DYNAMICS / NUMERICAL DESIGN STUDY

Skid vibration isolation

Sizing two support groups for a small rotating-equipment skid, including pitch, startup, and foundation motion.

120 kN/mSelected base pair stiffness

7.577 NWorst running-band force envelope

1.265 mmWorst combined transient relative motion

This is an executed numerical reference study. The geometry, unbalance, mount properties and foundation pulse are declared inputs. No physical skid or commercial isolator was measured.

Selected design

The finite search evaluates 30 stiffness/damping combinations against 32 combinations of mass, center-of-mass position, independent front/rear stiffness variation and damping variation. The selected base stiffness is 120 kN/m per support pair, with a nominal damping parameter of 0.10. Front and rear stiffness are 126 and 114 kN/m; pair damping is 528.3 and 542.8 N s/m. Each pair represents two side-by-side mounts. These are equivalent target properties for a later component comparison, not a selected vendor part.

The choice minimizes the worst sampled running-band sum of individual support-force amplitudes among grid candidates that satisfy static compression, sensor acceleration and pole-decay limits. All 32 selected-design corners are then integrated through start/stop and foundation-pulse scenarios. The combined transient bound adds the separate peak magnitudes at each support, so it covers arbitrary relative timing without claiming a time history was run for every phase. A 1% peak allowance is included. The rejected 600 kN/m retrofit has lower sag but transmits 46.25 N in the operating band, exceeding the 18 N criterion.

Check over the stated envelope	Computed value	Limit
static compression mm	7.4406	8.5
operating reaction envelope N	7.5769	18.0
operating sensor acceleration m s2	1.4148	1.8
decay 2percent s	1.4138	2.0
transient relative motion mm	1.2648	2.0
total compression mm	8.6774	11.0
group load min N	318.37	0.0
group load max N	923.6	1100.0

Model and calculus

Vertical and pitch coordinates of a rigid skid with two asymmetric support groups and eccentric force — Coordinates are measured from the nominal center of mass. Each spring/damper symbol represents one support pair. Schematic is not to scale.

The coordinates q = [z, theta] describe downward translation and small pitch about the mass center. At support i, relative compression is delta_i = z + x_i theta - y, where y is common vertical foundation motion. Linear springs and viscous dampers act about the gravity-loaded equilibrium.

B = [[1, x_front], [1, x_rear]] M = diag(m, I); K = Bᵀ diag(k_i) B; C = Bᵀ diag(c_i) B M q̈ + C q̇ + K q = [1, x_force]ᵀ F(t) + Bᵀ(k y + c ẏ) R_i = k_i delta_i + c_i deltȧ_i Static: K q_static = [m g, 0]ᵀ c_i = 2 zeta_design sqrt(k_i m_i,static), where m_i,static = R_i,static / g Sensor acceleration: a_sensor = z̈ + x_sensor thetä

The matrices come from differentiating kinetic energy T = (m ż² + I thetȧ²)/2 and elastic energy V = sum(k_i delta_i²)/2. Virtual work at an offset force point supplies both force and moment. The undamped modes solve K phi = omega_n² M phi. Damping is not assumed proportional, so transient integration retains both coupled coordinates.

Frequency response and its stiffness derivative

D(omega) = K - omega² M + i omega C D Q = [1, x_force]ᵀ F_hat + Bᵀ(k + i omega c) Y_hat ∂Q/∂k_j = -D⁻¹ (b_j b_jᵀ) Q [fixed damping, fixed force, Y_hat = 0] ∂|Q_l|/∂k_j = Re(conj(Q_l) ∂Q_l/∂k_j) / |Q_l| [Q_l ≠ 0]

The derivative is obtained by differentiating DQ = F with respect to a spring parameter. It is checked against centered finite differences at five frequencies for each support, including frequencies near the modes. The search itself remains a transparent finite grid; it does not claim a continuous optimum.

Integrated startup and energy balance

During spin-up: s = t/T, f = f_max (3s² - 2s³) phase = 2 pi f_max T (s³ - s⁴/2); omega = phasė; alpha = omegȧ F(t) = U [omega² sin(phase) - alpha cos(phase)] E = (q̇ᵀ M q̇ + sum(k_i delta_i²))/2 Ė = F (ż + x_force thetȧ) - ẏ sum(R_i) - sum(c_i deltȧ_i²) E(t) - E(0) = integral(input power dt) - integral(damper loss dt)

The angular-acceleration term is retained during the smooth ramps. Work and viscous loss are integrated as additional ODE states and compared with independently calculated kinetic and relative spring energies. A separate sampled trapezoidal integration checks the dissipation integral. The foundation pulse is y = A sin²(pi t/T_b) for 0 < t < T_b; displacement and velocity return to zero at both ends.

