Predicted Accelerations of Surface-Mounted Electron Devices during Spacecraft Launch

Nonlinear dynamic response of a thin-and-flexible horizontally-oriented rectangular printed circuit board (PCB), whose non-deformable support contour is subjected to a suddenly applied constant acceleration during spacecraft vertical launch, is considered. The general concept is illustrated by a numerical example. It is shown that the accelerations of the board’s inner points could be much higher than the acceleration of its support contour. The results of the analysis could be used, particularly, when deciding if the acceleration-sensitive electron devices are robust enough and where to place them on the board, so that the induced acceleration will still be below the allowable level.


Introduction
Elevated accelerations could possibly cause mechanical and/or functional failures of the surface-mounted electron devices. Dynamic response of electronic and photonic systems to shocks and vibrations was addressed in numerous publications (see, e.g., [1][2][3][4][5][6][7][8][9][10]). In this analysis the nonlinear dynamic response of a spacecraft PCB, with surface-mounted IC devices on it, is considered and analyzed. The spacecraft vertical launch condition is addressed, and the engineering theory of bending of plates is employed to analyze the response of a thin-and-flexible board, whose non-deformable support contour experiences a suddenly applied constant acceleration during spacecraft launch. The objective of the analysis is to predict the accelerations at inner points of the board. These accelerations might exceed significantly the acceleration of the board's support contour. The cases of simply supported and clamped boards are considered. The analysis is an extension and a modification of the author's earlier work [2,3]. The current (for the past five years or so) state-of-theart could be found in numerous NASA website information and publications.
Consider a printed circuit board (PCB) whose non-deformable support contour experiences suddenly applied and constant vertical acceleration ( Figure 1). Using the engineering theory of thin plates (see, e.g., [11,12]), the normal in-plane stresses In the case of a non-deformable contour, the function φ(x,y) must satisfy also the conditions Where υ is Poisson's ratio of the board's material, and a and b are the board's dimensions in the x and y directions, respectively. These conditions indicate that, for a non-deformable contour, the in-plane displacements caused by the in-plane ("membrane") tensile stresses 0 x σ and 0 y σ should be equal to the displacements caused by the board's bending. The conditions (4) are based on assumptions that the board's material is isotropic and homogeneous.
Our analysis is limited to the first mode of vibrations. The derivation in Figure 2 justifies such a limitation even for a linear system, in which the role of the higher modes is expected to be considerably higher than in a nonlinear system, mostly because the strain energy of the first mode of vibrations in linear systems does not contain the energy caused by the in-plane strains, which result in nonlinear effects. The situation, when the surface mounted device is mounted on an elongated PCB, is addressed, and it is shown that, as far as the kinetic energy of the PCB experiencing shock-excited linear vibrations is concerned, the higher modes represent only about 23% of the total PCB energy. When deflections are significant and the board's vibrations become nonlinear, this percentage is even smaller, so that, in an approximate analysis, consideration of the first mode of vibrations seems to be justified.
So, the functions w = w(x,y,t) and φ(x,y) could be sought as Substituting (5), with consideration of (6), into the continuity equation (2) and into the conditions (4) of non-deformability of the board's support contour, we conclude that that the function φ 1 (x,y) should satisfy the equation And the conditions Then the following expression for the static stress function φ 1 (x,y) could be obtained: Although the situation might be much more complicated for a clamped board [13] experiencing nonlinear vibrations, in an approximate analysis it could be assumed that for a clamped board, whose aspect ratio b a is between 1.0 and 1.
For a square board this formula yields:

Equation of motion
The kinetic, T, and the strain, V, energies of the board are expressed as (see, e.g., [11,12]) Here A = ab is the board area, m is its mass per unit area (with consideration of the masses of the mounted components, assuming that, in an approximate analysis, these masses can be uniformly spread out over the board's surface), 3 2 12(1 ) is the board's flexural rigidity (we assume that the sizes of the mounted components are small, and therefore they affect only the board's mass, but not its flexural rigidity), h is the board's thickness, and x y is the Laplace operator. The first term in the second formula in (14) is due to bending, and the second term is due to the tensile membrane stresses.
Substituting (5) into (14), the following formulas for the kinetic and the strain energies can be obtained: Introducing the formulas (15) into the Lagrange equation (see, e.g., We obtain the following nonlinear differential equation for the principal coordinate z(t): is the excitation force caused by the acceleration c w   of the board's support contour, is the factor that considers the effect of the coordinate function on the magnitude of this force, and c w is the vertical displacement of the support contour. For a simply supported board For a square simply supported board (a = b) For a clamped board For a square clamped board (a = b) Note that the parameter α of non-linearity is a little higher for a clamped board than for a simply supported one. For v = 1/3, e.g., the ratio of these parameters for square boards is 1.306.

