The optimum functional characteristics of suspension components, namely, linear/nonlinear spring and nonlinear damper characteristic functions are determined using simple lumped parameter models. A quarter car model is used to represent the front independent suspension, and a half car model is used to represent the rear solid axle suspension of a light commercial vehicle. The functional shapes of the suspension characteristics used in the optimisation process are based on typical shapes supplied by a car manufacturer. The complexity of a nonlinear function optimisation problem is reduced by scaling it up or down from the aforementioned shape in the optimisation process. The nonlinear optimised suspension characteristics are first obtained using lower complexity lumped parameter models. Then, the performance of the optimised suspension units are verified using the higher fidelity and more realistic Carmaker model. An interactive software module is developed to ease the nonlinear suspension optimisation process using the Matlab Graphical User Interface tool.

Vehicle suspension design and performance problems have been studied extensively using simple car models such as two degrees-of-freedom (d.o.f.) quarter car, four or six d.o.f. half car, or seven d.o.f. full car models. Usually, the suspension design methodologies are based on analytical methods where a linear vehicle model is investigated by solving linear ordinary differential equations. Laplace and Fourier transforms are valuable tools that are used while investigating suspension units with linear characteristics. The performance functions represented by transfer functions in Laplace and/or Fourier domains are considered to be related to ride comfort, tire forces, and handling criteria versus road roughness input to achieve an optimum design. On the other hand, the investigation of nonlinear suspension characteristics must be based more on numerical methods rather than analytical methods due to the more complicated nature of the problem. In this investigation, both linear and nonlinear spring and damper characteristics of a light commercial vehicle are considered and used in an optimisation study.

Lumped parameter suspension models are used in this paper. Mass, stiffness, and damping are distributed spatially throughout a mechanical system like a suspension. Mass, stiffness, and damping are therefore functions of spatial variables (

The optimisation requirements of suspension systems and the state-of-the-art of suspension research in the last decade are reviewed first. It should be noted, however, that the available literature is vast and only a small portion of it can be presented here. This paper includes the well-known ride, handling trade-off optimisation, and geometrical optimisation of light commercial vehicle suspension systems. Some heavy vehicle suspension optimisation papers are also reviewed due to their conceptual contribution to the subject.

Vehicle suspensions can be regarded as interconnections of rigid bodies with kinematic joints and compliance elements such as springs, bushings, and stabilizers. Design of a suspension system requires detailed specification of the interconnection points (or so called hard points) and the characteristics of compliance elements. Tak and Chung [

Maximizing tractive effort is essential to competitive performance in the drag racing environment. Antisquat is a transient vehicle suspension phenomenon which can dramatically affect tractive effort available at the motorcycle drive tire. Wiers and Dhingra [

The design of suspension systems generally demands a compromise solution to the conflicting requirements of handling and ride comfort. The following examples demonstrate this compromise.

For example, for better comfort a soft suspension and for better handling a stiff suspension is needed.

A high ground clearance is required on rough terrain, whereas a low centre of gravity height is desired for swift cornering and dynamic stability at high speeds.

It is advantageous to have low damping for low force transmission to the vehicle frame. On the contrary, high damping is desired for fast decay of oscillations.

Goncalves and Ambrosio [

Eskandari et al. [

Li et al. [

The roll steer of a front McPherson suspension system is studied, and the design characteristics of the mechanism are optimized by Habibi et al. [

An optimum concept to design “road-friendly” heavy vehicles with the recognition of pavement loads as a primary objective function of vehicle suspension design was investigated by Sun [

This paper concentrates on optimisation of the nonlinear shape of the spring and damper of a light commercial vehicle. The work reported in the paper is motivated by the fact that the spring and damping characteristics in an actual road vehicle are designed to be nonlinear on purpose by the automanufacturer. Larger spring forces are used in the rebound motion of the wheel in order to keep forcing an appropriate level of tire-road contact at all times, for example. When the automanufacturer starts working on a new model, a previous, successful suspension design is used as the base characteristic which is modified to fit the characteristics of the new vehicle model. The work presented here follows the same approach as spring, and damper characteristics of an existing base design are used as the starting point in the optimisation. The optimisation procedure is embedded into an interactive Matlab Graphical User Interface to offer ease of use to suspension designers.

