Nonlinear Response
In a typical situation, when the PCB contour remains nondeformable during the board’s deformations caused by the postimpact vibrations, and when the impact is significant, as it should be in FOAT, HALT or even in qualification tests (QT), appreciable reactive in-plane (membrane) stresses arise in the board. Because of that, the board vibrations become (geometrically) nonlinear: the deflections are not proportional anymore to the applied load. It has been determined [35-41] that appreciable nonlinearity manifests itself even for small drop heights. It has also been determined that it is sufficient to consider the fundamental mode only, and that the nonlinear response of a PCB with a nondeformable support contour could be described by the Duffing equation
for the principal coordinate /(f). In this equation, to is the frequency of the linear vibrations,
is the parameter of nonlinearity (for the case of an elongated board), a is half the board’s length, D is its flexural rigidity, m is board’s mass per unit area, and h is its thickness. The Duffing equation addresses nonlinear vibrations with the rigid cubic characteristic of the restoring force (nonlinearity) and lends itself to an exact solution [35-39] (see also the Appendix to this chapter), no matter how significant the loading and, hence, the nonlinearity level might be, provided, of course, that the elasticity limit of the PCB material is not exceeded.
Here is how the amplitude A of the induced vibrations could be obtained in a rather elementary way, without even solving nonlinear equation (9.5). This equation can be written as
Hence, the expression in the brackets should be constant:
It has to be this way, because the sum of the kinetic energy (first term) and the strain energy (the second and the third terms), whether linear or nonlinear, should be constant, as long as there no dissipation in the system. At the initial moment of time, the principal coordinate f(t) is zero and the initial velocity is f(t) = /(0) = yj2gH. Hence, the constant C is C = 2gH. The velocity should be zero when the displacement reaches its amplitude value A. Then, the biquadratic equation
that can be obtained from (9.8) yields
where
is the linear amplitude,
is the factor of the nonlinear amplitude, and
is the dimensionless parameter of nonlinearity. The factor (12) changes from one to zero, when the linear amplitude A0 changes from zero to infinity. The nonlinear acceleration (deceleration) can be found from (9.5). The obtained relationships indicate that the nonlinear amplitude can be substantially lower than the linear amplitude, while the induced acceleration (deceleration) can be significantly higher.
It is natural to assume that the mechanical behavior of the solder joint interconnections has a minor effect on the PCB vibrations, as long as the size of the SMDs is small in comparison with the size of the board and, because of that, these devices do not change the flexural rigidity of the board. It has been suggested, therefore [39], that the nonlinear drop impact excited vibrations of a “flexible-and-heavy” PCB are evaluated assuming that the mounted devices do not affect its flexural rigidity. The SMD masses could be considered in an approximate way, by spreading their total mass over the PCB surface, thereby making the board both “flexible” and “heavy.” The attributes of the particular SMD (package) and, more importantly, its solder joint interconnections could be considered at the next step from the known (calculated or measured) tensile forces and bending moments acting in the PCB cross sections located in the close proximity of the particular SMD of interest. Figures 9.4 and 9.5 illustrate the significance of the in-plane stresses and how the induced bending moment and the membrane force can be experimentally determined. Of course, these force and moment can also be evaluated theoretically, and the results can and should be compared.
FIGURE 9.4 Experimental setup for drop tests. The PCB experiences shock excited vibrations that are transmitted to the SMD and particularly to the solder joint interconnections that attach the SMD to the PCB. The gages placed at the upper and the lower surfaces of the PCB show different maximum strains. This is an indication on the occurrence of in-plane (membrane) stresses.
FIGURE 9.5 The in-plane tensile forces and bending moments acting in the PCB cross sections can be determined from the computed or measured strains at the upper and the lower surfaces of the PCB experiencing drop impact-induced vibrations.
Nonlinear dynamic response of flexible structural elements experiencing periodic shock loads distributed over their length (surface) and idealized as a train of instantaneous impacts (impulses) was addressed, apparently for the first time, in connection with the observed stochastically unstable (“chaotic”) motions of such elements [1, 35, 42]. Distributed dynamic loading might occur, for example, as a result of acoustic (sonic) pulses acting on the plate (membrane). Linear and nonlinear, steady state and transitional, and undamped and damped vibrations were addressed and analyzed. The approach and the obtained solutions were later on extended and applied to PCBs employed in electronics [36-39].
