Vibrations of a rolling tyre

Journal of Sound and Vibration 331 (2012) 1669–1685

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Vibrations of a rolling tyre I.F. Kozhevnikov n Institution of Russian Academy of Sciences, Dorodnicyn Computing Centre of RAS (CC RAS), Vavilov st. 40, 119333 Moscow, Russia

a r t i c l e i n f o

abstract

Article history: Received 25 June 2011 Received in revised form 19 November 2011 Accepted 21 November 2011 Handling Editor: H. Ouyang Available online 21 December 2011

We investigate vibrations of an unloaded and loaded tyre rolling at constant speed without slipping in the contact area. A previously proposed analytical model of a reinforced tyre is considered. The surface of the tyre is represented by flexible tread, combined with parts of two tori (sidewalls of the tyre). The contact between the wheel and the ground plane occurs by the part of the tread. The natural frequencies (NF) and mode shapes (MS) are determined analytically for unloaded tyre and numerically for loaded tyre. The results were compared with experiments for the non-rotating tyre. In the case of loaded rotating tyre, the increasing of the angular velocity of rotation implies that NF decrease. Moreover, a phenomenon of frequency loci veering is visible here: NF as functions of angular velocity approach each other and then veer away instead of crossing. The MS interact in veering region and, as a result, interchange. & 2011 Elsevier Ltd. All rights reserved.

1. Introduction The vibrations of tyres were studied by many authors. The vibration transmission properties of the tyre between the tyre– road contact point and the wheel axle are analysed using a flexible ring model [1]. Precession of the standing waves in a thin elastic ring rotating with constant angular velocity was first observed in [2]. It was shown that if the standing waves were excited in the ring and if the ring rotates with a constant angular velocity, then the wave turns on a different angle with respect to inertial space depending on the number of MS. The inertial properties of elastic waves during rotation were discussed in [3] for inextensible ring rotating with variable angular velocity and in [4] for an axisymmetric shell rotating with constant angular velocity. The vibrations of flexible extensible rotating ring are considered in [5] taking into account the geometrical nonlinearity. Paper [6] presents a survey of wheel vibrations in a complex dynamic vehicle suspension system and their influence on the forces transduced in a high frequency area from the tyre to the vehicle’s body. Secondly, it also presents the transient evolution of tyre models used for prediction and understanding high frequency movements in the tyre contact area, producing the guiding forces and torques during vehicle handling. The effects of rotation on the NF of a loaded tyre have been studied in [7] using FE model. In particular, the phenomenon of a frequency loci veering is considered. This phenomenon consists of mutually repulsive behaviour of the NF, and is induced by the aperiodicity of the tyre resulting from its deformations under the load. A model of a reinforced tyre was proposed in [8]. In the case of wheel rolling without slipping in an unknown in advance contact area, the complete system of equations of motion was obtained. The steady-state regime of rolling at constant speed was investigated. In this paper, we study the vibrations of an unloaded and loaded tyre rolling at constant speed. This tyre model was also used in studying the vibrations of an unloaded and loaded non-rotating tyre [9].

n

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0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.11.019

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Current problems of investigating the dynamics require an implementation of fast calculating models. Therefore, the problem of constructing models, which are capable of both simulating complex dynamic processes with reasonable accuracy and not requiring significant computational resources, is very important. There are two approaches: model approach and phenomenological approach. Within the phenomenological approach, the relations characterising the dependence of forces and moments applied to the wheel disc on the parameters of motion are empirical. The internal structure of the deformable periphery and the details of interaction are not usually considered. Thus, the object of study is represented in the form of a ‘‘black box’’ which can be viewed solely in terms of its input and output characteristics. On this approach a number of modern researches are based. In contrast to the phenomenological approach, a model approach is used in this paper. This approach allows mathematical description of a deformable periphery. That is, we propose a possible mathematical model of radial tyre. The paper is structured as follows. The model of a wheel with a reinforced tyre will be considered first. Then, the equations of motion and the conditions on the boundary of an unknown in advance contact area will be obtained. Next, the vibrations of unloaded non-rotating (UNR), unloaded rotating (UR), loaded non-rotating (LNR) and loaded rotating (LR) tyre will be considered. The results for the unloaded non-rotating and loaded non-rotating cases will be compared with experiment. The results will be discussed before concluding remarks.

2. Modelling of a wheel with a reinforced tyre Assume that the wheel with a reinforced tyre consists of disc (0) joined to the sidewalls (1,2) and of tread (3) (Fig. 1(a)). The wheel disc is a rigid body with six degrees of freedom. In the undeformed state, the sidewalls are represented by parts of two tori. The elastic sidewalls are described by the Mooney–Rivlin model [10] of incompressible rubber. The tread is reinforced with inextensible cords. In the undeformed state, the tread is represented by a circular cylinder of radius r and height 2l (tread width). Introduce an inertial frame OX 1 X 2 X 3 (IF), such that the tyre contacts by its tread with the ground plane OX 1 X 2 and a moving frame Cx1 x2 x3 (MF) with its origin C coincident to the mass centre of the disc (Fig. 1(a)). Denote ðX 1 ,X 2 ,X 3 Þ the coordinates of point C in the IF. The position of the tread is determined by the rotation angles b0 , y about the axes OX3 and Cx2, respectively. We define the surface of tread for b0 and y fixed in the form ! 3 3 X X j mod 2p X i li þ C3 ðb0 ÞC2 ðWÞ re1 þ lxe2 þr U i ðj, x,tÞei , r3 ðj, x,tÞ ¼ x 2 ½1; 1 i¼1

2

C2 ðWÞ ¼ 6 4

i¼1

cos W

0

sin W

0

1

0

sin W

0

cos W

3 7 5,

2

cos b0 C3 ðb0 Þ ¼ 6 4 sin b0 0

sin b0 cos b0 0

0

3

7 0 5,

W ¼ yþj

(1)

1

x3 ϑ

X3

θ

l0

ϕ

x3 2

3

θ ϕ

0

lξ

l0 β0

C

x2

θ

C

O

θ

X2

ϕ

r

ϕ

ru

rv ϕ1

X1 α2

α1

x1 ϕ2

Fig. 1. The model of a wheel with a reinforced tyre.

ϑ

x1

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Here li ,ei are unit vectors of axes OXi and Cxi, respectively; the value of parameter x ¼ 1 (x ¼ 1) corresponds to the linking line of the tread and the first (second) sidewall, the value x ¼ 0 corresponds to median line l0 of the tread; rU i ðj, x,tÞ are components of displacement vector of points of tread in the MF. The tread deformations are considered taking into account the exact nonlinear conditions of inextensibility of reinforcing cords [8]    2  2  2 @r3    ¼ 1 ¼) 2 l @U 2 þ @U 1 þ @U 2 þ @U 3 ¼0  lqx  r @x @x @x @x    @r3    r@j ¼ 1

¼)

@r3 @r3  ¼0 @x @j

¼)

       2 @U 3 @U 3 2 @U 2 2 @U 1 þ þ þ U3 ¼0 þ U1  2 U1  @j @j @j @j       @U 1 @U 1 @U 2 l @U 2 @U 3 @U 3 þ þ U3 þ 1 þ U1   ¼0 @x @j @j r @x @x @j

(2)

We assume that the functions Uk and their derivatives are small. Neglecting their squares we obtain the linearised equations @U 2 ¼ 0, @x

@U 3 ¼ U1, @j

l @U 2 @U 3 ¼ r @j @x

(3)

The solution of system (3) reads l U 1 ¼ xw00 ðj,tÞ þ uðj,tÞ, r

U 2 ¼ wðj,tÞ,

l U 3 ¼ xw0 ðj,tÞvðj,tÞ, r

u ¼ v0 ,

0

¼

@ @j

(4)

