Frequency response — VehicleLab Documentation

Modelvehiclelab.quarter-car.linear-passive v0.2.0

Frequency solvervehiclelab.linear-frequency-response v0.1.0

Numeric pathDirect complex 2 × 2 · Float64

Steady-state harmonic road input

Frequency response asks how the settled sinusoidal response changes as excitation frequency changes. It does not show startup, a finite bump, or the decay of initial conditions. Road displacement is normalized with Zr = 1; the core response therefore does not assume an arbitrary road amplitude.

z_r(t)=\operatorname{Re}\!\left\{Z_r e^{j\omega t}\right\},\qquad \omega=2\pi f

z_s(t)=\operatorname{Re}\!\left\{Z_s e^{j\omega t}\right\},\qquad z_u(t)=\operatorname{Re}\!\left\{Z_u e^{j\omega t}\right\}

User-facing frequency f is in hertz (cycles per second). The equations use angular frequency ω in radians per second, with ω = 2πf.

Derivation from the unchanged quarter-car equations

Substitute the harmonic forms into the documented sprung and unsprung force balances, using one time derivative as multiplication by jω and two as −ω². Collecting the complex displacement amplitudes gives:

\begin{bmatrix}k_s+j\omega c_s-m_s\omega^2&-(k_s+j\omega c_s)\\-(k_s+j\omega c_s)&k_s+k_t+j\omega c_s-m_u\omega^2\end{bmatrix}\begin{bmatrix}Z_s\\Z_u\end{bmatrix}=\begin{bmatrix}0\\k_tZ_r\end{bmatrix}

At each requested frequency VehicleLab solves this 2 × 2 system for Zs and Zu with normalized Zr. A finite determinant is required. There is no FFT, time sampling, interpolation between transient simulations, or RK4 in the production frequency solver.

The four displayed magnitudes

\left|\frac{Z_s}{Z_r}\right|,\quad \left|\frac{-\omega^2Z_s}{Z_r}\right|,\quad \left|\frac{Z_s-Z_u}{Z_r}\right|,\quad \left|\frac{Z_u-Z_r}{Z_r}\right|

Sprung displacement transmissibility: Body displacement divided by road displacement; dimensionless.
Sprung acceleration response: Magnitude of −ω²Zs/Zr in m/s² per m.
Suspension travel transmissibility: Relative spring/damper displacement (Zs − Zu)/Zr; dimensionless.
Dynamic tire deflection transmissibility: Wheel-side displacement relative to road (Zu − Zr)/Zr; dimensionless.

Phase is retained in the completed record and CSV in degrees, while the primary Studio remains focused on suspension-response magnitudes. Connected plot segments join exact direct evaluations on the requested dense grid; no smoothing is applied.

Grid, peaks, and interpretation

The default grid contains 240 logarithmically spaced points from 0.2 to 30 Hz. Accepted definitions are bounded to 0.05–200 Hz and 16–2,000 points; the minimum must be lower than the maximum. Endpoints and point count are exact, and invalid requests are rejected rather than clamped.

VehicleLab reports the dominant interior local maximum found independently for each response quantity. A missing or ambiguous maximum is left unreported. These samples can help locate response regions often interpreted as body-dominated or wheel-side behavior, but the algorithm does not hard-code such bands or promote those informal names to mathematical modal identities.

A lower response in one plot does not automatically make a suspension superior. Body response, acceleration, available suspension working space, and tire response can move differently; intended use and constraints remain outside this normalized analysis.

Independent verification

Canonical V&V checks deterministic structure and analytical low/high-frequency limits. A validation-only path builds the documented four-state matrices and solves the separate 4 × 4 state-space expression below using pivoted complex elimination. It does not call the production 2 × 2 solver.

\mathbf{H}(j\omega)=\mathbf{C}(j\omega\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}+\mathbf{D}

A second cross-domain check drives the unchanged production quarter-car derivative and RK4 step with a validation-only sinusoidal road. It discards at least 24 cycles and eight seconds, then estimates steady-state amplitudes from eight complete cycles using orthogonal sine/cosine projection. The selected frequencies come from the actual computed response, not fixed resonance bands.

Review computed frequency-response evidence →

Evidence boundary

Numerically verified is not physically correlated.

The evidence supports the documented linear equations and their implementation. It does not represent nonlinear suspension behavior, contact loss, transient startup, a measured vehicle, certification, optimization, or a design recommendation. Physical correlation remains not performed.