MEEG 311-010/011: Homework 6
Fall 2017
Notes: Read the textbook chapter 3.3, 3.4 and 3.5.
1
Problem 1: 3.29
A certain servomechanism system has dynamics dominated by a pair of complex poles and no fin
Notes
+2 x=0
x n
x ( t )=C 1 e i t +C 2 e i t
x ( t )= Acos ( n t )
n
n
x ( t )= A1 cos ( n t ) + A 2 sin ( n t )
A= A2 + A 2 ; =tan 1
1
2
A2
A1
( )
A 1=C 1+ C2 . A 2=i ( C 1C 1 )
x ( t )= A o sin (
Notes
Example 1
Balancing equation:
F=mx -> -kx -cx + F(t) = mx
m + c +kx=F ( t )
x x
The general solution is the sum of the homogenous solution
(Xh(t) capturing the component of the motion when F(t)
Notes
+ 25 n + 2 x=0
x
x n
n=natural frequency
n=
=damping ratio
In this case,
k
c
; = ; c cr =2 ( mk )
m
c cr
Solution types:
1) Underdamped, <1
[
x ( t )=e t C 1 ei 1
I)
Charged values : S 1,2=(
Notes
Model-
m ( t ) + c ( t ) +kx ( t ) =F0 cos ( t )
x
x
X=
F0
( k m ) +( c )
2 2
sin ( )=
2
cw
2 2
( k m ) + ( cw )
2
cos ( ) =
k m 2
( km ) +( cw )
2 2
2
+ 2 n x + 2 x =m + c +kx
x
x
n
In nond
Notes
Laplace Transform Objectives: Conclude how a linear system evolves without explicitly stating the solution of the
ODE that describes the system.
Ingredients :
Some complex variable s= +i
Funct
Notes
L cfw_ output
=transfer function at zero initial conditions Example (cont.)
L cfw_input
M 1 x 1+ ( b1 +b2 ) x 1b1 x 2=r ( 1 )
M 2 2+ b1 x 2 +k 2 x 2b1 x 1=0 ( 2 )
x
Output :could be x 1 , x 1x
Notes
Example 1: Positioning the reading head of a disk drive.
Plant:
( Js +b ) ( La s + Rac ) + K m K e
s
K
G p ( s )= m
2
Values: J =1 Nm
s
rad
b=20
Nms
rad
R=1 ohm
La=1 mH
K m=
5 Nm
A
Control of th
Notes
Electric Motors-
T m ( t )=K 1 K f i (t ) i a ( t )
1)Field (current) Controller-
T m ( t )=K m , f i f ( t )
InputsV f ( t ) , d ( t )
Output
s
J + bs T d
K m ,f
1
( s )=
V f ( s)
( J s +bs
Notes
Characterization of Transients
( i ) T p=
A Swiftness of response
( Exact for 2 nd order , underdamped )
n 1 2
( i ) PO=100 e 1
PO
( 100 )
=
PO
+ ln (
100 )
2
2
C Characterization of SteadyS
Notes
Example 1:
Deriving equations for this system:
>System is Frictionless
>Define a variable (with respect to the origin).
>Displacement (), equilibrium (),
>Coordinate with equil. Position and dis
Motor with shaft:
Rotor : J 1 1=B 1b ( 1 2 ) k ( 1 2 ) +T m
Load : J 2 2=b ( 2 1 )k (2 1)
d
Circuit n: v aKn1
e 1=Lan2 i a + Ra i a
p ( s )=s +a1 s +a2 s dt+ an1 s+ an
Routh Array:
(
1
n
s
(s n1) a1
MEEG 311-010/011: Homework 5
Fall 2017
Notes: Read the textbook chapter 3.2 and 3.4.
1
Problem 1: 3.14
Figure 1: Tape drive schematic.
A simplified sketch of a computer tape drive is given in Fig. 1.
MEEG 311-010/011: Homework 1
Fall 2017
1
Problem 1: 1.26
Figure 1: Springs connected in series-parallel.
Find the equivalent spring constant of the system shown in Fig. 1.
2
Problem 2: 1.51
Figure 2:
MEEG 311-010/011: Homework 2
Fall 2017
Notes: Read Textbook Chapter 2.6, 3.2, 3.3, 3.4.
1
Problem 1: 2.132
Figure 1: Railroad car stopped by spring-damper system.
A railroad car of mass 2, 000 kg trav
MEEG 311-010/011: Homework 3
Fall 2017
Notes: Read Textbook (Control Part) Chapter 1.1, 1.2, 3.1, Appendix A.
1
Problem 1: 1.5
Draw a block diagram of the components for temperature control in a refri
MEEG 311-010/011: Homework 4
Fall 2017
Notes: Read Textbook Chapter 2.1, 2.2, 2.3, 2.4.
1
Problem 1: 2.8
Figure 1: Schematic of a system with flexibility.
In many mechanical positioning systems there
A second order approximation of a satellite control system is shown in figure 1
A. Determine the closed loop transfer function of the system
B. Select the gain K so that the percent overshoot of the s
Position
Small Angles
cos ( )isin ()sin ( )=
Step Input =
1
s
1
X ( s )= ( sX ( s ) )
s
Velocity
sX (s )
2
(
)
cos =1
P-Integral
2
cos ( cos
1(
2)1 ) cos ( 2 ) +sin ( 1 ) sin ( 2)
K
Proportional
Gc (
VIBRATION AND CONTROL (MEEG311)
Equation of Motion
2
meq x + c eq x + k eq x=F ( t ) x +2 n x + n x=F(t)
Note: divide by meq before continuing
The above equation can be derived through three methods
M
Notes
Y ( s )=
G (s )
R (s )
1+G ( s ) H ( s )
The poles of the closed loop transfer function
above are the roots of the denominator, or the
characteristic polynomial.
Question: How do the closed loop