The spring solenoid has the following linear characteristics Torque (T) 'w = 0 Tspring = -...

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The spring solenoid has the following linear characteristics Torque (T) 'w = 0 Tspring = - Ke While the damper behaves as above Tdamper = - bw A schematic of an installed door is shown below, where: +T WALL 0 = angle of the door (in radians) ( = 0 door closed) I moment of inertia of the door (kg. m) =w= angular velocity (rad/sec) = w = a = angular acceleration (rad/sec) K = spring torque constant (Nm/rad) b = damping factor (kg.mrad sec) T = total torque at hinge (Nm = kg.m/sec) Using the relations: Tspring-k0 Tdamping = -bw Tapplied = 0 T = la It can be shown that for x = 0 and x2 = the system is described by the equations = x, x1 T applied 0 = [1 X a) Write the above differential equations in non-matrix form. b) Using the following block diagram, find the transfer function of this door: H(s) = 0(s) T(s) The following diagram may be helpful: 1100 k c) The ace Door Company buys its dampers from Fly-By-Night Industries' surplus to cut down on costs. Therefore, b is not variable. However, they do manufacture their own springs, and by varying the number of turns, k can be changed. Given a door with a moment of inertia I and damping coefficient b, find k in terms of b and I such that the system is critically damped. That is, the poles are both real and as "fast" as possible. d) What is the general relation (in terms of I, k, and b) for the location of the poles of this system? e) Can this system ever be made unstable? Marginally stable? (Remember: the values of k and b must always be non-negative.) 3 The spring solenoid has the following linear characteristics Torque (T) 'w = 0 Tspring = - Ke While the damper behaves as above Tdamper = - bw A schematic of an installed door is shown below, where: +T WALL 0 = angle of the door (in radians) ( = 0 door closed) I moment of inertia of the door (kg. m) =w= angular velocity (rad/sec) = w = a = angular acceleration (rad/sec) K = spring torque constant (Nm/rad) b = damping factor (kg.mrad sec) T = total torque at hinge (Nm = kg.m/sec) Using the relations: Tspring-k0 Tdamping = -bw Tapplied = 0 T = la It can be shown that for x = 0 and x2 = the system is described by the equations = x, x1 T applied 0 = [1 X a) Write the above differential equations in non-matrix form. b) Using the following block diagram, find the transfer function of this door: H(s) = 0(s) T(s) The following diagram may be helpful: 1100 k c) The ace Door Company buys its dampers from Fly-By-Night Industries' surplus to cut down on costs. Therefore, b is not variable. However, they do manufacture their own springs, and by varying the number of turns, k can be changed. Given a door with a moment of inertia I and damping coefficient b, find k in terms of b and I such that the system is critically damped. That is, the poles are both real and as "fast" as possible. d) What is the general relation (in terms of I, k, and b) for the location of the poles of this system? e) Can this system ever be made unstable? Marginally stable? (Remember: the values of k and b must always be non-negative.) 3