Bouc-Wen model of hysteretic damping

Available online at www.sciencedirect.com

ScienceDirect Available online at www.sciencedirect.com Procedia Engineering 00 (2017) 000–000

ScienceDirect

www.elsevier.com/locate/procedia

Procedia Engineering 201 (2017) 549–555

3rd International Conference “Information Technology and Nanotechnology”, ITNT-2017, 25-27 April 2017, Samara, Russia

Bouc-Wen model of hysteretic damping A.M. Solovyova*, M.E. Semenova,b,c, P.A. Meleshenkoa,b, A.I. Barsukovc a

Voronezh State University, Universitetskaya sq.1, 394006 Voronezh, Russia Zhukovsky-Gagarin Air Force Academy, Starykh Bolshevikov st.54“A”, 394064 Voronezh, Russia c Voronezh State Technical University, Dvadtsatiletiya Oktyabrya st.84, 394006 Voronezh, Russia

b

Abstract In this work we consider the dynamics of a mechanical system under external force with a damping part taking into account the hysteretic nature of the damper. As a mathematical base of the hysteretic damper we use the Bouc-Wen model. The simulation results in the form of the force-transfer function and demonstrates the “efficiency” of the hysteretic damper in comparison with the nonlinear viscous damper. © 2017 The Authors. Published by Elsevier Ltd. © 2017 The Authors. Published by Ltd. committee of the 3rd International Conference “Information Technology and Peer-review under responsibility ofElsevier the scientific Peer-review under responsibility of the scientific committee of the 3rd International Conference “Information Technology and Nanotechnology. Nanotechnology”. Keywords: linear and nonlinear viscous damper; hysteretic damper; Bouc-Wen model; force transfer function

1. Introduction Dampers and damping processes have a long history and have their relevance (both from the fundamental and applied points of view), especially in the present days due to the development of modern impact-vibrational systems (see, e.g., [1]). The damper is a device which can be applied for damping the mechanical, electrical and other modes of vibration arising in the machines and mechanical systems during their operation. The problem of damping is important and has a wide range of applications. In the general case, damping is a process whereby the energy is taken from the vibrating system and absorbed by the surroundings. Examples of damping include:

* Corresponding author. Tel.: +7-908-131-4174. E-mail address: [email protected] 1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the 3rd International Conference “Information Technology and Nanotechnology.

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the 3rd International Conference “Information Technology and ­Nanotechnology". 10.1016/j.proeng.2017.09.605

A.M. Solovyov et al. / Procedia Engineering 201 (2017) 549–555 A.M. Solovyov et al. / Procedia Engineering 00 (2017) 000–000

550 2

   

internal forces of a spring; viscous force in a fluid; shock absorber in a car; anti-seismic damping devices in buildings.

It should also be noted that damping devices are widely used in modern avionics (e.g. the damper of aero-elastic vibrations; such damper is an electronic system for automatic cancelation of short aircraft vibrations that inevitably arise when the flight modes change). It is well known that for a wide range of oscillations of mechanical systems the model of linear viscous damping (which is based on the energy dissipation due to viscous friction) is widely used. However, this type of damping has low “efficiency” (this efficiency can be understood in terms of the observable mechanical characteristics such as a force-transfer function) outside the region of resonance of the system. One of the way to solve this problem is to use a nonlinear viscous damper [2,3,4,5,6,7] or a damper with hysteretic properties [8,9]. The main purpose of the present work is to study the dynamics of a mechanical system under various external perturbations (forced oscillations) in the case of the presence of a damping block. We would be particularly interested in a damper with hysteretic properties. As a mathematical model of such hysteretic damper we consider the phenomenological Bouc-Wen model [10,11,12]. 2. Physical system: equation of motion In this section we consider a mechanical system under an external force in the presence of the damping part fd(t) as shown in Fig. 1. This system can be presented as a cylinder with mass M under external force f(t) of harmonic nature. In the cylinder there is a car of mass m (moving without friction in the horizontal plane) connected to the border by a spring with stiffness k.

