An Enhanced Random Vibration and Fatigue Model for Printed Circuit Boards

Braz, Bruno de Castro; Bussamra, Flávio Luiz de Silva

doi:10.1590/1679-78253163

Bruno de Castro Braz ^** Corresponding author

Space System Division, Instituto Nacional de Pesquisas Espaciais, São José dos Campos, Brazil. Email: bruno.braz@inpe.br

Flávio Luiz de Silva Bussamra

Aerospace Engineering Division, Instituto Tecnológico de Aeronáutica, São José dos Campos, Brazil. Email: flaviobu@ita.br

* Corresponding author

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Figure 1:
Simplified (polygonal) lead geometry.

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Figure 2:
Lead free-body diagrams: (a) concentrated lead P; (b) bending moment M; (c) torsional moment T.

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Figure 3:
PCB with an electronic component (EC).

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Figure 4:
Kinematics of the PCB and EC: (a) underformed position; (b) displacement and rotations.

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Figure 5:
Double-beam model for EC.

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Figure 6:
Test setup - PLCC and LCCC (Liguore and Followell 1995Liguore, S., Followell, D., (1995). Vibration fatigue of surface mount technology (SMT) solder joints, IEEE Proceedings Annual Reliability and Maintainability Symposium.).

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Figure 7:
Test setup - PBGA (Yu et al. 2011Yu, D., Al-Yafawi, A., Nguyen, T. T., Park, S., Chung, S., (2011). High-cycle fatigue life prediction for Pb-free BGA under random vibration loading, Elsevier Microelectronics Reliability 51: 649-656.).

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Figure 8:
Test setup (Genç 2006Genç, C., (2006). Mechanical fatigue and life estimation analysis of printed circuit board components, Thesis of Science Master Degree, Middle East Technical University, Beirut, Lebanon.): a) PDIP; b) Tantalum Capacitor.

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Table 1:
Cycles to failure for PLCC with 30.6 G_rms, 68 J-Lead.

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Table 2:
Cycles to failure for PLCC with 80.7 G_rms, 68 J-Lead.

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Table 3:
Cycles to failure for LCCC with 30.6 G_rms, 32 pins.

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Table 4:
Cycles to failure for LCCC with 30.6 G_rms, 68 pins.

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Table 5:
Cycles to failure for LCCC with 30.6 G_rms, 84 pins.

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Table 6:
Minutes to failure for PBGA SAC 305 with 0.25 g²Hz^-1.

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Table 7:
Minutes to failure for PBGA SAC 305 with 0.15 g²Hz^-1.

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Table 8:
Minutes to failure for PBGA SAC 405 with 0.25 g²Hz^-1.

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Table 9:
Minutes to failure for PBGA SAC 405 with 0.15 g²Hz^-1.

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Table 10:
Failure step (minutes to failure) for PDIP.

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Table 11:
Failure step (minutes to failure) for tantalum capacitor.

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Table C.1:
PCB properties.

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Table C.2:
EC properties.

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Table C.2:
EC properties (continuation).

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15a)

(15b)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(A.1)

(A.2)

(A.3)

(A.4)

(A.5)

(A.6)

(A.7)

(A.8)

(A.9)

(A.10)

(A.11)

(B.1)

(B.2)

(B.3)

(B.4)

(B.5)

(B.6)

(B.7)

(B.8)

Model	U1 and U3 components	U2 component	Fail Point (Fig. 1)
Complete / 9 modes	4.79×10⁶	14.73×10⁶	a
Acceleration off	5.14×10⁶	16.47×10⁶	a
Moment off	5.22×10⁶	16.00×10⁶	a
Complete / 1 mode	4.78×10⁶	14.73×10⁶	a
Steinberg (2000Steinberg, D. S., (2000). Vibration analysis for electronic equipment, John Wiley Sons (New York).)	1.96×10⁶	0.21×10⁶	not available
Liguore and Followell (1995Liguore, S., Followell, D., (1995). Vibration fatigue of surface mount technology (SMT) solder joints, IEEE Proceedings Annual Reliability and Maintainability Symposium.)	1×10⁶ to 9×10⁶		not available

