Subtopic: Human engineering and safety considerations in design of tractor and agricultural implements
Question Statement:
In a tractor seat system, the chassis frequency and seat suspension damping rate are 20 rad s⁻¹ and 400 N m⁻¹ s, respectively. The critical damping rate of tractor seat system is 1600 N m⁻¹ s. If the combined mass of the seat and operator is 80 kg, the transmissibility of vibration is __________ (rounded off to 2 decimal places).
Official Answer Key:
0.43 to 0.46
Step-by-Step Solution & Explanation
Damping ratio, ζ = c/c_c = 400/1600 = 0.25.
Critical damping c_c = 2√(km) ⇒ stiffness k = (c_c/2)²/m = (800)²/80 = 8000 N/m.
System's own natural frequency, ω_n = √(k/m) = √(8000/80) = √100 = 10 rad/s.
Forcing (chassis) frequency ω = 20 rad/s, so frequency ratio r = ω/ω_n = 20/10 = 2.
Transmissibility, TR = √[1+(2ζr)²] / √[(1−r²)²+(2ζr)²] = √[1+(1)²]/√[(1−4)²+(1)²] = √2/√10 = 0.4472 ≈ 0.45. Within official range 0.43–0.46.
Section 4 — Soil and Water Conservation Engineering
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