20.4 Hysteresis — More Details

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Hysteresis and creep are common to many systems, such as electromechanical actuators, especially when

used at high drive levels. Their transfer function is influenced by “rate-independent memory effects.” The

state of the actuator depends not only on the present value of the input signal but also on the nature of

their past amplitudes, especially the extremum values, but not on rates of the past (Visintin, 1996). This

statement is in support of the author’s secondary creep model of hysteretic damping, where the

amplitude of the previous turning point determines the magnitude of the internal friction force for

the half-cycle that follows. One of the most dramatic examples of a memory effect is the demonstration

mentioned above, by Gross in the 1950s, concerning a twisted wire.

Damping complexities derive from the defect structures that are found in real materials and which give

rise to hysteresis, which in the Greek language means to “come late.” Although, almost everybody seems

to appreciate magnetic hysteresis at some level, too few individuals (at least in physics) have been trained

in the mechanisms of mechanical hysteresis responsible for damping. Dislocations, for example, are

usually an add-on chapter to a solid-state physics text — even though they are known to be indispensable

with regard to actual, as opposed to idealized, properties of materials.

In the case of ferrous materials, the magnetization of a specimen lags behind the field generated by an

electric current, to which the specimen responds. In the case of real springs that do not obey Hooke’s Law

F ¼ 2kx; the displacement x lags behind the spring’s restoring force F: It is convenient to express the

resulting hysteresis in terms of “intrinsic” variables instead of x and F: Thus, the strain 1 (fractional

change in the spring’s length if it were a wire in tension) lags the stress s (force per unit area). Usually in

engineering practice, the stress is reckoned with respect to the external force (negative of the spring F), so

that the equivalent to Hooke’s Law is s ¼ E1; where E is an elastic modulus descriptive of the material

from which the spring is fabricated. In the case of a straight wire, E would be Young’s modulus but, for

coil springs, E is determined primarily by the shear modulus. Some of the ways in which hysteresis can be

represented for a freely decaying oscillator are shown in Figure 20.3. The generalized coordinate q would

be spring elongation for the force case shown, or it would be strain when the ordinate quantity is stress.

The graph of velocity vs. displacement is referred to as a phase-space plot. It is commonly used in

describing chaotic systems and, if “strobed” at the frequency of the oscillator, becomes the Poincare´

section. Notice that the circulation is of opposite sign when using external force as opposed to spring

force, in addition to the curves occupying different quadrants. It is important to recognize this difference,

particularly when discussing negative damping where the oscillation amplitude builds in time, as

illustrated in the right hand part of the figure.

Although not very common in mechanical oscillators, it is possible to realize negative damping.

One example is that of an optically driven pendulum, because of the LiF crystals that were placed in its

support structure (containing a high density of color centers produced by radiation) (Coy and Molnar,

1997). An interesting feature of this pendulum was its unwillingness to entrain to the driving laser.

FIGURE 20.3 Three different ways to represent hysteresis damping for an oscillator in free-decay. Cases of both

positive and negative damping are illustrated.

Damping Theory 20-19

© 2005 by Taylor & Francis Group, LLC

There are also examples of negative damping from aerodynamics, such as flutter. Since buildings and

bridges can experience negative damping in catastrophic manner (Tacoma Narrows bridge as an

example), it is not a subject to be ignored.

Another example of hysteresis that is very much like negative damping (though not usually labeled as

such) is to be found in a heat engine (Peters, 2001a, 2001b). The motion is not simple harmonic; rather,

the speed with which the hysteresis curve is traversed (in pressure vs. volume) increases as the size of the

hysteresis loop increases. A larger loop (greater work done by the gas) results in higher revolutions per

minute (r/min) of the engine as opposed to a larger amplitude of the motion at constant period. The gas

pressure provides a force similar to the Hooke’s Law force of the spring in a mass/spring oscillator.

It is usually assumed that hysteresis loops are “smooth,” which is not necessarily true. For example, in

the case of magnetic hysteresis, the “jerky” parts known as the Barkhausen effect (Barkhausen, 1919) are

well known. The equivalent jerky behavior in metallic alloys is known as the Portevin – LeChatelier effect

(Portevin and LeChatelier, 1923). Although we have historically avoided these cases that appear to be

intractable in a mathematics sense (not obeying the fundamental theorem of calculus), their presence is

undeniable testimony of the complex nature of hysteresis.

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