Here ruðj,tÞ, rvðj,tÞ and rwðj,tÞ are, respectively, the radial, tangential and lateral components of the displacement vector ruðj,tÞe1 þ rwðj,tÞe2 rvðj,tÞe3 of points of l0 in the MF (Fig. 1(b)). Substituting the expressions (4) in (2) we obtain that w0 ¼ w00 ¼ 0. Eqs. (4) can be represented as U 1 ¼ uðj,tÞ,

U 2 ¼ wðtÞ,

U 3 ¼ vðj,tÞ

Thus, due to nonlinear geometric constraints in the deformed state, the tread retains its cylindrical shape, which is not circular for a typical configuration. Eqs. (2) imply the condition of the inextensibility of l0 2Z 6 ¼ ð1 þ u þ v0 Þ2 þðvu0 Þ2 ¼ 1

or

2ðu þv0 Þ þðu þ v0 Þ2 þðvu0 Þ2 ¼ 0

(5)

It was assumed [8] that the contact area of the tyre and the plane OX 1 X 2 can be represented by a rectangle of constant width 2l, equals to the tread width, and of variable length rðj2 ðtÞj1 ðtÞÞ. The length is defined by two functions of time j1 ðtÞ, j2 ðtÞ, which are unknown in advance (Fig. 1(b)). These functions can be obtained from the equations of motion. The values j 2 L1 ¼ ½j1 ðtÞ, j2 ðtÞ correspond to contact area, and the values j 2 L2 ¼ ½j2 ðtÞ,2p þ j1 ðtÞ correspond to the free surface of the tyre. In the contact area L1 the holonomic constraint is given by  p (6) r3 ¼ r j l1 þlxl2 , j 2 L1 2 In this relation, without loss of generality, we have assumed that the rolling of the wheel occurs along the axis OX1, when the median line l0 of the tread coincides with this axis (the angle b0 ¼ 0). The conditions (1) and (6) allow the determination of displacements of points of l0 in the contact area     p X1 X3 p X1 X3 X2 j cos W þ sin W1, v ¼  j sin W þ cos W, w ¼  , W ¼ yþj (7) u¼ 2 r r 2 r r r We assume that X 2 ¼ 0 and therefore (the penultimate relation of (7)) w ¼0, i.e. the mass centre of the disc does not move laterally. 3. The equations of motion Let us project the velocity field of points of the tread r_ 3 onto the first and the third axes of the IF. Its projections ðZ_ 1 , Z_ 2 , Z_ 3 ÞT ¼ r_ 3 can be represented by the expressions Z_ 1 ¼ X_ 1 r y_ ½ð1þ uÞsin W þ v cos W þr u_ cos Wr v_ sin W Z_ 3 ¼ X_ 3 r y_ ½ð1 þuÞcos Wv sin Wr u_ sin Wr v_ cos W

(8)

Suppose that the wheel rolls without slipping and without jumping. This means that the velocity of points of l0 in the contact area L1 are equal to zero, i.e. Z_ 1 ¼ Z_ 3 ¼ 0. The virtual displacements satisfy the equations dZ 1 ¼ dZ 3 ¼ 0, obtained from (8) by replacing the time derivatives on a variation of the corresponding functions.

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The equations of motion and conditions on the unknown in advance boundary of contact area were obtained [8] from the Hamilton–Ostrogradsky variational principle for nonconservative systems Z t2 ðdT þ dAÞ dt ¼ 0 t1

The kinetic energy of the wheel T consists of kinetic energy of the disc and the kinetic energy of the tyre assuming that the whole mass of the tyre is distributed uniformly along l0 with linear density r Z 2p 2 1 1 1 2 2 2 2 T ¼ md ðX_ 1 þ X_ 3 Þ þ J2d y_ þ rr ðZ_ 1 þ Z_ 3 Þ dj 2 2 2 0 Here md, J2d are the mass and moment of inertia of the disc about the axis Cx2. The work dA at virtual displacements has the following structure:

dA ¼ dAF þ dAP þ dN 1 þ dN3 þ dN 6 Here dAF is the work performed by the longitudinal force F1, by the vertical load P and by the wheel torque M2 applied to the wheel disc

dAF ¼ F 1 dX 1 P dX 3 þ M2 dy The work dAP is the work performed by the potential forces. It comprises the work performed by the pressure and the variation of potential energy of the rubber stretching in the Mooney–Rivlin model when the sidewalls and the tread are deformed  Z 2p  1 1 1 1 dAP ¼  n0 du þ n01 du2 þ n11 du02 þ n02 dv2 þ n12 dv02 þ m21 v0 du þ m12 u0 dv þ m20 vdu0 þm02 udv0 dj (9) 2 2 2 2 0

Remark 1. The constant coefficients n0, n01, n11, n02, n12, m21, m12, m20, m02 are determined analytically (see Appendix A) in elementary functions by evaluating definite integrals (by integrating over sidewalls and over tread of the tyre) and depend on the geometric parameters of the tyre and on the internal tyre pressure p. Usually, in the phenomenological approach the coefficients are unknown and are found from experiment. The works dN 1 , dN 3 , dN6 are performed by the reactions of the constraints: Z Z dN 1 ¼ m1 ðj,tÞdZ 1 dj þ m11 dZ 11 þ m12 dZ 12 , dN3 ¼ m3 ðj,tÞdZ 3 dj, L1

dN6 ¼

L1

Z

lðj,tÞdZ 6 dj

(10)

L2

The Lagrange multiplier m1 ¼ m1 ðj,tÞ describes the longitudinal component of the reaction of the constraint, m11 , m12 , dZ 11 , dZ 12 are the longitudinal components of the reaction and the virtual displacements at the boundary points (j ¼ j1 , j ¼ j2 ) of the contact area. The Lagrange multiplier m3 ¼ m3 ðj,tÞ describes the normal component of the reaction of the constraint, reduced to unit length of the tread in the contact area. The second relation of (10) does not contain the terms m31 , m32 , since it is assumed that the components of the reactions of the constraints along the axis OX3 vanish at the boundary points of the contact region. The Lagrange multiplier l ¼ lðj,tÞ determines the tension of l0, Z6 was determined in (5). The complete system of 14 equations in 14 unknowns X 1 ðtÞ, X 3 ðtÞ (the coordinates of the mass centre of the disc), yðtÞ (the rotation angle about the axis Cx2), ruðj,tÞ9j2L1 , rvðj,tÞ9j2L1 , ruðj,tÞ9j2L2 , rvðj,tÞ9j2L2 (the radial and tangential components of the displacement vector of points of l0 in the MF in the contact area L1 and in the free surface of the tyre L2), j1 ðtÞ, j2 ðtÞ (the functions describing the boundaries of the contact area), m1 ðj,tÞ9j2L1 , m3 ðj,tÞ9j2L1 , m11 ðtÞ, m12 ðtÞ, lðj,tÞ9j2L2 (the Lagrange multipliers) has the following structure: three Lagrange’s equations of motion with Lagrange multipliers (a feature of these equations is containment of the integral terms of the functions u,v and their derivatives) Z Z d d  rX_ 1 T þ F 1 þ m1 dj þ m11 þ m12 ¼ 0,  rX_ 3 TP þ m3 dj ¼ 0 dt dt L1 L1

ry T

d r _ T þ M2 þ dt y

Z L1

@Z_

@Z_

!

