Question

A machine weighing 750 N is mounted on springs of 1200kN/m stiffness with assumed damping factor of ξ=0.2. A piston within the machine weighing 20 N ha reciprocating motion with a stroke of 0.075 m and a speed of 3000rev/min. Assume the motion of the piston to be harmonic, determine:

a. The amplitude of motion of the machine.
b. Its phase angle with respect to the exciting force.
c. The transmissibility and the force transmitted to the foundation.
d. The phase angle of the transmitted force with respect to the exciting force.
e. Has vibration isolation been achieved, and if so how?

Answer 1

The amplitude of motion of the machine is 0.075 m. The phase angle with respect to the exciting force can be calculated using the given values. The transmissibility and the force transmitted to the foundation can be calculated using the provided formulas and values.

Step-by-step explanation:

To determine the amplitude of motion of the machine, we need to calculate the displacement of the piston. Since the motion of the piston is assumed to be harmonic, we can use the formula A = x_max, where A represents the amplitude and x_max represents the maximum displacement. Given that the stroke of the piston is 0.075 m, the amplitude of motion of the machine is 0.075 m.

To find the phase angle with respect to the exciting force, we can use the formula tan(θ) = (2ξω)/ω_n, where θ represents the phase angle, ξ represents the damping factor, ω represents the angular frequency, and ω_n represents the natural angular frequency. We are given that the speed of the piston is 3000 rev/min, so we can convert it to the angular frequency using the formula ω = (2πf)/60, where f represents the frequency. Substituting the given values, we can calculate the phase angle.

To calculate the transmissibility, we can use the formula T = (1/√(1-ξ^2))*√((1+(2ξω/ω_n)^2)), where T represents the transmissibility. Substituting the given values, we can find the transmissibility. The force transmitted to the foundation can be calculated by multiplying the transmissibility by the exciting force.

To find the phase angle of the transmitted force with respect to the exciting force, we can use the formula tan(θ) = (2ξω)/ω_n, where θ represents the phase angle. Substituting the given values, we can calculate the phase angle.

Whether vibration isolation has been achieved or not can be determined by comparing the transmissibility values with the desired criteria. If the transmissibility is below the desired criteria, vibration isolation has been achieved. If the transmissibility is above the desired criteria, vibration isolation has not been achieved.