Reference input	Value
Skid mass / pitch inertia	120 kg / 18 kg m²
Support coordinates	[-0.42, 0.36] m
Force / sensor location	0.22 m / 0.25 m
Unbalance / running band	0.002 kg m / [20.0, 35.0] Hz
Mass factors / CG shifts	[0.9, 1.1] / [-0.04, 0.04] m
Independent stiffness factors / damping factors	[0.9, 1.1] / [0.8, 1.2]
Startup and stop	2 s up and down, 1 s hold at 30 Hz, 2 s settle
Foundation pulse	1 mm, 0.15 s

Nominal undamped modes are 6.697 and 7.608 Hz. The pole-based 2% decay estimate is a modal envelope diagnostic, not a guarantee for arbitrary initial conditions. The actual start/stop traces are checked separately.

Computed results

Reaction-force and acceleration frequency responses for selected and stiff support systems — The shaded band is the declared running range. Frequency response outside that band explains startup resonance; operating limits apply within the band.

Thirty grid candidates and their worst force transmission across uncertainty corners — Cells marked reject fail at least one static, operating or decay criterion. The lowest nominal force candidate is not necessarily feasible at the uncertainty corners.

Numerical force, support travel and transmitted force during start, hold and stop — Actual computed transient for the selected nominal design. The CSV retains every 2 ms output sample.

Foundation pulse, skid response and sensor acceleration — The pulse response is evaluated separately and combined with startup using a conservative peak bound.

Verification

Analytical limits, matrix force/moment equilibrium, a complex steady-state solution, parameter derivatives, energy integrals, output refinement, invalid inputs and the selected uncertainty envelope are checked. A candidate is rejected when a declared operating limit is exceeded; a successful verification does not make the rejected candidate acceptable.

Executed check	Observed	Acceptance
Complex FRF derivative for support group 1	1.79e-09	relative norm error < 1e-6
Complex FRF derivative for support group 2	1.405e-09	relative norm error < 1e-6
Time-domain solution agrees with complex frequency response	1.003e-12	relative displacement error < 1e-7
Symmetric centered forcing agrees with the scalar oscillator	1.934e-16	relative response error < 1e-12
Initial energy becomes viscous heat	6.544e-15	relative energy balance < 1e-7
Independent sampled integral agrees with ODE dissipation	1.443e-10	relative integral error < 1e-4
Foundation input work balances dynamic spring energy and heat	2.096e-10	relative energy residual < 1e-7
Peak travel is stable when sampled four times more finely	5.381e-06	relative peak change < 0.005
Limiting corner 27 peak/travel/load refinement	8.944e-05	maximum relative metric change < 0.005
Limiting corner 25 peak/travel/load refinement	0.0005494	maximum relative metric change < 0.005
Limiting corner 13 peak/travel/load refinement	0.0003657	maximum relative metric change < 0.005
Running-band peak agrees after local scalar maximization	0	refined peak minus grid peak < 0.01 N

Verification record · Regression test log · Additional acceptance report

Files and reproduction

Inputs used for this run · Frequency sweep CSV · Candidate grid CSV · Corner outcomes CSV · Startup trace CSV · Foundation trace CSV · Summary JSON

python -m pip install -r requirements.txt python run.py python complete_validation.py python -m unittest -v test_project.py

To evaluate another declared scenario: python run.py --inputs inputs.json --output alternate-run. That command writes its report, copied inputs and output files under the chosen directory. The library functions use SI units. The display converts displacement to millimeters and pitch to milliradians.

Model limits

Pitch inertia scales with the mass factor. It is defined about each shifted center of mass, with the reference radius of gyration retained.
The body is rigid and has vertical/pitch motion only. Roll, yaw, lateral modes, flexible piping, frame bending and soil/floor flexibility are excluded.
Spring and damping coefficients are constant equivalent values. Elastomer frequency dependence, temperature, creep, hysteresis and contact lift-off are not simulated. A negative minimum reaction is a rejection, not a solved separation event.
The two mount pairs share load equally left-to-right. The stated group load limits apply to each pair; individual mount ratings need a separate lateral-load split and vendor data.
The input corners are a finite design sweep, not a probabilistic confidence level or proof over every interior parameter combination. Frequency peaks use a dense grid and local scalar refinement for the nominal selection.
The 1 mm pulse, rotating unbalance and equivalent mount properties are synthetic assumptions. No accelerometer record, fabricated hardware, fatigue life, seismic qualification or machinery safety certification is claimed.

References

MIT OpenCourseWare, Dynamics and Control I, Lecture 24: forced response of multi-degree-of-freedom systems. Matrix frequency-response formulation.
SciPy solve_ivp reference. Initial-value integration, tolerances and DOP853 implementation.