Parameter influences on the static behavior of the Bouc- Wen model

The Bouc-Wen model, (3.7) contains four parameters, which have a strong influence on the hysteresis behavior. It is therefore very important to have a clear view on the influence of each parameter. Wong et al. [29] carry out a parametric study for the Bouc-Wen model. A similar approach is used here to gain a good understanding of the complete model for quasi-stctic in both loading-unloading and unloading-reloading is equal to 2 0

I

z

In.

Hence the parameter j3 controls the stiffness change when the sign of k alters.

Table 3.2: The slope of the hysteresis loop

The shape of the hysteresis curve is mainly determined by the parameters ,B and y. Dif- ferent combinations of ,6 and y lead to various hysteretic loops with different stiffness char- acteristics. A closer look is taken at the fashion of the slope in the case of loading, 5

>

0 and unloading x

<

0 respectively. This leads to five types of hysteresis loops which are physically meaningful, corresponding to five combinations for ,B and y , which are summarized below.

(P +

^{y )}

^>

^{0 and}

^{( p}

^{- y )}

^>

⁰ will be shown that these combinations do not yield a hysteretic curve.

For the above five cases simulations have been carried out in Matlab, using a Simulink model, see Appendix C. The hysteresis loops for these five cases are shown in figures 3.1- 3.5, where the arrow indicates the direction of motion. The parameters a and n are both kept constant at 1.0. The legend shows the values for the amplitude of the displacement.

As mentioned in section 2.4 hysteresis is a rate-independent effect and therefore the velocity 2 has no influence on the shape of the hysteretic curve. In the first three cases softening nonlinear behavior occurs during loading due to the fact that

(P +

^{y )}

^>

0. In the fourth case, under loading, the system exhibits hardening non-linearity due to the fact that

(P +

^{y )}

^<

^{0 ,}

whereas in case five, where ( p

+

^{y ) =} 0 the system behaves quasi-linearly during loading.

16 3.2 Parameter influences on the static behavior of the Bouc-Wen model

Figure 3.1: Hysteresis loop for

(Pf

^y)

>

0 and ( p - 7)

>

0

Figure 3.3: Hysteresis loop for(P

+

^y)

^>

⁰

and

( p

- 7) = 0

Figure 3.2: Hysteresis loop for (P

+

^y)

^>

0 and (p - y)

<

0

5 4 3 2

-

_z 1

- - 0

-

+4 hl -1 -2 3 -4 -5

-2 -15 -1 -05 0 0 5 1 1 5 2

~ ( t ) [ml

Figure 3.4: Hysteresis loop for (P

+

^y)

^<

⁰

and

(p

^{- y)}

>

0

Figure 3.5: hysteresis loop for

(P +

^{y) = 0 and (P - y)}

>

0

Figure 3.6: Hysteretic curves for

P

⁼ 0.8, y = 0.2 and

P

⁼ 0.5, y = 0.5

It has to be noted that the figures 3.1 and 3.3 produce almost identical hysteretic curves.

This rises the question whether a redundancy is present in the model and whether y may not be omitted from the model. Figure 3.6 depicts the hysteretic curves for the combinations ,B = 0.8, y = 0.2 and

P

⁼ 0.5, y = 0.5. Note that for both scenarios the following equation holds

Indeed the hysteretic curves resemble each other, although a difference is present due to the different value of ,B since a higher value for ,B leads to a larger stiffness change when the sign of 2 changes. A generalised form for equation (3.10) looks like

,B

+

^{y = constant} (3.11)

By substituting relation (3.11) into equation (3.8) a general conclusion can be drawn. If equation (3.11) holds and ,B and y are both larger than zero, the slope of the hysteretic curve will be the same as long as sgn(2) = sgn(z). Therefore, the figures 3.1 and 3.3 produce almost the same hysteretic curve.

This "redundant parameter" phenomenon has to be dealt with in the identification process.

Imposing one of the constraints ,B

>

7 , ,B = 7 and ,B

<

y during the identification determines which softening scenario the final solution will be part of. For instance, imposing the second constraint ,8 = y leads to the third case, as depicted in figure 3.3. However, when the system is a priori known to display softening hysteresis loops, an altered version of the Bouc-Wen model omitting the parameter y may be adopted for identification. If the parameter y is omitted, the Bouc-Wen model reduces to the Ozdemir-model [19]. Bhatti and Pister [7] and Fujita et al. [ll] have used the latter to model nonlinear damping devices with softening hysteretic behavior.

Except from the combination of the parameters

P

and y, also the influence of each param- eter separately has to be investigated. Subsequently, simulations are performed with different values for ,B and y. Figure 3.7 shows the results of these simulations. The parameters cr and n are again set equal to 1.0. The values for y in (a) and ,B in (b) are 0.2 and 0.8 respectively.

The same trends can be seen in the case of varying ,B as well as the case of varying y if another

18 3.2 Parameter influences on the static behavior of the Bouc-Wen model

Figure 3.7: The influence of the parameters ,B (a) and y (b) on the hysteresis loop

value for y or ,B is used, even when the system changes from hardening into softening when the value for ,B or y changes.

A remark must be made. Earlier research shows that the wire rope spring possess a kind of hysteretic behavior which cannot be described by the Bouc-Wen model. Ni et al. [2] and Leenen [14] show that a wire rope spring possess softening stiffness for small amplitudes, which via quasi-linear behavior, gradually changes into hardening stiffness for increasing amplitudes.

This is called soft-hardening and is shown in figure 3.8. Figure 3.8 has been contructed based upon simulation results with a modified version of the Bouc-Wen model as proposed by Ni et a1. [2].

Figure 3.8: Soft-hardening hysteresis Figure 3.9: Hardening overlap It can be concluded from table 3.2 that for z

>

0 the slope from the hysteretic curve cannot manifest itself first as decreasing and then as increasing, without changing the value for ,B or y and therefore soft-hardening cannot be described.

Ni et al. [2] also observe that the loading path for various amplitudes is the same, as depicted in figure 3.9. Since the used wire rope spring possess hardening behavior in tension they call this phenomenon hardening overlap, which cannot be described by the Bouc-Wen model. This is confirmed by figure 3.4 as for increasing amplitudes the hysteretic curve becomes somewhat thicker and hence no hardening overlap is present. As we will see in

chapter 4 the used wire rope spring exhibits this hardening overlap behavior as well.

The quantity n governs the smoothness of the transition from linear to nonlinear range.

Increasing values for n lead to more elasto-plastic behavior. In the limiting case, where n = CQ,

the system exhibits true elasto-plastic behavior. When n ⁼ oo equation (3.8) approximates infinity for z > 0 and for z

<

0. However, as will be proven in section 3.4 z has a maximum,

Summarizing, the Bouc-Wen model can describe various stiffness characteristics by a proper choice of ,!3 and y. When the system exhibits softening behavior different values for ,8 a,nd y can lead to almost identical hysteresis loops. Therefore, a constraint for /3 and 7 can be added or an altered version of the Bouc-Wen model, the Ozdemir model, can be used.

The Bouc-Wen model fails in describing soft-hardening hysteresis and hardening overlap, phenomena that are found in the hysteretic behavior of wire rope springs. To cope with these phenomena Ni et al. [2] propose a modified version of the Bouc-Wen model, which will be discussed in section 3.4. First however in section 3.3 the influence of the various parameters on the dynamic behavior is adressed.

In document Eindhoven University of Technology MASTER Modelling and identification of the dynamic behavior of a wire rope spring Schwanen, W. (pagina 26-30)