Figure 4 shows a delicate instrument placed on a table that moves as a result of a vertical force acting on it. Springs and dampers connect the table to the ground to limit the table's movement. The combined mass of the table and instrument is m. The total stiffness of the springs is k and the total damping is c.
a. Express the mechanical system into mathematical model by deriving the differential equation
b. Find expressions for the steady-state gain K, the damping ratio Ī¶, and the natural frequency Ļn in terms of the physical parameters m, c, and k.
c. Numerical values of the physical parameters are m = 40 lbm, k = 45 lbf/ft, and c = 4 lbf s/ft. Find the response of the table when the platform is subjected to a sudden deflection due to a force of 12 lbf.
d. Graph the solution and estimate the duration of the transient.
e. The instrument is not usable if it is moving faster than 0.04 ft/s. How long a period of time must pass after the force is applied before the instrument will function properly?
Simulation of Dynamic Systems with MATLABĀ® and SimulinkĀ®
m
M=MT + Mā
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Figure 4 Mechanical system
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