When a compressor of 75 kg directly sits on a factory floor, any harmonic force generated by rotating unbalance in the compressor during its operation can be transmitted to the floor and cause disturbance to other equipment. To prevent the transmission of this unwanted force to the floor, an isolation system that includes both stiffness and damping needs to be designed to support the compressor. It is expected that 85% of vibration isolation must be achieved. When the compressor is undergoing free vibration (no harmonic force generated), its amplitude is first measured to be 40 mm and 5 cycles of vibration later, the amplitude is measured to be 5 mm.
1. When the compressor is running at 200 Hz, determine the stiffness of the isolation system. (70 points)
Solution must include: (1) A mass-spring-damper system diagram with all parameters labeled and sign convention labeled, (2) a free-body diagram, (3) apply Newton's Law of Motion to derive the equation of motion based on the FBD. EOM must be consistent with the FBD. (4) steps of solving the EOM and working through the problem. Directly using any transmissibility equation is not accepted.
2. Before the compressor starts running (i.e. no harmonic force is generated), if it is being disturbed by an initial displacement of 20 mm and an initial velocity of 25 m/s, find its free response. (30 points) Solution must include the equation of motion and steps of determining the free response.
Bonus (10 points): Let's consider an undamped system for simplicity, i.e. the foundation only provides stiffness with no damping. Consider the same stiffness as determined in question 1 above. If the compressor is given a zero initial displacement and an initial velocity of 25 m/s while having a harmonic force F(t) = 220 cos(t) N applied to it, find the complete solution representing the motion of the compressor.
Must present the equation of motion and the solution process and work through all steps. No partial credits if (1) simply describing the solution method without working through it numerically, or (2) simply writing down some equations without working through the problem.