@Z_

@Z_

m1 _1 þ m3 _3 dj þ m11 _11 þ m12 _12 ¼ 0 @y @y @y @y

(11)

four equations of motion in partial derivatives N 1 ðu,vÞ þ m1 r cos Wm3 r sin W ¼ 0,

ru T

d ru_ T þ N1 ðu,vÞ þ lð1 þu þv0 Þ½lðu0 vÞ0 ¼ 0, dt

N 2 ðu,vÞm1 r sin Wm3 r cos W ¼ 0,

rv T

N 1 ðu,vÞ ¼ n0 n01 u þ n11 u00 ðm21 m20 Þv0 ,

j 2 L1

d rv_ T þN 2 ðu,vÞ½lð1 þ u þv0 Þ0 lðu0 vÞ ¼ 0, dt N2 ðu,vÞ ¼ n02 v þ n12 v00 ðm12 m02 Þu0

j 2 L2 (12)

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three constraint equations: (5) for j 2 L2 and the first two equations of (7) for j 2 L1 . And four conditions imposed on the jumps in the functions at the boundary points of the contact area (dynamic boundary conditions) _ kj _ k ð1Þk ½lðu0 vÞlðkÞ þ n11 ½u0 k þ r m1k cos Wk ¼ 0, Wk ¼ y þ jk rr3 ½u _ kj _ k ð1Þk ½lð1 þ u þ v0 ÞlðkÞ þ n12 ½v0 k r m1k sin Wk ¼ 0, k ¼ 1; 2 rr3 ½v

(13)

Here ½hðWÞk ¼ hðWk þ0ÞhðWk 0Þ is the jump in the function at the final point of the contact area. The index l(k) means that: ½hðWÞlð1Þ ¼ hðW1 0Þ, ½hðWÞlð2Þ ¼ hðW2 þ 0Þ. Remark 2. In determining the functions u, v taking into account the continuity conditions at the boundary points of the contact area (kinematic boundary conditions) is also necessary. The existence of potential energy of pressure allows writing them as ½uk ¼ ½vk ¼ 0, k¼1,2. Remark 3. The subsystem (11) is related to subsystems (12)–(13), because it comprises the unknowns, which we determine from the subsystems (12)–(13). 4. Unloaded tyre Suppose that the unloaded wheel rotates with constant angular velocity O. Then X 1 ¼ const,

X 3 ¼ const,

y_ ¼ O

The equations of motion (the third and fourth equations of system (12)) and the condition of inextensibility of l0 (5) read _ þn11 u00 n01 u þðm20 m21 Þv0 n0 þ lð1 þu þ v0 Þ½lðu0 vÞ0 ¼ 0 € O2 ð1þ uÞ2OvÞ rr 3 ðu € O2 v þ2OuÞ _ þðm02 m12 Þu0 þ n12 v00 n02 v½lð1 þu þv0 Þ0 lðu0 vÞ ¼ 0 rr 3 ðv 2ðu þ v0 Þ þ ðu þ v0 Þ2 þðvu0 Þ2 ¼ 0

¼)

u ffi v0

(14)

In the case of an unloaded tyre, the boundary conditions for the functions uðj,tÞ, vðj,tÞ, lðj,tÞ are to be replaced by the periodicity conditions of these functions uð0Þ ¼ uð2pÞ,

vð0Þ ¼ vð2pÞ,

lð0Þ ¼ lð2pÞ

Suppose that

lðj,tÞ ¼ n0 g 0 þ l1 ðj,tÞ, g 0 ¼ rr 3 O2 Here l1 is of first order of smallness. In this case, the tension of rotating tread increases (compared with non-rotating tyre) due to the centrifugal forces of inertia added to the pressure inside the tyre. Differentiating both sides of the first equation of (14) and adding it to the second equation, using the linearised condition of the inextensibility of l0, we obtain the equation for the function vðj,tÞ and the condition of its periodicity

rr 3 v€ 00 þ 4rr3 Ov_ 0 rr 3 v€ þ ða0 g 0 Þvð4Þ þ ða1 3g 0 Þv00 þ a2 v ¼ 0, vð0Þ ¼ vð2pÞ a0 ¼ n0 n11 ,

a1 ¼ 2n0 þ n01 þn12 þm20 m21 m02 þm12 ,

a2 ¼ n0 n02

(15)

The natural vibrations of the tyre, described by Eq. (15), can be obtained in the form v ¼ eiot XðjÞ,

u ¼ v0 ¼ eiot X 0 ðjÞ

where o is an angular frequency. Then ða0 g 0 ÞX ð4Þ þ ða1 3g 0 rr 3 o2 ÞX 00 þ 4rr 3 OoiX 0 þ ða2 þ rr 3 o2 ÞX ¼ 0 The solution of this equation can be represented in the form XðjÞ ¼ G1 ep1 j þG2 ep2 j þ G3 ep3 j þ G4 ep4 j

(16)

where pj , j ¼ 1, . . . ,4 are the roots of the characteristic equation ða0 g 0 Þp4 þða1 3g 0 rr 3 o2 Þp2 þ4rr 3 Ooip þða2 þ rr 3 o2 Þ ¼ 0

(17)

Since the function XðjÞ must be 2p-periodic, only exponents with pure imaginary indices need to be retained in (16), i.e. p ¼ in, n 2 Z. The characteristic Eq. (17) reads yðoÞ ¼ Aðn2 Þo2 þ Bðn, OÞo þ Cðn2 , O2 Þ ¼ 0 Aðn2 Þ ¼ rr 3 ð1þ n2 Þ,

Bðn, OÞ ¼ 4rr 3 nO,

Cðn2 , O2 Þ ¼ ða0 g 0 Þn4 ða1 3g 0 Þn2 þa2

(18)

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Table 1 The input data used for comparison. Input

Tyre size

m (kg)

p (MPa)

I II

205/55 R16 205/60 R15

9.37 8.5

0.25 0.16

Table 2 The NF of an unloaded non-rotating tyre. Comparison with experimental data. Model, Input/ Frequency (Hz) Experiment/ Deviation n1

n2

n3

n4

n5

n6

I I %

101.5 97.35 4.26

117.69 122.93 4.26

143.1 149.47 4.26

173.0 176.64 2.06

205.28 205.7 0.2

238.93 235.9 1.29

273.43 – –

II II %

94.31 – –

110.77 111.66 0.79

134.57 134.98 0.3

162.29 161.56 0.45

192.19 190.62 0.82

223.38 223.41 0.02

255.38 252.61 1.1

n7

4.1. Unloaded non-rotating tyre If O ¼ 0, i.e., the wheel does not rotate, then Bðn,0Þ ¼ 0, g 0 ¼ 0. In this case, the infinite frequency spectrum can be found analytically from Eq. (18) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a0 n4 þ a1 n2 a2 o0n ¼ 7 , n2Z (19) rr3 ð1 þ n2 Þ in agreement with results of [9]. The corresponding MS are represented as X n ðjÞ ¼ Gn1 cosðnjÞ þ Gn2 sinðnjÞ,

8Gn1 ,Gn2

Thus, each MS is a linear combination of cosðnjÞ and sinðnjÞ. 4.1.1. Comparison of results with experimental data for UNR tyre The quantities of obtained NF nn ¼ o0n =ð2pÞ were compared with results of two experiments: Experiment I1 [11], Experiment II [12]. Making comparison we used the same input data as in the experiments (the tyre size,2 the mass of tyre m, the quantity of internal pressure p) (Table 1). Remark 4. In Experiment II the data concerning the mass of the tyre was not represented. Therefore, the mass of the tyre was estimated based on the perceptions of mass of tyre of similar size. The corresponding NF are represented in Table 2. Thus, the maximum deviation, expressed as a percentage, is 4.26 percent (n1 , n2 , n3 ) for Experiment I and 1.1 percent (n7 ) for Experiment II. 4.2. Unloaded rotating tyre The infinite frequency spectrum of an UR tyre can be found analytically from the characteristic Eq. (18). Thus, the NF of an UR tyre are expressed in terms of NF of an UNR tyre sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2n n2 ðn2 1Þ2 2 on ¼ O 7 ðo0n Þ2 þ O , n2Z 2 1þ n ðn2 þ 1Þ2

1 TMPT (Tyre Model Performance Test) was developed by an international group of experts consisting of members of vehicle industry, tyre ¨ manufacturers, tyre model developers, multi-body-system program suppliers and universities. It was organised by: Prof. P. Lugner and Prof. M. Plochl (Institute of Mechanics and Mechatronics, Division of Vehicle System Dynamics and Biomechanics, Vienna University of Technology, Austria). 2 205/55 R16: the first number means the ‘‘nominal section width’’ of the tyre in millimetres (the widest point from both outer edges of the inflated tyre); the second number is the ‘‘aspect ratio’’ of the sidewall height to the total width of the tyre, as a percentage; letter R indicates construction of the fabric carcass of the tyre (radial); the third number is diameter in inches of the wheel that the tyres are designed to fit.

