Nonlinear Dynamics, Volume 1 pp 159-165 | Cite as

# Application of Nonlinear Displacement-Dependent Dampers in Suspension Systems

## Abstract

Dampers are frequently used for vibration reduction and isolation. While passive dampers are still being used, semi-active dampers such as MR and ER dampers have found their ways to expensive commercial applications. They use magnetorheological (MR) or electrorheological (ER) fluids as the damper fluid, subjected to a controllable field to obtain variable damping. These dampers are more efficient; however, due to the high cost of MR and ER fluids, they are too expensive to be used in the suspension systems for passenger cars Nonlinear Displacement-Dependent (NDD) damper has been recently developed for vibration reduction and control in mechanical systems. The damping coefficient of the NDD damper increases as the velocity reduces which compensates for the reduced velocity. This low-cost damper results in a smoother and more consistent damping force and energy dissipation and resolves the major drawback of the linear dampers, which is their poor performance, and the semi-active ones, which is their high cost. It also causes smaller force transmission in vibration isolation applications. In this paper the performance of the NDD damper in suspension systems has been investigated. The suspension system equipped with the NDD damper is modeled and its performance is compared to that of the conventional models.

## Keywords

Suspension systems Nonlinear displacement-dependent damper Vibration reduction Ride comfort Handling## 17.1 Introduction

Vibration control is an issue in many engineering disciplines, mechanical, civil, aerospace, and automotive to pick a few [1, 2, 3, 4]. Heavy machinery, bridges, landing gear, suspension systems are all types of problems where damping of vibration is an integral part of the product working properly [5, 6, 7]. Automobiles use dampers as a means to control road vibration. If not for suspension dampers, riding in a vehicle would be extremely uncomfortable and dangerous. The objective of a good suspension damper is to be affordable and to reduce vibration the safest, most comforting way. This paper will focus on investigating the performance of the NDD damper in suspension systems. We model the suspension system equipped with the NDD damper and compare its performance to that of the conventional models.

Generally, there are three distinct types of damping systems, passive, semi-active, and active damping systems [9, 10, 11, 12]. Passive damping is the simplest, most cost effective way of damping. This method is very robust with respect to structural uncertainties. However, the damping values that can be obtained are moderate. The intended increase of performance however is accompanied with a loss of robustness, with respect to suboptimal tuning due to structural uncertainties [13]. Adaptive-passive and semi-active vibration isolation involve changing the system properties, such as damping and stiffness as a function of time [14]. Magnetorheological (MR) dampers have, over the last several years, been recognized as having a number of attractive characteristics for use in vibration control applications [15]. Semi-active control strategies can maintain the reliability of passive devices using a very small amount of energy, yet provide the versatility, adaptability and higher performance of fully active systems [14, 16, 17]. MR fluids were developed in the 1940s, and consist of a suspension of iron particles in a carrier medium such as oil [18]. This type of system costs more money however produces higher performance of damping vibrations as they can recalculate hundreds of times per second. The final damping system is active damping. Active dampers are operated by using an external power which in most cases is provided by hydraulic actuators. The main disadvantages of active dampers are their high power consumption, size, heavy weight, and cost [19]. Semiactive dampers are a compromise between the active and passive dampers [20].

Current passive systems are not the most effective because of the limited variability of the system. These dampers are mostly linear. Semi-active dampers are great with performance, however the costs are high. The costs of MR dampers can be justified in high performance, exotic automobiles but not in highly manufactured cars [19]. Active dampers are not prevalent in the automotive industry because of the size, weight and energy needed.

This paper is organized as follows. In Sect. 17.2 the mechanism of the NDD damper is described. Mathematical formulation of a suspension system equipped with the NDD damper is performed in Sect. 17.3. The results of numerical simulations are discussed and concluded in Sects. 17.4 and 17.5.

## 17.2 Mechanism of the NDD Damper

*c*is the damping coefficient.

*c*can be obtained by

*D*and

*d*are the cylinder diameter and the opening fluid gap diameter, respectively. Also,

*λ*= 8

*π μ L*where

*μ*denotes for dynamic viscosity of the fluid and

*L*is the piston width. For a set of parameters

*D*,

*d*and

*L*, the damping coefficient has a constant value. We manipulate the mechanism to make the linear damper into nonlinear and displacement dependent. We consider the following function in Cartesian

*r*-

*u*coordinates:

*u*-axis. The function-dependent shaped part must be assembled into the linear damper, so that the origin of the coordinates is located on the center of the piston opening. The nonlinear part must be fixed such that the fluid will travel on its outer surface and the inner surface of the orifice as shown in Fig. 17.2. According to Fig. 17.2, during the motions of the piston on the

*u*-axis, the area of the fluid gap is inconstant and the damping coefficient is consequently varied. Therefore, the ordinary linear damper with a constant damping coefficient is converted to the nonlinear damper with a variable displacement-dependent damping coefficient.

