A steel block of mass m = 8 kg attached to a horizontal spring with k = 64 N/m m
ID: 1473231 • Letter: A
Question
A steel block of mass m = 8 kg attached to a horizontal spring with k = 64 N/m moves on a rough surface The force of friction on the moving block is F_fric = - gamma x. with gamma = 48 kg/s Show that this system is overdamped, and find the value (including units) of q = The block is resting in its equilibrium position, .t = 0, when it is hit at time t = 0 by a bullet (of negligible mass), which sets the block in motion with an initial speed of +2 5 m/s. Find the subsequent position and velocity of the block as functions of time. Find the time when x is a maximum, and sketch x(t).Explanation / Answer
The damping is called viscous because it models the effects of an object within a fluid. The proportionality constant c is called the damping coefficient and has units of Force over velocity (lbf s/ in or N s/m).
Fd = - c*v = -c dx/dt
m dx^2/dt + c dx/dt'+ kx = 0
The solution to this equation depends on the amount of damping. If the damping is small enough the system will still vibrate, but eventually, over time, will stop vibrating. This case is called underdamping – this case is of most interest in vibration analysis. If we increase the damping just to the point where the system no longer oscillates we reach the point of critical damping (if the damping is increased past critical damping the system is called overdamped). The value that the damping coefficient needs to reach for critical damping in the mass spring damper model is:
Cc =2* sqrt(km)
To characterize the amount of damping in a system a ratio called the damping ratio (also known as damping factor and % critical damping) is used. This damping ratio is just a ratio of the actual damping over the amount of damping required to reach critical damping. The formula for the damping ratio () of the mass spring damper model is:
= c /(2*m)
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