Model	U1 and U3 components	U2 component	Fail Point (Fig. 1)
Complete / 9 modes	0.86×10⁴	2.65×10⁴	a
Acceleration off	0.92×10⁴	2.96×10⁴	a
Moment off	0.94×10⁴	2.87×10⁴	a
Complete / 1 mode	0.86×10⁴	2.65×10⁴	a
Steinberg (2000Steinberg, D. S., (2000). Vibration analysis for electronic equipment, John Wiley Sons (New York).)	0.37×10⁴	0.038×10⁴	not available
Liguore and Followell (1995Liguore, S., Followell, D., (1995). Vibration fatigue of surface mount technology (SMT) solder joints, IEEE Proceedings Annual Reliability and Maintainability Symposium.)	2×10⁴ to 10×10⁴		not available

Model	U5 and U6 components	Fail Point (Fig. 1)
Complete / 9 modes	5.19×10⁵	solder
Acceleration off	5.44×10⁵	solder
Moment off	11.52×10⁵	solder
Complete / 1 mode	5.19×10⁵	solder
Steinberg (2000Steinberg, D. S., (2000). Vibration analysis for electronic equipment, John Wiley Sons (New York).)	0.90×10⁵	not available
Liguore and Followell (1995Liguore, S., Followell, D., (1995). Vibration fatigue of surface mount technology (SMT) solder joints, IEEE Proceedings Annual Reliability and Maintainability Symposium.)	10×10⁵ to 40×10⁵	solder

Model	U4 and U7 components	Fail Point (Fig. 1)
Complete / 9 modes	1.87×10⁵	solder
Acceleration off	1.90×10⁵	solder
Moment off	2.95×10⁵	solder
Complete / 1 mode	1.87×10⁵	solder
Steinberg (2000Steinberg, D. S., (2000). Vibration analysis for electronic equipment, John Wiley Sons (New York).)	1.17×10⁵	not available
Liguore and Followell (1995Liguore, S., Followell, D., (1995). Vibration fatigue of surface mount technology (SMT) solder joints, IEEE Proceedings Annual Reliability and Maintainability Symposium.)	2×10⁵ to 20×10⁵	solder

Model	U8 and U9 components	Fail Point (Fig. 1)
Complete / 9 modes	5.94×10⁵	solder
Acceleration off	6.00×10⁵	solder
Moment off	8.52×10⁵	solder
Complete / 1 mode	5.95×10⁵	solder
Steinberg (2000Steinberg, D. S., (2000). Vibration analysis for electronic equipment, John Wiley Sons (New York).)	0.63×10⁵	not available
Liguore and Followell (1995Liguore, S., Followell, D., (1995). Vibration fatigue of surface mount technology (SMT) solder joints, IEEE Proceedings Annual Reliability and Maintainability Symposium.)	2.0×10⁵ to 10×10⁵	solder

Model	x= ā/4; y= $\bar{b}$ /2 (Fig. 3)	Fail Point (Fig. 1)
Complete / 9 modes	1.27×10²	solder
Acceleration off	1.33×10²	solder
Moment off	1.71×10²	solder
Complete / 1 mode	1.35×10²	solder
Steinberg (2000Steinberg, D. S., (2000). Vibration analysis for electronic equipment, John Wiley Sons (New York).)	0.15×10²	not available
YU et al (2011Yu, D., Al-Yafawi, A., Nguyen, T. T., Park, S., Chung, S., (2011). High-cycle fatigue life prediction for Pb-free BGA under random vibration loading, Elsevier Microelectronics Reliability 51: 649-656.)	0.1×10² to 2.20×10²	solder
	average: 1.13×10²	solder

Model	x= ā/4; y= $\bar{b}$ /2 (Fig. 3)	Fail Point (Fig. 1)
Complete / 9 modes	7.61×10²	solder
Acceleration off	7.99×10²	solder
Moment off	10.24×10²	solder
Complete / 1 mode	8.08×10²	solder
Steinberg (2000Steinberg, D. S., (2000). Vibration analysis for electronic equipment, John Wiley Sons (New York).)	0.92×10²	not available
YU et al (2011Yu, D., Al-Yafawi, A., Nguyen, T. T., Park, S., Chung, S., (2011). High-cycle fatigue life prediction for Pb-free BGA under random vibration loading, Elsevier Microelectronics Reliability 51: 649-656.)	1.0×10² to 4.0×10²	solder
	average: 3.95×10²	solder