It has been found [1, 35] that
- 1. the linear transient response of a single-degree-of-freedom (SDOF) linear system with viscous damping can lead, for particular combinations of the frequency ratio and the level of damping, to higher amplitudes and accelerations than in the subsequent steady-state vibrations; that
- 2. in linear SDOF systems, low level damping might be responsible for the very fact that the vibrations stabilize, such as become steady-state, but might not affect the nonresonant response, and this response can therefore be determined by simply imposing the conditions of periodicity on the system (these conditions state that the sequential impulse does not change the position of the system, but changes its velocity in a step-wise fashion, with the step defined as the ratio of the magnitude of the impulse to the system’s mass); that
- 3. the nonlinear response of a PCB can be reduced if the fundamental mode of vibrations is considered to the Duffing type of equation (9.5) for the principal coordinate, that is, to the equation for a SDOF oscillator with the rigid cubic characteristic of the restoring force; and that
4. an exact closed-form solution to equation (9.5) can be obtained for time- independent practically important types of excitations (instantaneous impulses, or suddenly applied and then suddenly removed constant loading), no matter how significant these excitations and the corresponding nonlinear effects might be.
The loading on a PCB in electronic equipment of an aircraft, spacecraft, or a missile during the take-off of the flying apparatus can be idealized as a suddenly applied (in a step-wise fashion) constant load whose duration considerably exceeds the linear period of vibrations of the PCB, not to mention its nonlinear period. Figures 9.6 and 9.7 illustrate the response of a large and flexible PCB to such a load. It has been shown [36] that the actual (nonlinear) accelerations in the midportion of the PCB can be significantly larger than those predicted on the basis of the linear theory, even if the combined effect of the “external” acceleration (experienced by the PCB support contour) and the induced accelerations caused by the PCB vibrations with respect to the contour is considered. While the external acceleration applied in the carried out example to the PCB support contour was only 25 g, the predicted linear acceleration at the center of the PCB, because of the PCB (linear) vibrations, was about 65.5 g, while the maximum nonlinear acceleration turned out to be as high as 118.5 g. If a certain device was designed to the highest allowable acceleration of 100 g, this device could not be placed inside the shaded rectangular in Figure 9.7. Thus, the application of a linear approach would be misleading in the situation in question.
FIGURE 9.6 Response of a flexible PCB to a constant instantaneously applied load acting on the PCB’s support contour. All the response characteristics, but the induced accelerations, decrease with an increase in the level of the nonlinearity (i.e., with the level of the induced inplane stresses). The maximum nonlinear acceleration could be by a factor of three higher than the linear acceleration. (From E. Suhir, “Nonlinear Dynamic Response of a Flexible Thin Plate to Constant Acceleration Applied to its Support Contour, with Application to Printed Circuit Boards Used in Avionic Packaging,” International Journal of Solids and Structures, vol. 29, No. 1. 1992.)
FIGURE 9.7 If the maximum allowable acceleration of an electronic device of interest is restricted to the acceleration of the 100 g level, it has to be placed on the PCB outside the shaded rectangular, because the predicted induced nonlinear acceleration at the PCB center is as high as 118.45 g and is by the factor of 4.74 larger than the acceleration of the contour. The maximum linear acceleration at the PCB center is “only” 65.5 g. Although this value is by the factor of 2.52 greater than the acceleration of the PCB contour, it is still below the allowable maximum acceleration of the device, and, if the design decision is based on the linear approach, an erroneous decision will be made. The maximum nonlinear acceleration is in this example by the factor of 1.81 greater than the maximum linear acceleration. The theory based on the exact solution to the Duffing equation for the principal coordinate indicates that this factor could be as high as three.
The method of principal coordinates (known in the theory of differential equations of mathematical physics as Fourier method) was employed in the aforementioned solutions to separate in the equation of motion the (time-dependent) principal coordinate from the (location-dependent) mode (coordinate function), and it has been assumed that the nonlinear effects could be accounted for by considering their impact on the principal coordinate only, while the coordinate function was assumed to be the same as in a linear approach. The only crucial requirement was that this function should satisfy the physically justifiable boundary conditions at the PCB support contour. Suhir and Arruda [40] have shown that this assumption is valid for a simply supported PCB, while the “linear coordinate function” assumption for PCBs with other boundary conditions could lead to an appreciable error. This is especially true for response characteristics (angles of rotation or curvatures) that require differentiation of the coordinate function with respect to one of the coordinates in the x-y plane. It should be pointed out that although the edges of an actual PCB seem to be closer to the clamped ones, rather than to simply supported ones, the conclusion obtained in [41] is less restrictive than one might imagine. This is because the analysis is intended to be applied to an experimental setup rather than to a PCB of an actual device.
