Answer Happy

A hoist control system is shown in Figure P8.29, where a pulley with an unextendible rope carries a load (weight Mg); the pulley is connected to a shaft by a torsional spring (k); and the shaft that is supported by a bearing (b) is driven by a torque t applied by a motor for lifting or lowering the load. The height y of the weight is measure by a sensor, with output y, =u,y, where u, is a gain constant. A controller determines a control signal u based on error e=r-y, , where r is a reference input, representing a desired height of the weight. The amplifier provides a motor voltage v by v=u_u, where u, is a gain constant. The motor torquet is governed by the differential equation dt +at=By, where a and ß are constants. Let the mass moment of inertia of the motor dt rotor-shaft assembly be J. Consider PID control. a (a) Write down the equations for all the components of the control system. (b) Derive the transfer function of the control system from the reference input r to the height y. (c) Draw a block diagram of the control system, with r and Mg as the inputs and y