A test rig shakes a car's suspension assembly with a hydraulic actuator while an…
PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Electronics, Instrumentation & Communication · Scientific programming for physicists
Scientific Computing for Physicists, Ch Spectra with error bars: the fast transform, convolution and spectral estimation
Problem
A test rig shakes a car's suspension assembly with a hydraulic actuator while an accelerometer bolted to the wheel hub records the vertical acceleration at fs=2000Hz for a total of 12s.
((a)) Over the whole recording the accelerometer's output drifts by about 0.05g as its electronics warm up, well modelled as linear in time. Explain, using the low-frequency power produced by an unremoved linear ramp, why this drift must be removed by subtracting a least-squares straight-line fit before any part of the record is used to estimate a power spectral density, and state what spurious feature would appear if it were left in.
((b)) For the first 8s the actuator drives a linear frequency sweep from 2Hz to 40Hz. Choose a spectrogram segment length M (a power of two, in samples) that balances time and frequency resolution for this sweep, using the balanced-segment rule τ≈1/ΔF/ΔT for a sweep of ΔF across a time ΔT. State the resulting frequency resolution Δf=fs/M and time resolution τ=M/fs.
((c)) For the following 4s the actuator drives the wheel with band-limited random excitation, a stationary random record, which is Welch-averaged with K=8 non-overlapping segments to estimate the hub's acceleration response PSD. State the segment length M in samples, the resulting frequency resolution, and the relative uncertainty of the PSD estimate at each frequency. The accelerometer is ideal (no aliasing, no instrument noise beyond the stated drift), and the eight random-excitation segments used for Welch averaging are statistically independent.