As mentioned earlier; our model impact system shows hysteretic behavior. There is a `catastrophic' jump produced when one of the system parameters is changed `adiabatically'. From the practical point of view such discontinuous dynamical response of the system can lead to damage or wear of mechanical parts as in case of transition from nonimpacting to impacting state at the grazing bifurcation point.

The practical importance of the phenomenon of hysteresis motivates the following work - in the low amplitude region, a qualitative analysis on the origin of hysteresis is carried out, and based on analytical results obtained, a quantitative analysis of the dependence of the hysteretic region (in the parameter space) on other system parameters is done.

Figure 1 shows the three period-1 solutions at the Poincaré section, taken when the drive phase goes through p, as A is varied from A » 0.40 to A = 2.61. The other system parameters in the plot are set at K = 0.7, b = 0.05, and w = 1.9. The color coding of the orbits are done according to the following convention:

If the system is on the P₁₀₀ orbit, then, as A is increased, the system stays on the nonimpacting solution until A = 2.61, at which point the orbit makes a grazing collision with the wall and jumps to the already existing symmetric impacting P₁₁₁ orbit. Now if A is decreased the system stays on the impacting solution. The value of A has to be decreased considerably (to A = 0.4) before the system jumps back to the nonimpacting solution.

The orbits, shown in figure above, at the constant phase Poincaré section and, in figure below, phase plane projection are obtained using results derived here. For fixed values of the system parameters, we first determine if there exists a solution to Eq.11. For w = 1.9, b = 0.05, and K = 0.7, there exists no solutions for A < 0.4. For 0.4 £ A£ 2.61, two symmetric solutions coexist (shown in red and green). Using Eq.11, we find t₀, which is used in Eq.9 to determine a₀. Finally, using any one of the equations, q₀ is determined. Having determined these unknowns, we use the general solution to determine the value of x and v for one complete cycle (for plotting Fig.2) or their value at some given phase (for plotting Fig.1). One of the symmetric solutions is stable and the other is unstable. Stability of the orbits is determined by calculating the eigenvalues of the Jacobian of the Poincaré map taken at the constant phase Poincaré section. The nonimpacting solution is trivially obtained by the steady state solution given by the equation 2.

ORIGIN OF HYSTERESIS: QUALITATIVE ANALYSIS

To understand the phenomenon of hysteresis shown by our rigid wall model system, a frequency response characteristic was plotted for the bilinear model of the system described below. Instead of treating collision as an instantantaneous velocity reversal process, we assume collision forces act in a continuous manner. In this approach the analysis of the colliding bodies is done by including the continuous forces in the system equations of motion during the finite time interval of the collision with the wall which we call the contact period. The simplest continuous force model, during the contact period, treats the wall as a linear spring with a much larger spring constant, k₂, as compared to the spring constant of the linear oscillator which is normalized to unity. When the mass is between the walls, the system is modeled by Eq. , but the impact is modeled as a linear spring with a high spring constant k₂ >> 1 with its equilibrium position located at 1 for the right wall (and -1 for the left wall) and a viscous damping b₂ effective only during the contact process. For k₂ ® ¥, the bilinear model reduces to a stiff wall model with instantaneous impact. Thus, the equation of motion during the contact process is given by

æ ç ç ç ç ç ç ç ç ç ç è . x   . v   . f   ö ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ø = æ ç ç ç ç ç ç è v -2b2 . x   - k2*(x-xs) - x + A cos(f) w ö ÷ ÷ ÷ ÷ ÷ ÷ ø ,

(1)

where x_s is the equilibrium position of the `wall'; x_s = 1 when x > 1 and x_s = -1 when x < -1.

Figure 3 shows the resonance plot for A = 0.2, b₂ = 0.1, (a) k₂ = 10 and (b) k₂ = 50. The plot is made in the manner similar to the way bifurcation plot was obtained except that, instead of plotting a value of x at some constant phase, we plot the maximum value of x that occurs during a drive cycle for each value of w and then w is changed in steps of 0.005.