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in agreement with results of [13,14], where the tyre was modelled as inextensible ring. In contrast to the spectrum (19) in these studies the expression for the NF of an UNR tyre has the following form: 2

ðo0n Þ2 ¼

n2 ðn2 1Þ2 Eh , n2 þ1 12rR4

n2Z

300

300

250

250 Frequency (Hz)

Frequency (Hz)

Here R is the mean radius of the ring, h is a thickness of the ring, r is a density of the material, and E is Young’s modulus. In our work, analytical expressions for the coefficients a0, a1, a2, determining the NF of an UNR tyre, have a fairly complicated form (the coefficients depend on the geometric parameters of the tyre and on the internal tyre pressure) and are calculated in terms of elementary functions. The plot of NF as a function of angular velocity is shown in Fig. 2(a) for Input I. The experimental NF, corresponding to an UNR tyre (Experiment I), are plotted as black squares. On the y-axis the two branches grow from each point corresponding to the NF of an UNR tyre (the top branch corresponds to on 4 0, the bottom branch corresponds to on 4 0, where n 40), i.e. each NF of an UNR tyre corresponds to two NF of an UR tyre. We turn to Fig. 3 to comment on this effect. Let us fix a number n 4 0. The parabola yðoÞ ¼ Aðn2 Þo2 þCðn2 ,0Þ, corresponding to UNR tyre is represented in the figure by the dotted line. In this case, the coefficients Aðn2 Þ 4 0, Cðn2 ,0Þ ¼ a0 n4 a1 n2 þ a2 o0. This parabola intersects the x-axis at the point o0n 40 (the frequency o0n o0 will not be considered). Now, if Oa0 (UR tyre), then the original parabola splits into two parabolae. One of these parabolae yðoÞ ¼ Aðn2 Þo2 þ Bðn, OÞo þ Cðn2 , O2 Þ situated on the right, corresponds to the case Bðn, OÞ o0, and the other parabola yðoÞ ¼ Aðn2 Þo2 þ Bðn, OÞo þ Cðn2 , O2 Þ situated on the left, corresponds to the case

200 150 100 50

200 150 100 50

0

0 0

50

100 150 200 Rotational velocity (rad/s)

250

300

0

50

100 150 200 Rotational velocity (rad/s)

250

300

Fig. 2. The NF of an unloaded rotating tyre (Input I) as a function of angular velocity, (a) Lagrangian specification, (b) Eulerian specification. The black squares ’ are the experimental NF (Experiment I).

Fig. 3. The split of NF due to rotation, (1) yðoÞ ¼ Aðn2 Þo2 þ Cðn2 ,0Þ, Bðn,0Þ ¼ 0, (2) yðoÞ ¼ Aðn2 Þo2 þ Bðn, OÞo þ Cðn2 , O2 Þ, Bðn, OÞ 4 0, (3) yðoÞ ¼ Aðn2 Þo2 þ Bðn, OÞo þ Cðn2 , O2 Þ, Bðn, OÞ o 0.

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Bðn, OÞ 40. In this case, the coefficient Cðn2 , O2 Þ ¼ Cðn2 ,0Þ þ g 0 n2 ð3n2 Þ r Cðn2 ,0Þ o 0 for n 41. Thus, the original NF o0n splits into on and on . One can show that o0n o on , on 4 0 and, starting with a certain nn, 8n 4 nn o0n o on . Thus, starting with a certain nn, both branches growing from a point corresponding to NF of an UNR tyre increase monotonically (Fig. 2(a)). If O-1 and n 4 1 the behaviour of branches is described as nO and nðn2 3Þðn2 þ1Þ1 O. Note that if n¼1, then the top branch corresponds to o1 ¼ O þ o01 , and the bottom branch corresponds to o1 ¼ O þ o01 . In this case, o1 vanishes at O ¼ o01 ¼ 2pn1 ¼ 637:74 rad s1 . Making the change of variables j by a ¼ j þ Otp=2 we pass from the Lagrangian specification to the Eulerian specification. In this case, one should use Eq. (22) (as we shall see below) instead of the characteristic Eq. (17). An infinite spectrum of NF can be represented in the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn2 1Þ n2 ðn2 1Þ2 2 on ¼ O 7 ðo0n Þ2 þ O , n2Z 1þ n2 ðn2 þ 1Þ2 The plot of NF as a function of angular velocity is shown in Fig. 2(b). The top branch corresponds to on 4 0, the bottom branch corresponds to on 4 0, where n 4 0. Note that if n¼ 1, then o21 ¼ ðo01 Þ2 does not depend on the angular velocity of rotation. If O-1 and n 4 1 the behaviour of branches is described as ðn2 þ 1Þð2nðn2 1ÞOÞ1 ðo0n Þ2 and 2nðn2 1Þðn2 þ1Þ1 O. Thus, neither frequency vanishes, in agreement with results of [15], where the free nonlinear vibrations of a rotating thin ring were studied and with [16], where it was ascertained by theoretical and experimental investigations for the in-plane flexural bending modes of a rotating ring that the frequencies never become zero over the whole range of the rotational speed. 5. Loaded tyre Consider the problem of vibrations of a tyre about the steady-state regime of rolling at constant speed without slipping in the contact area. The steady-state regime of rolling of a loaded tyre was considered in [8]. Suppose that X_ 1 ¼ r O,

y_ ¼ O

X 3 ¼ const,

Making the change of variables j by a ¼ j þ Otp=2 we pass from the Lagrangian specification to the Eulerian specification. We represent the functions determining the shape of the deformed tread, the functions determining the contact area and the Lagrange multipliers in the form uðj,tÞ ¼ UðaÞ þU vib ða,tÞ,

vðj,tÞ ¼ VðaÞ þV vib ða,tÞ,

lðj,tÞ ¼ l0 ðaÞ þ lvib ða,tÞ, m1k ðtÞ ¼ m01k þ mvib 1k ðtÞ

ak ðtÞ ¼ a0k þ avib k ðtÞ, a0k ¼ j0k ðtÞ þ Ot

p 2

j_ 0k ðtÞ ¼ O

¼ const,

The boundaries of contact area are defined by two functions a1 ðtÞ, a2 ðtÞ (Fig. 1(b)), which are unknown in advance. We take into account the differentiation rules 0 _ j,tÞ ¼ U 0 ðaÞO þ U vib ða,tÞO þ U_ vib ða,tÞ uð

@ 00 0 € j,tÞ ¼ U 00 ðaÞO2 þU vib uð ða,tÞO2 þ 2U_ vib ða,tÞO þ U€ vib ða,tÞ, _¼ , @t

0

¼

@ @a

0

Then, the solution describing the steady-state motion UðaÞ, VðaÞ, l ðaÞ, m01k , a0k and satisfying the equations and boundary conditions can be found [8] from the system 0

0

g 0 ð1þ UU 00 þ2V 0 Þn0 n01 U þ n11 U 00 ðm21 m20 ÞV 0 þ l ½l ðU 0 VÞ0 ¼ 0 0

0

g 0 ðVV 00 2U 0 Þn02 V þn12 V 00 ðm12 m02 ÞU 0 ðl Þ0 l ðU 0 VÞ ¼ 0 2ðU þ V 0 Þ þ ðU þV 0 Þ2 þ ðVU 0 Þ2 ¼ 0

¼)

