2005 SLOVAK UNIVERSITY OF TECHNOLOGY48

1. INTRODUCTION

Many of workers deal with dynamic analysis of structures and soil. Some workers are oriented on the problems of building structures (Ivánková, O., 2002) some workers on the problems of transport structures (Smutný, J. – Pazdera, L., 1999), or they pay attention to the problems of soil (Králik, J., 2000). The analysis can be carried out in a time or frequency domain, depending upon the goals and character of the task. The analysis of the characteristics defining the dynamic individuality of a dynamic system is an integral part of a complex dynamical analysis. Natural frequencies, natural modes or frequency response functions presented as amplitude-frequency or phase-frequency characteristics are allocated to such characteristics. Frequency response functions are advantageously applied to the sensitivity analysis of dynamic systems, especially if the system is excited by force or kinematic excitation with a variable frequency composition. The submitted paper introduces

some vehicle – road interaction problems, especially solutions in the frequency domain. It is dedicated to a sensitivity analysis of vehicle vibration. It evaluates the interaction forces occurring between a tire and road surface by means of amplitude-frequency and phase-frequency characteristics in relation to kinematic excitation with a variable frequency composition. The source of the kinematic excitation is the road surface’s unevenness.

2. THEORETICAL BASIS

The frequency response of a linear dynamic system (frequency response function H(p) for p = i.ω) is defined as the ratio of the steady state response to the harmonic excitation

(1)

J. MELCER

VEHICLE – ROAD INTERACTION, ANALYSIS IN A FREQUENCY DOMAIN

KEY WORDS

• Vehicle-road interaction, • dynamics, • frequency response,• MATLAB

ABSTRACT

This paper is dedicated to the solution of some problems of vehicle - road interaction in a frequency domain. The one-dimensional so-called quarter model of a vehicle is considered. The vibration of the model is described by the system of ordinary differential equations that are changed by Laplace´s transformation into a frequency domain. The frequency response functions as functions of the angular frequency ω of the individual kinematic and power quantities are derived. The amplitude-frequency and phase-frequency characteristics of the frequency response functions are calculated numerically in a frequency range of ω from 0 to 100 rad/s. The expressions for the critical speeds or critical lengths of the periodically repeated waves of a road’s unevenness are tested with respect to the occurrence of possible resonance states.

Prof. Ing. Jozef Melcer, DrSc. Professor, Department of Structural MechanicsVice-Dean, Faculty of Civil EngineeringResearch field:Statics and dynamicsDiagnostics of civil engineering structures Full address: University of ZilinaFaculty of Civil EngineeringDepartment of Structural MechanicsKomenského 52010 26 ŽilinaE-mail: melcer@fstav.uniza.skPhone: ++421-41-513 5612

2006 SLOVAK UNIVERSITY OF TECHNOLOGY

2006/3 – 4 PAGES 48 – 52 RECEIVED 18. 9. 2006 ACCEPTED 27. 11. 2006

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49VEHICLE – ROAD INTERACTION, ANALYSIS IN A FREQUENCY DOMAIN

If the entering value is periodical with a united amplitude

(2)

we can write for the output quantity

. (3)

The graphic representation of the frequency response is the frequency characteristic. The graphic representation of the absolute value (modulus) of the frequency response function versus the frequency of harmonic excitation is an amplitude characteristic. The phase characteristic is a graphic representation of the argument (phase) of the frequency response function versus the frequency of the harmonic excitation. The frequency response function H(i.ω) is a complex function and can be expressed as a vector sum or real part R(ω) and imaginary part I(ω), Fig. 1

, or , (4)

where |H(iω)| is an absolute value, or the magnitude of the frequency response function, and it represents the amplitude of the response r(t) of the system excited by the simple harmonic function according to (2). In equation (4) ϕ represents the phase of the frequency response function, more precisely, the phase of the response r(t). Substituting (4) for (3), we obtain

(5)

The next relations are

and . (6)

3. COMPUTING MODEL OF A VEHICLE

In this paper the so-called quarter model of a vehicle is analyzed, Fig. 2.

