University of Toledo
umped Parameter !ystems
umped Parameter !ystems
E+P&)&TIO) P#EDI%TIO)
SENSE
FORMULATE
INTERPRET
TEST
Understand the pro-lem What are the factors and relevant relationships
What assumptions can -e made
What e/uili-rium conditions e0ist
Dra$ and la-el an engineering s1etch 2ree -ody diagram
*ydraulic schematic
Electrical schematic
Write the e/uili-rium e/uations 3usually di4erential or di4erence5 )e$ton 6nd a$
7ircho4 a$s for current and voltages
2lo$ continuity la$s
%hec1 the validity of the results
 
Understand the Pro-lem
Do the results
System8 a functional group of interrelated things
State8 & condition 3$hich may or may not -e physical5 of the system regarding form, structure, location, thermodynamics or composition
State vector8 a collection of varia-les that fully descri-e the o-9ect over time
Input8 an e0ternal o-9ect provide to the system
Output8 a dependent varia-le 3often a state5 from $ithin the system that can -e measured or /uanti:ed
 
et 0 -e a vector formed of the state varia-les
 The num-er of components of the state vector is called the order
2ormulate the system as
 The matri0 & is the called the !tate Dynamics Matri0
 The matri0 ' is called the Input or %ontrol Matri0
 The matri0 % is called the Output or !ensor Matri0
 The matri0 D is called the Pass Through or Direct term
1 2{ ( ), ( ),...}T  x x t x t =
  State Transition Equation
et 0 -e a vector formed of the state varia-les
 The num-er of components of the state vector is called the order
2ormulate the system as
 The matri0 & is the called the !tate Dynamics Matri0
 The matri0 ' is called the Input or %ontrol Matri0
 The matri0 % is called the Output or !ensor Matri0
 The matri0 D is called the Pass Through or Direct term
1 2{ ( ), ( ),...}T  x x t x t =
( )
( ) ( ) ( ) State Transition Equation
 y t Cx t Du t 
=
+ = +
= +
 
!tate !pace 2ormulation Procedure8 Develop the e/uations of e/uili-rium Put the e/uili-rium e/uations in the form of
the highest derivative e/ual the remainder of the terms
Ma1e a choice of states, the input and the outputs
Write the e/uili-rium e/uations in terms of the state varia-les
%onstruct the dynamics, the input, the output and the pass through matrices
Write the state space formulation
 
material element level
Partial di4erential e/uations descri-e the transfer of force from the constitutive e/uations
2EM'EM often used
component level
ODEDi4 E -ased on lin1ing component parameters
E/uations solved analytically or numerically
 
umped parameter systems
simpler /uic1er results
'oth can -e used in -uilding controls
 
?
 
 
E$%ii&ri%m E$%ation" Needed'
(. En)ine to c%tch
*. C%tch to tran"mi""ion
coordinate" /θe,θd,θa,and θ0
( )c d c e d  T J bθ θ θ = + −;; ; ;
( ) 0c d d d tf a J k N θ θ θ + − =;;
 
c d c e d c d c e d  
c d c e d tf a d d  
c d d d tf a c d d tf a d d  
t a d tf a d a a w t a d d d tf a a a w
w w t w a w a t w w
T J b J T b b T b k N k  
 J k N J k N k 
 J k N k J k k N k k 
 J b k k J 
θ θ θ θ θ θ   θ θ θ θ  
θ θ θ θ θ θ  
θ θ θ θ θ θ θ θ θ  
θ θ θ θ θ  
= + − ⇒ = − − ⇒ = − + + −
+ − = ⇒ = −  
+ − + − = ⇒ = − + +
+ + − + = ⇒
;; ; ; ;; ; ;
; ;
;; ;;
;; ;;
;; ; ;; ( )
{ }
State !ariab"es are , , , ,
0 0 0
0 0
0 0
w
d tf  d 
t t t  w
dt    J J J 
k k k b
θ 
θ 
θ 
θ 
θ 
d 
c
a
ea
w
w
d 
a
a
w
w
b
T 
 y
θ 
θ 
 
2
2
b b
a a a
dt 
dt dt  
dt L L dt L
 Kid b d 
a
i
V 
dt    K b
 
hat i" the "2eed3
Note ho the mechanica
(0 45L for the eectrica
*0 NSL for the mechanica
+0 Reation"hi2 or co%2in)
e$%ation &eteen the to
In a contro" 2ro&em,
"ometime" caed
for motor angle?
b b
a a a
dt 
dt dt  
dt L L dt L
 Kid b d 
a
i
V 
dt    K b
 
2
2
b b
a a a
dt 
dt dt  
dt L L dt L
 Kid b d 
Same 2roce"",
different $%e"tion,
hat i" the motor an)e3
If the ind%ctance La i" "ma "%ch
that it can &e ne)ected, then
another "im2er form%ation i"
2
2
2
dt dt  
dt J J dt  
'alance !ystems  & large num-er of control pro-lems are
 
<eneral Dynamics E/uation form is
 This e/uation is usually nonlinear
( )( , , ) , ( , , , ) M q q q C q q B q q q u+ =;;; ; ;;;
Ener)! Con"er6in) Term"
E7terna Forcin) term"
  ( sin os )
  sin os
os
os
dt 
 M m p ml ml bp F 
 J ml mlp ml 
 M m ml 
ml J ml  
0 sin 0 0 0
0 0 0 sin 0
ere is te !isous frition at te /ee"s and is te !isous frition in te pin
 p ml b p F 
ml 
γ  
2
-ssumin$ and are sma"", ten sin , os 1 and 0
/itout te frition terms,
os 0 sin 0 0
os 0 0 sin 0
 M m ml p ml F 
ml J ml ml  
 M m ml p
θ θ 
 J ml ml  F ml   F ml   p
 M m  M m ml 
 J ml ml  ml    M m F  p
 J M m Mml ml 
 F M m p
ml J ml  
 p m l J ml F  p    J M m Mml 
θ 
( ) 0 0 0
 p m l F J ml   J M m Mml 
ml    M m F 
 J M m Mml 
 p p   J ml m l  
 p p J M m Mml d    J M m Mml  
dt 
θ 
Distri-uted parameter systems8
Material element level
Partial di4erential e/uations descri-e the transfer of force from the constitutive e/uations
umped Parameter !ystems %omponent level
%omponent properties are self contained and complete $ith ODEDi4 E -ased on lin1ing component parameters for e/uili-rium e/uations
Mechanical system e/uations