Circular Stirling engine concept
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1. Circular Stirling engine concept 2. Construction of the epitrochoid form of the displacer. 3. Displacer rotation with gear wheel. 4. Mathematics 5. Crankshaft replacement. 6. Stirling engine with trochoid displacer and ball bearing mechanism. 7. Combining two trochoid pistons/displacers with cylinders. 8. Combining two trochoid pistons/displacers with cylinders, Siemens type. 9. Combining two trochoid pistons/displacers with cylinders, Gamma type. 10. Combining two trochoid pistons/displacers with cylinders, Alpha type. Synchronized with a connecting rod. 11. Ball engine or pump. 12. Ball engine or pump 2. end of the page |
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Construction of the epitrochoid form of the displacer.Limaçon or limacon, also known as a Limaçon of Pascal, is defined as a roulette, formed when a circle rolls around the outside of a circle of equal radius. Thus, it belongs to the family of curves called centered trochoids; more specifically, it is a epitrochoid. In this case, the point that describes the limacon is on half the radius of the circle. |
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Displacer rotation with gear wheel.The yellow gear is fixed to the displacer, and the white gear is fixed to the cylinder. In this picture you only see one moving part. A fork with one pin in the middle of the yellow gear, rotating round the center of the cylinder, can drive the displacer exactly in the desired path. The green line on the displacer is only included in the animation to make it easier to see the movement of the displacer. Note that the spot where the green line meets the yellow gear moves in a straight line through the center to the opposite side. This is the case at every point of the pitch circle of the yellow gear. |
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MathematicsFor the displacer I found: 52142 pixels=14.48389*r^2 ~ 14.5*r^2 ~ (7.611541*crank radius)^2 for the displacer cylinder: 82802 =23.00056*r^2 ~ 23*r^2 ~ (9.591779*crank radius)^2 where r= 60 = the radius of the circles to construct the trochoid form of the displacer. For the hot volume area I found this for every degree turn of the crankshaft: 15473,15570,15941,16067,16439,16562,16934,17067,17431,17563, 17934,18054,18422,18546,18907,19043,19402,19527,19892,20008, 20370,20493,20850,20969,21333,21446,21796,21921,22278,22388, 22742,22848,22961,23311,23421,23761,23864,24207,24310,24409, 24748,24852,25192,25285,25376,25714,25796,25877,26220,26300, 26381,26700,26773,26853,27180,27247,27330,27625,27701,27772, 27835,28100,28142,28189,28236,28513,28556,28599,28653,28919, 28970,29016,29042,29084,29115,29151,29398,29417,29451,29471, 29501,29502,29516,29531,29546,29552,29561,29563,29568,29573, 29576,29573,29570,29563,29562,29551,29546,29533,29519,29505, 29500,29470,29453,29420,29400,29149,29115,29083,29041,29015, 28968,28919,28650,28596,28551,28510,28231,28183,28132,28085, 27834,27771,27699,27624,27329,27246,27177,26852,26772,26699, 26380,26299,26219,25877,25796,25714,25376,25285,25192,24851, 24747,24408,24309,24206,23864,23760,23419,23311,22960,22847, 22499,22387,22277,21921,21795,21445,21332,20968,20850,20492, 20369,20007,19891,19526,19401,19043,18906,18545,18421,18053, 17933,17562,17430,17066,16933,16561,16438,16066,15940,15569, 15169,15071,14701,14573,14208,14078,13709,13581,13208,13080, 12716,12584,12219,12089,11723,11605,11241,11125,10756,10638, 10272,10146,9786,9666,9308,9203,8848,8733,8368,8261, 7906,7798,7682,7327,7239,6877,6778,6428,6326,6232, 5887,5811,5454,5350,5272,4933,4849,4760,4432,4364, 4254,3939,3864,3809,3466,3404,3332,3022,2944,2890, 2815,2517,2481,2404,2371,2048,2022,1942,1906,1609, 1554,1512,1500,1445,1415,1373,1125,1076,1060,1016, 986,979,971,951,935,918,915,896,898,892, 882,887,893,893,909,912,927,945,962,973, 986,1013,1056,1071,1117,1373,1410,1444,1492,1510, 1554,1607,1894,1938,2010,2044,2355,2397,2466,2513, 2816,2883,2944,3016,3331,3394,3465,3794,3864,3939, 4254,4365,4431,4759,4849,4931,5272,5350,5454,5811, 5889,6233,6325,6429,6781,6878,7241,7328,7683,7799, 8150,8263,8368,8735,8849,9204,9310,9667,9787,10147, 10274,10639,10758,11126,11242,11606,11724,12090,12219,12586, 12718,13081,13209,13582,13710,14079,14209,14574,14702,15073, 15473,With a maximum of 29576 and a minimum of 882. So the changed area between 90 degrees and 270 degrees= (29576-882)/60^2 =7.970556*r^2 ~ 8*r^2 I have plotted it in the next picture: 
The white line is the counted number of red pixels in a 480*640 animation, like the one you can see above.