U ¼ V 0

0

g 0 ½U 0 k þ ð1Þk ½l ðU 0 VÞlðkÞ n11 ½U 0 k þ r m01k a0k ¼ 0 0

g 0 ½V 0 k þð1Þk ½l lðkÞ n12 ½V 0 k þ r m01k ¼ 0 ½Uk ¼ ½Vk ¼ 0, 0

0 n0 g 0 þ l1 ,

k ¼ 1; 2

l01

where is of first order of smallness. The variables U vib ,V vib , defining the vibrations of tyre about the Here l ¼ steady-state motion, satisfy following system: 0 00 0 2rr 3 OU_ 0vib þ 2rr 3 OV_ vib rr 3 U€ vib þ g 0 ðU vib þ V vib Þ þðn11 n0 ÞU vib n01 U vib þðm20 m21 þ n0 ÞV vib þ lvib ¼ 0 00 0 0 00 þU vib Þ þ ðm02 m12 n0 ÞU vib þ n12 V vib þðn0 n02 ÞV vib l0vib ¼ 0 2rr 3 OV_ 0vib 2rr 3 OU_ vib rr 3 V€ vib g 0 ðV vib 0 0 0 2ðU vib þV vib Þ þ ðU vib þ V vib Þ2 þ ðV vib U vib Þ2 ¼ 0

¼)

0 U vib ¼ V vib

ð1Þk ½rr 3 OU_ vib ðn0 g 0 ÞV vib þðn0 n11 ÞU 0vib lðkÞ þ r m01k avib k ¼ 0

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ð1Þk ½rr 3 OV_ vib ðn12 g 0 ÞV 0vib þ lvib lðkÞ þ r mvib 1k ¼ 0 ½U vib k ¼ ½V vib k ¼ 0,

k ¼ 1; 2

(20)

Here taking into account that Lagrange multipliers lvib , mvib 1k , as well as, avib k are of first order of smallness is necessary. We have also used the following property ½U 0vib k ¼ ð1Þk ½U 0vib lðkÞ (the functions determining the vibrations are equal to zero in the contact area). Differentiating both sides of the first equation and adding it to the second equation, using the linearised condition of the inextensibility of l0, we obtain the equation and the boundary conditions ð3Þ

00 0 rr3 V€ vib rr 3 V€ vib þ 2rr 3 OV_ vib þ 2rr 3 OV_ vib þ a0 V ð4Þ þ a1 V 00vib þ a2 V vib ¼ 0 vib

V vib ða01 þ2p þ avib 1 Þ ¼ 0,

V 0vib ða01 þ2p þ avib 1 Þ ¼ 0,

V vib ða02 þ avib 2 Þ ¼ 0,

V 0vib ða02 þ avib 2 Þ ¼ 0

(21)

Taking into account the dynamic boundary conditions (the fourth relation in (20)) one can determine avib k

avib k ¼

ð1Þk ðn0 n11 Þ½V 00vib lðkÞ r m01k

Thus, the boundaries of the contact area vibrate at the same frequency as the function V 00vib . However, in determining the frequency of the vibrations of the tyre, the length of the contact area can be taken as constant, since within the model chosen its variation determines a second order of smallness correction to the frequency. Hence, the boundary conditions in problem (21) are equivalent to the following: V vib ða01 þ 2pÞ þ V 0vib ða01 þ 2pÞavib 1  V vib ða01 þ2pÞ ¼ 0,

V vib ða02 Þ þ V 0vib ða02 Þavib 2  V vib ða02 Þ ¼ 0

V 0vib ða01 þ 2pÞ þ V 00vib ða01 þ 2pÞavib 1  V 0vib ða01 þ2pÞ ¼ 0,

V 0vib ða02 Þ þ V 00vib ða02 Þavib 2  V 0vib ða02 Þ ¼ 0

For simplicity, we will write ak instead of a0k . The Lagrange multipliers lvib , mvib 1k determining, respectively, the correction to the tension and the corrections to the longitudinal reactions at the boundary points of the contact area during vibrations, are expressed from the system (20) in terms of V vib 0 0 lvib ¼ rr3 V€ vib 2rr 3 OV_ 00vib 2rr 3 OV_ vib þ ðn11 n0 ÞV ð3Þ ðm20 m21 þ n0 þn01 ÞV vib vib 00 r mvib 1k ¼ ð1Þk ½2rr 3 OV_ vib þ ðn0 n11 ÞV ð3Þ  vib lðkÞ

We will represent the functions determining the vibrations in the form V vib ¼ eiot XðaÞ,

U vib ¼ V 0vib ¼ eiot X 0 ðaÞ

Substituting the expression for V vib into Eq. (21), we obtain the equation a0 X ð4Þ þ 2rr 3 OoiX ð3Þ þ ða1 rr 3 o2 ÞX 00 þ2rr 3 OoiX 0 þða2 þ rr 3 o2 ÞX ¼ 0 with solution of form (16) (replacing j on a). The characteristic equation reads a0 p4 þ 2rr 3 Ooip3 þ ða1 rr 3 o2 Þp2 þ2rr 3 Ooip þ ða2 þ rr 3 o2 Þ ¼ 0

(22)

The coefficients Gi in the solution (16) are determined from the boundary conditions G1 ep1 ða1 þ 2pÞ þG2 ep2 ða1 þ 2pÞ þ G3 ep3 ða1 þ 2pÞ þ G4 ep4 ða1 þ 2pÞ ¼ 0 G1 ep1 a2 þ G2 ep2 a2 þ G3 ep3 a2 þG4 ep4 a2 ¼ 0 G1 p1 ep1 ða1 þ 2pÞ þG2 p2 ep2 ða1 þ 2pÞ þ G3 p3 ep3 ða1 þ 2pÞ þ G4 p4 ep4 ða1 þ 2pÞ ¼ 0 G1 p1 ep1 a2 þ G2 p2 ep2 a2 þ G3 p3 ep3 a2 þG4 p4 ep4 a2 ¼ 0

(23)

The homogeneous system (23) has a non-zero solution if its determinant is equal to zero: f ðoÞ ¼ eðp1 þ p2 þ p3 þ p4 Þða1 þ 2pÞ ½ðp3 p1 Þðp4 p2 Þðeðp2 þ p4 ÞðDa2pÞ þ eðp1 þ p3 ÞðDa2pÞ Þ ðp3 p2 Þðp4 p1 Þðeðp1 þ p4 ÞðDa2pÞ þ eðp2 þ p3 ÞðDa2pÞ Þðp2 p1 Þðp4 p3 Þðeðp3 þ p4 ÞðDa2pÞ þeðp1 þ p2 ÞðDa2pÞ Þ ¼ 0

(24)

Here Da ¼ a2 a1 determines the length of the contact area. Remark 5. Using the Lagrangian specification the term of the third degree p3 disappears (the characteristic Eq. (17)). Then, taking into account Vie te’s formulae p4 ¼ ðp1 þ p2 þ p3 Þ, the determinant (24) can be represented as f ð oÞ ¼ ½ðp1 þ p2 Þ2 ðp2 þ p3 Þ2 chððp1 þp3 Þð2pDjÞÞ þ½ðp1 þ p3 Þ2 ðp1 þ p2 Þ2 chððp2 þ p3 Þð2pDjÞÞ 2 þ ½ðp2 þ p3 Þ2 ðp1 þ p3 Þ2 chððp1 þ p2 Þð2pDjÞÞ ¼ 0 where Dj ¼ j2 j1 determines the length of the contact area.