Fig. 1. Considered physical system.

Let us assume that force f(t) is described by the relation (harmonic force):

2 f (t )  Y  sin(t ),

(1)

where Yω2 is the amplitude and ω is the frequency of the force. The equation of motion can be written as (we consider a one-dimensional motion of the car along x axis)

 My  kz  f d (t )  f (t ),  mx  kz  fd (t )  0,   2 t ) Y  sin(t ) , z (t ) y (t ) x(t ) .  f (

(2)



A.M. Solovyov et al. / Procedia Engineering 201 (2017) 549–555 A.M. Solovyov et al. / Procedia Engineering 00 (2017) 000–000

551 3

Here the overdot is used for the time derivative and z(t) is the relative displacement. 3. Damping: linear and nonlinear viscous damping Let us consider the case of viscous damping. In the general case, the viscous friction can be described as follows:

n  fd (t ) c 1 z (t ) z , n0,

(3)

where c is the damping coefficient. The case n=0 corresponds to a linear viscous friction. For n>0 the nonlinear viscous friction takes place [4,7]. Remembering (3), the equation of motion for relative displacement z(t) becomes

z (t ) 

M m Mm

Y 2  sin(t ). c[1  z(t )]n z(t )  kz(t )  M

(4)

We introduce the dimensionless variables

 



mM Y  ,  ,A , 0 mM M

u z  ,  0t ,  0

k



 ,,

c

0 

(5) .

Using these variables, the final equation of motion for the mechanical system under consideration, taking into account the nonlinear viscous friction, has the form: 2 d u n du 2   (1  u )  u  A sin( ). 2 d  d

(6)

4. Hysteretic damping: Bouc-Wen model In this section we consider the nonlinear hysteretic damper based on the Bouc-Wen model. Bouc-Wen model has been widely used to describe the nonlinear hysteretic systems and can be formulated in the form of a nonlinear differential equation which describes the dependence of the output on the local history via introducing an extra state variable. Using appropriate choices of parameters in this model, allows us to represent a wide variety of softening or hardening of smoothly varying or nearly bilinear hysteretic behavior. This model has been generalized to include the hysteresis pinching and stiffness/strength degradation. The Bouc-Wen model can be described by the following differential equation



g (t )  u (t ) B    sign  g (t )u (t )     g (t )

p

,

(7)

where B, β>0, γ and p are dimensionless quantities controlling the behavior of the model (p=∞ refers to an elastoplastic hysteresis). It should be noted that the Bouc-Wen model has pure phenomenological nature. For small values of the positive exponential parameter p, the transition from elastic to the post-elastic branch is smooth, while for large values of this parameter this transition is abrupt. Parameters B, β and γ control the size and

A.M. Solovyov et al. / Procedia Engineering 201 (2017) 549–555 A.M. Solovyov et al. / Procedia Engineering 00 (2017) 000–000

552 4

shape of the hysteretic loop. Wen assumed that p are integer while in fact any real positive values of p could be used. Parameter β is positive by definition, while the range of possible values of γ, [-β,β], can be derived from thermodynamics analysis [13]. We consider the hysteretic damper based on the phenomenological Bouc-Wen model. In this case (using the notations introduced above) the damping force can be presented as:

 fd ( )  kb g ( ),   p g ( )  u ( ) B   sign  g ( )u ( )     g ( ) . 





(8)

In this case the equation of motion for the mechanical system under consideration becomes: 2 u  kb g  u    k   A sin    ,  u B    sign  g u     g  g    



p



(9) ,

where kb is the stiffness of the damper. 5. Mechanical characteristics: transmission function Let us consider the main observable characteristics describing the “efficiency” of the damper in the resonance and beyond. These characteristics are force transmission function and “force-displacement” transmission function. The force transmission function is determined by the ratio of the force applied to the cylinder M and the force applied to the car m (Fig. 1). This function reflects the “efficiency” of suppression of the external perturbations by the force transmission from an external source to the load. This characteristic is described as:

