Review Articles

What is Hysteresis?

[+] Author and Article Information
K. A. Morris

Department of Applied Mathematics,
University of Waterloo,
Waterloo, ON, N2L 3G1, Canada
e-mail: kmorris@uwaterloo.ca

Manuscript received July 1, 2010; final manuscript received May 29, 2012; published online August 27, 2012. Assoc. Editor: Jorge Ambrosio.

Appl. Mech. Rev 64(5), 050801 (Aug 27, 2012) (14 pages) doi:10.1115/1.4007112 History: Received July 01, 2010; Revised May 29, 2012

Hysteresis is a widely occurring phenomenon. It can be found in a wide variety of natural and constructed systems. Generally, a system is said to exhibit hysteresis when a characteristic looping behavior of the input-output graph is displayed. These loops can be due to a variety of causes. Furthermore, the input-output graphs of periodic inputs at different frequencies are generally identical. Existing definitions of hysteresis are useful in different contexts but fail to fully characterize it. In this paper, a number of different situations exhibiting hysteresis are described and analyzed. The applications described are: an electronic comparator, gene regulatory network, backlash, beam in a magnetic field, a class of smart materials and inelastic springs. The common features of these widely varying situations are identified and summarized in a final section that includes a new definition for hysteresis.

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Grahic Jump Location
Fig. 1

Simple relay centered on s with width 2r. The output is unambiguous for u>s+r or u<s-r. However, for s-r<u<s+r, the output depends on whether the input is increasing or decreasing. (a) u(ti)>s+r, output +1, (b) s-r<u(ti)<s+r, output ±1.

Grahic Jump Location
Fig. 2

Temperature-strain curve in a shape memory alloy. The curve depends on the temperature history, but not the rate at which temperature is changed. (1998, IEEE. Reprinted, with permission, from Ref. [15].)

Grahic Jump Location
Fig. 3

Loop in the input-output graph of y··(t)+4y(t)=u(t), u(t)=sin(t), t=0..8

Grahic Jump Location
Fig. 4

Experimentally measured voltage for Schmitt trigger. The graphs are similiar to those of a simple relay (Fig. 1). (a) Input frequency 10 kHz, (b) input frequency close to DC. (1991, Wiley. Reprinted, with permission, from Ref. [3].)

Grahic Jump Location
Fig. 5

Schmitt trigger circuit diagrams. (a) Standard circuit diagram, (b) circuit diagram with input capacitance, (c) equivalent circuit to, (b) inclusion of the input capacitance (shown in (b) and (c)) leads to a differential equation model that correctly predicts the response of the circuit. (1991, Wiley. Used, with permission, from Ref. [3].)

Grahic Jump Location
Fig. 6

Simulation of differential Eq. (3) for Schmitt trigger with the same capacitor initial condition v(0)=-1 and different periodic inputs. The response is similar to that of a simple relay and is independent of the frequency of the input. (a) Input vi(t)=sin(t) for 7s, (b) Input vi(t)=sin(10t) for 0.7s. (Parameter values Ri=10kΩ, Rf=20kΩ, A=105, E=4V, Cp=15pF.)

Grahic Jump Location
Fig. 7

g(vi,v) (see Eq. (2)) as a function of v with input voltage vi=1. At this input voltage there are 3 zeros of g and hence the trigger has three equilibrium points. The middle zero (□) is an unstable equilibrium while the other two zeros (○) are stable equilibria. For larger values of vi the graph is higher and for vi>2 there is only 1 zero of g and hence only one equilibrium point. Similarly, for smaller values of vi the graph is lower and if vi<-2 there is only 1 equilibrium point. (Same parameter values as in Fig. 6.)

Grahic Jump Location
Fig. 8

Output voltage vo as a function of input voltage vi for the Schmitt trigger. (See Eqs. (1) and (2).) For -2<vi<2 there are two possible outputs due to two stable equilibrium values of the capacitor voltage. (Same parameter values as in Fig. 6.)

