Free Piston Beta drive engine

Referring to the section on Sinusoidal Volume Variations we find that both Beta and Gamma type machines have more complex relations in that the movement of both the piston and the displacer affects the volume variation of the compression space Vc.






In the phasor diagram following:
φ is the phase advance of the displacer with respect to the piston,
δ is the phase advance of the compression space volume with respect to the piston, and
α is the phase advance of the expansion space volume with respect to the compression space volume.




In the following MATLAB program we assume that both the piston and displacer motions are sinusoidal.

function betadrive
% Beta drive engine configuration
% Israel Urieli 1/22/08
global vclc vcle % compression,expansion clearence vols [m^3]
global vswc vswe % compression, expansion swept volumes [m^3]
global alpha % phase angle advance of expansion space [radians]
global new fid % new data file
fprintf('beta drive engine configuration\n')
if (strncmp(new,'y',1))
     xpa = input('enter piston amplitude (m): ');
     xda = input('enter displacer amplitude (m): ');
     phid = input('enter displacer phase angle advance [degrees]: ');
     dp = input('enter piston diameter (m): ');
     dd = input('enter displacer diameter (m): ');
     dr = input('enter displacer rod diameter (m): ');
     vclc = input('enter compression space clearance volume [m^3]: ');
     vcle = input('enter expansion space clearance volume [m^3]: ');
     fprintf(fid, '%.3e\n', xpa);
     fprintf(fid, '%.3e\n', xda);
     fprintf(fid, '%.1f\n', phid);
     fprintf(fid, '%.3e\n', dp);
     fprintf(fid, '%.3e\n', dd);
     fprintf(fid, '%.3e\n', dr);
     fprintf(fid, '%.3e\n', vclc);
     fprintf(fid, '%.3e\n', vcle);
else
     xpa = fscanf(fid,'%e',1);
     xda = fscanf(fid,'%e',1);
     phid = fscanf(fid,'%f',1);
     dp = fscanf(fid,'%e',1);
     dd = fscanf(fid,'%e',1);
     dr = fscanf(fid,'%e',1);
     vclc = fscanf(fid,'%e',1);
     vcle = fscanf(fid,'%e',1);
end
ap = pi*dp*dp/4; % piston area (m^2)
ad = pi*dd*dd/4; % displacer area (m^2)
ar = pi*dr*dr/4; % displacer rod area (m^2)
vpa = xpa*(ap - ar); % (piston - rod) volume aplitute {m^3)
vda = xda*(ad - ar); % (displacer - rod) volume amplitude(m^3)
vea = xda*ad; % displacer volume aplitute {m^3)
phi = phid*pi/180; % radians
delta = atan2(vda*sin(phi),(vda*cos(phi) - vpa));
   % compression space volume to piston amplitude phase advance
vca = sqrt(vpa*vpa - 2*vpa*vda*cos(phi) + vda*vda);
   % compression space volume amplitude (m^3)
vswc = 2*vca; % compression space swept volume (m^3)
vswe = 2*vea; % expansion space swept volume (m^3)
alpha = pi + phi - delta; % expansion phase angle advance (radians)
fprintf('\nbeta drive engine data summary:\n');
fprintf(' comp clearence,swept vols %.1f, %.1f [cm^3]\n', vclc*1e6,vswc*1e6);
fprintf(' exp clearence,swept vols %.1f, %.1f [cm^3]\n', vcle*1e6,vswe*1e6);
fprintf(' expansion phase angle advance %.1f[degrees]\n\n', alpha*180/pi);
%==============================================================



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Stirling Cycle Machine Analysis by Israel Urieli is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License