SDOF Vibration: Method

Theory, Conventions, Newmark Integration and the Shock Response Spectrum

The single-degree-of-freedom model

A mass m on a linear spring k with a viscous damper c, one coordinate x measured from the static equilibrium position (gravity is already balanced by the static spring deflection δ_st = W/k, so it drops out of the equation): m ẍ + c ẋ + k x = F(t). Many machines on mounts, instruments on isolators, a floor slab, a tower, a vehicle on its suspension and the first mode of almost any structure are analysed this way. The mass may be entered as a weight (m = W/g, g = 9.80665 m/s² = 386.09 in/s²); the stiffness may be entered as a measured natural frequency (k = m (2π f_n)²); the damping may be entered as the coefficient c, the ratio ζ, a logarithmic decrement read from a free-decay record or the half-power points of a measured response curve.

Natural circular frequency ω_n = √(k/m), natural frequency f_n = ω_n/2π, period T_n = 1/f_n; critical damping c_c = 2√(km) = 2mω_n, damping ratio ζ = c/c_c; for ζ < 1 the damped frequency ω_d = ω_n√(1 − ζ²) and the logarithmic decrement δ = ln(x_i/x_i+1) = 2πζ/√(1 − ζ²) (Rao eqs. 2.84 to 2.86). From a record in which the amplitude falls from x₁ to x_n+1 in n cycles, δ = (1/n) ln(x₁/x_n+1) and ζ = δ/√(4π² + δ²). From a frequency-response curve the frequencies f₁, f₂ at which the amplitude is 1/√2 of the peak give ζ ≈ (f₂ − f₁)/(2f_n) (Rao eq. 3.45, good for ζ below about 0.1; the page also prints the exact relation ζ√(1 − ζ²) = (f₂² − f₁²)/(4f_n²), from M(r)² = M_peak²/2).

Units

Mass in lbm or kg (or a weight in lb or N), stiffness in lb/in or N/m, lengths in inches or millimetres, forces in lb or N: these are converted by the unit switch. The damping coefficient c (lb-s/in or N-s/m), the initial velocity (in/s or m/s), frequencies (Hz, rad/s, rpm), accelerations (g) and the pulse duration (ms) are plain numbers whose unit is named in the label; they are not converted when the system is switched, so re-enter them. Internally everything is in SI base units.

Free vibration

Underdamped (ζ < 1): x(t) = e^−ζω_nt[x₀ cos ω_dt + ((v₀ + ζω_nx₀)/ω_d) sin ω_dt] = X₀ e^−ζω_nt cos(ω_dt − φ₀) (Rao eq. 2.72); the envelope ±X₀e^−ζω_nt is drawn with the response and successive positive peaks are listed with ln(x₁/x₂), which equals δ. Critically damped (ζ = 1): x(t) = [x₀ + (v₀ + ω_nx₀)t] e^−ω_nt (Rao eq. 2.80), the fastest non-oscillatory return. Overdamped (ζ > 1): x(t) = C₁e^s₁t + C₂e^s₂t, s_1,2 = (−ζ ± √(ζ² − 1))ω_n (Rao eq. 2.81). The largest displacement of the record and its time are located by a scan of the closed form refined by golden-section search. Rao's Example 2.11 (a motorcycle shock absorber: amplitude reduced to a quarter in a half cycle, damped period 2 s, ζ = 0.4037) and Example 2.12 (a critically damped gun recoil, m = 500 kg, k = 10,000 N/m: a 0.4 m recoil needs v₀ = 4.8626 m/s and the barrel is back to 0.1 m at 0.826 s) are reproduced by the test script.

Harmonic force

For F(t) = F₀ sin ωt the steady state is x_p = X sin(ωt − φ) with X = (F₀/k)·M, the magnification factor M = 1/√[(1 − r²)² + (2ζr)²] and tan φ = 2ζr/(1 − r²), r = ω/ω_n (Rao eqs. 3.30, 3.31). Below resonance the mass moves in phase with the force (φ → 0), at r = 1 the lag is 90°, far above resonance 180° and X → F₀/(mω²). The undamped peak is unbounded; with damping the peak is 1/(2ζ√(1 − ζ²)) at r = √(1 − 2ζ²). The force transmitted to the support through the spring and damper is F_T = F₀·T_r, T_r = √[(1 + (2ζr)²)/((1 − r²)² + (2ζr)²)]: every curve passes through T_r = 1 at r = √2, so a mount isolates only when r > √2, and there more damping transmits more force. The isolation efficiency is 1 − T_r. The default example is Rao's Example 3.1: a 150 lb pump on a 1,200 lb/in plate under 50 cos 62.832t lb; r = 1.13, M = 3.60 and X = 0.150 in with the response 180° out of phase (Rao prints X = −0.1504 in with g = 386.4 in/s²; the page uses 386.09 and gives 0.1498 in).

Rotating unbalance

An unbalanced mass m_u at eccentricity e turning at ω inside a machine of total mass M (the system mass) applies m_ueω² sin ωt. The amplitude is X = (m_ue/M)·r²M, so MX/(m_ue) = r²/√[(1 − r²)² + (2ζr)²] (Rao eq. 3.80): zero at rest, 1/(2ζ√(1 − ζ²)) at r = 1/√(1 − 2ζ²), and 1 at high speed, where the machine's centre of mass stands still and the housing orbits with amplitude m_ue/M. The transmitted force is m_ueω²·T_r, which grows with ω² even when T_r falls, the reason stiff mounts and high speeds are a poor combination.

