Page 105 - Robotics Designing the Mechanisms for Automated Machinery

94 Dynamic Analysis of Drives

For the first case we have to integrate Equation (3.118) in the limits from p 0 to some
pressure p* which is less than that at ^. Thus, in place of Equation (3.120) we obtain

For the second case, we have to continue our investigation for the subcritical
regimes. For a subcritical regime,

Now, in Equation (3.111) the value of/3 varies from the initial value of 0.528 to 1. In
this case we must substitute in the differential Equation (3.118) G, which is not con-
stant and is defined by Equation (3.111). Thus,

Since p=fip r,

Therefore,

To integrate this equation, we introduce an auxiliary function,

which gives

After substituting Equations (3.124) and (3.125) in Equation (3.123), we obtain

The limits of integration are determined by the initial value of/? 0 and the critical value
of ft cr. Thus,

In the general case, the time t* required to reach a pressure sufficient to move the
piston and overcome the load and the forces of resistance may be written in the form