A motor of weight m is supported by a mounting that has spring constant k. If the unbalance of the motor is equivalent to a force F=F0cos(ωt) and damping can be neglected, calculate the amplitude of the response, X0, when m = 150 kg, k = 5 MN/m, F0 = 450 kN and ω=120 rad/s.

Question

A motor of weight m is supported by a mounting that has spring constant k. If the unbalance of the motor is equivalent to a force F=F0cos(ωt) and damping can be neglected, calculate the amplitude of the response, X0, when m = 150 kg, k =  5 MN/m, F0 = 450 kN and ω=120  rad/s.

BlackTom AI · Accepted Answer

We start by identifying the system: a mass m attached to a mounting with spring constant k, subjected to an external force F = F0 cos(ωt) with damping neglected. For a single-degree-of-freedom mass-spring system under harmonic forcing and zero damping, the steady-state displacement has amplitude X0 given by X0 = F0 / |k − m ω^2|. This comes from balancing the inertia term m x'' with the restoring force k x in the frequency domain, yielding a dynamic stiffness of (k − m ω^2) for the forced response.

Now plug in the numbers:
- m = 150 kg
- k = 5 MN/m = 5 × 10^6 N/m
- ω = 120 rad/s
- F0 = 450 kN = 4.5 × 10^5 N

Compute the term m ω^2:
m ω^2 = 150 × (120)^2 = 150 × 14,400 = 2,160,000 N/m.

dynamic stiffness = k − m ω^2 = 5,000,000 − 2,160,000 = 2,840,000 N/m.

Finally, the amplitude X0:
X0 = F0 / |k − m ω^2| = 450,000 / 2,840,000 ≈ 0.1585 m.

Converting to centimeters: 0.1585 m × 100 cm/m ≈ 15.85 cm, which rounds to about 15.8 cm.

If any of the inputs were different (for example, including damping), the expression for X0 would include a damping term, and the amplitude would be F0 / sqrt((k − m ω^2)^2 + (c ω)^2) with c representing damping. Here, since damping is neglected, the denominator reduces to the absolute difference (k − m ω^2). Note that you should confirm the sign inside the absolute value, but only the magnitude matters for amplitude.

Regarding the answer choices: the provided data show a single numerical result for X0, and no alternative options are given to compare against. The calculation steps above lead to the numeric value in consistent units.

A motor of weight m is supported by a mounting that has spring constant k. If the unbalance of the motor is equivalent to a force F=F0cos(ωt) and damping can be neglected, calculate the amplitude of the response, X0, when m = 150 kg, k = 5 MN/m, F0 = 450 kN and ω=120 rad/s.

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A motor of weight m is supported by a mounting that has spring constant k. If the unbalance of the motor is equivalent to a force and damping can be neglected, calculate the amplitude of the response, X0, when m = 150 kg, k = 5 MN/m, F0 = 450 kN and rad/s.

A motor of weight m is supported by a mounting that has spring constant k. If the unbalance of the motor is equivalent to a force and damping can be neglected, calculate the amplitude of the response, X0, when m = 150 kg, k = 5 MN/m, F0 = 450 kN and rad/s.

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