Linear motion, with constant acceleration

If the linear gas velocity approaches the speed of sound, the simple mathematical model used in equation (4) breaks down. The acceleration term must be taken into account, and the steady-state equation of motion for a straight pipeline with constant diameter may be written (8) ... 

The variation of total energy is zero because the system, in contradistinction with the linearly accelerated motion seen previously, is isolated. Now, in a uniform motion the velocity and the radius have constant modules and the angle they form is a constant right angle, so the integration of the Gibbs equation provides the following equality between dot products... 

The solution for the intake, compression and exhaust strokes is very straightforward. The optimal piston velocity in each of these is constant, with a brief acceleration or deceleration at the maximum allowed rate at the juncture of each stroke with the next. The analysis was done both with no constraint on the maximum acceleration and deceleration, and with finite limits on the acceleration. The power stroke required numerical solution of the optimal control equations, in this case a set of non-linear fourth-order differential equations. Figure 14.3 shows the optimal cycle with limits on the acceleration and deceleration, both in terms of the velocity and position as functions of time. The smoother grey curves show the sinusoidal motion of a conventional engine with a piston linked by a simple connecting rod to the drive shaft that rotates at essentially constant speed. The black curves show the optimized pathway. 

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