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5.1. Loaded non-rotating tyre In the case of a LNR tyre the characteristic Eq. (22) is biquadratic and its roots have the property p3 ¼ p1 , p4 ¼ p2 . Eq. (24) reads f ð oÞ ¼ 2p1 p2 ½1chðp1 ð2pDaÞÞchðp2 ð2pDaÞÞ þ ðp21 þ p22 Þshðp1 ð2pDaÞÞshðp2 ð2pDaÞÞ ¼ 0 4 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u urr 3 o2 a 8 ða rr 3 o2 Þ2 4a ða þ rr 3 o2 Þ t 1 1 0 2 p1;2 ðoÞ ¼ 2a0

(25)

in agreement with results of [9]. Let us solve Eq. (25) numerically for Input I. Three graphs of function f ðoÞ for three values of length of the contact area Da ¼ 0:1, 0.3, 0.5 rad are represented in Fig. 4. The points of intersection of the graphs with the x-axis define an infinite spectra of NF (Table 3). The figure shows that increasing of the contact area (due to the increasing of the vertical load) implies that NF also increase. In fact, the mass of the vibrating part of the tyre decreases with the increasing of the contact area, as the tyre does not vibrate in the contact area. Moreover, this result correlates well with the theorem from the small vibrations theory on the NF behaviour (interfusion) at imposing the constraint. The corresponding MS (Fig. 5) (Input II, Da ¼ 0:3 rad) are represented as XðaÞ ¼ G1 ep1 a þ G3 ep1 a þ G2 ep2 a þG4 ep2 a where the constants Gi are determined from boundary conditions (23) G1;3 ¼ e 8 p1 ða1 þ 2pÞ ½p1 shðp2 ð2pDaÞÞ 7 p2 ðe 7 p1 ð2pDaÞ chðp2 ð2pDaÞÞÞGn5 G2;4 ¼ e 8 p2 ða1 þ 2pÞ ½ 7p1 ðe 7 p2 ð2pDaÞ chðp1 ð2pDaÞÞÞp2 shðp1 ð2pDaÞÞGn5 ,

8Gn5

The MS are represented in Fig. 5 about the equilibrium of a LNR tyre. Each mode of the UNR tyre is double and for each NF two identical MS exist. The fixed contact points of the tyre cause a loss of the circular symmetry. Thus, for each NF of an UNR tyre there are two different NF of a LNR tyre. This is explained by the disturbance of free wave motion due to contact. The identical modes split into two not identical ones. The MS subdivides into a symmetric and an antisymmetric shapes. For symmetric MS the

50000 40000 30000

Δα=0.1 Δα=0.3 Δα=0.5

f(omega)

20000 10000 0 -10000 -20000 -30000 -40000 -50000 100 120 140 160 180 200 220 240 260 280 300 Frequency (Hz) Fig. 4. The function f ðoÞ for loaded non-rotating tyre (Input I) and for Da ¼ 0:1, 0.3, 0.5 rad.

Table 3 The NF of a loaded non-rotating tyre (Input I).

Da (rad)

0.1 0.3 0.5

Frequency (Hz)

n1

n2

n3

n4

n5

n6

n7

n8

100.4 100.58 100.8

105.82 106.53 107.33

115.51 116.92 118.49

127.72 129.85 132.18

141.71 144.57 147.67

156.76 160.32 164.18

172.65 176.92 181.53

189.06 194.01 199.35

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Fig. 5. The basic mode shapes of a loaded non-rotating tyre (Input II, Da ¼ 0:3 rad), (a) n1 ¼ 92:15 Hz, (b) n2 ¼ 99:35 Hz, (c) n3 ¼ 109:72 Hz, (d) n4 ¼ 122:05 Hz, (e) n5 ¼ 135:81 Hz, (f) n6 ¼ 150:47 Hz.

corrections r mvib 11 , r mvib 12 to the longitudinal reactions at the boundary points of the contact area have opposite signs. As for the antisymmetric MS the corrections carry the same sign. The mass centre of the tyre does not move in the longitudinal direction for the symmetric MS, and it moves for the antisymmetric MS (Fig. 5). Thus, the antisymmetric MS ‘‘sways’’ from side to side. Remark 6. Eq. (25) may give extraneous roots. In particular, one easily sees that this equation has zero roots: p1 ¼ 0 and p2 ¼ 0. However, the corresponding MS XðaÞ is equal to zero. Remark 7. When p1 ¼ p2 one must search the MS in the form XðaÞ ¼ ðG1 þG2 aÞep1 a þ ðG3 þ G4 aÞep1 a The coefficients Gi are determined from boundary conditions ðG1 þG2 ða1 þ 2pÞÞep1 ða1 þ 2pÞ þ ðG3 þ G4 ða1 þ 2pÞÞep1 ða1 þ 2pÞ ¼ 0 ðG1 þ G2 a2 Þep1 a2 þ ðG3 þ G4 a2 Þep1 a2 ¼ 0 ðG1 p1 þ G2 þ G2 p1 ða1 þ2pÞÞep1 ða1 þ 2pÞ ðG3 p1 G4 þG4 p1 ða1 þ 2pÞÞep1 ða1 þ 2pÞ ¼ 0 ðG1 p1 þG2 þ G2 p1 a2 Þep1 a2 ðG3 p1 G4 þG4 p1 a2 Þep1 a2 ¼ 0

(26)

The determinant of system (26) reads f ðoÞ ¼ 1þ 2p21 ð2pDaÞ2 chð2p1 ð2pDaÞÞ, 2

p21 ¼

rr3 o2 a1 2a0

The equation f ðoÞ ¼ 0 has a unique root corresponding to p1 ¼ 0. But in this case, we again obtain that XðaÞ ¼ 0. 5.1.1. Comparison of results with experimental data for LNR tyre The quantities of obtained NF nn ¼ on =ð2pÞ of a LNR tyre for Input II and Da ¼ 0:3 rad were compared with results of experiment [12]3 at vertical load P ¼4000 N (Experiment III). The corresponding NF and MS are represented in Table 4 and in Fig. 5. Thus, for Experiment III the maximum deviation, expressed as a percentage, is 9 percent (n3 ). 3 A similar experiment was also performed by the author in 2005 in the laboratory LAMI (Laboratoire d’Analyse des Mate´riaux et Identification) ENPC (L’Ecole Nationale des Ponts et Chausse´es) with tyre Michelin X Classic 165/65 R13. The wheel placed under the press was loaded with a special frame (Fig. 6(a)). Electromagnetic vibration exciter, suspended using two ropes and glued to the tyre, applies a force to the tyre and makes it vibrate. Information about the applied force is transmitted to the analyser (Fig. 6(b)). The parameters of vibration are measured by an accelerometer, glued to the tyre and are also transmitted to the analyser. The analyser is equipped by two channels and allows the measurement of a transfer function. The NF of vibrations correspond to the maximums of transfer function.

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Amplifier

Accelerometer

Amplifier

Analyzer FFT and signal generator

Exciter

Amplifier

Force sensor Tyre

Fig. 6. (a) The test rig for measurements of NF of a loaded non-rotating tyre, (b) the scheme of the test rig.

Table 4 The NF of a loaded non-rotating tyre. Comparison with experimental data (Experiment III). Model, Input/ Experiment/ Deviation

II III %

Frequency (Hz)

n1

n2

n3

n4

n5

n6

n7

n8

n9

n10

n11

n12

n13

n14

92 – –

99 – –

110 101 9

122 122 0

136 138 1

150 151 1

166 166 0

182 178 2

198 199 1

214 207 3

231 232 0

248 243 2

265 262 1

282 276 2

5.2. Loaded rotating tyre In the case of a LR tyre the spectrum of NF is determined from Eq. (24), where p1, p2, p3, p4 are the roots of (22). This quartic equation can be solved by Ferrari’s method ! ! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffi 2A2 rr3 Oo 1 2A2 rr3 Oo 2g0 7 2ðg0 þ A1 Þ pffiffiffiffiffiffiffiffi  2g0 8 2ðg0 þ A1 Þ þ pffiffiffiffiffiffiffiffi i , p3;4 ðoÞ ¼ i p1;2 ðoÞ ¼ 2 2 a0 a0 2g0 2g0

g0 ¼ 

A1 þ C1 þ C2, 3

A1 ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 3 t B2 B22 B31 7 þ , C 1;2 ¼  2 4 27

