Application of Nonlinear Displacement-Dependent Dampers in Suspension Systems

Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)


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.


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

In Fig. 17.1 we depict a schematic of a simple viscous damper consists of a moving piston having one or more orifices inside a cylinder filled with a viscous fluid. Velocity of the piston and the damping force are correlated linearly as follows:
$$\displaystyle{ F = c\frac{\partial u} {\partial t} }$$
where c is the damping coefficient.
Fig. 17.1

Schematic of a linear viscous damper

Assuming the piston has only one orifice, by using the Hagen-Poiseuille equation for the laminar flow, c can be obtained by
$$\displaystyle{ c =\lambda \left [\left (\frac{D} {d} \right )^{2} - 1\right ]^{2} }$$
where D and d are the cylinder diameter and the opening fluid gap diameter, respectively. Also, λ = 8π 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:
$$\displaystyle{ u = nr^{s} }$$
$$\displaystyle{ r = \left (\frac{u} {n}\right )^{\frac{1} {s} } }$$
A solid cone shaped part can be created by rotating the area below the curve of the previously mentioned functions around the 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.
Fig. 17.2

Schematic of the NDD damper

For the designed displacement-dependent damping mechanism [4, 5, 6] shown in Fig. 17.2, the fluid gap diameter is equal to d − 2r. Therefore, we substitute d − 2r for d into Eq. (17.2) as follows:
$$\displaystyle{ c_{\mathrm{NDD}} =\lambda \left [\left ( \frac{D} {d - 2r}\right )^{2} - 1\right ]^{2}. }$$
Substituting for r from Eq. (17.4) leads to
$$\displaystyle{ c_{\mathrm{NDD}} =\lambda \left [\gamma ^{2}\left ( \frac{1} {1 -\beta u^{\frac{1} {s} }}\right )^{2} - 1\right ]^{2}. }$$

17.3 Mathematical Formulation of Suspension System

Figure 17.3 shows a schematic of the proposed suspension system. The system is a quarter car suspension model. It consists of two masses m1, for the chasis, and m2, for wheel, connected to each other by a spring and a damper, with the stiffness k1 and damping coefficient c1, respectively. The wheel is in contact to the ground by a tire which is modeled by a linear spring, k2, and a linear damper, c2. The road condition is modeled as an input u(t) shown in the figure. The coordinates x1 and x2 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 \(\mathbf{z} = \left [x_{1},\,x_{2},\,\dot{x_{1}},\,\dot{x_{2}}\right ]\).
Fig. 17.3

Schematic of the suspension system

For the conventional linear suspension system, k1 and c1 are constant. We obtain the governing differential equation in state-space form:
$$\displaystyle{ \dot{\mathbf{z}} = \mathbf{A}\mathbf{z} + \mathbf{B}u(t) + \mathbf{C}\dot{u}(t) }$$
where \(\mathbf{A} = \left [\begin{array}{*{10}c} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \\ \frac{-k_{1}} {m_{1}} & \frac{k_{1}} {m_{1}} & \frac{-c_{1}} {m_{1}} & \frac{c_{1}} {m_{1}} \\ \frac{k_{1}} {m_{2}} & \frac{-(k_{1}+k_{2})} {m_{2}} & \frac{c_{1}} {m_{2}} & \frac{-(c_{1}+c_{2})} {m_{2}} \end{array} \right ]\), \(\mathbf{B} = \left [\begin{array}{*{10}c} 0\\ 0 \\ 0 \\ \frac{k_{2}} {m_{2}}\\ \end{array} \right ]\), and \(\mathbf{C} = \left [\begin{array}{*{10}c} 0\\ 0 \\ 0\\ \frac{c_{ 2}} {m_{2}}\\ \end{array} \right ]\).
We obtain the proposed nonlinear suspension system by replacing the linear spring and damper which connects the chasis to the wheel, by a nonlinear spring and the NDD damper. Therefore, k1 and c1 are no longer constant. We obtain the following state-space model for the NDD suspension model:
$$\displaystyle{ \dot{\mathbf{z}} = \mathbf{A}_{\mathrm{n}}\mathbf{z} + \mathbf{f}_{\mathrm{n}}(\mathbf{z}) + \mathbf{B}u(t) + \mathbf{C}\dot{u}(t) }$$
where \(\mathbf{A} = \left [\begin{array}{*{10}c} 0& 0 & 1 & 0\\ 0 & 0 & 0 & 1 \\ 0& 0 & \frac{-c_{\mathrm{NDD}}(\mathbf{z})} {m_{1}} & \frac{c_{\mathrm{NDD}}(\mathbf{z})} {m_{1}} \\ 0& \frac{-k_{2}} {m_{2}} & \frac{c_{\mathrm{NDD}}(\mathbf{z})} {m_{2}} & \frac{-(c_{\mathrm{NDD}}(\mathbf{z})+c_{2})} {m_{2}} \end{array} \right ]\), 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 c1 is replaced by the NDD damper inconstant coefficient cNDD 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 .

Now we evaluate the performance of both systems by applying an input. We consider a small road bump with the height of 0. 1 m shown in Fig. 17.4. We measure the response of both system to this input. Figure 17.5 shows the vibrations of the chassis. As we can see in this figure, both systems begin to vibrate at 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.
Fig. 17.4

Road condition as the input to the suspension system

Fig. 17.5

Displacement of the chasis for the linear and the NDD system

In addition to the displacement, acceleration of the chassis is another important factor in the performance of the suspension system. In Fig. 17.6 we compare the acceleration of the linear and the NDD suspension system. Once the wheel reaches the bump (and the input is applied) the acceleration of both systems increase with a similar slope. However, the peak acceleration of the NDD system is smaller than that of the linear system. Also, as soon as the wheel passes the bump, the NDD system acceleration dies out quickly, while that of the linear system reduces slowly in an oscillating manner.
Fig. 17.6

Acceleration of the chasis for the linear and the NDD system

Figure 17.7 illustrates the contact force between the tire and the ground. As long as the contact force is above zero the car has good handling. The handling reduces as the contact force goes below zero. As shown in the figure, the minima of the contact force for the Linear system is smaller than that of the NSDD suspension system. Therefore, we observe a bigger reduction in the handling of the linear system. Also, once the wheel passes the bump, the handling of the NDD system becomes stable immediately. For the linear system, however, the handling continues to fluctuates before becoming stable.
Fig. 17.7

Contact force between the tire and the road for the linear and the NDD system

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.


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Copyright information

© The Society for Experimental Mechanics, Inc. 2017

Authors and Affiliations

  1. 1.Department of Mechanical, Industrial, and Systems EngineeringUniversity of Rhode IslandKingstonUSA

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