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ACHEV eq. questions
| Question | Answer |
|---|---|
| Step-by-step Answer Derive damper force law | Start from proportionality between force and relative velocity → F ∝ Δż → introduce coefficient c → F = -c·Δż |
| (negative sign = opposition to motion) Derive spring force law | Assume linear elastic behavior → force proportional to displacement → F ∝ Δz → introduce stiffness k → F = -k·Δz |
| Derive quarter-car equations (sprung mass) | Apply Newton 2nd law: ΣF = Mz̈ → forces: damper -c(ż - żt), spring -k(z - zt - Δs), gravity -Mg → sum → Mz̈ = -c(ż - żt) - k(z - zt - Δs) |
| Mg Derive quarter-car equations (unsprung mass) | Apply Newton: m z̈ t → forces: +c(ż - żt), +k(z - zt - Δs), tire -kt(zt - zr - Δt), gravity -mg → m z̈ t = c(ż - żt) + k(z - zt - Δs) - kt(zt - zr Δt) - mg |
| Explain sign convention in equations | Forces between masses appear with opposite signs due to action-reaction principle |
| Derive system order | Two second-order equations → each gives position and velocity → total 4 states → 4th order system |
| Define state vector | Choose positions and velocities → x = [z, z ̇, zt, żt] |
| Explain SIMO structure | One input (zr), two outputs (z, zt) |
| Derive actuator dynamics equation | Assume first-order response → rate proportional to difference → ċ = -βc + βcin |
| Derive actuator transfer function | Take Laplace: sC(s) = -βC(s) + βCin(s) → (s+β)C(s) = βCin(s) → C(s) = β/(s+β) Cin(s) |
| Interpret actuator transfer function | First-order low-pass filter with gain 1 and bandwidth β |
| Derive equilibrium equations | Set derivatives = 0 and zr=0 → obtain algebraic equations from dynamic ones |
| Solve equilibrium system | Write in matrix form Ax = B → solve x = A⁻¹B |
| Explain why damping disappears at equilibrium | Velocity = 0 → damper force = 0 |
| Derive linearized model | Define perturbations: z = z̄ + δz, etc → substitute → cancel equilibrium terms → obtain linear equations |
| Show linearized equations | Mδz̈ = -c(δż - δżt) - k(δz δzt); mδz̈ t = c(δż - δżt) + k(δz - δzt) - kt(δzt - δzr) |
| Explain why system becomes LTI | Parameters constant and no nonlinear terms (c fixed) |
| Show nonlinearity with controllable damping | Term c(t)(ż - żt) → product of input and state → nonlinear system |
| Derive Fz expression | Sum contributions → Fz = (M+m)g + aerodynamic load + dynamic load |
| Explain loss of contact condition mathematically | If Fz → 0 then tire force = 0 → no interaction with road |
| Derive coil spring stiffness formula (conceptual) | Based on torsion of wire → stiffness depends on material (G) and geometry (d, D, n) |
| Input zr, output z → high-frequency attenuated due to inertia and damping → behaves like 2 low-pass filter | |
| Derive gas spring transfer function | From internal dynamics: c·dx2/dt = kg(x1 - x2) → Laplace → x2 = kg/(kg + sc) x1 |
| Interpret gas spring TF | First-order low-pass filter with bandwidth kg/c EXAM TIP |
| First-order low-pass filter with bandwidth kg/c EXAM TIP | |
| Practice writing full derivations without looking EXAM TIP | Input zr, output z → high-frequency attenuated due to inertia and damping → behaves like |
| Always justify each term physically (not only mathematically) EXAM TIP | |
| Be ready to derive both equations quickly and cleanly | From internal dynamics: c·dx2/dt = kg(x1 - x2) → Laplace |
| → x2 = kg/(kg + sc) x1 Interpret gas spring TF | First-order low-pass filter with bandwidth kg/c |
| EXAM TIP | Practice writing full derivations without looking EXAM TIP |
| (not only mathematically) EXAM TIP | Be ready to derive both equations quickly and cleanly |
Created by:
Filotì