The following numeric characteristics correspond to the Tatra T148 lorrym1 = 2 514.138 kg m2 = 440.0 kgk1 = 197 965.0 N.m-1 k2 = 1 200 000.0 N.m-1

b1 = 11 423.6 kg.s-1 b2 = 1 373.4 kg.s-1

The equations of motion are as follows:

, (7)

. (8)

Respecting that

, (9)

, (10)

equations (7) and (8) can be transformed into

, (11)

. (12)

Fig. 1 Frequency response function

Fig. 2 Quarter model of a vehicle

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And the relation for the interactive force occurring between the tire and road surface is

. (13)

4. FREQUENCY RESPONSE FUNCTIONS

The transformation of equations (11) and (12) into the frequency domain can be carried out by the use of Laplace´s transformation

, (14)

, (15)

where p = i . ω. Let us introduce the signification

and . (16)

The equations (14) and (15) through the use of (16) can be rewritten as

, (17)

. (18)

After solving equations (17) and (18), we obtain two unknown complex variables Hr1 and Hr2. These are the frequency response functions of deflections r1 and r2, defined by the relation (16). From the frequency response functions Hr1 and Hr2, it is possible to derive the frequency response function of the interaction force

. (19)

Fig. 3 Amplitude-frequency (a) and phase-frequency (b)characteristics of Hr1

Fig. 4 Amplitude-frequency (a) and phase-frequency (b)characteristics of Hr2

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5. RESULTS OF THE SOLUTION

The amplitude-frequency and phase-frequency characteristics of the frequency response functions Hr1, Hr2, and HF in Figs. 3, 4, and 5.The amplitude-frequency and phase-frequency characteristics shown were numerically calculated in the environment of the MATLAB program language as functions of the angular frequency ω [rad/s] in the frequency range from 0 to 100 rad/s. The amplitude-frequency characteristics have one or two local extremes. The numeric values of the mentioned local extremes and corresponding frequencies are seen in Tab. 1.

6. CONCLUSIONS

The frequency response functions, respectively the amplitude-frequency and phase-frequency characteristics, are dynamic characteristics frequently used for assessment of dynamic systems in a frequency domain. These characteristics express the sensitivity of the dynamic system very well in answer to the excitation with a variable frequency composition. If the results obtained are employed on a road surface’s unevenness and on the speed of a vehicle’s motion for every speed, of the vehicle’s motion, it is possible to calculate the critical wave length l0 of the periodically repeated unevenness. Upon deriving these relations, we start from

Tab. 1 Numeric values of extremes of amplitude-frequency characteristics

The1st extreme The 2nd extremefollowed value ω [rad/s] followed value ω [rad/s]

r1/u [-] 2,80164 8,0 - -r2/u [-] 1,27681 7,7 1,68426 51,4F/u [N/m] 519 048,11396 8,7 2 326 470,16788 60,09

Tab. 2 Critical wave length l0 at critical speeds of a vehicle’s motionV = 20 V = 40 V = 60 V = 80 V = 100 V = 120

r1/u for ω = 8.00 l0 = 2.18 l0 = 4.36 l0 = 6.54 l0 = 8.72 l0 = 10.90 l0 = 13.09r2/u for ω = 7.70 ω = 51.4

l0 = 2.26 l0 = 0.34

l0 = 4.53 l0 = 0.67

l0 = 6.80 l0 = 1.01

l0 = 9.06 l0 = 1.35

l0 = 11.33 l0 = 1.69

l0 = 13.60 l0 = 2.03

F/u for ω = 8.70 ω = 60.0

l0 = 2.00 l0 = 0.29

l0 = 4.01 l0 = 0.58

l0 = 6.01 l0 = 0.87

l0 = 8.02 l0 = 1.16

l0 = 10.03 l0 = 1.45

l0 = 12.03 l0 = 1.74

Fig. 5 Amplitude-frequency (a) and phase-frequency (b) characteristics of the HF

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the equality of exciting and natural frequencies ω = ω(j), (Melcer, J.,1997).

(20)

and from there

. (21)

The results of the little parametric study concerning the critical wave length l0 in [m] of the periodically repeated unevenness at the speeds of a vehicle’s motion V = 20, 40, 60, 80, 100, and 120 km/h are seen in Tab. 2.

REFERENCES

• Ivánková, O. (2002). Staticko - dynamická analýza oceľovej priestorovej haly (Static and Dynamic Analysis of a Steel Hall). In.: New Trends in Statics and Dynamics of Buildings, SvF STU Bratislava, October, 2002, p. 273. (in Slovak)

• Králik, J. (2000). Dynamics of Subsoil. Dynamic Interaction of Structures with Subsoil. Seismic Stress of Overground and Underground Structures. In.: Postgraduate study of Dynamics of Structures. SvF STU Bratislava, November, 2000.

• Smutný, J.–Pazdera, L.(1999). Philosophy of Measurement and Time-Frequency Analysis of Vibration from Rail Transport. Railway Engineering – 99, 2nd International Conference, London, UK, Engineering Techniques Press, Edinburgh, 6/1999, ISBN 0-947644-39-3, p.10.

• Melcer, J.(1997). Dynamický výpočet diaľničných mostov (Dynamic calculation of highway bridges) . EDIS, ŽU Žilina, 1997. (in Slovak)

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