Types of Nonlinearities

Submitted by:Nida Quazi

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Content

•Nonlinearity•Classification of nonlinearity•Brief description of types

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Nonlinearity

•Nonlinearity is the behavior of a circuit , in which the output signal strength does not vary in direct proportion to the input signal strength. E.g.: diode.

•Nonlinear system do not follow superposition property.

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Classification of Nonlinearities•Nonlinearity can be classified in to : • On the basis of magnitude ○ Incidental nonlinearity : present inherently in the system. ○ Intentional nonlinearity : inserted in the system to modify system characteristics. • On the basis of frequency

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On the basis of

Magnitude Frequency1) Dead zone2) Saturation3) Friction4) Backlash5) Relay

1) Limit cycle2) Jump resonance3) Frequency entrainment4) Beat frequency production5) Self excitation6) Harmonics7) Chaotic behavior

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1) Saturation •Output of the system is proportional to input in a

limited range of input signal.•On exceeding range , output tends to become nearly

constant.

Eg: amplifiers, torque and speed saturation in motors

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•Limits max. achievable magnetic field in system containing electromagnetic components

• In electronic circuits ferromagnetic core inductor- on applying AC signal , nonlinearity can cause harmonics and intermodulation distortion so we have to limit the level of signal accordingly.

•Used to limit the current in saturable core transformer, arc welding and ferroresonant transformer which serve as voltage regulator.

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2) Friction

•Viscous friction •Coulomb friction

•Stinction friction

•It comes in to existence when mechanical surface comes in sliding contact.

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E.g.: HDDs servo systems-•It uses a voice-coil-motor (VCM) actuator to actuate the R/W recording arm assembly, which consists of a pivot with a ball bearing, a metal arm and a rigid suspension that holds the R/W headerand slider.•The presence of friction in rotatory actuator pivot bearingresult in large residual errors and high frequency oscillation;which may produce positioning error and deteriorate the performance of the servo system.

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3)Dead zone•Kind of nonlinearity in which the system doesn’t

respond to the given input until the input reaches a particular level

Or•Output becomes zero when input crosses certain

limiting value .

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•Effect of dead zone: • system performance degradation • reduced positioning accuracy • may destabilize system•E.g.: • Dead zone in actuators, such as hydraulic servo valves, give rise to limit cycle and instability. • Electronic devices like diode.

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4) Backlash •The difference between the tooth space and tooth

width in mechanical system , which is essential for rapid working gear transmission is known as backlash.

• It is present in most of the mechanical and hydraulic systems.• Increase with wear.

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• When the backlash is traversed, no torque is transmitted through the shaft and when the contact is achieved, the resulted impact can destroy the gear and cause a high frequency noise.

•Effect of backlash : • gear backlash may cause sustained oscillation or chattering phenomenon • system may turn unstable for large backlash

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5) Relay • It is intentional nonlinearity.• Ideal relay is the extreme case of saturation. • occurs when the linearity range is shrunken to zero and slope in linearity range become vertical .• It can lead to chattering due to discontinuity.• Ideal characteristic can be nearly achieved by SCR switching .

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•Practical relay has a definite amount of dead zone . •Relay coil require a finite amount of current to actuate (cause of dead zone).•To close the relay larger coil current is needed , so relay characteristic exhibit hysteresis also .

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•Relay can be switched abruptly in any of these 3 states :

•off •full forward •full reverseTherefore are used widely in control field .•Application: • temperature control system • aircraft and missile control system • space-vehicle attitude control system • power system

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1) Limit cycle

• It corresponds to an oscillation of fixed amplitude and period.

•Limit cycle is a closed trajectory; other trajectories about the limit cycle are spirals from various points of phase plane.

•Limit cycle divides the phase plane in to 2 zones.

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•Stable limit cycle : • signify – system will approach an oscillatory state regardless of the initial condition.

•Semi stable limit cycle : • in this case maintenance of oscillation depends on initial condition.

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•Unstable limit cycle :

•Limit cycles are usually less sensitive to system parameter variations.

•Although L.C. can be sustained over a finite range of system parameters.

•E.g.: squealing door hinges, electric wires whistling in the wind ,whirling shafts.

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2) Jump resonance• It is a discontinuous change in the steady state

response of the system when a parameter (forcing frequency) is slowly varied.

• Its produced around the system’s resonant frequency.•Consist of multiple values of the amplitude and phase

of input signal when the amplitude or frequency of the harmonic input signal is varied slowly.

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•E.g. : hard spring•As input frequency is increased gradually from 0 ,response follow curve ABC .•At ‘C’- increment in frequency resulted in discontinuous jump down to ‘D’.•Further increase – response curve follows through

DE.• If frequency is now decreased – curve EDF is

followed with a jump up to B from the point F and then to A .

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3) Frequency entrainment• It is because of the physical absorption of the

particular frequencies.•Frequency absorbed by that system is not seen or

observed in the spectrum .•E.g.: water in pneumatic system

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4) Beat frequency• The system is sensitive to some

frequency only.• E.g.: guitar sonometer

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5)Self excitation

• It may occur without any external or internal periodic forcing.

• It appears due to specific internal properties of a system.

• It is of 2 types: • soft self excitation • hard self excitation

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•Soft self excitation : •represented in phase space by a stable limit cycle.•Hard self excitation : •represented by unstable limit cycle. •depending on initial conditions trajectory tends to equilibrium point or to infinite; therefore this

type is called catastrophic .•E.g.: fluttering of airplane wings, shimming of

vehicle wheel , unwanted vibration during machining processes (chatter).

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6) Harmonics •Nonlinear system may contain frequencies other than

forcing frequency in input•These frequencies are multiple of forcing frequency.• If f(t)= A sin ωt then oscillatory phenomenon of frequency ᾤ may

appear at certain points here ω not equal to ᾤ

when ᾤ > ω it is superharmonic when ᾤ < ω it is subharmonic

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7) Chaotic behavior and bifurcation

•Chaos – complex , irregular motion that are extremely sensitive to initial conditions.

•Bifurcation – qualitative or topological change in system dynamics produced by varying system parameters .

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•Bifurcation – periodic doubling•chaos – series of periodic doubling•The illustration above shows a bifurcation diagram of

the logistic map obtained by plotting as a function of   a series of values for   obtained by starting with a random value  ,

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iterating many times, and discarding the first points corresponding to values before the iterates converge to the attractor. In other words, the set of fixed points of   corresponding to a given value of   are plotted for values of   increasing to the right.

Xn = r(n-1) (1-x(n-1))

•E.g.: Spur gear system with backlash ; vibration in it have bifurcation and chaotic behavior depending upon initial conditions .

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