.[phi]z]s + [k.sub.[phi]z]) ([ms.sup.2] + [c.sub.s] + [k.sub.x]) (6.3)

[A.sub.22] = -([ms.sup.2] + [c.sub.z]s + [k.sub.z]) ([Js.sup.2] + [c's.ub.[phi]]s + [k'.sub.[phi]) + [([c.sub.[phi]z]x + [k.sub.[phi]z].sup.2] (6.4)

[A.sub.23] = -([c.sub.[phi]x]s + [k.sub.[phi]x]) ([ms.sup.2]) + [c.sub.z]s + [k.suzb.x]) (6.6)

From the general form of the transfer function described in Eq. (4), H(s) can always be written in partial fraction form as

H(s) = [2n.summation over (i=1)][[alpha].sub.i]/s--[p.sub.i]] (7)

where n denote number of degree of freedom; [[alpha].sub.i] is the residue matrix for the kit root; [p.sub.i] =--[[sigma].sub.i] [+ or -] j [[tau].sub.i] denote the ith root of the equation obtained by setting the determinant of the matrix

B(s) equal to zero (note that minus sign correspond to the complex conjugate pole of transfer function)--with [[sigma].sub.i] denote the modal damping coefficient, [[tau].sub.i]is the natural frequency, [[omega].sub.i] = [square root of [[sigma].sup.2.sub.i] + [[tau].sup.2.sub.i]] is the resonant frequency and [[xi].sub.i] = [[sigma].sub.i]/[[omega].sub.i] is the percent of critical damping.

3. DISCUSSIONS AND PARTIAL RESULTS

This formulation, in complex domain, have two main advantages, such as: first, it allow the access to partial transfer functions and this fact can lead to identify the modal shapes, and second, it enable the utilization of more complex excitation signals--stochastic transient signals, stationary or nonstationary random signals--by means of theirs spectral composition (Harris et al., 2002; Josephs, 2002). It had to be noted that is relative simple to synthesize a composite input signals with specified but arbitrary power spectrum. This can be stationary or slowly time varying and it can be described with any mix of deterministic and stochastic attributes. Conceptually, generating stochastic signals with a specified power spectra (or correlation function) is accomplished by filtering a white noise signal with a filter exhibiting the desired power spectrum.

The expression of transfer matrix provide a partial transfer functions and, supposing the general matrix form of the motion --Eq. (4), it can be shown that the modal parameters can be identified from any row or column of the transfer matrix H(s) ,except those corresponding to components known as node points. In other words, it is impossible to excite a mode by forcing it at one of its node point (a point where no response is present).

Regarding partial fraction form of H(s)--Eq. (7), it had to be noted that each complex conjugate pair of poles corresponds to a mode of vibration of the structure. The transfer matrix completely defines the dynamics of the system. In addition to the poles of the system (which define the natural frequency and damping), the residues from any row or column of H(s) define the system mode shapes for the various natural frequencies. In general, a pole location will be the same for all transfer functions in the system because a mode of vibration is a global property of an elastic structure. The values of the residues, however, depend on the particular transfer function being measured. The values of the residues determine the amplitude of the resonance in each transfer function and, hence, the mode shapes for the particular resonance.

4. CONCLUDING REMARKS

The conclusions turn around the dynamics evaluation suitability of the isolation devices response close to the real behaviour. Working with the spectral composition of the input and output signals, the chance to simulate a close configuration to the real signals grows up. On the other hand, using the spectral composition, it is possible to consider only the components or the spectral area that assure the dominant dynamic character, or to extend the components influences for over-charging and resonance analysis.

Final remark is that the suitability of these approaches is, in fact, given by the advantages of computational dynamics in complex space relative to the time domain.

As a further research the authors will propose to generalize the previous theoretical model with excitation on both directions, and, in a second step, to consider a spatial structure, insulated from the ground with four set of orthogonal nonlinear visco-elastic devices and subjected to a spatial excitation.

5. REFERENCES

Axinti, G. (2008). Mechanics Digest, Editura Tehnica-Info, ISBN 978-9975-63-107-5, Chisinau

Bratu, P. (1990). Insulation Elastic Systems for Machines and Equipments, Editura Tehnica, ISBN 973-31-0234-2, Bucharest, Romania

Bratu, P. (2000). Elastic Systems Vibrations, Editura Tehnica, ISBN 973-31-1418-9, Bucharest, Romania

Bratu, P. (2001). Structural requirements imposed to vibration systems, International Journal of Acoustics and Vibration, Vol. 5, No. 2

Harris, C.M. & Piersol, A.G. (2002). Shock and Vibration Handbook, 5th Edition, McGraw Hill, ISBN 0-07-137081-1

Josephs, H, & Huston, R.S. (2002). Dynamics of Mechanical Systems, CRC Press, ISBN 0-8493-0593-4