Translational System State Variables

Work the problem shown in the figure below (a modified version of Figure P2.11 from your text).

• use the definitions shown in the figure for directions and external forces,
• draw complete free-body diagrams (always!)
• assume displacements measured from unstretched spring positions
• write the state variable equations for the entire system
• mark your final answers – double underline or "box" them

!

Rotational System State Variables

Work the problem shown in the figure below (aFigure P4.19 from your text).

• use the definitions shown in the figure for directions and external forces,
• draw complete free-body diagrams (always!)
• assume displacements measured from unstretched spring positions
• write the state variable equations for the entire system
• mark your final answers – double underline or "box" them

Combined Translational/Rotational System State Variables

Electrical System State Variables

For the problem below,

• define positive directions for currents & voltages (equivalent to defining positive forces & velocities)
• write all necessary node equations and "loop" equations (equivalent to applying Newton’s 2^nd law)
• determine necessary state variables ( = independent energy storing elements)
• write state variable equations - NOT required to use MapleÔ for these problems
• write an output equation for the specified e_o

First Order System Response

The response of the translational system to a constant (step) input in velocity, v_i(t) is shown below.

1. if the mass is M = 0.45 kg, what is B in N-sec/m?
2. if the viscous damping is B = 3.7 lbf-sec/inch, what is M in lbm?

Second Order System Response

A second-order electrical system is shown below..

1. find expressions for the natural frequency and damping ratio in terms of system parameters
   Hint: find the state variable equations, take the derivative of one of them, then substitute the other equation
2. if the parameter values are R₁ = 470 ohms, L=5.6 mH, and C = 2.2 mF, and the system roots are at -587 ± 8990 rad/sec, what is the value of the other resistance, R₂?

Laplace Transforms

Use the Laplace Transform to find the solution to the following differential equations:

Transfer Functions

(a). Write the state variable equations for the system shown below.

(b). Find the transfer functions with "hand" calculations

(c). Use Maple to find the two transfer functions given in (b)

Frequency Response

For each of the problems below, determine the analytical expressions for magnitude ( M(w) ) and phase angle, f(w).

For each of the problems below, manually compute 20 log₁₀( M(w) ) and f(w) for w = 3 rad/sec.

For each of the problems below, plot both 20 log₁₀( M(w) ) and f(w) vs. w on semi-log (x) axes.

1.           2.       3.

Control System

a) Determine the K_p value that will give the overall system transfer function a set of complex roots with z = 0.707

b) Determine the steady-state error to a unit step input for the system with this value of K_p.

c) Replace the proportional controller K_p with a PD controller (K_p + K_ds) that will have half of the steady-state error (to a unit step input) of part (b) and the same damping ratio of z = 0.707

d) Replace the proportional controller K_p with a PI controller (K_p + K_i/s) and determine the steady-state error to a step input for the system.