The organization of the rest of the paper is as follows. The vehicle models used in this study are introduced in Section

In this section, the scope of the current investigation is summarized by presenting the vehicle models that are used in this study. The use of a complex three-dimensional model of the vehicle, with a detailed description of all suspension systems and road/tire interaction, is necessary to fully investigate the problem. However, such models are computationally expensive especially when used in an iterative optimisation design process. A good alternative which is used here is the optimisation of a subsystem of a complex model. The suspension subsystem is very important in terms of vehicle dynamics. Its spring and damper load deflection characteristics are treated as the basic design variables here.

The ride comfort optimisation is achieved by finding the optimum of a ride comfort index which results from a metric that accounts for the linear and the angular accelerations of the model’s suspended mass centre and properly combined in a cost function, considering their importance for the comfort of the occupant. Two lumped parameter models are built in Matlab considering the independent front suspension unit (a quarter car model) and the rear axle suspension unit (a half car model) of a light commercial vehicle. Vertical displacement

The dynamics of the quarter car and the half car rear axle suspension models are governed by nonlinear differential equations of motion. The road profile described in Cebon [

Basic shapes representing the spring and the damper characteristics (force versus deflection for the spring and force versus velocity for the damper) are used according to automotive manufacturer’s specifications. Basic functional shapes in each operating mode (extension or compression regions of the spring and the damper) are predetermined and the functional fits to these shapes are obtained. These functions and their linear combinations are then scaled searching for the optimum characteristics. The emphasis of this investigation is placed on finding nonsymmetric optimum nonlinear functions of the spring and the damper force characteristics. Optimised functional relations are then incorporated into a model built in a high fidelity, realistic commercial vehicle dynamics software to evaluate the performance of the vehicle model with the optimised suspension. Carmaker software [

The mathematical models of the quarter car and the half car representing one of the front quarters and the rear axle suspension unit of a light commercial vehicle are presented in this section.

The quarter car model subject to road disturbances is shown in Figure

Quarter car model.

Similarly the equation of motion of the vehicle wheel may be written as

The rear solid axle suspension unit of the light commercial vehicle considered here is represented with a half car model (see Figure

Half car model.

The half car model can represent the bounce (

The choice of subobjective functions and their weights in the combined (main objective) function plays a very critical role in the optimisation process. In this section, the literature review of the vehicle suspension objective functions and the performance indices are summarized, and our approach to the objective function formulation is presented. The nonlinear stiffness and damping characteristics are optimized by Koulocheris et al. [

Geometrical parameters of the suspension were considered in Mitchell et al. [

The objective function in Eskandari et al. [

A global performance index is considered in Tak and Chung [

In order to optimize ride characteristics, human sensitivity to vibrations needs to be considered. For that purpose, the vertical motion is weighed according to the ISO 2631 [

Frequency weights as specified in ISO 2631-1 standard.

Figure

Body acceleration signal is weighed according to ISO2631-1 standards.

The target accelerations of the vehicle models are weighed according to the ISO 2631 standards in this investigation [

In this section, the optimisation procedure is explained in detail. In the current investigation, the complexity of the optimisation problem is reduced by deciding on the basic shape/behaviour of the force versus displacement and force versus velocity characteristics and then scaling them up or down during optimisation. This is motivated by the suspension design procedure used in the automotive industry where a previous, successful design is used as the base design, modified for the vehicle model being developed. The procedure used here has two steps.

A function like a polynomial, rational, or an exponential function that can fit the basic initial data of force versus displacement/velocity profile is chosen first.