T ff 

2 2d x max m0 . 2 2 Y d 1

(10)

The “force-displacement” transmission function is determined by the motion of car m relative to the cylinder M and the force applied to the cylinder. It reflects the “efficiency” of vibration absorption by ability of the damper to reduce the relative motion of the car under external forces. This characteristic is described as

T fd 

max x ( ) . 2 Y

(11)

In order to compare the linear viscous, nonlinear viscous and hysteretic dampers, based on Bouc-Wen model, these characteristics will be used in our simulations. 6. Simulation results In order to compare viscous and hysteretic dampers we focus on the numerical results for two characteristics of the system: the force transmission function and “force-displacement” transmission function. For the nonlinear viscous damper we use the following set n={0,2,4}. For the hysteretic damper we use the following values of parameters kb=100, B=1, β=0.7, γ=0.3, p=6 (as follows from numerical experiments, the characteristics with these values of parameters correspond to the optimal operation



A.M. Solovyov et al. / Procedia Engineering 201 (2017) 549–555 A.M. Solovyov et al. / Procedia Engineering 00 (2017) 000–000

553 5

of the hysteresis model). The following dimensionless characteristics of the mechanical system will be used: M=1, m=1, ζ=0.8, ω0=10; the external force will be described by dimensionless parameters A=1, ω=0,…,30 (with a step 0.2). The simulation results are shown in Fig 2. As can be seen from these figures, the linear viscous damper has high “efficiency” in the resonance region of the system. However outside the resonance region the damping properties sharply decrease. At the same time, although the nonlinear viscous damper can be applied for a wide range of parameters, it turned out to be less “efficient” than the linear damper in the resonance region of the system.

Fig. 2. Force (left) and “Force-displacement” (right) transmision functions.

The hysteretic damper based on the Bouc-Wen model is highly “efficient” both in and outside the resonance region. The disadvantage of the hysteretic damper lies in the fact that its ability to dump the relative motion of the car under external forces is reduced outside the resonance region of the system. Let us now consider the phase portraits for the system for viscous and hysteretic dampers. As the origin of the phase plane we use the instantaneous values of the car's coordinate inside the cylinder x(τ) and its relative speed x   . The phase portraits of the system under consideration are presented in the figures 3-6.

Fig. 3. Phase portraits of the system in the case of linear viscous damper n=0 at Ω=3 (left) and Ω =30 (right).

6554

A.M. Solovyov et al. / Procedia Engineering 00 (2017) A.M. Solovyov et al. / Procedia Engineering 201000–000 (2017) 549–555

Fig. 4. Phase portraits of the system in the case of nonlinear viscous damper n=2 at Ω=3 (left) and Ω =30 (right).

Fig. 5. Phase portraits of the system in the case of nonlinear viscous damper n=4 at Ω=3 (left) and Ω =30 (right).

Fig. 6. Phase portraits of the system in the case of hysteretic damper at Ω=3 (left) and Ω =30 (right).

Figures 3-6 show phase portraits of the system with viscous (linear and nonlinear) and hysteretic damping. The left panels show phase portraits in the resonance region, the right panels refer to high frequency regions. As follows



A.M. Solovyov et al. / Procedia Engineering 201 (2017) 549–555 A.M. Solovyov et al. / Procedia Engineering 00 (2017) 000–000