2a0 ða1 rr 3 o2 Þ þ 3g 0 rr 3 o2 , 2a20 A3 ¼

A2 ¼

B1 ¼ A3 

A21 , 12

B2 ¼

A1 A3 A22 A3   1 3 8 108

rr3 Ooð2a20 a0 ða1 rr 3 o2 Þg 0 rr3 o2 Þ a30

16a30 ða2 þ rr 3 o2 Þ þ 4a0 g 0 rr 3 o2 ð4a0 a1 þ rr 3 o2 Þ3g 20 ðrr 3 o2 Þ2 16a40

i

(27)

Since all non-zero complex numbers have three distinct complex cube roots, then the third relation in (27) gives three values for g0 . One can take any of these three values, but one cannot combine arbitrary value of the first cube root C1 with another arbitrary value of the second cube root C2. If some value for the first cube root was chosen, then one must choose the value for the second cube root such that C 1 C 2 ¼ B1 =3. The plot of the first 20 NF as a function of angular velocity for the Input I, Da ¼ 0:3 rad is shown in Fig. 7(a). The figure shows that an increase of the angular velocity implies that NF decrease, in agreement with results of [17], where FE-based approach was used. The previously observed split of NF of an UR tyre caused by rotation disappears under rolling conditions due to the disturbed symmetry, that also agrees with the results of [17]. In addition, an interesting phenomenon of frequency loci veering (Fig. 7(b)) is visible here: the NF-lines approach each other and suddenly veer away instead of crossing [7]. One might note that as we change the frequency o, the expressions under the roots in (27) might change sign. Thus, the function f ðoÞ is a complex-valued function, but it assumes real or purely imaginary values and hence is calculated by various formulae (Fig. 8). Searching the roots of the equation f ðoÞ ¼ 0, one ought to consider only those points of intersection of the graph of the function with the x-axis such that the function f ðoÞ is calculated by the same formula in neighbourhoods of these points. Starting with a certain on , the infinite spectrum of NF is calculated for the following

I.F. Kozhevnikov / Journal of Sound and Vibration 331 (2012) 1669–1685

1681

300

Frequency (Hz)

250 200 150 100 50 0

50

100

150

200

250

300

Frequency (Hz)

Rotational velocity (rad/s)

106 104 102 100 98 96 140

160

180 200 220 240 260 Rotational velocity (rad/s)

280

300

Fig. 7. Loaded rotating tyre (Input I, Da ¼ 0:3 rad): the NF as a function of angular velocity in Eulerian specification.

120 100

+-CR --CC

80 f(omega)

60 40 20 0 -20 -40 -60 85

90

95

100 105 Frequency (Hz)

110

115

120

Fig. 8. Loaded rotating tyre (Input I, Da ¼ 0:3 rad): the various formulae for function f ðoÞ for O ¼ 273 rad s1 .

values of parameters: B22 B31 þ 4 0, 4 27

2A

2A

2 2 g0 o0, 2ðg0 þ A1 Þ pffiffiffiffiffiffiffiffi o 0, 2ðg0 þA1 Þ þ pffiffiffiffiffiffiffiffi Z 0 2g0 2g0

Let us denote this configuration by ‘‘ þ CR’’. And only a finite number of NF corresponding to frequency loci veering region is calculated for the following values of parameters: B22 B31 þ o0, 4 27

2A

2A

2 2 o0, 2ðg0 þ A1 Þ þ pffiffiffiffiffiffiffiffi o 0 g0 o 0, 2ðg0 þ A1 Þ pffiffiffiffiffiffiffiffi 2g0 2g0

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Fig. 9. The basic real (——) and imaginary (——) mode shapes of a loaded rotating tyre (Input I, Da ¼ 0:3 rad, O ¼ 175 rad s1 ), (a) n1 ¼ 90:13 Hz, (b) n2 ¼ 94:93 Hz, (c) n3 ¼ 100:46 Hz, (d) n4 ¼ 102:72 Hz, (e) n5 ¼ 112:78 Hz, (f) n6 ¼ 124:47 Hz.

Let us denote this configuration by ‘‘   CC’’. Fig. 8 shows the appearance of two close roots around 100 Hz corresponding to frequency loci veering region (Input I, Da ¼ 0:3 rad, O ¼ 273 rad s1 ). The MS reads XðaÞ ¼ G1 ep1 a þ G2 ep2 a þ G3 ep3 a þG4 ep4 a where the constants Gi are determined from boundary conditions (23) G1 ¼ ep1 ða1 þ 2pÞ ððp4 p3 Þep2 ðDa2pÞ ðp4 p2 Þep3 ðDa2pÞ þ ðp3 p2 Þep4 ðDa2pÞ ÞGn5 G2 ¼ ep2 ða1 þ 2pÞ ððp4 p3 Þep1 ðDa2pÞ þðp4 p1 Þep3 ðDa2pÞ ðp3 p1 Þep4 ðDa2pÞ ÞGn5 G3 ¼ ep3 ða1 þ 2pÞ ððp4 p2 Þep1 ðDa2pÞ ðp4 p1 Þep2 ðDa2pÞ þ ðp2 p1 Þep4 ðDa2pÞ ÞGn5 G4 ¼ ep4 ða1 þ 2pÞ ððp3 p2 Þep1 ðDa2pÞ þ ðp3 p1 Þep2 ðDa2pÞ ðp2 p1 Þep3 ðDa2pÞ ÞGn5 ,

8Gn5

The function V vib ða,tÞ is represented by the linear combination V vib ða,tÞ ¼ ðcosðotÞReðXðaÞÞsinðotÞImðXðaÞÞÞ þ iðcosðotÞImðXðaÞÞ þ sinðotÞReðXðaÞÞÞ of real MS ReðXðaÞÞ and of imaginary MS ImðXðaÞÞ corresponding to the same NF. Eventually the real MS is transformed into imaginary MS and vice versa  p  ¼ ImðXðaÞÞ þ i ReðXðaÞÞ V vib ða,0Þ ¼ ReðXðaÞÞ þi ImðXðaÞÞ, V vib a, 2o The MS are represented in Fig. 9 for the Input I, Da ¼ 0:3 rad, O ¼ 175 rad s1 (angular velocity of rotation below the frequency loci veering region 1802190 rad s1 ) and in Fig. 10 for O ¼ 200 rad s1 (angular velocity of rotation above the frequency loci veering region). The rolling occurs in the clockwise direction. In Fig. 9 the configuration ‘‘þ  CR’’ corresponds to the first, fifth and sixth MS and the configuration ‘‘  CC’’ corresponds to the second, third and fourth MS. In Fig. 10 the configuration ‘‘þ  CR’’ corresponds to the first and sixth MS and the configuration ‘‘  CC’’ corresponds to the second, third, fourth and fifth MS. One can note that the third and fourth MS interact in frequency loci veering region and finally interchange. The evolution of the third real and imaginary MS of a LR tyre (Input I, Da ¼ 0:3 rad) is represented in Fig. 11. For O ¼ 3, 100, 130 rad s1 the third MS of a LR tyre has, respectively, three, four, five nodes and is similar to the third, fourth, fifth MS of a LNR tyre. Thus, the third MS changes from a three-node to a five-node shape, while the NF decreases from 116.91 Hz to 102.81 Hz. 6. Conclusions We have thus completed the investigation of vibrations of an unloaded non-rotating, unloaded rotating, loaded nonrotating and loaded rotating tyre rolling at constant speed without slipping in the contact area. The natural frequencies and mode shapes of an unloaded and loaded tyre are determined.

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Fig. 10. The basic real (——) and imaginary (——) mode shapes of a loaded rotating tyre (Input I, Da ¼ 0:3 rad, O ¼ 200 rad s1 ), (a) n1 ¼ 85:89 Hz, (b) n2 ¼ 90:49 Hz, (c) n3 ¼ 97:53 Hz, (d) n4 ¼ 100:54 Hz, (e) n5 ¼ 106:92 Hz, (f) n6 ¼ 117:77 Hz.

