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2.004 Dynamics and Control II
Spring 2008
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Massachusetts Institute of Technology
Department of Mechanical Engineering
2.004 Dynamics and Control II
Spring Term 2008
Lecture 131
Reading:
• Class Handout: Modeling Part 2 – Summary of One-Port Primitive Elements
1 Generalized System Representation
In the electrical and mechanical systems we have defined, and used, two power variables
(variables whose product is power):
Electrical
Mechanical
P = vi, where
P = F v, where
v – voltage drop v – velocity drop
i – current
F – force
We have also developed a common graph notation to express the system topology, and have
used extensions to Kirchoff’s laws to draw analogies between pairs of system variables:
(1) voltage drop and velocity drop, and
(2) current and force.
In addition, we drew analogies between the primitive modeling elements in each domain:
(1) capacitors and masses
(2) inductors and springs
(3) resistors and dampers.
We now look at generalizing these associations.
Variables: In each energy domain we will class the two power variables as either:
(a) Those variables that are defined by measuring a difference, or drop, across an element,
that is between nodes on a graph (across one or more branches). These variables sum
to zero around any closed loop on the graph (they satisfy the compatibility conditions).
These variables are defined to be across-variables.
The two across variables we have defined so far are (i) velocity drop in mechanical
systems, and (ii) voltage drop in electrical systems.
Example: To measure voltage drop in an electrical circuit you must connect a volt­ meter across an element:
1
c D.Rowell 2008
copyright �
13–1
R
1
V
+
v R
V
R
2
2
A v o ltm e te r m e a s u r e s v o lta g e
" a c r o s s " a n e le m e n t
-
(b) Those variables that are measured through an element, that is are considered as being
transmitted through an element unchanged. These variables sum to zero at the nodes
on a graph, and are said to satisfy the continuity condition. They are defined to be
through variables.
The through variables defined so far are (i) current in electrical systems, and (ii) force
in mechanical systems.
Example: to measure force in a mechanical system, or current in an electrical system
a sensor mus be inserted in series with an element.
A
A n a m m e te r m e a s u re s c u rre n t
" th r o u g h " a n e le m e n t.
i
+
v R
V
F o r c e is m e a s u r e d b y in s e r tin g a n in s tr u m e n t in s e r ie s
w ith a n e le m e n t.
F (t)
R
F
K
-
m
Generalized Variables: In dealing with generalized modeling power variables, without
regard for a particular energy domain we now define
• A generalized across variable, designated v, and
• A generalized through variable, designated f .
We also generalize the constraints on across and through variables imposed by the structure
of a system’s graph:
• The compatibility condition
n
�
vi = 0
around the n elements in any loop on a graph
i=1
which is clearly a generalization of Kirchoff’s voltage law, and
• The continuity condition
n
�
fi = 0
in the n elements connected to any node on a graph
i=1
which is clearly a generalization of Kirchoff’s current law.
13–2
Generalized Sources: Along with the generalized variables we define a pair of ideal sources
(1) The Across Variable Source: An across variable source maintains the across vari­
able between the nodes it is connected to, regardless of the magnitude of the through
variable supplied to the system.
v
a
O n a n a c r
in th e d ir e
d ro p , th a t
v
V
v
o s s
c tio
is w
>
a
v
- v a r ia b le s o u r c e th e a r r o w p o in ts
n o f th e a s s u m e d a c r o s s - v a r ia b le
e a s s u m e
b
b
The examples we have seen so far are the voltage source in electrical systems, and the
velocity source in mechanical systems.
(2) The Through Variable Source: An through variable source maintains the through
variable into the nodes it is connected to, regardless of the magnitude of the across
variable it must generate to maintain its through variable.
O n a n th r o u g h - v a r ia b le s o u r c e th e a r r o w p o in ts
in th e d ir e c tio n o f th e a s s u m e d th r o u g h - v a r ia b le
" flo w " .
