# A.7. Parameter of Nonlinearity

Introducing formulas (A.3) and (A.6), with consideration of solution (A. 11), into the expressions (A.l) and (A.2), the following formulas for the kinetic and strain energies can be obtained:

Here,

is the generalized mass of the PCB,

is the frequency of its free linear vibrations, and

is the parameter of nonlinearity. This parameter increases with an increase in the Young’s modulus of the PCB material and its thickness, and decreases with an increase in the PCB mass (and, hence, the masses of the SMDs) and its size. Note that the PCB’s generalized mass *M* is only a quarter of its actual mass. This is because the expression (A. 16) for the generalized mass considers that different points on the PCB respond differently to the same initial velocity of different vibration modes at the moment of impact.

# A.8. Basic Equation and Its Solution

Introducing formulas (A.15) into the Lagrange equation (see, for example, [A.26],)

the following basic differential equation (of Duffing type) can be obtained for the principal coordinate:

This equation lends itself to an exact solution, no matter how significant the level of the nonlinearity is:

Here, *A* is the vibration amplitude, *спи* is the elliptic cosine (see, for example, [A.27]), о is the parameter of the nonlinear frequency, *t* is time, and e is the initial phase angle. Using formulas

for differentiating the elliptic functions, and formulas

of the “elliptic geometry,” we obtain the following expressions for the velocity and the acceleration (deceleration):

where *к* is the modulus of the elliptic function, *snu* is the elliptic sine, and *dnu* is the function of delta-amplitude. Introducing solution (A.21) for the displacement and the second formula in (A.24) for the acceleration into equation (A.20) of motion, one could conclude that equation (A.20) is fulfilled, if the relationships

take place. These relationships define the parameter, o, of the nonlinear frequency and the modulus, *k,* of the elliptic function, respectively.

# A.9. Vibration Amplitude

The amplitude, *A,* of the induced vibrations can be found, without even solving equation (A.20), based on the following simple reasoning. Equation (A.20) can be written as

and therefore

The maximum deflection *f„ _{m} = A* takes place, when the velocity

*f(t)*is zero, so that

On the other hand, at the initial moment of time, *t =* 0, the principal coordinate, /(0), is zero, while the initial velocity /(0) = *V _{u}* has its maximum value. This yields C =

*V„.*Hence, the amplitude,

*A,*can be found from the biquadratic equation

as

where

is the amplitude of the corresponding linear vibrations (p = 0), and the factor considers the effect of nonlinearity on the vibration amplitude. Here,

is the dimensionless parameter of nonlinearity. The factor (A.30) changes from one to zero when the linear amplitude changes from zero to infinity.

# A.10. Effective Initial Velocity

From (A.3), we find:

If the PCB’s vibrations are caused by a drop impact, the velocities of all the points of the PCB at the initial moment of time are the same and are equal to

for any point x,y of the PCB. Here, *H* is the drop height. Using Fourier transform for the expression (A.33), we obtain the following formula for the effective initial velocity:

The factor -!^ = 1.6211 in this formula reflects the effect of the coordinate function *n*

on the initial velocity.

# A.11. Nonlinear Frequency

Я/2

The period of the elliptic cosine cn(f,e) is 4 *K(k)* (A.27), where *K(k) =* f , ^

*{ yjl- k ^{2} sin^{2}1;*

is the complete elliptic integral of the first kind. Hence, the period of the function

4

cn(ot,e), that is, the period of the induced vibrations, is *—K(k),* and the corresponding frequency is °

In the linear case, *K(k) = ^,* and *p* = о = w. In the nonlinear case, о > *p >* со, however, even for highly nonlinear vibrations the parameter a of the nonlinear frequency, is not very far away from the nonlinear frequency *p* itself (see numerical example).

# A.12. Bending Moments

The bending moments can be found, using expression (A.3) for the deflection function, as (see, for example, [A.25])

and

The bending moments change from

in the middle of the PCB to zero at its edges.

# A.13. Equivalent Static Loading

The equation of bending in the case of static loading is

where *q* is the static loading and

is the function of the in-plane tensile stresses.

