PCB Vibration and Thermal Cycling Fatigue: Method

Board Frequency, Miles' Response, the Displacement Limit, Lead Frame, Isolator and Plated-Through Hole

What the tool estimates

A printed circuit board in a vibrating chassis bends at its own natural frequency, and the components riding on it are loaded through their leads and solder joints by the board's curvature. The components fail by fatigue of the leads, the joints or the plated-through holes long before the board itself does. Steinberg's screening method, in use since the 1970s, reduces the problem to a few relations: the board's fundamental frequency, its response at that frequency, the relative displacement between the board and the component body, an empirical allowable displacement for 20 million cycles, and a fatigue relation of slope 6.4 for everything else. The tool evaluates those relations as mathematics, with their constants as cited facts, together with the lead frame, the isolator and the barrel of a plated-through hole. It is a first-pass design and screening tool; a finite-element modal analysis and a test are the next step for a critical assembly, and the limits at the foot of the results page say where the closed forms should not be trusted.

Board natural frequency

The board is a thin rectangular plate of flexural stiffness D = k_rib E h³/(12 (1 − ν²)), where k_rib is an entered multiplier for stiffening ribs or a bonded frame (1 for a bare board), carrying a mass per unit area ρ_A = ρ h + m_parts/(a b) in which the mounted components are smeared over the area. Its fundamental frequency is f_n = (λ/(2π a²)) √(D/ρ_A), with the frequency parameter λ from Warburton's formula for rectangular plates (NASA SP-160, eq. 4.16 and Table 4.1): λ = π² √[G_x⁴ + G_y⁴ (a/b)⁴ + 2 (a/b)² (ν H_x H_y + (1 − ν) J_x J_y)], where G, H and J depend on the support pair of each direction (simply supported, clamped or free at either edge) and on the mode index. For four simply supported edges G = H = J = 1 and the formula reduces to the exact result λ = π² (1 + (a/b)²), the familiar f_n = (π/2) √(D/ρ_A) (1/a² + 1/b²); for other combinations Warburton's values are within about 1 to 3 % of the exact plate solutions (the Plate Vibration tool computes those by the Rayleigh-Ritz method). The tool scans the mode indices that can carry the fundamental, including the rigid-body indices of a free pair, and takes the lowest non-zero λ. A card edge in a guide is closer to simply supported than clamped; a connector edge is between simply supported and free. A measured or finite-element frequency, entered in the optional field, replaces the plate value in every response calculation, which is the right thing to do for a board with a heavy transformer or a stiffener.

Random vibration response

The input is an acceleration spectral density W(f) in g²/Hz entered as breakpoints joined by straight lines on log-log axes, the convention of MIL-STD-810H and GEVS; the overall level G_rms is the square root of the exact segment areas W₁ f₁ [(f₂/f₁)^m+1 − 1]/(m + 1). The board responds like a single-degree-of-freedom system at f_n with amplification Q, and Miles' equation (NASA-HDBK-7005 eq. 5.27) gives the rms acceleration of its centre, G_rms = √((π/2) f_n Q W(f_n)): the response is set by the input level at the resonance alone, the rest of the spectrum passing through almost unamplified. Steinberg's estimate for a plug-in board with ordinary damping is Q = √f_n, used when no measured value is entered; the exact white-noise value, larger by √(1 + 1/Q²), is printed beside it. The rms relative displacement of the board centre follows from the acceleration at the resonant frequency, Z₁ = G_rms g/(2π f_n)², which in inches is 9.78 G_rms/f_n², the 9.8 G/f² of the handbooks. The design value is the 3σ displacement Z₃ = 3 Z₁, the amplitude a Gaussian process exceeds 0.27 % of the time.

The displacement limit and the life relation

Steinberg's limit is the single-amplitude 3σ displacement of the board at which a component's leads and solder joints reach about 20 million stress reversals under random vibration: Z_lim = 0.00022 B/(c h r √L) inches, with B the length of the board edge parallel to the component, h the board thickness, L the component length, r the relative position factor (1 at the centre of a board supported on all sides, 0.707 at the half point of one axis and the quarter point of the other, 0.5 at the quarter points of both) and c a constant for the component type (0.75 for axial-lead and chip components, 1.0 for a standard DIP or a through-hole pin grid array, 1.26 for side-brazed DIPs and leaded ceramic chip carriers, 1.75 for a BGA, 2.25 for a leadless ceramic chip carrier). The constants and the tables are cited from Steinberg's text as reprinted by Irvine (2013); the tool evaluates the relation in inches and converts. The ratio Z₃/Z_lim and the margin Z_lim/Z₃ − 1 are the screening result. The life at any other displacement follows the relation N = N₀ (Z_lim/Z)^b with b = 6.4, Steinberg's exponent for lead wires and solder joints, and N₀ = 20 × 10⁶ at the limit; taking every cycle at the 3σ amplitude gives the conservative single-level life and the hours to failure at f_n cycles per second.

Three-band and Rayleigh Miner sums

Steinberg's three-band method applies Miner's rule to a Gaussian response by grouping the cycles at the 1σ, 2σ and 3σ amplitudes with the fractions of time a Gaussian process spends inside ±1σ, between 1σ and 2σ, and between 2σ and 3σ: the tool uses the exact fractions 68.27 %, 27.18 % and 4.28 % (Steinberg's rounded set is 68.3, 27.1 and 4.33 %). Each band's cycles are f_n T times its fraction, each band's life comes from the relation above, and the damage is D = Σ n_i/N_i over the exposure T; the hours to failure are T/D. For comparison the tool also prints the narrow-band Rayleigh sum: when the peaks of a narrow-band Gaussian response follow the Rayleigh distribution, the Miner sum has the closed form D = f_n T (√2 Z₁)^b Γ(1 + b/2)/(N₀ Z_lim^b), as derived in the Sandia reference-model paper (SAND2014-1768C, eqs. 10 to 13), which is the same as every cycle at the equivalent amplitude Z_eq = √2 Z₁ Γ(1 + b/2)^1/b. The two sums differ by a few percent. Design practice keeps D between 0.3 and 0.7 rather than 1.

