Nonlinear Systems
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Modeling: some nonlinear effects
Standard state equation description
Equilibrium points
Linearization about EPs
Simulation and insight
Equilibrium point design
Some nonlinear effects
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Aerodynamic drag on a vehicle
Rotodynamic pump
Nonlinear spring effect
Nonlinear geometry
Check valve modeling
Aerodynamic drag effect
Typical aerodynamic drag on a passenger vehicle can be
modeled by a drag force, Ra, given by
Ra  (  / 2) * CD * Af *Vr
where
ρ is mass density of air,
CD is drag factor due to vehicle shape,
Af is frontal (projected) area, and
Vr is vehicle speed
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Rotodynamic pump
A typical model for the outlet port of a rotodynamic
pump is given by
P  k3 * N  k 2 *Q
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where
P is outlet port pressure,
N is shaft speed, and
Q is outlet port flow.
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Hardening spring
A typical relation for the characteristic of a hardening
spring is given by
F  k1 *  k   
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where
F is the spring force and
δ is the spring deflection from free length.
Nonlinear geometry
Mass, m
L
Fmagnetic
θ
mg
(1) mL *  Fmag * L cos  mg * L sin 
(2)   
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Examples to illustrate nonlinear methods
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Pendulum with magnetic force applied
Spring-loaded pendulum
Hanging sign problem
Print-head mechanism
Formulation of standard equations
• Identify the state and input vectors X(t) and
U(t).
• Formulate a set of system equations.
• Reduce the equations to the form
X (t )  f ( X ,U )
(If this is not possible then a different approach
needs to be taken which we will not discuss here.)
Equilibrium points
• We seek equilibrium points (EPs) under the
following conditions:
– Assume all inputs are constant, U(t) =Uc.
– Assume all derivatives go to zero simultaneously.
• The equations become 0  f ( X , U )
• Solve the EP equations for Xep, given Uc.
(There may be zero, one or many EPs. Finding
them can be a daunting task on occasion. )
Linearization about an EP
• To gain insight about the nature of a given EP
we can linearize the model about the EP.
• We use a Taylor series method, expanding
about the EP.
• The resulting linearized model can be written
as
X (t )  A * X  B * U
See Linearization a la Taylor notes.
Simulation for insight
• To locate a stable EP in a difficult problem we
can sometimes simulate the dynamic response
and watch it go to the EP.
• Once such an EP has been located we can
simulate the behavior in the vicinity of the EP
to get a feeling for the local behavior.
• It is also possible to numerically approximate
the linearized A, B matrices at an EP.
See Hanging Sign example and Numerical linearization
notes.