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Still_More_Solutions - ME577 Homework#6 Problem 1 Here is a...

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Here is a simplified diagram of the pitchers arm: The following notation will be used for the solution: h x , h y , h z : Position of the hand in the x,y, and z directions w x , w y , w z : Position of the wrist in the x,y, and z directions e x , e y , e z : Position of the elbow in the x,y, and z directions s x , s y , s z : Position of the shoulder in the x,y, and z directions fa x , fa y , fa z : Position of the center of mass of the forearm in x, y, and z directions hu x , hu y , hu z : Position of the center of mass of the Humerus in x, y, and z directions r h , v h , a h : Position, Velocity, and Acceleration vectors of the hand r w , v w , a w : Position, Velocity, and Acceleration vectors of the wrist r e , v e , a e : Position, Velocity, and Acceleration vectors of the elbow r s , v s , a s : Position, Velocity, and Acceleration vectors of the shoulder r fa , v fa , a fa : Position, Velocity, and Acceleration vectors of the center of mass of the forearm r hu , v hu , a hu : Position, Velocity, and Acceleration vectors of the center of mass of the Humerus x, y, z : Unit vectors in the x, y, and z directions A) To determine the velocity of the center of masses of the hand, forearm and humerus, the velocity at each of the measured points (hand, wrist, elbow, and shoulder) must first be calculated. For the hand, the position vector from the fixed origin to the 3rd knuckle is given by the following equation: r h = h x x + h y y + h z z To calculate the velocity at this point, differentiate the position vector with respect to time. This gives the equation for the velocity of the hand. v h = ˙ h x x + ˙ h y y + ˙ h z z Using the provided position data for the hand, the first derivative of the position can be approximated using the forward difference equation: v h ( t 1 ) = ˙ r h ( t 1 ) r h ( t 2 ) r h ( t 1 ) t 2 t 1 This calculation is then repeated for each time interval. ME577 Homework #6 Problem 1 N S E W H
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Using the same calculation, but substituting the position data for the wrist, elbow, and shoulder, v w , v e , and v s can be found. Using the assumption that the position data given for the 3rd knuckle is roughly the position of the center of mass. Then v h is the velocity vector for the hands center of mass. To compute the velocity of the center of mass of the forearm, the vector from the wrist and the elbow must be determined. It can be calculated with the following equation: r e = r w + r e/w
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