*d*− 2

*r*. Therefore, we substitute

*d*− 2

*r*for

*d*into Eq. (17.2) as follows:

*r*from Eq. (17.4) leads to

## 17.3 Mathematical Formulation of Suspension System

*m*

_{1}, for the chasis, and

*m*

_{2}, for wheel, connected to each other by a spring and a damper, with the stiffness

*k*

_{1}and damping coefficient

*c*

_{1}, respectively. The wheel is in contact to the ground by a tire which is modeled by a linear spring,

*k*

_{2}, and a linear damper,

*c*

_{2}. The road condition is modeled as an input

*u*(

*t*) shown in the figure. The coordinates

*x*

_{1}and

*x*

_{2}measure the displacements of the chasis and the wheel, respectively. We obtain a mathematical description of the model in state-space form. We define a state vector as \(\mathbf{z} = \left [x_{1},\,x_{2},\,\dot{x_{1}},\,\dot{x_{2}}\right ]\).

*k*

_{1}and

*c*

_{1}are constant. We obtain the governing differential equation in state-space form:

*k*

_{1}and

*c*

_{1}are no longer constant. We obtain the following state-space model for the NDD suspension model:

**B**and

**C**are the same as defined above, and \(\mathbf{f}_{\mathrm{n}} = \left [\begin{array}{*{10}c} 0\\ 0 \\ -\frac{k_{1}} {m_{1}} (z_{1} - z_{2})^{3} \\ \frac{k_{1}} {m_{1}} (z_{1} - z_{2})^{3}\\ \end{array} \right ]\) is the function of the nonlinear terms. Please note that the linear damper coefficient

*c*

_{1}is replaced by the NDD damper inconstant coefficient

*c*

_{NDD}which is defined in Eq. (17.6) as a function of the state variables.

## 17.4 Numerical Simulations

For the purpose of numerical simulations and comparing the performance of the NDD suspension with the linear suspension system with derive the system properties from [14]. The properties for the linear model are given as: \(m_{1} = 466.5\,\mathrm{kg},\,m_{2} = 49.8\,\mathrm{kg},\,k_{1} = 5700\, \frac{\mathrm{N}} {\mathrm{M}},\,k_{2} = 135000\, \frac{\mathrm{N}} {\mathrm{M}},\,c_{1} = 1000\,\frac{\mathrm{Ns}} {\mathrm{M}}\) and \(c_{2} = 1400\,\frac{\mathrm{Ns}} {\mathrm{M}}\). For the NDD damper we consider the following parameters: \(s = \frac{1} {3},\,D = 0.12\,\mathrm{m}\) and *d* = 0. 03 m. We also define *r*_{ max } = 0. 0135 m as the maximum allowed radius of the nonlinear part in order the avoid high accelerations at the end of each cycle. This means that *r* of the nonlinear part will not exceed *r*_{ max }.

*t*= 1 s. However, the peak of the linear system vibration is twice that of the NDD system. Also, for the NDD system the chassis does not oscillate about the equilibrium and dies out quickly. On the other hand, for the linear suspension system, the chassis oscillates untill it dies out after a longer period of time.

## 17.5 Conclusions

In this paper we proposed the use of the NDD damper in suspension systems. We described the mechanism of the NDD damper and obtained its governing equation. We modeled a linear suspension system as well as a nonlinear one.In the nonlinear model we used a nonlinear spring and the NDD damper. We obtained the governing differential equations of both systems. We simulated the models using numerical methods. Our results describes a higher performance from the nonlinear systems in both ride comfort and handling.