Maximum deflection, velocity and acceleration
The maxima of the deflection, velocity and acceleration can be determined even without solving the equation (18) of motion (this is always useful, of course, and is done in Appendix A). Indeed, the equation (18) can be written as Hence, the expression in the parentheses (which is, in effect, the total energy of the board) should be constant: If the initial displacement and the initial velocity are zero, the constant C should be zero as well, so that the equation Should be fulfilled. This equation, written as Determines the s. c. phase diagram, which establishes the relationship between the displacement and the velocity for the given nonlinear system.
The maximum value z max of the displacement takes place for ż = 0 and, as evident from (27) and (28), can be found from the cubic equation Considers the effect of non-linearity on the maximum static displacement. The dynamic factor is in this case The distribution of the induced accelerations over the board's surface can be obtained from the first formula in (5) as follows: As evident from this formula, the initial acceleration 1 w   is the maximum on the board's contour (where w 1 = 0) and is the minimum At its center, where w 1 = 1. In the case of a simply supported board, using (6), we have: Is the maximum linear dynamic displacement (α = 0), and the factor η z that considers the effect of the nonlinearity on the maximum nonlinear dynamic displacement could be obtained from the equation (29) as 3 3 3 Is dimensionless parameter of nonlinearity of the dynamic system in question. The factor η z tends to one, when the parameter µ of nonlinearity tends to zero. It will be shown in the Appendix that the exact solution to the basic equation (18) leads to a different dimensionless parameter of nonlinearity, When μ changes from zero to infinity, δ changes from 1 to 3 . Some further details could be found in the Appendix.
Note that in Figure 1 both governing parameters μ and δ are indicated.
The maximum linear static displacement Comparing this result with (31), we conclude that the dynamic factor for the maximum linear displacement is If the same suddenly applied constant load q is applied to a linear system dynamically and stays on this system, the induced maximum displacement is twice as large as the displacement, when this load is applied statically. In a linear system this is also true for the maximum velocity and the maximum acceleration, but in a nonlinear system, as will be shown below, the effect of nonlinearity is different for the displacements, velocities and accelerations, and, as far as the accelerations are concerned, could be significantly greater than in a linear system. In the case of a clamped board, whose coordinate function is expressed by the formula (11), the initial acceleration is Thus, negative initial accelerations occupy a rather large portion of the board area and their absolute maxima are comparable with the magnitude of the acceleration of the board's contour. In elongated boards the region of the negative initial accelerations is -0.212a ≤ x ≤ 0.212a for a simply supported board, and -0.169a ≤ x ≤ 0.169a for a clamped board. The accelerations at the board center for a simply supported and clamped boards are Where, g is the acceleration due to gravity.

Conclusions
The following conclusions can be drawn from the carried out analyses: 1) Simple, easy-to-use and physically meaningful analytical model has been developed on the basis of the engineering theory of thin plates for the evaluation of the induced accelerations in a flexible PCB in a spacecraft during its launch.
2) It is shown that the accelerations of the inner points of the board could be significantly higher than of those on its support contour and that, for an immovable support contour, it is important to account for the nonlinearity of the board's vibrations.
3) The results of the analysis could be used particularly, when deciding where to place acceleration sensitive electron devices on the board. They can also be used when designing experiments and to build the most appropriate test vehicle.
4) The supporting structure could have some components to thy also that because of the structural damping the vibrations will fade in the course of time, and therefore at the moments of time sufficiently remote from the moment of loading, the board's accelerations will not be different of the accelerations c w   of its contour.

Numerical Example
Let an ASTM/NEMA Class G-10 fiber-glass PCB simply supported on its contour be subjected to a constant suddenly applied acceleration 25 c w g =  applied to its contour. Let the weight of all the surface-mounted devices are 20% of the board's weight, and let there exists certain flexibility in the location, where the given device could be mounted on the board. The device cannot safely withstand accelerations exceeding 100 g. Let us determine, where this device could be safely installed, so that its reliable operation and strength are not compromised. We use the following input data: density of the PCB material: The distribution of the total (absolute) maximum acceleration over the board's surface predicted by the formula (42) is absorb vibration as well, e.g., dynamic vibration absorber technique. Such a possibility will be considered as a future work.
The results of the analysis could be used when designing a suitable experiment, conducting computer-aided simulations, considering, if necessary, the appropriate absorber techniques, and, if there is a need, to optimize the structure for the anticipated loading conditions.