Harmonic base motion

For a support displacement y = Y sin ωt the absolute motion of the mass has amplitude X = Y·T_d with the displacement transmissibility T_d = √[(1 + (2ζr)²)/((1 − r²)² + (2ζr)²)] (the same function as the force transmissibility) and phase tan φ = 2ζr³/(1 − r² + (2ζr)²) (Rao eqs. 3.68, 3.69). The relative motion z = x − y, the spring stroke, has amplitude Z = Y·r²/√[(1 − r²)² + (2ζr)²] (Rao eq. 3.75) and the force on the base is F_T = kY·r²T_d (Rao eq. 3.72). Rao's Example 3.5, a 1,200 kg vehicle (k = 400 kN/m, ζ = 0.5) at 20 km/h on a road of 6 m wavelength and 50 mm amplitude, gives T_d = 1.101 and X = 55.0 mm; the test script reproduces it.

Shock pulses and Newmark-β integration

A shock is a short base acceleration (or force) pulse. The standard laboratory shapes are the half-sine A sin(πt/τ), the versed sine (haversine) ½A(1 − cos 2πt/τ), the rectangular pulse, the terminal-peak sawtooth At/τ (which rises linearly and drops abruptly) and the initial-peak sawtooth (abrupt rise, linear fall), each of amplitude A and duration τ; MIL-STD-810 uses the terminal-peak sawtooth and the trapezoid for its functional and crash-hazard shocks, and the classical 50 g, 11 ms half-sine is the second example on the page. Any measured history can be pasted as t–value pairs instead and is interpolated linearly. For a base pulse the relative coordinate z = x − y obeys m z̈ + c ż + k z = −m ÿ, and the absolute acceleration of the mass is ẍ = z̈ + ÿ.

The equation of motion is integrated by Newmark's method with average acceleration (β = ¼, γ = ½; Newmark 1959, Chopra Table 5.4.2): the acceleration is taken constant over each step at the mean of its end values, which is unconditionally stable for linear systems, introduces no numerical damping and elongates the period by about (ω_nΔt)²/12. The step is the smaller of T_n and τ divided by 200 (a period error below 10⁻⁴), the record runs over the pulse plus three natural periods so the residual free vibration shows its first peaks, and the maximum response and its time are read from the record. The maximum during the pulse is the primary response, the maximum of the free vibration after it the residual response; for short pulses (fτ small) the residual governs and the response depends only on the impulse (the area of the pulse), for long pulses (fτ large) the response follows the pulse quasi-statically and tends to the static value. The dynamic amplification is ω_n²z_max/A (or ẍ_max/A) for a base pulse and x_max/(F_peak/k) for a force pulse. For the half-sine the closed form during the pulse, x/x_st = (sin ωt − r sin ω_nt)/(1 − r²) with ω = π/τ, is used as a check of the integrator in the test script (agreement to 10⁻⁶ of the peak).

Shock response spectrum

The shock response spectrum (SRS) is the maximax response of a family of single-degree-of-freedom oscillators, all with the same damping ratio, to the same base input, plotted against their natural frequency. It is the standard way of specifying a shock environment for equipment and of comparing shocks of different shape: two pulses with the same SRS damage a structure of the given natural frequency alike. The tool integrates each oscillator (unit mass, k = ω_n², 100 Newmark steps per the smaller of its period and τ, over the pulse and two periods beyond it) and reports the maximum absolute acceleration divided by the input peak, the pseudo-acceleration ω_n²|z|_max/A (identical for ζ = 0, the equivalent static acceleration that produces the spring stroke), and the primary and residual parts. The frequencies run from fτ = 0.1 to 10, where the curve is the classical dimensionless spectrum of the shape (Harris, Shock and Vibration Handbook, Chapter 23; Rao Section 4.6): the half-sine peaks at about 1.77 near fτ = 0.8, the rectangular pulse reaches 2.0 for fτ ≥ 0.5 and stays there, the versed sine peaks at 1.70 near fτ = 1, the terminal-peak sawtooth at 1.26 near fτ = 0.65 and then settles to 1 (the reason it is preferred for test specifications: a flat spectrum), and the initial-peak sawtooth, which starts with a step, tends to 2 at high frequency like the rectangular pulse; every pulse without a step tends to 1 (the quasi-static value) at high frequency and to zero at low frequency, where the response is set by the impulse. Enter a frequency range instead to read the spectrum in Hz, and ζ = 0.05 (Q = 10) for the usual specification damping. The spectrum value at the system's own f_n equals the acceleration amplification printed for the transient response.

Reading the results

The system properties come first. The free-vibration section shows the decay from the entered initial conditions and checks the logarithmic decrement against two successive peaks. The harmonic section prints the operating point (r, M or T_d, X, φ, transmitted force, isolation efficiency) and draws M(r), φ(r) and the transmissibility for ζ = 0.05 to 1 together with the system's own damping (red); a point on a steep part of the magnification curve is sensitive to small changes of speed or stiffness. The shock section prints the maximum response and its time, the primary and residual maxima, the dynamic amplification and the spectrum; the maximum spring force (k·z_max) or the maximum acceleration is the load to carry into a stress check.

Limits

Linear spring and viscous damper only (no friction, clearance or hardening); one degree of freedom (a second mass, rotational coupling or a flexible foundation needs a multi-degree-of-freedom model); the steady-state formulas assume the transient has died away; the Newmark record is a numerical solution with the step error stated above; random vibration, fatigue and the strength of the parts are outside the tool.