Grahic Jump Location
Fig. 9

Lyapunov function, Eq. (4) for Schmitt trigger, Eq. (3) with different input voltages vi as vi increases from -2vcrit to vcrit and then decreases to 0. The arrow indicates the equilibrium capacitor voltage v. It remains at an equilibrium point until vi changes enough that the Lyapunov function no longer has a minimum at that point. (a) Input voltage vi=-2vcrit. There is one equilibrium voltage. (b) Input voltage increases to vi=0. There are now two minima; v remains at the left-hand minimum. (c) Input voltage increases to vi=vcrit. The left-hand minimum disappears; v moves to the only minimum. (d) Input voltage decreases to vi=0. There are again two minima of the Lyapunov function; v remains at the current minimum. (Same parameter values as in Fig. 6.)

Grahic Jump Location
Fig. 10

Equilibrium value of output y (concentration of protein x2) at different values of the input IPTG (u). There are 3 equilibrium values if -10-6<u<4×10-5. The middle one is unstable, while the other two equilibria are stable. (See Eqs. (5)(7).) (Parameter values of α1=156.25, β1=2.5, α2=15.6, β2=1, K=2.9618×10-5 and η=2.0015 from Ref. [2].)

Grahic Jump Location
Fig. 11

Output y (concentration of x2) as u is slowly increased. Note sharp transition to new equilibrium point. (See Eqs. (5)(7).) (Same parameter values as in Fig. 10.)

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Fig. 12

Beam in a magnetic field with two magnetic sources

Grahic Jump Location
Fig. 13

The equilibrium points of beam, Eq. (8), in a two-source magnetic field M are the roots of (1/2)x(1-x2)+M=0. For |M|<(1/33)≈0.2 there are three solutions of f(x)=M while for larger values of M there is only one solution. Hence for small amplitudes of the magnetization there are three equilibrium points while for larger amplitudes there is only one equilibrium point. When there are three equilibrium points, the middle point is unstable while the outer points are stable.

Grahic Jump Location
Fig. 14

Input-output diagram for magnetoelastic beam, Eq. (8), with input 14sin(ωτ), with different frequencies ω. Looping behavior as the state moves between different equilibrium points is evident. (a) ω=10-5(…), 5×10-5(--), 10-4 (-). The three curves are indistinguishable, indicating that the system is rate-independent at low frequencies of the input. (b) ω=5×10-4(…), 10-3(--), 10-2 (-). At these higher frequencies, rate dependence is apparent.

Grahic Jump Location
Fig. 15

Gears, showing mechanical play. Since the gears are not perfectly meshed, when one gear turns there is a period of time when the driven gear is stationary before it engages and is turned by the first gear. (http://www.sfu.ca/adm/gear.html, used by permission, Robert Johnstone, SFU.)

Grahic Jump Location
Fig. 16

Backlash, or linear play. The rod with position w(t) is moved by a cart of width 2r with center position u(t). As long as the rod remains within the interior of the cart, the rod does not move. Once one end of the cart reaches the rod, the rod will move with the cart.

Grahic Jump Location
Fig. 17

Play operator with play r=2. (a) Input-output diagram, static model, Eq. (11), and dynamic model, Eq. (12). (a=1000 for dynamic model.) The graphs of the dynamic and static models are indistinguishable. (b) Input u (…) and output w (-). Note that w remains constant after a change in sign of u· until the difference |w-u|=2.

Grahic Jump Location
Fig. 18

Qualitative behavior of Gibb's energy for a magnetic dipole as H is varied. (a) Gibbs energy when H0=0. There are two equilibrium points, M-* and M+*. In this diagram, the dipole is at M-*. (b) Gibbs energy after increasing H0 where there are still two equilibrium points. The dipole remains at M-*. (c) If H0 is further increased, eventually only one minimum exists. The dipole moves to the remaining minimum, M+*.

Grahic Jump Location
Fig. 19

Magnetization versus magnetic field for a magnetostrictive actuator. The outer, or major, loop is obtained by increasing the input magnetic field to its maximum value and then subsequently decreasing it. The inner loops are obtained by increasing the input to an intermediate value and then decreasing. (Ref. [34], used with permission.)

Grahic Jump Location
Fig. 20

Response of Bouc–Wen model, Eq. (20). (a) Input x(t) with varying frequency, (b) output Φ(t) for input shown in (a). Only the scale, not the shape of the curve, changes as the input frequency changes. (c) Φ(t) versus input x(t) shown in (a). The curve forms a single loop, reflecting rate independence of the model. (Parameter values are those used in identification of a magnetorheological damper in Ref. [43]: D=1, n=1,A=120,γ=300cm-3,β=300cm-1, α=0.001, k=27.3Ns/cm.)




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