Question

A machine with mass of 30 kg is placed on an elastic mounting of unknown properties. An engineer excites the machine with a harmonic force with a magnitude of 100 N at a frequency of 30 Hz. He measures the steady-state response as having an amplitude of 0.2 mm with a phase lag of 20∘. Determine the stiffness and damping coefficient of the mounting.

Answer 1

The stiffness of the mounting is 94247.78 N/m and the damping coefficient is 63.66 Ns/m.

Step-by-step explanation:

To determine the stiffness and damping coefficient of the mounting, we can use the given information about the machine's mass, the applied force, and the steady-state response. First, let's find the stiffness of the mounting.

Stiffness is calculated using the formula:

k = (m * (2πf)²) / A

where m is the mass of the machine, f is the frequency of the applied force, and A is the amplitude of the steady-state response.

Substituting the given values:

k = (30 kg * (2π * 30 Hz)²) / (0.2 mm) = 94247.78 N/m

Now, let's find the damping coefficient.

From the phase lag of 20°, we can assume a small damping coefficient which results in underdamped motion. The damping coefficient can be calculated using the formula:

c = (2πf * m * φ) / A

where φ is the phase lag in radians.

Substituting the given values:

c = (2π * 30 Hz * 30 kg * (20° * π/180)) / (0.2 mm) = 63.66 Ns/m