The blue line is a sinus, calibrated at the top with the 29576 top and the -1 is at the zero point. The difference visible at the bottom of the graph is the 882 pixels of dead space. Conclusion: With the trochoid form displacer the volume changes in the same way as in the classical crankshaft and piston configuration. So the volume = (sin(angle)+1)/2* 8*r^2 *length of the displacer. Important: r here is twice the distance the rotating axis on the displacer throws from the axis of the crankshaft. ( r here is the radius of both the large circles that are used to construct the form of the displacer) So the volume on one side = (sin(angle)+1)*(4*crank offset)^2 * length of the displacer This means that the Stirling engine design program on http://jordaan.info/greengasoline/stirling.html can be used to calculate all other aspects of the engine with just a little modification of some inputs. |
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Crankshaft replacement.a = distance between the middle of the power shaft (dark blue) and the center of the central ball bearing (light blue). b = shortest distance between the middle of the power shaft and circle through the middle of the balls of the central ball bearing. 2 * ( a + b ) = diameter of the circle through the middle of the balls of the central ball bearing. c = shortest distance between the circle through the middle of the balls of the central ball bearing and the circle through the middle of the outer ball bearing. 2 * ( 2 * a + b + c ) = diameter of the circle through the middle of the balls of the outer ball bearing. a = distance from the middle of the outer ball bearing and the middle of the hole for the central ball bearing. 4 * a = length of the stroke. In this example a = 30, b = 48, c = 48. The inner ball bearing is fixed to the power shaft. If you use a crankshaft with the axis on the dark blue point, and you turn the middle ball bearing 180 degrees, with the crank fixed in the hole where the blue point will be, you will have the same piston stroke as with a regular crankshaft and connecting rod configuration. In this mechanism the connecting rod can be fixed to the piston. It is even better if needle roller bearings are used in this mechanism. Click here if the Media Player does not start displaying. Here a = 48, b = 0, c = 48. The dark blue circle is the crankshaft. The crank moves through the dark blue point. |
Links on this page
1. Circular Stirling engine concept 2. Construction of the epitrochoid form of the displacer. 3. Displacer rotation with gear wheel. 4. Mathematics 5. Crankshaft replacement. 6. Stirling engine with trochoid displacer and ball bearing mechanism. 7. Combining two trochoid pistons/displacers with cylinders. 8. Combining two trochoid pistons/displacers with cylinders, Siemens type. 9. Combining two trochoid pistons/displacers with cylinders, Gamma type. 10. Combining two trochoid pistons/displacers with cylinders, Alpha type. Synchronized with a connecting rod. 11. Ball engine or pump. 12. Ball engine or pump 2. end of the page |
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Stirling engine with trochoid displacer and ball bearing mechanism.1. The crankshaft, with the rotating axis in the dark blue ring, and the crank in the light green point, fixed to the inner side of the small ball bearing. 2. The displacer, with the rotating axis in the light green point, fixed to the outer side of the small ball bearing, and the inner side of the big ball bearing. 3. The power pistons, fixed with a rod to the outer side of the big ball bearing. The piston on the right can be used as a power piston for a second engine on this side of the screen, with the crankshafts fixed, and the displacers fixed to each other. The hot side will be below and the cold side on top. (I am thinking about turning the whole machine through 90 degrees. Here at the top you see a heat pipe ( there will be more of them, with a burner there). The dark green lines are the walls of the power cylinder. The violet part on the left below is the regenerator. The line below is the cooling pipe ( there will be more of them, with cooling water that flows to a radiator). This animation shows that with this mechanism the displacer is forced in the desired motion, while moving the pistons in the desired motion. In the next model I will use two ball bearings, each at the exterior sides of the displacers, with one ball bearing in the middle connected to the pistons and with the rotating axis at the same place where it is now. The crank will pass through a tube through the ball bearings. In this way that ball bearing can be almost half the size it is now. The other two ball bearings can be larger than they are now. ( I have put the ball bearings in this animation inside each other to make the mechanism easier to understand.) ' In this model a=30 (crank), b=0 , c=45, d=20, k=10 (diam.balls) 
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Combining two trochoid pistons/displacers with cylinders.Click here if the Media Player does not start displaying. |
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Combining two trochoid pistons/displacers with cylinders, Siemens type.Click here if the Media Player does not start displaying. |
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Combining two trochoid pistons/displacers with cylinders, Gamma type.Click here if the Media Player does not start displaying. |
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Combining two trochoid pistons/displacers with cylinders, Alpha type. Synchronized with a connecting rod.Click here if the Media Player does not start displaying. |
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Ball engine or pump.Click here if the Media Player does not start displaying. |
Links on this page
top of the page. 1. Circular Stirling engine concept 2. Construction of the epitrochoid form of the displacer. 3. Displacer rotation with gear wheel. 4. Mathematics 5. Crankshaft replacement. 6. Stirling engine with trochoid displacer and ball bearing mechanism. 7. Combining two trochoid pistons/displacers with cylinders. 8. Combining two trochoid pistons/displacers with cylinders, Siemens type. 9. Combining two trochoid pistons/displacers with cylinders, Gamma type. 10. Combining two trochoid pistons/displacers with cylinders, Alpha type. Synchronized with a connecting rod. 11. Ball engine or pump. 12. Ball engine or pump 2. end of the page |
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Ball engine or pump 2.Click here if the Media Player does not start displaying. |
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