Vibration analysis of a wheel composed of a ring and a wheel-plate modelled as a three-parameter elastic foundation

Journal of Sound and Vibration 333 (2014) 6706–6722

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Vibration analysis of a wheel composed of a ring and a wheel-plate modelled as a three-parameter elastic foundation Stanisław Noga a,n, Roman Bogacz b, Tadeusz Markowski c a

Faculty of Mechanical Engineering and Aeronautics, Rzeszów University of Technology, Al. Powstańców Warszawy 12, 35-959 Rzeszów, Poland Faculty of Automotive and Construction Machinery Engineering, Warsaw University of Technology, ul. Narbutta 84, 02-524 Warsaw, Poland c Faculty of Mechanical Engineering and Aeronautics, Rzeszów University of Technology, Al. Powstańców Warszawy 12, 35-959 Rzeszów, Poland b

a r t i c l e in f o

abstract

Article history: Received 4 January 2013 Received in revised form 12 July 2014 Accepted 15 July 2014 Handling Editor: S. Ilanko Available online 10 August 2014

The free in-plane vibrations of circular rings with wheel-plates as generalised elastic foundations are studied using analytical methods and numerical simulations. The threeparameter Winkler elastic layer is proposed as a mathematical model of the foundation. The effects of rotary inertia and shear deformation are included in the analytical model of the system. The motion equations of systems are derived on the basis of the thin ring theory and Timoshenko's theory. The separation of variables method is used to find general solutions to the free vibrations. Elaborated analytical models are used to determine the natural frequencies and the natural mode shapes of vibrations of an arbitrarily chosen set of simplified models of aviation gears and railway wheels. The eigenvalue problem is formulated and solved by using a finite element representation for each simplified model. The results for these models are discussed and compared. The proposed solutions are verified by experimental investigation. It is important to note that the solutions proposed here could be useful to engineers dealing with the dynamics of aviation gears, railway wheels and other circular ring systems. & 2014 Elsevier Ltd. All rights reserved.

1. Introduction The problem of free in-plane vibration in circular rings with wheel-plates as elastic foundations is important theoretically and practically because the rings and plates have many engineering applications to the railway and aviation industries [1–4]. Rotating wheels can be modelled as rings with wheel-plates [1,3,5,6], where their shape and dimensions are linked to the design of such systems. The ring systems are typically modelled by simple or complex two-dimensional continuous systems. The simplest fundamental model of a composed two-dimensional system consists of a circular ring with an arbitrarily chosen cross-sectional shape and a wheel-plate modelled by a generalised Winkler-type elastic foundation. Ziemba [7], Rao [8], Solecki and Szymkiewicz [9], and others present the fundamental vibration theory of circular ring systems in a number of monographs. Intensive theoretical developments have generated numerous papers on n

Corresponding author. Tel.: þ48 17 8651639; fax: þ 48 17 8651150. E-mail address: [email protected] (S. Noga).

http://dx.doi.org/10.1016/j.jsv.2014.07.019 0022-460X/& 2014 Elsevier Ltd. All rights reserved.

S. Noga et al. / Journal of Sound and Vibration 333 (2014) 6706–6722

Nomenclature a1 ; b1 ; c1 ; h1 factors defined in Eq. (16) a0 ; b0 ; c0 ; d0 ; h0 factors defined in Eq. (18) A area of the ring cross-section C jn ; φjn constants defined in Eq. (22) d1 ; dw ; dz diameters of the simplified models (see Fig. 2) E Young's modulus f external radial force per unit of length F shearing force (see Fig. 1) G modulus of shear elasticity h radial depth of the ring hn ¼ ðP 0 R2 Þ=ðE I 1 Þ I1 area moment of inertia of the ring pffiffiffiffiffiffiffiffi cross-section i ¼ 1 imaginary number k shear correction factor kf ; kp respective radial and tangential stiffnesses per unit of length kS flexural rotational stiffness of the ring crosssection lr ; lw thicknesses of the simplified models (see Fig. 2) M1 in-plane bending moment m1 distributed external in-plane bending moment

n p P P0 R t U u un U0 w z εn ϕ ν θ ρ ρz Ω0 ω; ωn ωfn ; ωcn

6707

mode number external tangential force per unit of length tensile force (see Fig. 1) normal force due to a steady centrifugal action radius of the ring centreline time radial displacement defined in Eq. (19) radial displacement of an arbitrary point on the ring normal modes amplitude of the radial displacement circumferential displacement of an arbitrary point on the ring number of teeth of the gear wheel frequency error slope of the deflection curve Poisson's ratio angular position on the ring circumference mass density ring model equivalent mass density rotational speed natural frequencies respective natural frequencies of the FE and the analytical models