Fig. 11. The evolution of the third real (——) and imaginary (——) mode shape of a loaded rotating tyre (Input I, Da ¼ 0:3 rad), (a) O ¼ 3 rad s1 , n3 ¼ 116:91 Hz, (b) O ¼ 100 rad s1 , n3 ¼ 107:71 Hz, (c) O ¼ 130 rad s1 , n3 ¼ 102:81 Hz.

The quantities of obtained NF for the UNR tyre were compared with the results of two experiments. Each NF of an UNR tyre corresponds to two NF of an UR tyre. This is a well known effect: the two counter-rotating waves, superimposed onto a standing vibration in resting structures, are distinct in speed for a rotating system. Starting with a certain number, both branches of NF as functions of angular velocity growing from a point corresponding to NF of an UNR tyre increase monotonically (Lagrangian specification). The bottom branch of first NF vanishes at certain value of angular velocity of rotation. In Eulerian specification neither frequency vanishes, in agreement with results of other papers. In the case of LNR and LR tyre, in determining the frequency of the vibrations, the length of the contact area was taken as constant, since its variation determines a second order of smallness correction to the frequency, in the model chosen. The Lagrange multipliers, determining the correction to the tension of the median line of the tread and the corrections to the longitudinal reactions at the boundary points of the contact area during vibrations are found. For LNR tyre, the increasing of the contact area (due to the increasing of the vertical load) implies that NF also increase. In fact, the mass of the vibrating part of the tyre decreases with the increasing of the contact area, as the tyre does not vibrate in the contact area. Moreover, this result correlates well with the theorem from the small vibrations theory on the NF behaviour (interfusion) at imposing the constraint. Each mode of an UNR tyre is double and for each NF two identical MS exist. The fixed contact points of the tyre cause a loss of the circular symmetry. Thus, for each NF of an UNR tyre there are two different NF of a LNR tyre. This is explained by the disturbance of free wave motion due to contact. The identical modes split into two not identical ones. The MS subdivides to a symmetric and an antisymmetric shapes. For symmetric MS the corrections to the longitudinal reactions at the boundary points of the contact area have opposite signs. As for the antisymmetric MS the corrections carry the same sign. The mass centre of the tyre does not move in the longitudinal

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direction for the symmetric MS, and it moves for the antisymmetric MS. Thus, the antisymmetric MS ‘‘sways’’ from side to side. The quantities of obtained NF for the LNR tyre were compared with the results of experiment. In the case of LR tyre, the increasing of the angular velocity implies that NF decrease. The previously observed split of NF of an UR tyre caused by rotation disappears under rolling conditions due to the disturbed symmetry. In addition, an interesting phenomenon of frequency loci veering is visible here: NF-lines approach each other and suddenly veer away instead of crossing. The MS interact in frequency loci veering region and finally interchange.

Acknowledgements I wish to thank V.G. Vil’ke for advices and comments and A.A. Burov and S.F. Adlaj for help in preparing the English text. This work is supported by the Federal Targeted Program (Governmental Contract No. 14.740.11.0995). Appendix A. Model parameters The constant coefficients n0, n01, n11, n02, n12, m21, m12, m20, m02 in Eq. (9) are determined analytically and depend on the geometric parameters of the tyre and on the internal tyre pressure p 2

n0 ¼ 2I1 2plr ,

m21 ¼ 2rðI5 þ I87 Þ, 3

I1 ¼ pb

Z

c2

c b

c1

I41 ¼ pb

3

Z

 þ cos c D2 dc,

c2

c b

c1

2

n01 ¼ 2ðI9 þI25 þ I41 þI57 I91 Þ,

m12 ¼ 2rðI5 þI124 Þ, 2

I5 ¼ 

pb c1  c2

 þ cos c D1 D3 dc,

Z

I57 ¼ pb

ðcc2 ÞD2 dc,

3

Z

c2 c1

Z

c2

ðcc2 Þ2 c  dc, þcos c b

c b

I9 ¼ pb

3

m02 ¼ 2rI87 ,

c2

sin cD1 D2 dc,

I25 ¼ pb

c1

 þ cos c D21 dc,

ðc1 c2 Þ

D2 ¼

2

c1

I87 ¼ 

8bðk1 þk2 Þ c1 c2

Z

Z

n12 ¼ 2r 2 I78 ,

I77 ¼

3

Z

c2

c1

pb ðc1 c2 Þ2

Z

c2 c1

cos cD22 dc,

ðcc2 Þ2 cos c dc,

c2

ðcc2 Þðsin cD1 þcos cD2 Þ c  dc, þcos c b Z c2 ðsin cD1 þcos cD2 Þ2 2 c  dc, I91 ¼ 8b ðk1 þk2 Þ c1 þcos c b   c Z Z c2 D21 2bðk1 þk2 Þ c2 ðcc2 Þ sin c þ b þ cos c D1 2 c  dc, I124 ¼ c  dc, I108 ¼ 2b ðk1 þ k2 Þ c1 c2 c1 c1 þcos c þ cos c b b  2 Z c2 ðcc Þsin c þ c þ cos c 2 2ðk1 þ k2 Þ b I132 ¼ c dc, ðc1 c2 Þ2 c1 þ cos c b " # " # r ðcc2 Þ2 ðcc2 Þ3 r 3ðcc2 Þ2 2ðcc2 Þ3 sin  c  D1 ¼ cos c1  , 1 b b ðc1 c2 Þ ðc1 c2 Þ2 ðc1 c2 Þ2 ðc1 c2 Þ3 I78 ¼

8ðk1 þ k2 Þ

2

m20 ¼ 2plr þ 2rI124 ,

c2

c1

n02 ¼ 2r 2 ðI77 þI132 Þ,

n11 ¼ 2plr þ 2I108 ,

c1

" # " # r 2ðcc2 Þ 3ðcc2 Þ2 r ðcc2 Þ ðcc2 Þ2 cos c1 sin  c 6   , 1 b ðc1 c2 Þ ðc1 c2 Þ2 b ðc1 c2 Þ2 ðc1 c2 Þ3

" # " # r 1 3ðcc2 Þ r 1 2ðcc2 Þ   D3 ¼ cos c1 2  sin c1 6 b ðc1 c2 Þ ðc1 c2 Þ2 b ðc1 c2 Þ2 ðc1 c2 Þ3 The geometric parameters of the tyre are represented in Fig. A1. The point K is the centre of a circle of radius b, an arc of which is the sidewall. The potential energy of the rubber stretching in the Mooney–Rivlin model is represented by the functional Z 2p Z ½k1 ðIc 3Þ þ k2 ðIIc 3Þðc þ b cos cÞb dj dc, IIIc ¼ 1, s1 ¼ ½c1 , c2 , s2 ¼ ½c2 ,c1  E½V ¼ 0

s1 [s2

where k1, k2 are, empirically determined, material constants, whereas Ic , IIc , IIIc are the invariants of the Cauchy–Green deformation tensor [10]. The values used for model parameters are represented in Table A1. The parameters b, c, c1 , c2 depend on the parameters r1, a, l1, l, r.

I.F. Kozhevnikov / Journal of Sound and Vibration 331 (2012) 1669–1685

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Fig. A1. The geometric parameters of the undeformed tyre.

Table A1 The input data of the model. Input

m (kg)

p (Pa)

k1 þ k2 (N m1 )

r1 (m)

a (m)

l1 (m)

l (m)

r (m)

b (m)

c (m)

c1 (rad)

c2 (rad)

I II

9.37 8.5

250000 160000

108270 95700

0.2032 0.1905

0.09737 0.0979

0.1025 0.1025

0.0875 0.0875

0.3169 0.3135

0.1773 0.2145

0.2455 0.2347

1.1565 1.1946

1.812 1.7783

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