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.

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  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.

BlackTom AI · Accepted Answer

The problem describes a single-degree-of-freedom system where a mass-spring setup experiences a harmonic unbalance force F0 due to motor operation, with damping neglected. The standard steady-state displacement amplitude under harmonic forcing is given by X0 = F0 / sqrt((k - m ω^2)^2 + (c ω)^2). Since damping c = 0 in this scenario, the expression simplifies to X0 = F0 / |k - m ω^2|.

Now, let's examine the data provided. The mass m = 150 kg and the spring constant k = 5 MN/m = 5×10^6 N/m are clear. The force amplitude is F0 = 450 kN = 450×10^3 N, which is also clear. However, the problem statement is missing the excitation angular frequency ω (it shows “and   rad/s,” with the numeric value omitted). Without ω, we cannot compute the denominator |k − m ω^2|, and therefore we cannot determine a numeric amplitude X0.

Here are the key checks for the given quantities:
- If ω is known, compute m ω^2, subtract from k, take the absolute value, and divide F0 by that result to get X0 in meters. If the result is 0 (i.e., k = m ω^2), the amplitude would be infinite in this idealized, undamped case, signaling a resonance condition with damping neglected.
- The units are consistent: F0 in newtons, k in newtons per meter, m in kilograms, ω in radians per second, so m ω^2 has units of N/m, matching k.

Because the excitation frequency is missing, any numerical evaluation of X0 (including a value like 15.8 cm) would be speculation. If ω were provided, one could proceed as follows: compute Δ = |k − m ω^2|, then X0 = F0 / Δ, and finally convert to centimeters if needed (X0 [m] × 100 to get [cm]).

In summary, the calculation hinges on the missing ω value. Once ω is supplied, substitute into X0 = F0 / |k − m ω^2| to obtain the amplitude, and convert to centimeters if the final answer should be in cm.

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. Single choice

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