The branch labeled E-F-S was obtained by scanning w upward from w = 0.1 and the branch labeled H -G was obtained by scanning w downward from 2.0. The branch E -F and branch H-G are the points on the nonimpacting orbit and the branch F -S are the points on the impacting orbit, at the Poincaré section defined by the surface x = maxx(t) for t = [0,2p]. The horizontal dotted line drawn at x_max = 1 indicates the position of the wall. Following the H-G branch, which represents the nonimpacting orbit, as w is decreased the only stable state for the system is the nonimpacting orbit until w = 1.43 for k₂ = 10 (shown in Fig. 3a) and until w = 1.70 for k₂ = 10 (shown in Fig. 3b). There are two coexisting stable periodic orbits, nonimpacting and impacting solution, as the value of w is decreased to 1.08 (point labeled G). This is the value of w for which the nonimpacting solution begins to `graze' the wall. This bifurcation is analogous to the grazing bifurcation for the case of rigid wall. From this value of w to the point labeled F, the only stable existing solution is the impacting solution. For values of w between points labeled F and E, the only stable solution is the nonimpacting solution. Following the impacting solution branch F-S, in the direction of increasing w, the impacting solution suddenly disappears and the system settles down to the nonimpacting branch G-H. The arrow indicates the transition from the impacting to nonimpacting solution. This is analogous to the destruction of stable impacting solution as a result of saddle-node collision for the case of a rigid wall model system. The point labeled S and G must be connected by an unstable impacting solution which is created as a result of saddle-node bifurcation at w = 1.43 and w = 1.7 for k₂ = 10 and k₂ = 50, respectively. Thus, the grazing of nonimpacting solution and its destruction at point G is due to stable-unstable orbit collision. It should be noted that as k₂ ® ¥ the impacting solution branch (F- S) would become flat and lie on top of the dotted line shown in Fig.1c d. This is consistent with the fact that for the case of the rigid wall the x_max for impacting solution is one.

To obtain a qualitative understanding of the phenomenon of hysteresis observed, when the drive amplitude is varied, for the case of rigid wall model system, we draw a schematic of frequency response plots, for the case of bilinear model, for different values of the drive amplitude. Figure 4 shows the frequency response plot for four different values of the drive amplitude, where A₁ < A₂ < A₃ < A₄. The position of the right wall is shown as a horizontal dotted line. Stable solutions are represented as solid line curves and the unstable orbits are represented in dashed lines. For fixed value of the drive frequency, the w = w₁ line shows the change in the dynamic response of the system as the drive amplitude (A) is increased from A₁ to A₄ (the points labeled P₁, P₂, P₃, P₄). For A₁ the system is at point P₁, the nonimpacting solution and as the amplitude is increased to A₂ the system is still at the nonimpacting solution, the point labeled P₂. As the A is increased to A₃ the oscillator just begins to graze with the wall (point labeled as G) and jumps to the impacting solution represented as point P₃ in the plot. The line w = w₂ shows the effect of decreasing A from A₄ to A₁ - the jump from impacting solution to nonimpacting solution. At point labeled S the impacting solution is destroyed as a result of collision between stable and unstable impacting solution and jumps to the nonimpacting solution.

PARAMETRIC STUDY OF HYSTERESIS

Using Eq. 3 and 13, we present results from a parametric study of the phenomena of hysteresis. Figure 5 shows the locus of points in the parameter space of (A-w) where the impacting symmetric P₁₁₁ orbit is created as a result of saddle-node bifurcation, and the locus of points where the impactless P₁₀₀ orbit goes through grazing bifurcation.

The curves labeled GB are plotted using 3, and the other three curves are plotted using Eq. 13. The plots are made for K = 0.7 and three different values of b = 0.6, 0.4, 0.05. The symbol plots are for data obtained from numerical solution.

The set of three curves (for different values of b) labeled GB are locus of points (in A-w space) where the nonimpacting orbit goes through the grazing bifurcation and the set of other three curves are locus of points where the symmetric impacting solution is created as a result of saddle-node bifurcation. The region in the parameter space between the two curves, given by Eq.  and for fixed values of b and K, are the values of A and w for which both the impactless P₁₀₀ and symmetric P₁₁₁ orbits coexist.

For fixed values of b and K, as we go further away from the resonant frequency, for fixed value of w (or A) there is a wider range of A (or w) for which we observe bistability. It is also clear from the plot that as b increases or K decreases, the area in A- w space for which the P₁₀₀ and P₁₁₁ orbits coexist decreases. Thus, we can conclude that as damping increases, the area between the curves given by Eq.  and decreases. It is also clear from these plots that for fixed values of b and K, the two curves defined by these equations meet each other at approximately the zero damping resonant frequency, w = 1. Equation  reduces to Eq.  as W® 0. For b = 0, W® 0 at w = 1, as b is increased, W® 0 for values of w which are slightly less than 1. This can be explained qualitatively in terms of the dependence of the resonant frequency on b.