How to Reduce Vibration Amplitude in a Lab Setup
What is the best way to reduce vibration amplitude in a lab setup?
How can the insertion of a soft massless cushion affect the vibration amplitude in a lab setup?
Solution to Reduce Vibration Amplitude
To reduce the vibration amplitude in a lab setup, especially when a delicate crystal of mass M is supported by four massless springs in parallel with spring constant k, a soft massless cushion can be inserted between the crystal and the table. This cushion produces a resistive force -b times the velocity of mass M, effectively dampening the vibrations.
Explanation of Solution
When graduate students in a lab move the table across the floor, the setup vibrates, producing an effective vertical force F = MA0 cos(ωdt) on the mass M. To achieve a tenfold reduction in vibration amplitude, the damping coefficient b needs to be increased. This increases the dampening effect on the system, reducing the oscillation amplitude significantly.
The equation of motion in this damped harmonic system takes into account restoring force from the springs, damping force from the cushion, and driving force from table vibration. Increasing the damping coefficient b in the equation results in heavier damping and decreased oscillation amplitude, meeting the criteria for heavy damping, which is b² > 4*M*k or b > 2*√kM.
By adding a soft massless cushion with the appropriate damping coefficient, the vibration amplitude in the lab setup can be effectively reduced to provide a more stable environment for delicate experiments.