Heave and pitch across frequency.

Choose the engineering question

Transient Study gives response versus time for a road event. Frequency Study gives steady-state harmonic response versus excitation frequency. They use the same passive linear 4-DOF model and answer complementary questions.

1. Open Studio → Study → Half Car · 4 DOF → Frequency.
2. Set physical parameters, explicit rear delay, frequency limits and point count.
3. Choose Run Frequency Study. Inspect acceleration, pitch, suspension travel, tire deflection, heave and phase.
4. Export completed CSV, each plot as PNG, or an Engineering Report. Export project saves the desired definition; import requires rerunning.

Trusted equations and coordinates

Upward heave z, nose-up pitch θ, front attachment z + aθ, rear attachment z − bθ. Coordinates are q = [z, θ, zf, zr]. Mass is diagonal: sprung mass, pitch inertia, front and rear unsprung mass.

The stiffness and damping matrices are extracted by eight basis evaluations of the unchanged transient derivative. A row-equilibrated complex solve uses partial pivoting, checks the inverse-norm condition estimate (limit 10¹²), and verifies the backward residual (relative limit 10⁻¹⁰). Singular or unresolved systems fail visibly and retain previous completed evidence.

Road delay and front reference

Both axles receive equal-amplitude harmonic road displacement. The rear arrives τ seconds later, consistent with transient input evaluated at t − τ. All output phases reference the front road. Zero delay means same-phase roads.

The unit front road is a transfer normalization, not a physical one-metre bump. Explicit delay is authoritative. The optional speed helper applies wheelbase / speed only when selected; later geometry edits do not silently change delay.

Units and retained evidence

Completed evidence retains real and imaginary parts of all nine responses. Magnitude is hypot(real, imaginary); phase is atan2(imaginary, real), wrapped to [−180°, +180°]. Exactly zero response uses zero phase by convention. The phase plot connects stored samples; jumps at the wrapping boundary do not indicate a discontinuous complex response.

Grid, peaks and modes

Use 16–2,000 logarithmically spaced samples between 0.01 and 100 Hz; maximum must exceed minimum. Defaults are 240 points, 0.2–30 Hz. Hz is converted to rad/s internally.

Each reported peak is the largest strict interior sampled local maximum. Equal-height ties keep the first sample; plateaus and endpoints are excluded. An absent peak means none was found on this grid. Natural frequencies come from the undamped M/K eigenproblem; damping, excitation phase and sampling can shift or hide forced-response peaks. They are not interchangeable labels.

Validation boundary

All nine complex outputs are checked against independently assembled energy/state-space matrices solved as a 16×16 real system at ten frequencies from 0.01 to 100 Hz. A second check drives the unchanged transient derivative and RK4 with sinusoidal roads at 0.4, 1.2, 1.8 and 11 Hz.

The transient reference discards at least 30 seconds and 30 cycles, then projects eight complete cycles onto sine and cosine. Steps are no longer than 0.25 ms and include at least 2,000 samples per cycle. Tests require relative magnitude error below 0.02% and wrapped phase error below 0.02°. Symmetry, pitch excitation, analytic symmetric modes, parameter trends and low/high-frequency limits are also checked.

These are numerical verification checks. No matched published numerical frequency fixture or measured-vehicle correlation is claimed. The existing 5,003-point Quarter-Car literature check remains a separate evidence boundary.

Freshness and reporting

Editing a definition retains the last completed response and marks it stale. Restoring the exact numerical definition returns it to Current without rerunning. CSV, PNG and PDF use completed evidence, including when stale. Projects contain definitions only. Successful completion marks Half-Car Frequency explored in Journey; errors do not.

Assumptions and scope

Small-angle pitch, linear springs and viscous suspension dampers, continuous tire contact, no tire damping, roll, lateral dynamics or active control. Steady-state harmonic analysis excludes startup transients. Frequency Compare, parameter exploration, optimization, measurement import and frequency video are deferred.

See Half-Car Transient Study, Quarter-Car Frequency, and Engineering Reports.