Real-time programming Newton’s law for translation: F = m a F in Newtons, m in kg, a in m/s2. Acceleration a = dv / dt Kinetic energy E = ½ m v2 E in Joules, m in kg, v in m/s So even though an object wobbles when there is no external force, the angular momentum is conserved: q = I | Design Realization lecture 20 John Canny 10/30/03 Last time Real-time programming This time Mechanics – Physics and Motors Review of physics Newton’s law for translation: F = m a F in Newtons, m in kg, a in m/s2. Acceleration a = dv / dt Kinetic energy E = ½ m v2 E in Joules, m in kg, v in m/s. Physics of translation Momentum p = m v and so F = dp / dt In the absence of force, momentum is conserved. Momentum conservation implies energy conservation. Physics of rotation Rotation is more complex; Euler’s equation: T = I + x I T (torque) in N-m, in radians/sec, in radians/sec2, I in kg-m2, = d / dt I is a 3x3 matrix, not necessarily diagonal. If T = 0, then I = - x I which is usually non-zero. So is non-zero, changes with time, and the object wobbles. Physics of rotation Angular momentum is q = I The rotation equation simplifies to T = dq / dt because dq/dt = I d /dt + dI/dt = I + x I So even though an object wobbles when there is no external force, the angular momentum is conserved: q = I Physics of rotation Kinetic energy of rotation is ½ T I In the absence of external torque, kinetic energy of rotation is conserved. But angular momentum conservation does not imply energy conservation. Work Work done by a force = F x (Joules) where x is the distance (m) through which the force acts. Work done by a torque = T (Joules) Power Power is rate of doing work. Power of a force = F v (Watts). Power of a torque = T (Watts). Power often expressed in horsepower = 746 Watts Motors Motors come in several flavors: DC motors Stepper motors (AC) induction motors (AC) Single-phase motors (AC) Synchronous motors The first two are highly controllable, and usually what you would use in an application. But we quickly review the others. 3-phase AC Three or four wires that carry the same voltage at 3 equally-spaced phases: Single phase AC requires two wires (only 1/3 the current or power of 3-phase). AC induction Motors Induction motors – . | Design Realization lecture 20 John Canny 10/30/03 Last time Real-time programming This time Mechanics – Physics and Motors Review of physics Newton’s law for translation: F = m a F in Newtons, m in kg, a in m/s2. Acceleration a = dv / dt Kinetic energy E = ½ m v2 E in Joules, m in kg, v in m/s. Physics of translation Momentum p = m v and so F = dp / dt In the absence of force, momentum is conserved. Momentum conservation implies energy conservation. Physics of rotation Rotation is more complex; Euler’s equation: T = I + x I T (torque) in N-m, in radians/sec, in radians/sec2, I in kg-m2, = d / dt I is a 3x3 matrix, not necessarily diagonal. If T = 0, then I = - x I which is usually non-zero. So is non-zero, changes with time, and the object wobbles. Physics of rotation Angular momentum is q = I The rotation equation simplifies to T = dq / dt because dq/dt = I d /dt + dI/dt = I + x I So even though an object wobbles when there is no external force,
