About the dynamic stresses induced by the vibrations of an electronic circuit plate.

Dragomirescu, Cristian George ; Iliescu, Victor ; Patrascu, Gabriela 等

1. INTRODUCTION

The study is done for the most representative situations:

1) We consider an electronic circuit with the components set parallel to the vibration direction. Due to the high stiffness of the plate, its transmissibility, [tau] = 1. This means that the perturbation is acting directly upon the electronic components, each of them becoming an oscillating system.

2) If we assume that the connections are the only deformable parts, the model is reduced to a static undetermined frame.

3) The system damping may be considered linear for stresses in the elastic domain and highly nonlinear in the elastic and plastic domain, due to the hysteretic phenomena.

4) If the circuit is including massive components with transversal connections, a rotation tendency occurs. In this case, two natural frequencies occur, for transversal and torsional vibrations. One may observe that the mechanical model is determined by the placement of the component on the plate, in respect with the stress direction and by the mass and the form of the component (Pascu, 1992).

2. THE ANALITIC STUDY

2.1 The perturbation is acting directly on the components

The stresses in the connectors and connections depend on the force acting on the component (Deciu & Dragomirescu, 2001):

F = m x a = m x [a.sub.p] x [tau], (1)

where m is the mass of the component [kg], a--the acceleration [m/[s.sup.2], [a.sub.p]--the acceleration at the plate level [m/s2], t - the transmissibility of the vibrating system (the ratio between the transmitted force and the acting force).

From the above expression one may deduce:

a = [a.sub.0]/[square root of [(1 - [[eta].sup.2]).sup.2] + 4[D.sup.2][[eta].sup.2]] (2)

where [eta] = [OMEGA]/[omega] is the relative frequency, [omega] = [square root of c/m]--the natural beat [[s.sup.-1]], [OMEGA]--the beat of the acting force [[s.sup.-1]] , D = b/[b.sub.[alpha]]--the damping factor, C - elasticity coefficient [N/m], b--the damping coefficient [kg/s], [b.sub.cr] = 2[square root of cm] = 2m[omega]--the critical value of the damping coefficient [kg/s], [[alpha].sub.0]--the acceleration of the acting force[m/[s.sup.2]].

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

2.2 The deformation of the connections

The most usual ways of placing the components on the plate, considering the direction of the stressing force (Hauger, 1990) are the ones in Fig. 1.

One may observe in Fig. 1 that if only the connections are deformed, the system is reduced to a static undetermined frame for which the deformations, forces and torques may be deduced as known (Voinea et al., 1998). In this respect, the Fig. 2 is presenting the mechanical model and the Tab. 1 the results.

2.3 Stresses in the elastic and plastic domain

The analysis is made on the mechanical model presented in Fig. 3. For the differential equation of the free undamped vibrations of the beam, considering harmonic sollutions and using the usual known limit conditions, we obtain the solution:

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Y(x) = 2[F.sub.0]/EI[[alpha].sup.3](1 + ch [alpha]1 x cos [alpha]l) [T([alpha]l)U([alpha]x)- S([alpha]l)V([alpha] x)], (3)

where S([alpha]l), T([alpha]l), U([alpha]x), V([alpha]x) are Krilov functions, [[alpha].sup.4] = [rho]A[[omega].sup.2]/EI, A--the section of the beam, E--the elasticity module, [rho]--the density of the beam, I--the geometric inertia momentum. In this case, the solution for the forced undamped vibrations is:

Y(x) = 2[F.sub.0]/EI[[alpha].sup.3](1 + ch [alpha]1 x cos [alpha]a) [T([alpha]l)U([alpha]x) - S[alpha]l)V([alpha]x)]cos[omega]t., (4)

For 1 + ch[alpha]1 x cos[alpha]1 = 0, the vibration amplitude is increasing towards infinite. This phenomenon of resonance is possible when the natural frequency of the system and the force frequency are identical.

