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Nonlinear Systems

Nonlinear Systems. Modeling: some nonlinear effects Standard state equation description Equilibrium points Linearization about EPs Simulation and insight Equilibrium point design. Some nonlinear effects. Aerodynamic drag on a vehicle Rotodynamic pump Nonlinear spring effect

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Nonlinear Systems

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  1. Nonlinear Systems • Modeling: some nonlinear effects • Standard state equation description • Equilibrium points • Linearization about EPs • Simulation and insight • Equilibrium point design

  2. Some nonlinear effects • Aerodynamic drag on a vehicle • Rotodynamic pump • Nonlinear spring effect • Nonlinear geometry • Check valve modeling

  3. Aerodynamic drag effect Typical aerodynamic drag on a passenger vehicle can be modeled by a drag force, Ra, given by 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

  4. Rotodynamic pump A typical model for the outlet port of a rotodynamic pump is given by where P is outlet port pressure, N is shaft speed, and Q is outlet port flow.

  5. Hardening spring A typical relation for the characteristic of a hardening spring is given by where F is the spring force and δ is the spring deflection from free length.

  6. Nonlinear geometry Mass, m L Fmagnetic θ mg

  7. Examples to illustrate nonlinear methods • Pendulum with magnetic force applied • Spring-loaded pendulum • Hanging sign problem • Print-head mechanism

  8. 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 (If this is not possible then a different approach needs to be taken which we will not discuss here.)

  9. 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 • 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. )

  10. 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 See Linearization a la Taylor notes.

  11. 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.

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