The analysis in [41] was applied for building a Nokia experimental setup with an objective to evaluate the response not so much of the PCB itself but rather of the BGA interconnections of the devices mounted onto the board. It is noteworthy that a test vehicle does not have to necessarily mimic the one in an actual structure, but has to be simple and lend itself to an easy and straightforward interpretation of the test data. Because of that, it is recommended that a simply supported PCB is used in an adequate experimental setup.
As has been indicated, the assumption that the analysis of the impact exited vibrations could be restricted to the consideration of the fundamental mode is even more applicable to the nonlinear response than to the linear one. The validity of this statement was confirmed by FEA [39]. In Figure 9.8, the effect of higher modes of vibration is manifested by just small “white caps” on the “crests” of the vibration “waves.”
As has been indicated, it has been found [1, 35, 42] that a train of impact shortterm loads can lead to the phenomenon of “stochastic instability” (“chaos”) even in the case of a weak nonlinearity and in the simplest case of a SDOF system: the system behaves in a quasi-random fashion, although neither the excitation, nor the system itself, are characterized by a random loading or by random properties of
FIGURE 9.8 Analytical versus finite element method (FEM) analysis of the shock-exited nonlinear vibrations. While analytical solution considers the fundamental mode of vibrations only, the FEM accounts for all the modes. As one could see from the diagram, the role of the higher modes has manifested itself in the little “wavelets” near the first mode maxima. Not only these maxima, but even the phase angles were predicted correctly by the analytical solution.
the system. Stochasticity occurs in situations when the period of the train of impulses substantially exceeds the period of natural linear vibrations (so that the stochastic phase approximation is applicable), but, on the other hand, is low enough, so that the responses caused by the sequential impulses still overlap. Certainly, the damping in the system does not have to be high either, otherwise the vibrations caused by sequential impacts will not overlap. The random phase approximation (stating that the phase angles at the moments of impact are evenly distributed between zero and 2n) and the Smoluchowski equation [42] can be used to obtain the condition of the “stochastic instability” of the vibrations and to characterize the quasi-random behavior of the “chaotic” vibrations. He and Fulton [37] and He and Stallybrass [38] have extended Suhir’s work in application to PCBs used in electronic packaging.
An additional way to enhance the loading on the solder material and solder joint interconnections when conducting accelerated reliability testing on the board level, is to attach heavy concentrated masses to the PCB. This measure increases the inertia forces and, hence, the total accumulated strain energy in the board, the induced lateral accelerations, and the PCB in-plane deformations. Suhir, Vujosevic, and Reinikainen [39] have carried out an analysis of the dynamic response of a “flexible-and-heavy” PCB experiencing an impact load applied during drop tests to the board’s support contour. Only a simply supported board was considered, for the sake of making the analysis as simple as possible and to be suitable for an experimental setup and for a straightforward interpretation of the results. It was assumed in the analysis that the attached heavy concentrated masses, if any, were sufficiently small in size, so that their effect on the flexural rigidity of the board was insignificant and did not have to be considered, while their total mass did have an appreciable effect on the induced inertia forces, the accelerations and the stresses, and should be accounted for. Elliptic functions were used to find the exact solution to the basic equation for the principal coordinate and to establish, on the basis of this solution, the amplitudes, frequencies, accelerations (decelerations), and stresses experienced by the board. Unlike in linear systems, the dynamic factors for different response characteristics (amplitudes, accelerations, frequencies, stresses) are different for different characteristics and are highly dependent on the level of the dynamic loading.
As is known, it is considerably easier to carry out the analysis of the dynamic response of a system when the actual short-term impact loading can be substituted by an instantaneous impulse. Having in mind a nonlinear system with the rigid cubic characteristic of the restoring force, Suhir and Arruda carried out the corresponding analysis for a nonlinear system [40]. The important conclusion is that, as far as the amplitudes of vibrations are concerned, the nonlinear system is much less sensitive to the dynamic nature of the applied load than a linear system, but, when the induced accelerations are important, the situation is just the opposite: the induced accelerations in the nonlinear system might be substantially larger than in a linear one. It is concluded that modeling based on the analysis carried out in [41] should always precede the construction of the experimental setup, the design of the experiments, and the analysis (interpretation) of the experimental data.
Numerous problems associated with the random loading on PCBs idealized as thin plates were addressed in the monograph [42] and in [44] and [45].