The amplitude of the free end of the beam vibrations is:

Y(1) = [F.sub.0]/EI[alpha]3 x ch[alpha]1 x sin[alpha]1 - sh[alpha]1 x cos[alpha]1/1 + ch[alpha]1 x cos[alpha]1. (5)

For the model considered, using a computerised algorithm, we obtain the natural forms of vibration shown in Fig. 4a and 4b (Craifaleanu et al., 2003). As one may see, an increasing force F(t) will produce at the beginning elastic deformations, proportionally with the stress. If the force is increasing too much, the deformations become plastic and the function describing the deformation becomes nonlinear. When the force is decreasing to zero, the deformations disappear if the plastic limit was not over passed. Otherwise, a plastic deformation of the beam will be visible (Fig. 4c and 4d).

The laboratory experiment lead to the following diagrams (Fig. 5).

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

2.4 The dynamic behavior of a massive component

If the circuit plate is vertical and the component is horizontally placed, due to the two connections, a rotation tendency around the mass centre axis may occur. In this case, the mechanical model in Fig. 3 must be transformed into the one shown in Fig. 6a (Deciu et al., 2002). In this situation, two natural frequencies, one for each type of motion, may be highlighted. The natural form vibration is shown in Fig. 6b.

In order to calculate the natural frequencies of the two types of vibration, for the translation we consider [l.sub.2] = 0 (similar to Fig. 3). The two values are:

[[omega].sub.trans] = [square root of c/m], [[omega].sub.rot] = [square root of c[phi]/[m[phi], (6)

where c = 3EI /[l.sup.3.sub.1][N/m], [c.sub.[phi]] = 3EI /[l.sub.l] [Nm/rad], I is the geometric inertia momentum of the connections section, [J.sub.[phi]]--the torsion mechanic inertia momentum considering the mass centre axis.

For [l.sub.1]=5 mm, [l.sub.2]=8 mm, d=0,5 mm (for the connections), m=1 g, E=[10.sup.5] N/[mm.sup.2] (copper connections), we obtained the following natural frequencies: [f.sub.trans] = 185Hz; [f.sub.rot] = 1218Hz .

3. CONCLUSION

The studied models are pointing out the physical phenomena that take place, showing also the ways to avoid them if necessary. The experimental study carried out is showing the risk domains. It is thus possible to better understand the harmful consequences that may occur if one is neglecting the mechanical vibrations of an electronic circuit plate, as it happens in practice in most of the situations.

4. REFERENCES

Craifaleanu, A.; Iliescu, V.; Craifaleanu, I. (2003), Laborator virtual de vibrafii mecanice (Virtual Laboratory of Mechanical Vibrations), Prima Conferin|a de eLearning, Educate [section]i Internet, Universitatea din Bucuresti, Departamentul de Inva|amant la Distan|a CREDIS, Bucuresti

Deciu, E. & Dragomirescu, C. (2001). Maschinendynamik (Dynamics of Machines), Editura Printech, ISBN 973-652438-8, Bukarest

Deciu, E.; Bugaru, M. & Dragomirescu, C. (2002). Vibrapi neliniare cu aplicafii in Ingineria Mecanica (Nonlinear Vibrations Applied in Mechanical Engineering), Editura Academiei Romane, ISBN 973-27-0911-1, Bucuresti

Hauger, W.; Schnell, W. & Gross, D. (1990). Technische Mechanik (Technical Mechanics), Springer Verlag, ISBN 3-540-53019-3, Belin Heidelberg

Pascu, A. (1992). Structura mechanica a aparatelor electronice (Mechanical Structure of Electronic Devices), OIDCM, Bucuresti

Voinea, R.; Voiculescu, D. & Simion, Fl. (1989). Introducere in mecanica solidului cu aplicafii in inginerie (Introduction in Solid Mechanics Applied in Engineering), Editura Academiei Romane, ISBN973-27-0000-9, Bucuresti

Tab. 1. The results for the adopted model

Fig. 1a [V.sub.A] = 3Fhc/1(1 + 6c); [H.sub.A] = F/2;
 [M.sub.A] = Fh/4 (1 + 1/6c + 1)

Fig. 1b [V.sub.A] = F/2; [H.sub.A] = 3F1/2h(8 + 4c);
 [M.sub.A] = F1/8c + 16

Fig. 1c [V.sub.A] = [F.sup.2][h.sup.2]/[8E.sub.2][I.sub.2];
 [H.sub.A] = F/2; [M.sub.A] = Fh/2;
 [M.sub.tB] = [F1.sup.2][GI.sub.2p]/
 8([2hE.sub.1][I.sub.1] + [1GI.sub.2p])