The static displacement vr(;c,)’) and the static stress function

where *f _{sl}* is the static displacement at the PCB center and the function Ф(лу) is expressed by formula (A. 12). Then, equation (A.39) yields

Using Galerkin’s method, we obtain the following relationship between the static loading *q* and the induced displacement *f _{s},: *

This relationship can be particularly helpful when one considers replacing drop or shock tests with static bending tests to generate similar curvatures and strains in the PCB. In the case of small deflections,

Comparing expressions (A.42) and (A.43), we conclude that the factor

considers the effect of the nonlinearity on the equivalent static loading. When *A = h *and v = the factor *%* is as high as *x* = 3.

# A.14. Numerical Example

See also (A.28).

Input data:

Drop height *H =* 1.273 m; Board size *2a* x *2a =* 300 mm x 300 mm; Board thickness *h =* 1.5 mm;

Young’s modulus of the PCB material *E =* 22.75 GPa = 2321 kg/mm^{2}; Poisson’s ratio v = 0.3;

Distributed mass of the PCB (including the mass of the SMDs and solder material) /и = 7.878x 10 ^{10} kgxsec^{2}/mm^{3}.

Computed data:

Initial velocity:

Flexural rigidity of the PCB:

Actual mass of the PCB:

Generalized mass of the PCB:

Linear frequency

(Note that the FEA prediction [A.28] was to = 209.48sec"^{1})

Linear amplitude of the induced vibrations

(The FEA prediction [A.28] was a>=38.13 mm).

Maximum linear acceleration (deceleration)

Parameter of nonlinearity Dimensionless parameter of nonlinearity

Factor of the nonlinear amplitude (the ratio of the linear amplitude, when the membrane forces are neglected, to the nonlinear amplitude, obtained with the consideration of these forces)

Nonlinear amplitude

Parameter of the nonlinear frequency

Maximum nonlinear acceleration (deceleration)

Modulus of the elliptic function

Elliptic integral [A.27] Zf(k) = 1.58.

Nonlinear frequency

Note that the nonlinear frequency *p* is not very much different of the parameter о of the nonlinear frequency.

Nonlinear period of vibrations is *T* = — = 0.0010198 sec.

*P*

The duration of the equivalent shock impact should be as low as

*t _{0}* = — = 0.0001275 sec, so that it could be replaced in the analyses by an instanta- 8

neous impulse. Note that in the case of linear vibrations, the duration of the equivalent shock impact load would be significantly (by a factor of 29.4) larger:

The magnitude of the impact pulse

The maximum force for a half-sine impulse

The maximum value of the corresponding acceleration

The maximum in-plane stress occurs at the PCB center

The in-plane stress at the PCB contour:

The corresponding strains:

The total tensile in-plane force at the PCB center The total tensile in-plane force at the PCB contour

The maximum curvatures occur at the PCB center and, in an approximate analysis, can be found as

The maximum bending moment (per unit PCB width)

The maximum bending stress

Maximum bending strain

Total maximum stress on the convex side of the PCB

Total maximum stress on the concave side of the PCB

that is, also tensile (because of the very high in-plane forces).

Total maximum strain on the convex side of the PCB

Total maximum strain on the concave side of the PCB

The equivalent distributed static loading resulting in the maximum deflection *f _{sl}* that is equal to the nonlinear amplitude

*A*of impact induced vibrations is

This loading, high as it is, is considerably lower than the maximum distributed inertia load

Thus, a simple, easy-to-use, and physically meaningful predictive analytical (mathematical) stress model is suggested for the evaluation of the dynamic response of a PCB subjected to a drop impact during board-level drop testing. An exact solution is obtained for the Duffing-type equation for the principal coordinate of the induced vibrations. This means that this solution is applicable no matter how significant the level of the nonlinearity might be. Evaluation of the stresses and strains in the solder joints of the second level of interconnections (package to the PCB) is intended to be considered as the next step. Particularly, the advantage of CGA technology, as far as the possible stress relief in the solder material is concerned, is intended to be considered at this step. The induced membrane forces and the bending moments could be either predicted, or determined from the measured strains on the upper and the lower surfaces of the PCB. Experimental verification of the suggested model is certainly required. Particularly, the predicted and the measured strains on the concave and the convex surfaces of the PCB should be compared. Future work will include a methodology for evaluating the drop impact induced dynamic shearing and peeling stresses in the solder material of BGA and CGA designs.