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Plate supporting a pump: steady-state amplitude of a single-degree-of-freedom system under a harmonic force (Rao Example 3.1)

Tool	SDOF Vibration
Source	Rao, S. S. (2011). Mechanical Vibrations, 5th ed. Prentice Hall, Upper Saddle River, NJ. Example 3.1 (plate supporting a pump), eqs. 3.30 and 3.31; Harris, C. M., and Piersol, A. G. (eds.) (2002). Harris' Shock and Vibration Handbook, 5th ed. McGraw-Hill, New York. Chapter 23 (shock response spectra of the half-sine pulse).
Level	Undergraduate

Problem

A reciprocating pump weighing 150 lb is mounted at the middle of a steel plate clamped along two edges whose stiffness at the pump is k = 192 E I / l³ = 1,200 lb/in. During operation the plate is subjected to the harmonic force F(t) = 50 cos 62.832t lb. Find the natural frequency, the frequency ratio and the amplitude of vibration of the plate; then find the response of the same system to the classical 50 g, 11 ms half-sine shock and the shock response spectrum of that pulse.

Inputs to Enter

Weight W = 150 lb (mass given as weight), stiffness k = 1,200 lb/in, damping ratio ζ = 0 (undamped)
Harmonic force F_0 = 50 lb at ω = 62.832 rad/s (10 Hz, 600 rpm); initial displacement 1 in for the free-vibration figure
Shock: half-sine base pulse, A = 50 g, τ = 11 ms, SRS damping 0, SRS over f τ = 0.1 to 10

Open the SDOF Vibration tool (loaded with this example)

Answers

ω_n = √(k g / W) = √(1,200 · 386.09 / 150) = 55.58 rad/s (Rao 55.598 with g = 386.4), f_n = 8.845 Hz, c_c = 2 √(k m) = 43.18 lb-s/in
r = ω/ω_n = 1.1306, M = 1/|1 − r²| = 3.595, X = (F_0/k) M = 0.1498 in with φ = 180° (Rao prints X = −0.1504 in, the sign showing the response out of phase; the 0.4 % difference is Rao's rounded g = 386.4 in/s²)
Force transmissibility T_r = 3.595, transmitted force 179.8 lb (r < √2: no isolation)
Half-sine 50 g, 11 ms shock (f_n τ = 0.0973): maximum relative displacement 2.411 in at 0.260 s (residual), acceleration amplification 0.386, maximum absolute acceleration 19.3 g; SRS peak 1.768 at f τ = 0.79 (Harris 1.76 near 0.8)

Notes

The tool's default inputs are this example; press Submit. The shock response spectrum values reproduce the classical half-sine curve (π/2 at f τ = 0.5, √3 at 1.0) within 0.3 % by Newmark-β integration (test_sdof_vibration.php, 160 checks, with a numpy Duhamel check in References/SDOF-Vibration/).