F
The examples we have seen so far are the current source in electrical systems, and the
force source in mechanical systems.
Generalized Elements: We have seen that in each of the energy domains studied so
far we have defined three primitive modeling elements: two energy storage elements and
a dissipative element. We now classify these elements according to (i) the variable that
accounts for the stored energy in energy storage elements, and (ii) we group together the
dissipative elements.
A-Type Elements These are the energy storage elements in which the stored energy is a
function of the across-variable.
Electrical: For a capacitor, let v = va − vb be the across variable:
v
a
C
i
v
C
v
b
dv
i = C
� t
� dt
t
E =
vi dt =
Cv dv
−∞
=
13–3
1 2
Cv
2
0
which defines the capacitor as an A-type element.
Mechanical: For a mass element
v
F
dv
F = m
� tdt
� t
E =
vF dt =
mv dv
m
m
−∞
v
re f
0
1 2
mv
=
2
which defines the mass element as an A-type element.
Define a generalized capacitance C that represents A-type elements inde­
pendent of the energy domain, and write its elemental equation in terms of
generalized across and through variables:
d v
f = C
g e n e r a liz e d a c r o s s v a r ia b le
d t
g e n e r a liz e d th r o u g h v a r ia b le
g e n e r a liz e d c a p a c ita n c e
A-Type elements may be summarized as in the following table:
Element
Elemental equation
Energy
dv
dt
dv
F =m
dt
dv
i=C
dt
1
E = Cv2
2
1
E = mv 2
2
1
E = Cv 2
2
Generalized A-type
f=C
Translational mass
Electrical capacitance
T-Type Elements These are the energy storage elements in which the stored energy is a
function of the through-variable.
Electrical: For a inductor, let v = va − vb be the across variable:
v
a
L
i
v
L
= v
a
v
- v
b
b
di
v = L
� dt
� t
t
E =
vi dt =
Li di
−∞
=
13–4
1 2
Li
2
0
which defines the inductor as an T-type element.
Mechanical: For a spring element
v
v
K
a
F
b
F
v
= v
K
- v
a
b
v =
1 dF
�K dt
t
1
E =
vF dt =
K
−∞
1 2
=
F
2K
�
t
F dF
0
which defines the spring element as an T-type element.
Define a generalized inductance L that represents T-type elements indepen­
dent of the energy domain, and write its elemental equation in terms of
generalized across and through variables:
d f
g e n e r a liz e d a c r o s s v a r ia b le
v = L d t
g e n e r a liz e d th r o u g h v a r ia b le
g e n e r a liz e d in d u c ta n c e
T-Type elements may be summarized as in the following table:
Element
Elemental equation
Generalized T-type
v = Ldf/dt
Translational spring
v=
1 dF
K dt
di
v=L
dt
Electrical inductance
Energy
1
E = Lf 2
2
1 2
E=
F
2K
1
E = Li2
2
D-Type Elements These are the dissipative elements (non-energy storage, the power
flow is always into the element).
Electrical: For a resistor, let v = va − vb be the across variable:
v
a
R
i
v
R
= v
a
v
- v
b
b
v = iR
P = vi = i2 R = v 2 /R
≥ 0
13–5
which defines the resistor as an D-type element.
Mechanical: For a damper element
v
v
B
a
b
F
F = Bv
P = vF = Bv 2 = F 2 /B
≥ 0
F
v
B
= v
a
- v
b
which defines the damper element as an D-type element.