Then, a scaling factor

First, the simulations of the quarter and the half car models subjected to a road excitation are carried out in the time domain. Note that while running the optimisation routines of the vehicle suspension models, two aspects could significantly affect results and might cause errors in the optimisation process as follows:

it is preferable to consider the steady state response of the vehicle run. Since a constant vehicle speed is assumed, it takes some time for the vehicle to reach the steady state conditions. Therefore, the beginning part of the time domain simulation containing the transient response is omitted,

attention should be paid to the static deflections (due to weights) and the initial conditions considering the static equilibrium points for the springs.

Then, the target point’s accelerations are weighed according to ISO-2631 standards in the frequency domain and used as part of the objective function.

A suitable global objective function is established according to the needs of automotive manufacturers. This is the most subjective step of the methodology. Since the choice of the objective function and weighing of the particular objectives will result in different optimum outcomes. Our choice of objective functions (performance index) for the current paper is explained in the previous section and in the following section on optimisation results.

As the final step, the optimisation type and algorithm are selected, and the optimisation step is performed in Matlab. The optimisation toolbox SQP algorithm with Quasi-Newton line search is implemented. The SQP algorithm like Simplex, Complex, and Hook-Jeeves belongs to the family of local search algorithms. The local search algorithms converge to the nearest optimum, since they depend upon the starting values of the design variables. Examples in the following section illustrate the simulation result for quarter car and half car vehicle models used here. Numerical simulation results show that the SQP algorithm can efficiently and reliably find the optimum in the neighbourhood of the initial point.

The nonlinear damper characteristic of the front independent suspension and rear solid axle of a light commercial vehicle are presented in normalized form in Figures

Normalized damper characteristic and its functional fit (front).

Normalized damper characteristic and its functional fit (rear).

For the nonlinear modelling of the spring of the rear axle, a lookup table which represents the nonlinear characteristic of the spring is used (see Figure

Normalized spring characteristic (rear).

The optimisations results for the front and rear suspension units of the quarter and the half car models are presented in this section. The performance indices used in the optimisation process are presented in the following Sections (

Optimisation runs are performed using the quarter car model in this part of the investigation considering the vehicle with a nonlinear damper and linear spring unit subjected to a generated road. It is required to have road surface input profiles for the realistic response of the vehicle dynamic simulations. These input profiles may come directly from the measurements made by a test vehicle. It is also possible to artificially generate the random road profiles like Robson’s Method, presented in Kamash and Robson [^{3}/cycle. For different classes of roads

Different road classes.

Road class | C/10^{−8} m^{0.5} cycle^{1.5} |
---|---|

Motorway | 3–50 |

Principal road | 3–800 |

Minor road | 50–3000 |

In the conventional spectral analysis the process is simply squaring the spectral coefficients which are determined by using discrete Fourier transform. It is required to have uniformly distributed random phases between 0 and 2

Road profile PSD versus wavenumber.

Generated road profile versus time.

The objective function has two components which are the weighed body acceleration RMS value and the penalty function of tire force difference. The objective function is defined as

Objective function’s run history.

Change of indices throughout optimisation process.

Optimum normalised linear spring stiffness and nonlinear damper characteristic scaling coefficients.

Performance indices throughout the optimisation process.

Default and optimized damper characteristic curves.

Default and optimised spring stiffness.

Values of the objective throughout two-dimensional search domain.

The half car model embodying a nonlinear rear suspension unit of the light commercial vehicle incorporating nonlinear dampers and nonlinear springs whose basic characteristic curve shapes are given in the previous sections is considered in this subsection. The performance index used in the previous case (the quarter car) is also considered here with the addition of a function including body roll motion. The objective function (performance index) used in optimisation is made up from three functions, as described below:

The left and the right sides of the road being followed are shown in Figure

Left and right side of the road generated with Robson Method.

The optimisation results for this case are presented in Figures

Total objective function progress.

Change of indices throughout optimisation process.