Sine dwell and half-sine shock

A sinusoidal input G_in dwelling at f_n is amplified by Q, Z = Q G_in g/(2π f_n)², and Steinberg's limit then corresponds to about 10 million cycles, so N = 10⁷ (Z_lim/Z)^6.4 and the dwell time to failure is N/f_n. A half-sine base shock of peak A and duration τ is integrated in time: the relative displacement of the damped oscillator, z̈ + 2ζω_nż + ω_n²z = −ÿ with ζ = 1/(2Q), is followed by a fourth-order Runge-Kutta scheme through the pulse and three periods after it, and the maximax relative displacement, its pseudo-acceleration ω_n² z/g (the shock response spectrum ordinate at f_n, which the Shock Response Spectrum tool draws against frequency) and the absolute acceleration are reported. The classical undamped values are recovered: an amplification of π/2 at f_n τ = 0.5 and the peak of about 1.77 near f_n τ = 0.8. Steinberg's allowable for shock is six times the random 3σ limit, the displacement a lead or joint survives for a few hundred shocks.

Component lead stress and the octave rule

Steinberg treats the lead wire as a frame between the stiff component body and the bending board. The tool solves one lead as a plane frame of two Euler-Bernoulli members, the shoulder l_s from the body to the bend and the vertical leg h_L from the bend to the board surface, by the direct stiffness method: the body end is fixed against rotation and horizontal motion (symmetry of the two lead rows) but free to move toward or away from the board, and the foot is fixed to the board. Two loads act together at the 3σ instant. The board's first-mode curvature at the component, κ = Z₃ r (π/B)², rotates the board surface under each foot by θ₀ = κ L/2 relative to the body's axis, and the body's inertia pulls each of the n leads with P = m_c (3 G_rms r) g/n. The lead stress is the largest end moment on the extreme fibre plus the axial stress; for a lead that leaves the body straight down (l_s = 0) the foot rotation alone gives M = 4 E I θ₀/h_L, the fixed-end moment of a beam with one end rotated. The stress is compared with an entered allowable, the fatigue strength of the lead alloy at the expected cycles with any concentration at the bend included. The same frame, loaded by a unit force with the foot level, gives the vertical stiffness K of one lead at the body and the component's own frequency on its leads, f_c = (1/2π) √(n K/m_c). Steinberg's octave rule asks f_c ≥ 2 f_n so that the body does not resonate with the board and multiply the lead loads; the ratio and the pass/fail are printed.

Isolator

When the chassis rides on isolators of natural frequency f_iso and damping ratio ζ, the absolute-acceleration transmissibility T = √[(1 + (2ζr)²)/((1 − r²)² + (2ζr)²)], r = f/f_iso, exceeds 1 below f_iso√2 and attenuates above it; the board then sees W(f) T(f)² and the tool uses the attenuated density at f_n in Miles' equation and in the response spectrum chart. The isolated mass responds at its own frequency with Q_iso = 1/(2ζ): its 3σ relative displacement, 3 g G_rms,iso/(2π f_iso)², is the sway space to leave around the chassis, and the static sag under 1 g is g/(2π f_iso)². The SDOF Vibration tool gives the complete response of the mount; the octave rule applies between the isolator and the board too.

Plated-through hole thermal cycling

Through its thickness a glass-epoxy laminate expands at 50 to 70 ppm/°C below its glass transition, several times the 17 ppm/°C of the copper barrel plated in the hole; every thermal cycle therefore stretches the barrel. The barrel (area π d t_cu) and the column of laminate around it (area π (D_eff² − d²)/4, D_eff an entered effective diameter near the pad size) act in series, so the copper takes the share E_b A_b/(E_cu A_cu + E_b A_b) of the free mismatch strain (α_z − α_cu) ΔT; the elastic stress E_cu ε is printed against the copper's strength, and above it the copper yields and the strain governs. The cycles to barrel cracking are estimated by Manson's Method of Universal Slopes, Δε = 3.5 (S_u/E) N^−0.12 + ε_f^0.6 N^−0.6, in the form printed in NASA/TM-2004-212985 (eq. 6), with the copper's tensile strength and true fracture ductility ε_f = ln(1/(1 − RA)); the IPC-6012 minimum of 36 ksi for plated copper is the default strength. The solder joints of surface-mount components, with their creep-fatigue model, dwell-time dependence and acceleration transforms, are the subject of the Solder Joint Fatigue tool; the barrel model here is deliberately the simple series-stiffness estimate.

Sources and copyright

Steinberg's relations are used as mathematics with their constants cited as single facts; the book itself is cited only. Miles' equation, the Gaussian fractions, Warburton's formula (NASA SP-160), the Rayleigh Miner sum (Sandia SAND2014-1768C), the Method of Universal Slopes (NASA/TM-2004-212985) and the PSD profiles of MIL-STD-810H, GEVS, NASA-HDBK-7005 and NAVMAT P-9492 are public documents. The default board stiffness, E = 3.795 Msi and ν = 0.266, is the quasi-isotropic measurement of the Sandia test board; the other defaults are illustrative and say so in their tooltips. The checks behind the tool, against closed forms and an independent numpy implementation of every step including twenty randomly generated boards, are in test-scripts/test_pcb_vibration_fatigue.php.