555 7

from these figures, the hysteretic damper has greater “efficiency” than linear and nonlinear viscous dampers in and outside the resonance region. 7. Conclusions We have considered various damping processes occurring in the oscillations of real mechanical systems with damping parts. Particular attention have been paid on the hysteretic damper based on the phenomenological BoucWen model. We have presented the numerical results for the observable characteristics of the system under consideration such as force transmission function and "force-displacement" transmission function both for the cases of hysteretic and viscous (linear and nonlinear) dampers. The phase portraits for the mechanical system with various kinds of damping have been analyzed. The results allow us to compare the various types of viscous dampers (linear and nonlinear) and the hysteretic damper. Based on the numerical results we can conclude that:  Linear viscous damper has high efficiency in the resonance region of the system, while outside the resonance region its damping properties sharply decrease.  Nonlinear viscous damping can be efficient ina wide range of parameters, but loses its efficiency to in the resonance region of the system.  Hysteretic damper, based on the Bouc-Wen model, has high efficiency both in and outside the resonance region. The disadvantage of the hysteretic damper lies in the fact that its ability to dump the relative motion of the car under external forces is reduced outside the resonance region of the system. Acknowledgements This work has been supported by the RFBR grant No 16-08-00312, 17-01-00251. References [1] V. Babitsky, V. Krupenin, Vibration of Strongly Nonlinear Discontinuous Systems, Springer, Berlin, 2001. [2] E. Rigaud, J. Perret-Liaudet, Experiments and numerical results on non-linear vibrations of an impacting hertzian contact. part 1: harmonic excitation, Journal of Sound and Vibration. 265 (2003) 289–307. [3] Z. Lang, S. Billings, R. Yue, J. Li, Output frequency response function of nonlinear Volterra systems, Automatica. 43 (2007) 805–816. [4] Z. Milovanovic, I. Kovacic, M. Brennan, On the displacement transmissibility of a base excited viscously damped nonlinear vibration isolator, Journal of Vibration and Acoustics. 131 (2009). [5] J. Felix, J. Balthazar, R. Brasil, B. Pontes, On lugre friction model to mitigate nonideal vibrations, Journal of Computional and Nonlinear Dynamics. 4 (2009). [6] Z. Peng, G. Meng, Z. Lang, W. Zhang, F. Chu , Study of the effects of cubic nonlinear damping on vibration isolations using harmonic balance method, International Journal of Non-Linear Mechanics. 47 (2012) 1073–1080. [7] Q.Lv, Z. Yao, Analysis of the effects of nonlinear viscous damping on vibration isolator, Nonlinear Dynamics. 79 (2015) 2325–2332. [8] R. Richards, Comparison of linear, nonlinear, hysteretic, and probabilistic mr damper models, Master’s thesis, Faculty of the Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 2007. [9] M. Latour, Theoretical and Experimental Analysis of Dissipative Beam-to-Column Joints in Moment Resisting Steel Frames, BrownWalker Press, 2014. [10] Qibao Lv, Zhiyuan Yao, Analysis of the effects of nonlinear viscous damping on vibration isolator, Nonlinear Dynamics. 79 (2015) 2325–2332. [11] F. Ikhouane, V. Manosa, J. Rodellar, Dynamic properties of the hysteretic Bouc-Wen model, Systems & Control Letters. 56 (2007) 197–205. [12] F. Ikhouane, J. Rodellar, On the Hysteretic Bouc-Wen Model, Nonlinear Dynamics. 42 (2005) 63–78. [13] T.T. Baber, Y.K. Wen, Random vibrations of hysteretic degrading systems, Journal of Engineering Mechanics. 107 (1981) 1069–1089. [14] M.E. Semenov, A.M. Solovyov, P.A. Meleshenko, Elastic inverted pendulum with hysteretic nonlinearity in suspension: stabilization problem, Nonlinear Dynamics. 82 (2015) 677–688. [15] M.E. Semenov, A.M. Solovyov, A.M. Semenov, V.A. Gorlov, P.A. Meleshenko, Еlastic inverted pendulum under hysteretic nonlinearity in suspension: stabilization and optimal control, Proceedings of the 5th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering Held in Crete Greece. 2 (2015) 2995–3004. [16] .E. Semenov, P.A. Meleshenko, A.M. Solovyov, A.M. Semenov, Hysteretic Nonlinearity in Inverted Pendulum Problem, Springer Proceedings in Physics. 168 (2015) 463–507. [17] M.P. Mortell, R.E. O’malley, A. Pokrovskii, V. Sobolev, Singular perturbations and hysteresis, SIAM, Philadelphia, 2005.