GRAND VALLEY STATE UNIVERSITY Padnos School of Engineering PROJECT II ASTRONAUT CENTRIFUGE BY KELLY MARTIN THAO LAI KEVIN LABEAU BRIAN STILLEY EGR 312 - DYNAMICS PROFESSOR: DR ALI MOHAMMADZADEH DECEMBER 10, 2004Page 2 1.0 Introduction To simulate the flight conditions that an astronaut experiences in a space vehicle, engineers have developed a centrifuge. This centrifuge is shown in Figure 1.1. Figure 1.1 Centrifuge for astronaut performance testing The main arm of the centrifuge rotates about axis A-A with an angular velocity of 1ω and an angular acceleration 1ω&. The pilot’s cockpit revolves about axis C-C with constant angular velocity 2ω. Inside the cockpit, the astronaut sits on a seat that permits rotation about axis B-B with constant angular velocity 3ω. The astronaut’s head is located 3 feet above the seat’s axis of rotation. If the acceleration of the astronaut’s head exceeds a specific number of g’s, the astronaut will loose consciousness. A study of the pilot’s performance was conducted by following the schedule of rotations given in Table 1.1. Table 1.1: Schedule of rotation. Variable RPM Range # of Equal Steps 1ω 2 – 20 10 2ω 2 – 12 6 3ω 5, 10, or 15 Discrete values 1ω& 5, 8, and 10* Discrete Values * Units are RPM2, not RPM.Page 3 All of the centrifuge rotations are controlled by a computer program that simulates specific spacecraft maneuvers corresponding to the entry and exit from the earths atmosphere. 2.0 Calculations Engineers have decided to determine how many times the pilot’s head will experience magnitudes 2 and 3 times the acceleration of gravity. The equations below derive the acceleration of the astronaut’s head. Derivation of head acceleration equation Figure 2.1: Centrifuge Diagram The pilot’s head experiences a motion relative to origin of the fixed frame, O. The position of the head relative to the fixed axis is given by Equation 2.1. BHBHrrr/vvv+= (2.1) where, Hrv= position from the pilot’s head to the origin of the fixed frame Brv= distance from the origin of the rotating frame to the fixed frame BHrv= position from the pilot’s head to the origin of the rotating frame y x z O B Hrv RrOBvv= ρvv=BHr 2ω Y X Z 1ω&1ω3ωPage 4 Velocity is the derivative of position, therefore Equation 2.2 is the velocity equation of the astronaut’s head. BHBHrdtdrdtdrdtdvvv+= BHBHBHrvvvvrvvv×++=ω (2.2) where, Hvv= velocity of the pilot’s head with respect to the origin of the fixed frame Bvv= velocity of the rotating frame’s origin with respect to the origin of the rotating frame BHvv= velocity of the pilot’s head with respect to the origin of the rotating frame BHrvv×ω = angular velocity effect caused by rotating frame The derivative of velocity was computed to find Equation 2.3, the governing equation of acceleration. ()BHBHBHrdtdvdtdvdtdvdtdvrvvv×++=ω BHBHBHBHvdtdvaaavvvvvv+×++=ω BHBHBHBHBHvavaaavvvvvvvv×++×++=ωω ()BHBHBHBHBHavrraavvvvvvvv&vv+×+××+×+=ωωωω2 (2.3) where, Hav= acceleration of pilot’s head observed from the fixed frame Bav= acceleration of the origin of the rotating frame observed from the fixed frame BHrvv&×ω= angular acceleration effect caused by the rotating frame with respect to the fixed frame ()BHrvvv××ωω= angular velocity effect caused by the rotating frame with respect to the fixed frame BHvvv×ω2 = Coriolis Acceleration. Combined effect of the pilot’s head moving relative to the rotating frame and rotation of the rotating frame BHav= relative acceleration of the pilot’s head with respect to the rotating framePage 5 The five components of Equation 2.3 can be used to further derive the equation of motion for the pilot’s head. The general motion equation can be applied to the first term,Bav, so that ()OBOBOBrraavvvvv&vv××+×+=111ωωω where all of the ‘O’ subscripts refer to the origin of the fixed frame. Since the origin of the fixed frame,Oav, experiences no acceleration, the first term can be reduced to Equation 2.4. ()OBOBBrravvvvv&v××+×=111ωωω (2.4) The second term can be rewritten as Equation 2.5. BHBHrrvv&vv&×=×1ωω (2.5) The third term can be rewritten as Equation 2.6. ()()BHBHrrvvvvvv××=××11ωωωω (2.6) The velocity of the fourth term can be written in terms of 1ωv,2ωvand 3ωv, since the acceleration of the astronaut’s head is affected by all three velocities. This new term can be seen as Equation 2.7. ()()[]BHBHBHrrvvvvvvvv×+××=×32122ωωωω (2.7) The general motion equation can also be applied to the fifth term, BHav, so that ()()BHBHBHBHBBHrrrraavvvvvvvv&vv&vv××+××+×+×+=332232ωωωωωωPage 6 Since the acceleration of the origin of the rotating axis,Bav, does not move with respect to itself, it is zero. The angular velocities, 2ωvand 3ωv, are both constant, so their respective accelerations are also zero. Thus, the entire tangential component is zero. The resulting fifth term is shown in Equation 2.8. ()()()[]ρωωωωωωvvvrrrrrr××+×+×+=32/3232/BHBHra (2.8) The variable,BHrv, is also known asρvand the term,OBrv, is known as Rv. Substituting Equations 2.4 through 2.8 into Equation 2.3, the final equation for absolute acceleration of the pilot’s head results is shown in Equation 2.9. ()()()()[]...232111111+×+××+××+×+××+×=ρωρωωρωωρωωωωvvvvvvvvvv&vvvvv&vRRaH ()()()()ρωωρωωωωvvvvvvvv××+×+×+323232 (2.9) 3.0 Results & Discussion The MatLab program, provided in Appendix A, was written to implement Equation 2.9. This program counted the number of times the astronaut’s head experienced more than 2 and 3 times the acceleration of gravity. Out of the 540 different centrifuge test combinations given by the rotation schedule in Table 1.1, there were 270 instances when the astronaut’s head experienced more than 2 g’s and 174 instances when it experienced more than 3g’s. The 174 instances have the most intense combinations of angular accelerations that the astronaut would endure and have the greatest potential to cause the astronaut to black out. If this happened in a real space mission, the astronaut’s safety would be jeopardized. The MatLab program also plotted the acceleration acting on the astronaut’s head against 1ω(rad/s) as shown in Figure 3.1. During this