## R ex

f (spring)

f (spring)

--jrJTjC

### Flo. 11.2. Mass spring damper vectors.

work done by the mass, and that as a result, the applied force F must have a component normal to displacement (parallel to velocity) pointing up. If there were no damping force, then the applied force would have no normal (work) component and would l>e in the direction of the acceleration (spring direction). In the discussion to follow, it is useful to consider each force as a combination of work components normal to the displacement and spring components parallel to displacement of the component under consideration.

The necessary requirement for free-piston operation is the dynamic equilibrium of forces as required by Newton's Second Law. i.e. the sum of all forces acting on each moving component must equal the acceleration force of the component. If this situation exists, then the system may in fact oscillate at that condition. The forces to be considered here are damping forces, spring forces and pressure forces. These must all sum to the acceleration vector which has magnitude of MAw2, where M is the component mass, A is its amplitude of displacement and w is its angular frequency in radians per second. The direction of the acceleration force is opposite to the direction of the displacement vector.

Once a thermodynamic analysis has been carried out as described in the previous chapters, and from it the desired motions to the components have been decided upon, the dynamic analysis to permit this desired motion may follow. The procedure for the dynamic analysis is to compute the acceleration force from the desired operating frequency, amplitude, and the mass; the damping forces from the pressure-drop characteristics of the heat exchangers and the gas-llow velocities; the pressure forces from the previous thermodynamic analysis or from an isothermal-cycle analysis as described next; and the spring forces from the mechanical or gas spring characteristics, choosing the spring forces to permit the vector sum of all forces to equal the acceleration force required.

working space forces

Gas pressure forces The working space pressure in real Stirling engines is a complex function of thermodynamic and dynamic influences, and its accurate determination requires a high order computer analysis. However, for the purpose of dynamic analysis, much simpler isothermal estimates are quite useful. Even the isothermal pressure relation is not harmonic, however, but it can be used to get the phase and amplitude of an equivalent harmonic pressure wave as follows: (a) For the design under study, with known volumes, temperatures, and displacements, compute an isothermal (or higher order) pressure relation and plot a pressure-volume diagram. Find the cycle work as the integral of the P-V diagram.

(b) Equate the cycle work to the vector product of volume and a harmonic pressure wave of amplitude equal to the pressure wave previously computed and phase angle such as to give the same cycle work. VV = -77 V • P sin <!>'. V is the amplitude of the volume vector which may itself be the sum of two or more vectors representing volume changes due to piston motions, P is the amplitude of the harmonic pressure wave, and <// is the pressure phase lag behind the negative volume vector (Fig. 11.3).

I his equivalent pressure vector may then he used in the vector representation of the engine dynamics. Its accuracy is good for engines with larger dead volume, but not so good for tight. low dead-volume engines

## Solar Stirling Engine Basics Explained

The solar Stirling engine is progressively becoming a viable alternative to solar panels for its higher efficiency. Stirling engines might be the best way to harvest the power provided by the sun. This is an easy-to-understand explanation of how Stirling engines work, the different types, and why they are more efficient than steam engines.

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