Define a generalized resistance R that represents D-type elements indepen­
dent of the energy domain, and write its elemental equation in terms of
generalized across and through variables:
v = R f
g e n e r a liz e d a c r o s s v a r ia b le
g e n e r a liz e d th r o u g h v a r ia b le
g e n e r a liz e d r e s is ta n c e
D-Type elements may be summarized as in the following table:
Element
Elemental equations
Generalized D-type
f=
Translational damper
F = Bv
Electrical resistance
i=
1
v
R
Power dissipated
v = Rf
v=
1
v
R
1
F
B
v = Ri
1 2
v = Rf 2
R
1
P = Bv 2 = F 2
B
1
P = v 2 = Ri2
R
P=
Generalized Impedances: The generalized impedance of an is
Z=
V (s)
F (s)
where V (s) and F (s) are the Laplace domain representation of the generalized across and
through variables. The impedances are summarized in the following table:
A-Type
Generalized
Translational
Electrical
1
Cs
1
sm
1
Cs
13–6
T-Type
D-Type
sL
R
1
s
K
1
B
sL
R
2 Notes on Transfer Function Generation
The system order (highest order derivative on the l.h.s. of the differential equation, or the
highest power in s in the denominator of the transfer function) is determined by the number
of independent energy storage elements in the system.
R
v
I s
o n e id e p e n d e n t e n e r g y s to r a g e e le m e n t
1
H (s ) =
(o rd e r = 1 )
R C s + 1
K
m
F
C
m
B
v = 0
tw o id e p e n d e n t e n e r g y s to r a g e e le m e n ts
1
H (s ) =
(o rd e r = 2 )
m s 2 + B s + K
An independent energy storage element is one whose stored energy is not directly propor­
tional to the energy stored in other elements or completely defined by the system sources.
For example, in the following figure the two springs are not independent energy storage ele­
ments, because their stored energies are directly related. Therefore this system will generate
a second-order differential equation even though there are three energy storage elements.
v
K
F
1
K
m
K
2
F
1
m
K
1
K
2
K 1 a n d K 2 a r e e ffe c tiv e ly in p a r a lle l a n d
m a y b e r e p la c e d b y a n e q u iv a le n t s p r in g .
1 a n d K 2 a r e n o t in d e p e n d e n t e n e r g y s to r a g e
e l e m e n t s s i n c e E K µ E K 2.
1
T h is is a s e c o n d - o r d e r s y s te m .
similarly, in the following figure the velocities of the two masses are related by the lever, and
they are not independent energy storage elements.
v
v
2
m
1
m
m
2
1 a n d m 2 a r e n o t in d e p e n d e n t e n e r g y s to r a g e
e le m e n ts b e c a u s e v1 µ v 2 a n d th e r e fo r e th e ir
e n e r g ie s a r e a lg e b r a ic a lly r e la te d .
1
Another situation where an energy storage element does not increase the order of a system
is when the stored energy is completely defined by a source.
(a)
Any element connected directly in parallel with an across variable source will not
affect the system dynamics, and in fact will not appear in the differential equation (or
13–7
transfer function). The reason is that the source completely defines the across variable
at its two nodes, and therefore the element can be removed without affecting the rest
of the system. For example in the following system the output is the velocity of the
mass m2 . The system input is a velocity source V applied to the mass m1 . The velocity
at the left-hand node is completely defined by the source, and would not be affected if
m1 was removed from the system. The transfer function relating vm2 to V is
Vm2 (s)
K/m2
= 2
V (s)
s + K/m2
H(s) =
and does not involve m1 .
K
K
v
v e lo c ity
s o u rc e
V
m
K
m
1
m
2
m
V
2
T h e v e lo c ity o f m
1
m
m
V
2
2
is c o m p le te ly d e fin e d b y th e s o u r c e
1
(b) Any element connected directly in series with a through variable source will not affect
the system dynamics. The through variable transmitted from the source to the rest of
the system will be unchanged by the presence of the series element(s). In the electrical
system below the resistance R1 and the inductor L are in series with the current
source I. Since iL = I, the current supplied to the right-hand node is unaffected by
the presence if either L or R1 , and this is effectively a first-order system.
I
R
L
1
R
I
C
R
2
I
L
1
C
R
2
I
C
R
2
R 1 a n d L d o n o t a ffe c t th e s y s te m d y n a m ic s s in c e th e
s o u r c e d e fin e s th e c u r r e n t th r o u g h th e m .
13–8