Spring and damper characteristics scaling coefficient progress.

Objective function components’ progress.

Optimized and the default spring characteristics.

Optimised and default damper characteristics.

Values of the objective throughout two-dimensional search domain.

Finally, the ride handling and comfort performances of the optimized suspension parameters are confirmed by employing the high fidelity Carmaker model of the considered vehicle (see Figure

Snapshot of Carmaker animation.

Roll angle comparison double lane change.

Roll rate comparison double-lane change.

Yaw rate change during the manoeuvre.

Vertical vehicle body acceleration PSD plots at four corners with high order model.

Vertical wheel acceleration PSD plots at four corners with high order model.

Light commercial vehicle front and rear suspension units incorporating both linear or nonlinear springs and nonlinear dampers were optimized to improve vehicle ride and handling. The nonlinear equations of motions of the quarter car and the half car representing the front and rear suspension models were presented and simulated in the Matlab/Simulink environment. Several aspects of performance criteria were considered for ride comfort and handling such as RMS values of weighed body acceleration, the range of tire forces, and RMS values of body roll angle. For each aspect of performance, time-domain performance measures were evaluated after the optimisation run. A simple optimisation methodology of nonlinear suspension unit was presented, incorporating typical data provided by car manufacturers for the initial characteristics. The methodology was based on keeping the shape of the damper and spring properties and curve fitting a proper function to these data and then scaling it throughout the optimisation process. Finally, the advantage of the nonlinear optimised suspension unit compared to a default suspension unit was demonstrated using a double lane change maneuver with a high fidelity full vehicle model in Carmaker. In order to generalise the nonlinear suspension unit optimisation problem, an interactive Matlab toolbox was constructed and used in obtaining the results presented here.

The fact that the improvement between original performances and optimized ones is not too big in the results presented in the paper is due to the fact that we started with an already optimized suspension which had been designed earlier. We decided against creating a nonideal starting value for the suspension parameters and showing a large improvement in performance. We were indeed able to show that the suspension design provided to us was very close to its optimum configuration, and only small performance achievements were possible. This is also a very useful outcome as the suspension designer can analytically evaluate his design against the optimal one. The method presented in the paper is fully automated in the form of a Matlab program and its interactive graphical user interface. The results are also obtained much faster as compared to the conventional method of suspension design including a lot of trial and error.

Degree of freedom

Performance index,

Total index

Root mean square

Power spectral density.

Sprung mass, kg

Unsprung mass, kg

Sprung mass vertical displacement, m

Unsprung mass vertical displacement, m

Road irregularity, m

Linear tire damping coefficient

Linear spring stiffness

Linear tire stiffness

Nonlinear damper characteristic function.

Sprung mass, kg

Unsprung mass, kg

Sprung mass vertical displacement, m

Unsprung mass vertical displacement, m

Left and right road irregularities, m

Left and right linear tire stiffness,

Left and right nonlinear damper characteristic functions, Ns/m

Left and right nonlinear damper characteristic functions, N/m

Antiroll bar torsional stiffness, Nm/rad

Track width, m

Sprung mass roll angle, deg/sec

Unsprung mass roll angle, deg/sec

Sprung mass moment of inertia about

Unsprung mass moment of inertia about

Weighing,

Tire force change,

Optimised value of tire force change,

RMS of weighed sprung mass acceleration

Optimised value of the RMS of weighed sprung mass acceleration

RMS of sprung mass roll angle.

The authors thank Yıldıray Koray for his help in the road profile used. The authors thank Server Ersolmaz, Erhan Eyol, Mustafa Sinal. and Selçuk Kervancıoğlu of Ford Otosan for helpful discussions on suspension design. The authors also thank Ford Otosan and the EU FP6 project AUTOCOM INCO-16426 for their support. The authors dedicate this paper to the dear memory of their second coauthor Professor Ü. Sönmez who passed away so suddenly and unexpectedly.