Model	x= ā/4; y= $\bar{b}$ /2 (Fig. 3)	Fail Point (Fig. 1)
Complete / 9 modes	1.41×10²	solder
Acceleration off	1.46×10²	solder
Moment off	1.72×10²	solder
Complete / 1 mode	1.48×10²	solder
Steinberg (2000Steinberg, D. S., (2000). Vibration analysis for electronic equipment, John Wiley Sons (New York).)	0.62×10²	not available
YU et al (2011Yu, D., Al-Yafawi, A., Nguyen, T. T., Park, S., Chung, S., (2011). High-cycle fatigue life prediction for Pb-free BGA under random vibration loading, Elsevier Microelectronics Reliability 51: 649-656.)	average: 1.02×10²	solder

Model	x= ā/4; y= $\bar{b}$ /2 (Fig. 3)	Fail Point (Fig. 1)
Complete / 9 modes	4.77×10²	solder
Acceleration off	4.93×10²	solder
Moment off	5.80×10²	solder
Complete / 1 mode	4.99×10²	solder
Steinberg (2000Steinberg, D. S., (2000). Vibration analysis for electronic equipment, John Wiley Sons (New York).)	2.09×10²	not available
YU et al (2011Yu, D., Al-Yafawi, A., Nguyen, T. T., Park, S., Chung, S., (2011). High-cycle fatigue life prediction for Pb-free BGA under random vibration loading, Elsevier Microelectronics Reliability 51: 649-656.)	average: 2.82×10²	solder

Model	x= ā/2; y= $\bar{b}$ /2 (Fig. 3)	x= ā/10; y= $\bar{b}$ /10 (Fig. 3)	x= ā/4; y= $\bar{b}$ /4 (Fig. 3)	Fail Point (Fig. 1)
Complete / 9 modes	>step 14 (8.43×10²)	step 14 (6.64×10²)	step 14 (7.05×10²)	no fail
Acceleration off	>step 14 (8.70×10²)	step 14 (6.64×10²)	step 14 (7.06×10²)	no fail
Moment off	>step 14 (8.60×10²)	step 14 (6.64×10²)	step 14 (7.08×10²)	no fail
Complete / 1 mode	>step 14 (9.65×10²)	step 14 (6.67×10²)	step 14 (7.12×10²)	no fail
Steinberg (2000Steinberg, D. S., (2000). Vibration analysis for electronic equipment, John Wiley Sons (New York).)	step 10 (5.43×10²)	step 13 (7.27×10²)	step 11 (6.31×10²)	not available
Genç (2006Genç, C., (2006). Mechanical fatigue and life estimation analysis of printed circuit board components, Thesis of Science Master Degree, Middle East Technical University, Beirut, Lebanon.)	step 3 up to 2.5 min. of step 14 with no failure			no fail
	total time: 6.62×10²			no fail

Model	x= ā/2; y= $\bar{b}$ /2 (Fig. 3)	x= ā/10; y= $\bar{b}$ /10 (Fig. 3)	x= ā/4; y= $\bar{b}$ /4 (Fig. 3)	Fail Point (Fig. 1)
Complete / 9 modes	step 3 (1.52×10²)	step 6 (3.07×10²)	step 3 (1.59×10²)	a
Acceleration off	step 9 (5.22×10²)	>14 step	step 12 (6.62×10²)	a
Moment off	step 2 (1.01×10²)	step 5 (2.53×10²)	step >14 (9.90×10²)	a
Complete / 1 mode	step 4 (2.07×10²)	step 4 (1.89×10²)	step 7 (4.06×10²)	a
Steinberg (2000Steinberg, D. S., (2000). Vibration analysis for electronic equipment, John Wiley Sons (New York).)	step 11 (5.17×10²)	step 13 (7.20×10²)	step 11 (6.12×10²)	not available
Genç (2006Genç, C., (2006). Mechanical fatigue and life estimation analysis of printed circuit board components, Thesis of Science Master Degree, Middle East Technical University, Beirut, Lebanon.)	first fail: between step 3 and 4 (1.52×10² and 2.04×10²)			a
	eleventh fail: step 6 (3.50×10²)			a

N = 20 \cdot 10^{6} {(\frac{0.00022 \cdot {25.4}^{1.5} B_{P C B}}{C_{s t} t_{P C B} r \sqrt{L}}) \cdot \frac{1}{Z_{\max_r m s}}}^{b_{b}},

k_{l} = {(\frac{c_{}^{3}}{3 E I_{z' 2}} + \frac{b_{2} c_{}^{2}}{E I_{z' 2}} + \frac{b_{1} c_{}^{2}}{E I_{z' 1}} + \frac{(a_{}^{3} + 3 a_{}^{2} c + 3 a c_{}^{2})}{3 E I_{z' 1}} - \frac{{(a + c)}^{2} d}{E I_{z' 1}}) {+ (\frac{b_{1} + d}{E A_{1}} + \frac{b_{2}}{E A_{2}})}}^{- 1},

