In the automotive industry, numerous expensive and time-consuming trials are used to “optimize” the ride and handling performance. Ideally, a reliable virtual prototype is a solution. The practical usage of a model is linked and restricted by the model complexity and reliability. The object of this study is development and analysis of a refined quarter car suspension model, which includes the effect of series stiffness, to estimate the response at higher frequencies; resulting Maxwell's model representation does not allow straightforward calculation of performance parameters. Governing equations of motion are manipulated to calculate the effective stiffness and damping values. State space model is arranged in a novel form to find eigenvalues, which is a unique contribution. Analysis shows the influence of suspension damping and series stiffness on natural frequencies and regions of reduced vibration response. Increase in the suspension damping coefficient beyond optimum values was found to reduce the modal damping and increase the natural frequencies. Instead of carrying out trial simulations during performance optimization for human comfort, an expression is developed for corresponding suspension damping coefficient. The analysis clearly shows the influence of the series stiffness on suspension dynamics and necessity to incorporate the model in performance predictions.

In the vehicle suspension design, dependent on the usage pattern, handling and ride comfort performance have contradicting requirements [

Models of varying complexity, considering only linear dynamic elements to systems involving nonlinear elements, have been developed to predict, specifically, vehicle handling and, to some extent, ride comfort. Shock absorbers have been modelled to represent hydromechanical behaviour for example, [

Simplest vehicle model used to assess discomfort due to vibration is of two degrees of freedom (DOF) [

The series stiffness, as in 1 DOF model, may influence effective damping; two modal damping ratios of the quarter car model can be different from those based on the Kelvin-Voigt model and the variations can be a complex function of the suspension damping coefficient and the stiffness. Further, the effect of series stiffness may not be equal on the modes of vibration. The resonance frequencies are expected to be dependent on the damping coefficient. For optimization, sufficient damping is required for both modes. The modal damping ratio calculation is not straightforward; in practice, they are not calculated explicitly; instead cost functions are formed and trials are carried out to reduce the vibratory responses directly. The process does not provide insight into the complex problem. It is desirable to find a combination of parameters resulting in large damping ratios and small shift in resonance frequency, without having to perform numerous forced vibration analyses; the focus of optimization continues to be the response reduction but achieved through an alternative approach. Availability of an explicit formula for effective damping can minimize the optimization effort and provide critical information.

In what follows, the quarter car model dynamic response with and without series stiffness is compared to show the effect of series stiffness. For the harmonic input, equations of motion are rearranged to calculate effective stiffness and damping values. Using the knowledge of forces transmitted through the series stiffness, a novel form of state space equations is generated so as to calculate the natural frequencies and modal damping ratios. The effects of combination of suspension damping coefficient and series stiffness are analysed showing regions of reduced vibration response. A simplified expression is developed for optimum suspension damping with the aim of mainly reducing wheel hop frequency response; the wheel hop frequency response will be shown being more sensitive to the presence of series stiffness than the vehicle body mode response. The model clearly shows the influence of series stiffness on the modal damping ratios and the natural frequencies.

The influence of damping on shift in the resonance frequency and variation in the corresponding amplitude for 1 DOF Maxwell’s model is well documented. The vehicle suspension system with series stiffness may show similar influence, but the influence may not be a simple function of the damping value alone; relative values of series and the tyre stiffnesses may play a significant role. Further, the behaviour due to different forms of input may also be complex.

Initially, in the next section two models of the suspension system, with and without series stiffness, are compared for their effect on the steady-state frequency domain response. Later, a state space model is used to calculate the natural frequencies and the modal damping ratios. An expression is developed for optimum damping. In the model, stiffness and damping coefficients used are linear equivalent parameters.

Figure

Schematic of a quarter car model with the use of Maxwell’s model to represent the suspension dynamics.

The effective damping (

Using (

Parameter values used for the quarter car model.

System parameter | Parameter value |
---|---|

Vehicle parameters | |

Tyre stiffness (N/m) | 2 |

Suspension stiffness (N/m) | 3 |

Suspension damping coefficient (Ns/m) | Varying |

Hub mass, front (kg) | 40 |

Quarter of a vehicle body mass (kg) | 250 |

Stiffness in series (N/m) | 2 |

(a) Wheel hub response comparison and (b) vehicle body response comparison (damping coefficient is 1500 Ns/m), (

The shift in resonance frequency and the change in amplitude are complex functions of series stiffness, suspension damping coefficient, and suspension stiffness. For complete analysis, forced responses may have to be calculated for all combinations of these parameters. The process can be expensive and may not provide insight into the problem. The aim is to find a combination resulting in large damping ratios and small shift in resonance frequency, without having to perform numerous forced vibration analyses. In the next section, a state space representation is explored to find complex eigenvalues from which natural frequencies and damping ratios can be extracted.

Equations of motion can be rearranged for state space formulation as given below:

Let

In the state space form, equations of motion are written as

Parametric studies were carried out to find variation of damping ratios and natural frequencies. The base data listed in Table

Wheel hub natural frequency variation as a function of the suspension damping coefficient and the series stiffness.

The wheel hub modal damping ratio (Figure

Wheel hub modal damping ratio variation as a function of the suspension damping coefficient and the series stiffness.

Figure

Vehicle body mode natural frequency variation as a function of the suspension damping coefficient and the series stiffness.

Vehicle body mode damping ratios (Figure

Vehicle body modal damping ratio variation as a function of the suspension damping coefficient and the series stiffness.

Figure

Ratio of damping ratios varying as a function of the suspension damping coefficient and the series stiffness.

The effective damping relating to hub mode (see (

Before developing an expression for optimum damping coefficient, the effect of assuming a constant hub mode frequency is analysed. Figure

Effective damping coefficient variation as a function of the suspension damping coefficient and the series stiffness (

The maximum effective damping coefficient leading to an optimal modal damping ratio can be obtained by the following process:

The result of (

Optimal suspension damping coefficient variation as a function of the series stiffness (

A vehicle quarter car suspension model was refined to include the effect of series stiffness. A novel form of state space equations was used to calculate the natural frequencies and the modal damping ratios. The effect of the suspension damping and series stiffness was analysed, showing regions of reduced vibration response.

The inclusion of series stiffness reduces the effective damping; the damping ratios achieved at two modes of the quarter car model are smaller than those based on the Kelvin-Voigt model. The variation of the damping ratios is a nonlinear function of suspension damping coefficient and stiffness. In extreme cases, for larger suspension damping coefficients the resulting damping ratios could be negligibly small. The effect may not be equal on the modes of vibration.

The most significant effect of series stiffness was on the wheel hop frequency and the amplitude. Increase in damping beyond the optimal values increases the amplitude at resonance, having a negative impact on vehicle ride comfort. A simplified expression for the optimal suspension damping coefficient was developed eliminating the need for trial simulations. Overall, the model clearly shows the influence of series stiffness on the modal damping ratios, the natural frequencies, and hence the dynamic response.