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Modeling of Lumped-Parameter
Electromechanical Systems
Prof. R.G. Longoria
Fall 2010
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
Typical Energy-Storing Transducers
L(x)
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
C(x)
Department of Mechanical Engineering
The University of Texas at Austin
Lumped-Parameter EM
• Sufficient accuracy in most cases by building a
‘lossless model’ of the coupling system
• Energy methods are used to “provide simple
and expeditious techniques for studying the
coupling process” [WM=Woodson & Melcher].
• We will look at ways to derive the stored
energy and for obtaining the “forces of electric
origin”.
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
Electromechanical Coupling
“There are four technically important forces of electric
origin.
• The force resulting from an electric field acting on a
free charge.
• The force resulting from an electric field acting on
polarizable material.
• The force resulting from a magnetic field acting on a a
moving charge (a current).
• The force resulting from a magnetic field acting on
magnetizable material.” [WM]
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
EQS vs. MQS
• We focus on ‘quasi-static’ systems, where the fields that give
rise to forces are dominantly electric or magnetic, but not both.
• EM Quasi-static laws are formulated by neglecting the
coupling terms in Maxwell’s equations, so that electromagnetic
waves that would result from this coupling are neglected.
• We define [HM]:
EQS = Electro-Quasi-Static systems
MQS = Magneto-Quasi-Static systems
• These ‘model’ types are commonly used in electromechanical
dynamics.
• This assumption allows you to determine the electric and
magnetic field characteristics independently.
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
“Lossless” magnetic field coupling
‘Terminal relations’
λ = λ (i, x)
f e = f e (i, x)
‘Multiport relations’
Here force is
current analog.
Woodson and Melcher [WM]
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
“Lossless” electric field coupling
v = v ( q, x )
e
e
f = f ( q, x )
or
f e = f e ( v, x )
Here force is
current analog.
Woodson and Melcher [WM]
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Woodson and Melcher [WM]
Department of Mechanical Engineering
The University of Texas at Austin
Bond Graph Multiports
Later we’ll see
where this
comes from.
or
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
Mixed Energy-Storing IC-Multiport
• Reference: KMR, Chapter 7 and Chapter 8
• For devices that “act” like I-multiports on some ports and like
C-multiports on others.
n
j
n
i =1
i =1
k = j +1
E = E ( p, q) = ∫ ∑ ei fi dt = ∫ ∑ fi dpi + ∫ ∑ ek dqk
∂E
fi =
, i = 1, 2,… , j
∂pi
From Ch. 7 of KMR
∂E
ek =
, k = j + 1, j + 2,… , n
∂qk
∂fi
∂ek
∂2 E
=
=
⇐ Reciprocity
∂qk ∂qk ∂pi ∂pi
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
Summary of the Approach
1.
Identify the lossless energy-storing multiport
as an EQS or MQS system
2.
Model EQS systems with multiport C and
MQS with multiport IC
3.
Identify the relevant electrical constitutive
relation, typically as ‘electrically linear’
4.
Derive the respective energy expression for
the multiport model (potential or ‘mixed’).
This requires an implied integration path that
forces unknown terms to zero, but provides a
general state-determined energy expression.
5.
Use the energy expression to derive the force
expressions by taking partial derivatives with
respect to the related displacement variables.
EQS
v
qɺ
F
xɺ
C
v=
MQS
1
q
C ( x)
λɺ
i
i=
IC
F
xɺ
1
λ
L( x)
U qx = ∫ vdq + ∫ Fdx
Eλ x = ∫ id λ + ∫ Fdx
⇓
⇓
U qx = ∫ vdq + ∫ Fdx
=0
Eλ x = ∫ id λ + ∫ Fdx
=0
F=
∂U qx
∂x
F=
∂Eλ x
∂x
For rotational EM devices, replace F-x with T-θ.
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
The following example problems with EM
elements illustrate the use of these methods.
The idea is to see that you can integrate EM
device models into your bond graph
modeling toolbox with relative ease.
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
Example: Butterfly Condenser*
The capacitance of a ‘butterfly condenser’ (see figure below) varies with rotor position according to
the relation C(θ) = C0 + C1 cos(2 θ), where C0 and C1 are constants. The rotor has axial moment
of inertia J and turns on low-friction fixed bearings.
1.
Develop a bond graph model of this system and derive the state equations. We are interested in
determining the rotor dynamics, particularly when an ideal constant-voltage source E0 is
connected across the condenser.
2.
Use your equations to determine all equilibrium positions for the rotor. Which are stable?
3.
What is the natural frequency of small oscillations in the neighborhood of a stable equilibrium
position?
4.
What is the maximum electrical torque available if: C0 = 15 x 10-12 farad, C1 = 10 x 10-12 farad,
and E0 = 1,000 volts?
5.
If the rotor shaft is 1 millimeter in diameter and the rotor mass is 100 grams, estimate the
minimum coefficient of friction required to prevent motion.
Assume negligible
resistance on the
electrical side.
C (θ ) = C0 + C1 cos 2θ
*From Crandall, et al, “Dynamics of Mechanical and Electromechanical Systems,” McGraw-Hill, 1968.
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
Example: Electrostatic Motor
An electrostatic motor, shown in cross-section below, has a fixed cathode of radius r, a
rotor of thickness t comprising alternate 90° segments of high dielectric constant kHε0
and low dielectric constant kLε0, and an outer anode with four equal segments. A high
constant voltage Vo is attached to terminals A-A and a zero voltage to terminals B-B;
every quarter revolution of the rotor the voltages are switched. The unit has thickness w.
The clearances may be assumed to be very small.
1.
2.
3.
4.
5.
Estimate the potential energy stored in the
electric fields of the motor, valid for up to
90° of rotation.
Find the torque, T, generated by the motor in
terms of the information given.
Find the electric current required by the
motor.
Develop a bond graph model that couples the
electrical and mechanical dynamics,
including rotational inertia and friction.
Write state equations.
From F.T. Brown, “Engineering System Dynamics,” Marcel-Dekker, NY, 2001.
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
Example: Plunger-type solenoid (1)
A simple plunger-type solenoid for operation in
relays, valves, etc. is shown. Assume a
conservative energy-storing model, using an
electrically linear relation,
 L 
λ = L( x)i =  o  ⋅ i
1 + x 
a