k_{b} = {(\frac{c}{E I_{z' 2}} + \frac{b_{2}}{E I_{z' 2}} + \frac{b_{1}}{E I_{z' 1}} + \frac{a}{E I_{z' 1}} - \frac{d}{E I_{z' 1}})}^{- 1},

k_{t} = {(\frac{c}{G J_{2}} + \frac{b_{2}}{E I_{x' 2}} + \frac{b_{1}}{E I_{x' 1}} + \frac{a}{G J_{1}} - \frac{d}{E I_{x' 1}})}^{- 1},

W_{m n} = π^{2} [{(\frac{m}{\bar{a}})}^{2} + {(\frac{n}{\bar{b}})}^{2}] \sqrt{\frac{E_{P C B} t_{P C B}^{3}}{12 ρ_{P C B} (1 - υ_{P C B}^{2})}},

G_{m n} (W) = \frac{- \frac{2 \sqrt{ρ a b}}{π^{2} m n} (1 - c o s (π m)) (1 - c o s (π n))}{W_{m n}^{2} - W^{2} + 2 i ξ W_{m n} W} .

I_{x, c} = \frac{\bar{d} [t_{c}^{3} - {(t_{c} / 2)}^{3}]}{12} \begin{matrix}  \end{matrix}, \begin{matrix}  \end{matrix} I_{y, c} = \frac{\bar{c} [t_{c}^{3} - {(t_{c} / 2)}^{3}]}{12} .

δ_{c, x} = \frac{{\bar{c}}^{3} k_{l} n_{x}}{24 E I_{x, c}} δ_{l e a d, x} \begin{matrix}  \end{matrix}, \begin{matrix}  \end{matrix} δ_{c, y} = \frac{{\bar{d}}^{3} k_{l} n_{y}}{24 E I_{y, c}} δ_{l e a d, y}

δ = (\frac{{\bar{c}}^{3} k_{l} n_{x}}{24 E I_{x, c}} + 1) δ_{l e a d, x} + (\frac{{\bar{d}}^{3} k_{l} n_{y}}{24 E I_{y, c}} + 1) δ_{l e a d, y}

f a t_{c} (x, y) = {(\frac{R_{x} {\bar{c}}^{3} k_{l} n_{x}}{24 E I_{x, c}} + \frac{R_{y} {\bar{d}}^{3} k_{l} n_{y}}{24 E I_{y, c}} + 1)}^{- 1} .

\begin{array}{l} δ_{c, l o n g} = \sum_{m, n = 1,1}^{m, n = φ, φ} [δ_{c g, m n} + δ_{c a, m n}] f a t_{c} = \sum_{m, n = 1,1}^{m, n = φ, φ} {δ_{m n} (x, y, W) G_{m n} (W) \\ + [Z_{m n} (x, y) G_{m n} (W) (- W^{2}) + \frac{1}{φ^{2}}] G_{c, l o n g} (W)} f a t_{c} (x, y) α (W) . \end{array}

f a t_{c m 1} (x, y) = {(\frac{R_{x} (x, y) \bar{c} k_{b} n_{x}}{2 E I_{x, c}} + \frac{R_{y} (x, y) \bar{d} k_{b} n_{y}}{2 E I_{y, c}} + 1)}^{- 1} f a t_{c} (x, y),

f a t_{c m 2} (x, y) = {(\frac{R_{x} (x, y) \bar{c} k_{t} n_{x}}{2 E I_{x, c}} + \frac{R_{y} (x, y) \bar{d} k_{t} n_{y}}{2 E I_{y, c}} + 1)}^{- 1} f a t_{c} (x, y),

D = n_{p}^{+} T \frac{{(σ_{V M, r m s})}^{b_{b}}}{C_{b}} [G_{1} Q^{b_{b}} Γ (1 + b) + {(\sqrt{2})}^{b_{b}} Γ (1 + \frac{b_{b}}{2}) (G_{2} {| R |}^{b_{b}} + G_{3})],

PCB properties	ā (mm)	$\bar{b}$ (mm)	Material	t_PCB (mm)	E_PCB (Mpa)	ν _PCB
PLCC / LCCC	304.8⁽¹⁾	152.4⁽¹⁾	E-Glass⁽¹⁾	1.45⁽¹⁾	80000	0.3
PBGA	85⁽²⁾	45⁽²⁾	FR4⁽²⁾	0.6⁽²⁾	25000⁽²⁾	0.28⁽²⁾
PDIP/ Capacitor	318⁽³⁾	233⁽³⁾	not available	2.5*	17000⁽³⁾	0.3⁽³⁾

