A schematic representation of an electromechanical positioning system of cutter is shown in figure below. Motor...

    

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A schematic representation of an electromechanical positioning system of cutter is shown in figure below. Motor drive system Cutter carriage Cutter blade Nut Lead screw Cutter Brushless 000 de motor motor (ac) Inertia=/ Mass=m Conveyor Guideway (damping constant=b) Positioning of the cutter is achieved through a lead screw and nut arrangement, which is driven by a brushless de motor. The cutter carriage is integral with the nut of the lead screw and the ac motor that derives the cutter blade, and has an overall mass of m (kg). The carriage slides along a lubricated guideway and provides an equivalent viscous damping force of damping constant b (N/m/s). The overall moment of inertia of the motor rotor and the lead screw is J (N m²) about the axis of rotation. The motor is driven by a drive system, which provides a voltage v to the stator field windings of the motor. Note that the motor has a permanent magnet rotor. The interaction between the field circuit and, the motor rotor is represented by Supply voltage L V Rotor The magnetic torque Tm generated by the motor is given by Tm = kmis The force FL exerted by the lead screw in the y-direction of the cutter carriage is given by F₁ =eh TL, in which Rotatory motion of lead screw h Translatory motion of the nut and e is the mechanical efficiency of the lead screw-nut unit. a) Write the necessary equations to study the displacement y of the cutter in response to an applied voltage v to the motor. What is the order of the system? Obtain the input- output differential equation for the system and from that determine the characteristic equation. What are the roots of the characteristic equation? b) Using summation junctions, integration blocks, and constant gain blocks only, draw a complete block diagram of the system, with v as the input and y as the output. c) Obtain a state-space model for the system, using v as the input and y as the output. d) Assume that R/L ratio is very high. Obtain an expression for the response y of the system to a step input with zero initial conditions. Show from this expression that the behavior of the system is unstable in the present form. A schematic representation of an electromechanical positioning system of cutter is shown in figure below. Motor drive system Cutter carriage Cutter blade Nut Lead screw Cutter Brushless 000 de motor motor (ac) Inertia=/ Mass=m Conveyor Guideway (damping constant=b) Positioning of the cutter is achieved through a lead screw and nut arrangement, which is driven by a brushless de motor. The cutter carriage is integral with the nut of the lead screw and the ac motor that derives the cutter blade, and has an overall mass of m (kg). The carriage slides along a lubricated guideway and provides an equivalent viscous damping force of damping constant b (N/m/s). The overall moment of inertia of the motor rotor and the lead screw is J (N m²) about the axis of rotation. The motor is driven by a drive system, which provides a voltage v to the stator field windings of the motor. Note that the motor has a permanent magnet rotor. The interaction between the field circuit and, the motor rotor is represented by Supply voltage L V Rotor The magnetic torque Tm generated by the motor is given by Tm = kmis The force FL exerted by the lead screw in the y-direction of the cutter carriage is given by F₁ =eh TL, in which Rotatory motion of lead screw h Translatory motion of the nut and e is the mechanical efficiency of the lead screw-nut unit. a) Write the necessary equations to study the displacement y of the cutter in response to an applied voltage v to the motor. What is the order of the system? Obtain the input- output differential equation for the system and from that determine the characteristic equation. What are the roots of the characteristic equation? b) Using summation junctions, integration blocks, and constant gain blocks only, draw a complete block diagram of the system, with v as the input and y as the output. c) Obtain a state-space model for the system, using v as the input and y as the output. d) Assume that R/L ratio is very high. Obtain an expression for the response y of the system to a step input with zero initial conditions. Show from this expression that the behavior of the system is unstable in the present form.