Vehicle Dynamics And Damping Pdf
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- Vehicle Dynamics and Damping: First Revised Edition
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- Car suspension
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The goal of the present study is the development of a spectral method to obtain the frequency response of the half-vehicle subjected to a measured pavement roughness in the frequency domain. For this purpose, a half-vehicle dynamic model with a two-point delayed base excitation was developed to correlate with the spectral density function of the pavement roughness, to obtain the system spectral transfer function, in the frequency domain. The vertical pavement profile was measured along two roads sections.
The surface roughness was here expressed in terms of the spectral density function of the measured vertical pavement profile with respect to the evenness wave number of the pavement roughness. A frequency response analysis was applied to obtain the vertical and angular modal vehicle dynamic response with the excitation of the power spectral density PSD of the pavement roughness.
The results show that at low speed, the vehicle suspension mode is magnified due to the unpaved track signature. Keywords: vehicle, dynamic, pavement, roughness, random. In general, during the vehicle project and design development phase, the automotive industry utilizes a combination of design tools such as vehicle modal response from numerical simulation Costa, , laboratory tests with shaker rigs Boggs, and the results of experimental field road tests, to fine tune vehicle suspension Vilela and Tamai, Despite the efficiency of the numerical simulations, laboratory and experimental tests are still in use, even though being time-consuming, expensive and limited to the specific road conditions of the test track.
Quarter car vehicle model with single random input is traditionally used for spectral studies Barbosa, ; Sun, ; Cebon, ; Silva, The complete vehicle model is employed for modal and control purpose Vilela, ; Costa The motivation of the present work is to extend the power of the analytic tools for the design of vehicle suspension with the application of the frequency domain response technique to deal with random input of the pavement roughness.
One of the contributions of the present study is the development of a half-vehicle model with delayed two-point base excitation correlated with the spectral density function of a measured pavement roughness, in order to generate the system spectral transfer function in the frequency domain.
The surface roughness is expressed with the spectral density function of the measured vertical pavement profile with respect to the evenness wave number of the pavement roughness.
The dynamic vehicle behaviour was accomplished with the traditional half-car vehicle representation Sun, The four-degree of freedom lumped parameter model describing relevant motion was adopted as shown at Fig. The front and rear suspension connections are described by spring-damper properties k f , c f , k r and c r. Here m 1 and m 2 are the vehicle unsprung mass with the correspondent tyre stiffness and damping are described by k 1 , c 1 , k 2 and c 2 values.
The model is excited by the road evenness u 1 t and u 2 t , which induces out-of-phase front and rear suspension movements, respectively, with a time delay. The equations of motion are obtained using the Lagrange method applied to the lumped rigid bodies.
The kinetic, potential and the generalized energy dissipation functions are respectively given by the following equations:. Substituting the partial derivatives of the above equations to the Lagrange expression given by. Table 1 shows the adopted values for the vehicle inertia, suspension elasticity and dissipation.
These values are typical of a medium sized passenger car Barbosa, The modal system properties are described by four coupled vibration modes due to the non-diagonal constitution of the system matrix.
The vehicle modal response has four natural damped frequencies around 1. For the body modes front and rear end bounce , as shown in Fig. For the suspension modes, associated with the unsprung mass of hub and tyre elasticity, damping factors are around 0. It should be noted that the suspension frequency is about one decade above those from the vehicle modes.
The normalized modal Eigen-vectors obtained from the dynamic matrix are shown in the following tables. One of the contributions of the present work is the introduction of the delayed out-of-phase inputs into the vehicle front and rear wheels. Considering that the rear wheel runs on the same track right after the front wheel, the surface elevation that produces the vehicle vertical suspension displacement is given by the same function which describes the excitation of the front wheel delayed in time.
Taking a harmonic function u 1 t as the imposed vertical displacement of the front wheel, then the rear wheel delayed input u 2 t can be expressed as:. The Laplace transformation of the front wheel and the rear wheel input functions, considering the transformation of the delayed function are respectively given by:.
Upon the substitution of U s into Eq. The displacement frequency response function FRF is known as receptance H s. The acceleration frequency response function known as inertance I s can be obtained. The inertance function for the coupled vertical and angular vehicle body motions is shown in Fig. Therefore, the FRF shape will be speed-dependent as can be observed in Fig. Humps can be noticed in the PSD curve shown in Fig. For the vertical mode, these humps occur at every integer, resulting in peaks at around 1, 3, 6, 9 Hz.
The modal frequencies are identified with a circle in the figure front end bounce at 1. For the angular mode, the peaks occur at Two sections of road surface irregularities were actually measured in the present work. The first section was a 1. The second section was a 2. The pavement roughness was measured with the 3-point-middle-chord measuring device.
This system is composed of three wheels and a displacement sensor. The two external wheels are steered and the central one is articulated with regard to the others. A conventional car pulls the measuring system along the road measuring the track evenness. The central wheel vertical motion is sampled every centimeter by an analogic to digital sample board installed in a portable computer. The data acquired are stored in magnetic media for post processing purposes Pavimetro, The measured data are treated with the device system transfer function, to obtain the topographic vertical elevation of the road surface roughness, as shown in Fig.