EC properties	mass (g)	lead material	E_lead (Mpa)	body material	E_body (Mpa)	a (mm)	b₁ (mm)	b₂ (mm)	c (mm)
PLCC	10*	Copper⁽¹⁾	128000	Plastic⁽¹⁾	26000⁽⁵⁾	0.08⁽⁷⁾	1.53⁽⁷⁾	1.53⁽⁷⁾	0⁽⁷⁾
LCCC 32	10*	Pb solder⁽¹⁾	44000⁽⁶⁾	Ceramic⁽¹⁾	275700⁽⁴⁾	0⁽⁸⁾	0.305⁽⁸⁾	0.305⁽⁸⁾	0⁽⁸⁾
LCCC 68	10*	Pb solder⁽¹⁾	44000⁽⁶⁾	Ceramic⁽¹⁾	275700⁽⁴⁾	0⁽⁸⁾	0.425⁽⁸⁾	0.425⁽⁸⁾	0⁽⁸⁾
LCCC 84	10*	Pb solder⁽¹⁾	44000⁽⁶⁾	Ceramic⁽¹⁾	275700⁽⁴⁾	0⁽⁸⁾	0.425⁽⁸⁾	0.425⁽⁸⁾	0⁽⁸⁾
PBGA	0.21⁽²⁾	SAC 305⁽²⁾	51000⁽²⁾	Plastic⁽²⁾	20900⁽²⁾	0⁽²⁾	0.2⁽²⁾	0.01⁽²⁾	0⁽²⁾
PBGA	0.21⁽²⁾	SAC 405⁽²⁾	51000⁽²⁾	Plastic⁽²⁾	20900⁽²⁾	0⁽²⁾	0.2⁽²⁾	0.01⁽²⁾	0⁽²⁾
PDIP	1.08⁽³⁾	CDA194⁽³⁾	121000⁽³⁾	Plastic⁽³⁾	14500⁽³⁾	0.457⁽³⁾	1.715⁽³⁾	0.551⁽³⁾	0⁽³⁾
Cap. Tant.	6.04⁽³⁾	Nickel⁽³⁾	207000⁽³⁾	-	186000⁽³⁾	3.5⁽³⁾	4.5⁽³⁾	0⁽³⁾	0⁽³⁾

EC prop.	d (mm)	l₁ (mm)	l₂ (mm)	t_leads (mm)	$\bar{c}$ (mm)	$\bar{d}$ (mm)	t_c (mm)	C_b (MPa)	b _b
PLCC	0⁽⁷⁾	0.735⁽⁷⁾	0.43⁽⁷⁾	0.25⁽⁷⁾	25⁽⁷⁾	25⁽⁷⁾	4.125⁽⁷⁾	1.57e19⁽⁴⁾	6.5⁽⁴⁾
LCCC 32	0⁽⁸⁾	1.27⁽⁸⁾	1.27⁽⁸⁾	cylinder	13⁽⁸⁾	13⁽⁸⁾	1.83⁽⁸⁾	4.032e9⁽⁴⁾	4⁽⁴⁾
LCCC 68	0⁽⁸⁾	1.27⁽⁸⁾	1.27⁽⁸⁾	cylinder	24⁽⁸⁾	24⁽⁸⁾	2.55⁽⁸⁾	4.032e9⁽⁴⁾	4⁽⁴⁾
LCCC 84	0⁽⁸⁾	1.27⁽⁸⁾	1.27⁽⁸⁾	cylinder	29⁽⁸⁾	29⁽⁸⁾	2.55⁽⁸⁾	4.032e9⁽⁴⁾	4⁽⁴⁾
PBGA	0⁽²⁾	0.266⁽²⁾	0.22⁽²⁾	cylinder⁽²⁾	10.5⁽²⁾	10.5⁽²⁾	1⁽²⁾	4.454e12⁽²⁾	7⁽²⁾
PBGA	0⁽²⁾	0.266⁽²⁾	0.22⁽²⁾	cylinder⁽²⁾	10.5⁽²⁾	10.5⁽²⁾	1⁽²⁾	2.534e10⁽²⁾	4.75⁽²⁾
PDIP	0⁽³⁾	1.65⁽³⁾	1.65⁽³⁾	0.3045⁽³⁾	19⁽³⁾	9⁽³⁾	3.43⁽³⁾	4.345e20⁽⁴⁾	6.5⁽⁴⁾
Cap. Tant.	0⁽³⁾	0.64⁽³⁾	0.64⁽³⁾	cylinder⁽³⁾	9⁽³⁾	19⁽³⁾	9⁽³⁾	9.051e17⁽⁴⁾	6.5⁽⁴⁾

