Engineering extended problems (resonance framework)

Problem 4 – Resonant vibration amplitude#

A machine component vibrates with amplitude

$$ A = \frac{T_f^2}{D_3 + τ_r}. $$

1. If $$T_f$$ increases by 20%, how does the numerator change?
2. If $$τ_r$$ increases by 10%, how does the denominator change?
3. What is the qualitative net effect on $$A$$?

Problem 5 – Stress concentration under triadic loading#

Stress concentration is modeled as

$$ σ = D_9 τ_r - X. $$

1. If $$τ_r$$ doubles, how does the first term change?
2. If $$X$$ increases due to higher frequency elevation, what is the qualitative effect on $$σ$$?

Problem 6 – Resonant damping coefficient#

A damping coefficient is given by

$$ c = \frac{ΛΘ}{1 + e^{-D_6 τ_r}}. $$

1. Sketch the qualitative shape of $$c(τ_r)$$.
2. If $$τ_r$$ increases, does damping become stronger or weaker?

Problem 7 – Material fatigue under resonance#

Fatigue accumulation is modeled as

$$ F = X \ln(1 + D_3 τ_r). $$

If $$τ_r$$ increases by 30%, how does the fatigue accumulation change qualitatively?

Problem 8 – Resonant heat transfer coefficient#

A heat transfer coefficient is modeled as

$$ h = \frac{T_f + D_6}{τ_r}. $$

If $$T_f$$ increases by 10% and $$τ_r$$ increases by 20%, what is the qualitative net effect on $$h$$?