We can model this type of device two ways:
“Mixed energy-storing” multiport
λɺ
i
IC
F
xɺ
Direct energy-storing multiport
λɺ
i
..N M
G
ϕɺ
C
F
xɺ
Decision to use one of these versus the other depends on information available.
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
Example: Plunger-type solenoid (2)
Let’s apply the mixed-energy storing multiport
concept to the basic solenoid.
λɺ
i
IC
F
xɺ
1
i = i (λ , x ) =
λ
L( x)
F = F (λ , x ) = ?
As for any multiport, we can use energy,
E = E (λ , x ) =
E
o
assume = 0
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
λ
x
0
0
+ ∫ id λ + ∫ Fdx
Department of Mechanical Engineering
The University of Texas at Austin
Example: Plunger-type solenoid (3)
Once we find the energy, we can use the constitutive restrictions,
∂E
i=
∂λ
∂E
F=
∂x
We can choose an integration path, bring energy to value at λ while dx = 0,
and using the known inductance relation,
2
λ
λ
λ
1 λ
E = E (λ , x) = ∫ id λ = ∫
dλ =
2 L( x)
0
0 L( x)
Now, the force becomes
where,
dL( x)
L′( x) ≜
dx
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
∂E λ 2 d  1  λ 2 L′( x)
F=
=
=


∂x
2 dx  L( x)  2 [ L( x) ]2
What is the holding force, given x and i?
Department of Mechanical Engineering
The University of Texas at Austin
Example: Plunger-type solenoid (4)
Complete bond graph
Apply causality
Derive state equations
R:Rcoil
1
λɺ
i
R:friction
IC
F
xɺ
1
I:m
plunger
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
Example: Angular-Motion EM system
The device shown has a driving blade that is made of iron, and can
move in the air gap.
Given:
L(θ ) = A + B θ
As well as other parameters.
This system also lends itself to
an IC-multiport model.
λɺ
i
IC
T
θɺ
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
1
i = i (λ , θ ) =
λ
L(θ )
T = T (λ , θ ) = ?
Department of Mechanical Engineering
The University of Texas at Austin
Example: EM transducer systems from Crandall, et al (1)
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
Example: EM transducer systems from Crandall, et al (2)
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
Example: Woodson and Melcher
Note, when you see µ → ∞
in a problem, it indicates you can assume that
you can ignore the energy stored in the ‘core
material’. For example, in this case most of the
energy would be stored in the air gap.
Basically, when µ goes to zero, the ‘reluctance’
to setting up a magnetic flux in the material is
very small (i.e., highly permeable material)
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
Example: Woodson and Melcher
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin
Example: Alternator (from KMR)
The basic transduction mechanism for electrical alternators
and motors may be understood by studying the ideal energystoring transducer shown below, consisting of a fixed and a
moving coil. Assume the two currents and the torque t are
related to two flux-linkage variables λ1, λ2 and the angular
position θ,
i = i (λ , λ , θ )
1
1
1
2
i2 = i2 (λ1 , λ2 , θ )
τ = τ (λ1 , λ2 ,θ )
For the electrically linear case, it is conventional to specify self- and mutual- inductance parameters;
for example,
 L1

 L0 cos θ
L0 cos θ   i1   λ1 
= 



L2  i2  λ2 
where L0, L1, L2 are constants and:
1. What kind of bond graph element describes this device?
2. Derive the torque relation from the inductance matrix above by computing the stored energy at fixed θ
when λ1 and λ2 are brought from zero to final values and then differentiating the energy function with
respect to θ (see pp. 322-323 of Crandall, et al handout – attached).
ME 383Q – Prof. R.G. Longoria
Modeling of Physical Systems
Department of Mechanical Engineering
The University of Texas at Austin