Z_{0} = \frac{4 \bar{a} \bar{b}}{π^{2} m n \bar{c} \bar{d}} \frac{2 B_{m n} (W)}{\sqrt{ρ \bar{a} \bar{b}}} s e n (\frac{n π \bar{f}}{\bar{b}}) s e n (\frac{n π \bar{d}}{2 \bar{b}}) s e n (\frac{m π \bar{e}}{\bar{a}}) s e n (\frac{m π \bar{c}}{2 \bar{a}})

\begin{array}{l} θ_{c} = \frac{12}{\bar{d} {\bar{c}}^{3}} \frac{2 \bar{b}}{π m n} \frac{2 B_{m n} (W)}{\sqrt{ρ \bar{a} \bar{b}}} s e n (\frac{n π \bar{f}}{\bar{b}}) s e n (\frac{n π \bar{d}}{2 \bar{b}}) {\frac{2 {\bar{a}}^{2}}{π^{2} m} s e n (\frac{m π \bar{c}}{2 \bar{a}}) s e n (\frac{m π \bar{e}}{\bar{a}}) \\ - \frac{\bar{a}}{π} [\bar{c} c o s (\frac{m π \bar{e}}{\bar{a}}) c o s (\frac{m π \bar{c}}{2 \bar{a}}) - 2 \bar{e} s e n (\frac{m π \bar{e}}{\bar{a}}) s e n (\frac{m π \bar{c}}{2 \bar{a}})] - \frac{2 \bar{a} \bar{e}}{π} s e n (\frac{m π \bar{e}}{\bar{a}}) s e n (\frac{m π \bar{c}}{2 \bar{a}})} \end{array}

\begin{array}{l} ϕ_{c} = \frac{12}{\bar{c} {\bar{d}}^{3}} \frac{2 \bar{a}}{π m n} \frac{2 B_{m n} (W)}{\sqrt{ρ \bar{a} \bar{b}}} s e n (\frac{n π \bar{e}}{\bar{a}}) s e n (\frac{n π \bar{c}}{2 \bar{a}}) {\frac{2 {\bar{b}}^{2}}{π^{2} m} s e n (\frac{m π \bar{d}}{2 \bar{b}}) s e n (\frac{m π \bar{f}}{\bar{b}}) \\ - \frac{\bar{b}}{π} [\bar{d} c o s (\frac{m π \bar{f}}{\bar{b}}) c o s (\frac{m π \bar{d}}{2 \bar{b}}) - 2 \bar{f} s e n (\frac{m π \bar{f}}{\bar{b}}) s e n (\frac{m \bar{d}}{2 \bar{b}})] - \frac{2 \bar{b} \bar{f}}{π} s e n (\frac{m π \bar{f}}{\bar{b}}) s e n (\frac{m π \bar{d}}{2 \bar{b}})} \end{array}

θ_{P C B} (x, y, W) = \frac{\partial Z_{m n} (x, y, W)}{\partial x} = \frac{2 m π B_{m n} (W)}{\bar{a} \sqrt{ρ \bar{a} \bar{b}}} c o s (\frac{m π x}{\bar{a}}) s i n (\frac{n π y}{\bar{b}})

ϕ_{P C B} (x, y, W) = \frac{\partial Z_{m n} (x, y, W)}{\partial y} = \frac{2 n π B_{m n} (W)}{b \sqrt{ρ \bar{a} \bar{b}}} s i n (\frac{m π x}{\bar{a}}) c o s (\frac{n π y}{\bar{b}})

θ_{2} = P (\frac{c_{}^{2}}{2 E I_{z' 2}} + \frac{b_{2} c}{E I_{z' 2}} + \frac{b_{1} c}{E I_{z' 1}} + \frac{(a_{}^{2} + 2 a c)}{2 E I_{z' 1}} - \frac{(a + c) d}{E I_{z' 1}})

[1] * Corresponding author

Brasil

Brasil

An Enhanced Random Vibration and Fatigue Model for Printed Circuit Boards

Abstract