## References

- 1.Ilbeigi, S., Chelidze, D.: Model order reduction of nonlinear euler-bernoulli beam. In: Nonlinear Dynamics, vol. 1, pp. 377–385. Springer, New York (2016)Google Scholar
- 2.Ilbeigi, S., Chelidze, D.: Reduced order models for systems with disparate spatial and temporal scales. In: Rotating Machinery, Hybrid Test Methods, Vibro-Acoustics & Laser Vibrometry, vol. 8, pp. 447–455. Springer, New York (2016)Google Scholar
- 3.Karimi, M., Jahanpour, J., Ilbeigi, S.: A novel scheme for exible nurbs-based c2 ph spline curve contour following task using neural network. Int. J. Precis. Eng. Manuf.
**15**(12), 2659–2672 (2014)CrossRefGoogle Scholar - 4.Asl, M.E., Niezrecki, C., Sherwood, J., Avitabile, P.: Predicting the vibration response in subcomponent testing of wind turbine blades. In: Special Topics in Structural Dynamics, vol. 6, pp. 115–123. Springer, New York (2015)Google Scholar
- 5.Adhikari, S.: Damping models for structural vibration. Ph.D Thesis, University of Cambridge (2001)Google Scholar
- 6.Asl, M.E., Niezrecki, C., Sherwood, J., Avitabile, P.: Design of scaled-down composite i-beams for dynamic characterization in subcomponent testing of a wind turbine blade. In: Shock & Vibration, Aircraft/Aerospace, Energy Harvesting, Acoustics & Optics, vol. 9, pp. 197–209. Springer, New York (2016)Google Scholar
- 7.Behmanesh, I., Yousefianmoghadam, S., Nozari, A., Moaveni, B., Stavridis, A.: Effects of prediction error bias on model calibration and response prediction of a 10-story building. In: Model Validation and Uncertainty Quantification, vol. 3, pp. 279–291. Springer, New York (2016)Google Scholar
- 8.Hassan, S.A.: Fundamental studies of passive, active and semi-active automotive suspension systems. Ph.D Thesis, University of Leeds (1986)Google Scholar
- 9.Go, C.-G., Shi, C.-H., Shih, M.-H., Sung, W.-P.: A linearization model for the displacement dependent semi-active hydraulic damper. J. Vib. Control (2010)MATHGoogle Scholar
- 10.Ilbeigi, S., Jahanpour, J., Farshidianfar, A.: A novel scheme for nonlinear displacement-dependent dampers. Nonlinear Dyn.
**70**(1), 421–434 (2012)MathSciNetCrossRefGoogle Scholar - 11.Jahanpour, J., Ilbeigi, S., Porghoveh, M.: Resonant analysis of systems equipped with nonlinear displacement-dependent (ndd) dampers. In: Nonlinear Dynamics, vol. 1, pp. 67–82. Springer, New York (2016)Google Scholar
- 12.Jahanpour, J., Porghoveh, M., Ilbeigi, S.: Forced vibration analysis of a system equipped with a nonlinear displacement-dependent (ndd) damper. Sci. Iran. Trans. B Mech. Eng.
**23**(2), 633 (2016)Google Scholar - 13.Holterman, J., de Vries, T.J.: a comparison of passive and active damping methods based on piezoelectric elements (2001)Google Scholar
- 14.Liu, Y., Waters, T., Brennan, M.: A comparison of semi-active damping control strategies for vibration isolation of harmonic disturbances. J. Sound Vib.
**280**(1), 21–39 (2005)MathSciNetCrossRefMATHGoogle Scholar - 15.Pare, C.A.: Experimental evaluation of semiactive magneto-rheological suspensions for passenger vehicles. Ph.D Thesis, Virginia Polytechnic Institute and State University (1998)Google Scholar
- 16.Ghane, M., Tarokh, M.J.: Multi-objective design of fuzzy logic controller in supply chain. J. Ind. Eng. Int.
**8**(1), 10 (2012)CrossRefGoogle Scholar - 17.Asl, M.E., Abbasi, S.H., Shabaninia, F.: Application of adaptive fuzzy control in the variable speed wind turbines. In: International Conference on Artificial Intelligence and Computational Intelligence, pp. 349–356. Springer (2012)Google Scholar
- 18.Jansen, L.M., Dyke, S.J.: Semiactive control strategies for mr dampers: comparative study. J. Eng. Mech.
**126**(8), 795–803 (2000)CrossRefGoogle Scholar - 19.Preumont, A.: Vibration Control of Active Structures: An Introduction, vol. 179. Springer Science & Business Media, New York (2011)MATHGoogle Scholar
- 20.Alanoly, J., Sankar, S.: A new concept in semi-active vibration isolation. J. Mech. Trans. Autom. Des.
**109**(2), 242–247 (1987)CrossRefGoogle Scholar - 21.Zhuge, J., Formenti, D., Richardson, M.: A brief history of modern digital shaker controllers. Sound Vib.
**44**(9), 12 (2010)Google Scholar - 22.Hassaan, G.A.: Car dynamics using quarter model and passive suspension, part i: Effect of suspension damping and car speed. Int. J. Comput. Techniques
**1**(2), 1–9 (2014)Google Scholar