Problem 1
Real systems have nonlinearities, fatigue, saturation, changing damping, and safety constraints absent from the ideal linear ODE.
Faculty use
Frequency Response, Energy and Engineering Limits
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Free-decay data to estimate damping/natural frequency and forced-response amplitude/phase across frequencies.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind using gain, phase, bandwidth, and energy balance to judge design behavior. Use the formulas gain=output amplitude/input amplitude; average power depends on phase.
Problem 4
A strong error analysis identifies the first step that violates classification, algebra, theorem hypotheses, units, initial/boundary data, or numerical assumptions; then repairs that step and re-verifies the final result.
Problem 5
Application responses vary. Require defined variables, a compatible differential-equation model connected to machines, buildings, vehicles, sensors, acoustic systems, and electrical circuits, a complete computation, interpretation, and limitations.
Problem 6
Accept two independent checks appropriate to the task: differentiation/substitution, initial or boundary conditions, dimensional analysis, phase-line or stability evidence, residuals, step refinement, conservation/energy checks, data comparison, or a second method.
Scoring guidance
Six problems × 10 points: 5 mathematical accuracy, 2 structure/justification, 2 verification/interpretation, 1 notation and communication.