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Problem: Analyze the controls aspect of a VHS tape drive system

Shown below in Figure 1 is a simple schematic of the inner workings of a video home system (VHS) tape drive system. We will analyze the controls aspect of this system to determine its step response and stability.

Figure 1: Simple Schematic of a VHS Tape Drive System.

Given: D(s) = K, D(s) = K(1+s+1/4s), D(s) = K(s+0.1)/(s+0.7), G(s) = 1/Js², J=10 kg m²

Variables	Description/Values/Labels	Variables	Description/Values/Labels
a  b:	Angular Velocities (rad/s)	R:	Spring Constant (20010 N/m)
Ja:	Moment of Inertia (0.00005 kg-m2)	Jb:	Moment of Inertia (0.00002 kg-m2)
Ba:	Motor Damping (0.015 N-m-sec)	Bb:	Turn Damping (0.015 N-M-Sec)
ra:	Wheel 'a' Radius (0.0025 m)	rb:	Wheel 'b' Radius (0.0025 m)
Tm:	Motor Torque Constant (0.025 N-m/A)	B:	Damper Constant (25 N-sec/m)
I:	Motor Current: (A)	T:	Constant Tension Force (5 N)
1:	Velocity of the tape (m/s)

Objective: Once you determine the equations of motion, put them into first-order differential equations in state-space variable form with the following states, x = [x_{1 a} x_{2 b}]. Use MATLAB to determine the control characteristics of this system:

1. Find the Transfer Function
2. Determine the Poles and Zeros and plot them
3. Analyze its Step Response when I = 0.1 Amp.
4. Determine if the system is stable
5. Plot the Root-Locus

Calculated: State-Space variable form

See Solution