## 1

Harmonic pv (ellipse"!

Harmonic pv (ellipse"!

FlO. 11.3. Pressure forces.

such as small annular gap designs, in which the true pressure wave is quite far from simple harmonic form.

Gas spring forces

where y = the adiabatic or other appropriate exponent A = the area of the piston P = the gas average pressure V = the volume of the gas spring.

### Damping forces

Damping forces in the heat exchanger may be estimated from the steady-flow pressure-drop characteristics of the heat transfer elements and the volume-flow rate to the associated active space.

True damping forces are in general highly anharmonic, and an equivalent harmonic force giving the same energy dissipation should be computed in a manner similar to that described for the pressure forces. This method is definitely a severe approximation to the truth, but can, nevertheless, give reasonably good results because damping forces due to flow pressure drops are usually rather small in comparison with other forces.

rYPES of engine The two-piston free-piston engine Fig. 11.4 shows perhaps the simplest concept with which to exemplify a free-piston dynamic analysis—a two-piston arrangement with power taken from one or both pistons.

In the idealized case, piston I is a massless damper and piston 2 is an undamped resonating mass. The pressure vector is collinear with the displacement of piston 2 and at right angles to the displacement of piston I. Piston I sees a pure damping force and piston 2 a pure spring. The resonating frequency is fixed by the mass of piston 2 and the system spring constant, which is the sum of that tine to working space pressure Pt and the bounce space pressure P?.

In the more realistic example shown, the pressure wave is assumed to lag both pistons, hence its resulting force has work components on both, and both can produce work through a damping load such as a linear alternator. If the pressure wave were collinear with the displacement of piston 2 then that piston would not be able to produce work and would simply resonate. If the pressure wave lay between the two displacements, lagging I and leading 2, then the resulting forces would do work on 1 and fa) Idealized cuse M J — Cj — 0

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