In this case, ten points per meter were sampled one sample at every 0. Wavelengths up to m were considered. The results for the vertical elevation of the road surface roughness for the asphalted road are shown in Fig. In this case, two points per meter were sampled. Wavelengths up to m were considered in the anti-aliasing processes.
The mean IRI value for this road section is 1. The data treatment was performed up to points at a sample rate down to m. This range allows analyzing wavelengths as long as m and down to 0. This wide range is unprecedented in this area considering that traditional measuring devices have a restricted observable band.
The unpaved and the asphalted track elevation measurements were further treated to generate distributions wavelengths of the periodic irregularities. The spectral density function PSD in the range of wavelength between 0. This measured road section spectrum has its particular signature with intensified wavelength content between 0. The spectral density function of wavelength between 1 and meters of the asphalted vertical elevation road is presented in Fig.
The spectrum of this road section has its intensified wavelength content between 30 and 40 meters. These measured spectra of vertical elevation of road surface roughness will be used to calculate the vehicle vertical and angular spectral responses.
The intensity of the measured pavement roughness is classified, according to the magnitude of the power spectral pattern of the irregularities in an exponential fashion with a particular slope ISO international standard, Displacement power spectral density PSD for a road roughness class is obtained by a logarithm expression in units of m 3 :.
The spatial frequency dependence term Sd at n o is obtained from:. These values were used as vehicle excitation in the frequency domain.
Considering the pavement irregularities as an ergodic stationary random process, described by the normal distribution, the evenness density can be expressed by the roughness root mean square value rms-value. Therefore, by taking the square root of the previous expression, one gets:. The vehicle natural behavior is expressed by its frequency domain response function Barbosa, In the present analysis, the road pavement is considered a rigid surface.
By transforming S n into the frequency domain, one gets:. According to the theory of stochastic process, the output of a linear time-invariant system is a stationary random process if the input is also a stationary random process. In most cases, the pavement roughness could be described as a zero mean Gaussian ergodic random process Newland, Hence, the response of the half-car system is also a zero mean Gaussian stationary random process.
It can observed in this figure that the most severe wavelength content of the unpaved track section, which is between 0. This effect magnifies the expected acceleration proneness around 12 Hz, which may cause discomfort to passenger.
A vehicle will be very susceptive to pavement roughness with wavelength content in the range about 2. This methodology is based on the modal vehicle frequency response function and the statistical description of the geometry of the rough.
Firstly, a half-vehicle dynamic model with a two-point delayed base excitation was derived. Secondly, two-road sections surface elevations were measured with a special referenced measuring device. The vehicle inertance function was then obtained. The inertance function is related to the passenger comfort and can be used for design purposes.
The vertical and angular vehicle body transfer functions were calculated with the surface frequency irregularities function in the frequency domain as input. The first measured road was a 1. The geometrical data collected was then processed to obtain a special broadband distribution with wavelengths between m and 0. The wide range of wavelengths thus obtained is unprecedented, considering that the traditional measuring devices have a restricted observable band.
The measured asphalted road section has an IRI of 1. The measured unpaved road section has a spectral signature distribution with concentration between 0. A two out-of-phase delayed vehicle inputs was considered corresponding to the front and rear wheel positions, by applying the measured track roughness density functions. The inertance function was obtained for the vertical and angular vehicle body motions. Considering the low speed The developed methodologies extend the efficiency of vehicle numerical simulation tools, with the power of providing vehicle frequency response analysis due to the pavement roughness statistically described.
Results allow the evaluation of passenger discomfort.
Vehicle Dynamics and Damping: First Revised Edition
Suspension is the system of tires, tire air, springs , shock absorbers and linkages that connects a vehicle to its wheels and allows relative motion between the two. The tuning of suspensions involves finding the right compromise. It is important for the suspension to keep the road wheel in contact with the road surface as much as possible, because all the road or ground forces acting on the vehicle do so through the contact patches of the tires. The suspension also protects the vehicle itself and any cargo or luggage from damage and wear. The design of front and rear suspension of a car may be different.
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The goal of the present study is the development of a spectral method to obtain the frequency response of the half-vehicle subjected to a measured pavement roughness in the frequency domain. For this purpose, a half-vehicle dynamic model with a two-point delayed base excitation was developed to correlate with the spectral density function of the pavement roughness, to obtain the system spectral transfer function, in the frequency domain. The vertical pavement profile was measured along two roads sections. The surface roughness was here expressed in terms of the spectral density function of the measured vertical pavement profile with respect to the evenness wave number of the pavement roughness.
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It is ideal for vehicle dynamics investigations in early development phases. A user interface lets users intuitively configure the vehicles and define maneuvers and roads. For commercial vehicles, diversity is a standard.
By Jan Zuijdijk. No part of this book may be reproduced, stored in a retrieval system, or transmitted by any means without the written permission of the author.
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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.
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