the subject. Refs. [10–12] study the problem of free in-plane vibration of a circular ring. In Ref. [10], the effects of rotary inertia and shear deformation are presented, and the natural frequencies of the vibration of free rings and stiffened rings are determined. Ref. [11] presents theories for several distinct types of ring vibration, including flexural, torsional and shear vibration, which can be predicted for arbitrarily chosen cross-sectional shapes of the ring. The wave approach is used in Ref. [12] to analyse the in-plane inextensional vibration of a thin circular ring with equi-spaced, identical radial supports. The three-dimensional vibration analysis method is developed in Ref. [13] for determining the natural frequencies and the natural mode shapes of thick, circular rings with elliptical and circular cross-sections. The fundamental vibration theory of circular systems with elastic foundations is investigated for various types of compound two-dimensional continuous systems [8]. In Ref. [14], a comprehensive study of the transverse vibrations of the elastically connected circular membrane and plate systems is presented. The annular case of the elastically connected double membrane compound system is studied in ref. [15]. In ref. [14,15], an elastic layer is treated as a Winkler-type layer, and analytical solutions of the free transverse vibration are determined by the separation of variables method. In Ref. [16], the free in-plane flexural and coupled twisting/ out-of-plane vibrations of thick rotating rings are analysed. Introductory studies related to the ring connected to the elastic support and studies of shear deformation and centrifugal effects are presented. A more comprehensive study related to rotating thin and thick rings on elastic foundations is presented in Ref. [17]. A portion of the results presented in this paper are verified by experimental investigations and data reported in the available literature. In Ref. [1], the vibration theory of elastic rings with elastic foundations is applied to study a steady state response of the railway wheels subjected to a distributed moving load. Ref. [18] studies the problem of the modal properties of equally spaced planetary gears with elastic ring gears by using perturbation and a candidate mode method. In many of the articles mentioned above, investigations are restricted to the problem of rings with the springs distributed in radial and circumferential directions. A general model of the elastic foundation and thin ring theory is used in Ref. [3] to predict the dynamic behaviour of the planetary gears. The proposed elastic foundation consists of two orthogonal distributed springs oriented at arbitrary inclination angles. The finite element (FE) method is a useful technique for solving dynamic problems related to complex engineering structures [19,20]. The free vibration problem of the fatigue test rig for aviation gear boxes is analysed in Ref. [21] using finite element representations. In Refs. [5,6], the FE method is used to investigate problems with the transverse vibrations of toothed gears with complex geometries. Ref. [5] presents an algorithm to identify proper deformed normal modes of rotating toothed gears at low and high speeds. Ref. [2] effectively applies the FE method to predict the mode shapes of the planetary gear structures. In Ref. [22], the forced vibrations of ring gears in equal- and diametrical-spur planetary gears are investigated using analytical methods and the FE method. Ref. [23] analyses the mechanical properties of composite gear wheels used on aviation equipment. Ref. [24] studies the analytical methods and FE techniques used to solve the free in-plane vibration of an arbitrarily chosen circular ring with a wheel plate used as the elastic foundation. Identical problems are investigated in Ref. [25]. In both papers mentioned above, the use of the two-parameter Winkler elastic layer in the modelling of ring system vibrations is considered.

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This paper discusses an exact solution to the problem of free in-plane flexural vibrations of circular rings with wheelplates for the elastic foundations. This paper proposes the three-parameter Winkler elastic foundation as a mathematical model of the foundation. The motion equations of the systems are derived from the thin ring theory and Timoshenko's theory. The undamped free vibrations of these systems are solved analytically using the separation of variables method. The elaborated analytical models are used to obtain the natural frequencies and natural mode shapes of vibrations of an arbitrarily chosen set of simplified models of common aviation gear wheels and railway wheels. Then, the FE method is used to solve the eigenvalue problem of each simplified model. The results obtained from the models mentioned above are discussed and compared. The proposed three-parameter model of the elastic foundation provides satisfactory analytical solutions that are in agreement with the corresponding FE solutions for the ring system vibrations and for more natural frequencies than those reported in the literature that use a two-parameter model of the elastic foundation. Theoretical and numerical studies are verified by an experimental investigation. The concluding remarks are provided, and the adequate mode shapes corresponding to the appropriate natural frequencies of the systems are shown. 2. Theoretical formulation The purpose of this paper is to develop a new dynamic model of a circular ring with a wheel-plate as a three-parameter elastic foundation. A plane circular ring with a constant rectangular cross-section is considered (see Fig. 1). It is assumed that the ring is elastic, that the centreline of the ring has radius R, and that an element of the ring, determined by angle θ, displaces in the radial and circumferential directions. The small radial and circumferential displacements in these directions are designated as u(θ,t) and w(θ,t), respectively, where t is time. The length of the centreline is assumed to be constant in the disturbed state. The continuously distributed elastic foundation is assumed to interact with the ring. This paper proposes the three-parameter elastic foundation (see free body diagram in Fig. 1). The generalised Winkler model represents the proposed foundation, and the first two coefficients, kf and kp, represent the radial and the tangential stiffness per length unit, respectively. An additional coefficient representing the stiffness associated with the distortion of the foundation due to in-plane rotation of a cross-section of ring element, determined by angle θ, is introduced to achieve better agreement with the actual system. This coefficient, denoted as kS, is the flexural rotational stiffness of the elastic foundation. The general equation of motion according to the thin ring theory is derived using a procedure similar to the one used by Rao [8] for a free ring. The cross-sectional dimensions of the ring are assumed to be small compared to the radius of the neutral line of the ring. A typical differential element of the ring system is considered (see free body diagram in Fig. 1). The equation of motion in the radial direction takes the form ∂F ∂2 u þ P  kf R u þ f ðθ; tÞR ¼ ρAR 2 ∂θ ∂t

(1)

where F(θ,t) is the shearing force, P(θ,t) is the tensile force, f(θ,t) is the external radial force per unit of length, ρ is the mass density, and A is the area of the ring's cross-section. The equation of motion in the tangential direction is given by ∂P ∂2 w  F  kp Rwþ pðθ; tÞR ¼ ρAR 2 ∂θ ∂t

(2)

where p(θ,t) is the external tangential force per unit of length. The circumferential moment equation of motion, in the case when the rotary inertia is neglected, leads to ∂M 1 þFR kS Rϕ þm1 ðθ; tÞR ¼ 0 ∂θ

dθ

(3)

f(θ,t)

M1+dM1

ρ

P+dP

w

m1(θ,t)

u

ring

p(θ,t)

w F+dF

θ 0

u kf

kp

dθ/2 dθ/2 R

kS

0

foundation

foundation Fig. 1. Physical model of the system.

A

F P

M1

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where M1 is the in-plane bending moment, ϕ is the slope of the deflection curve, and m1(θ,t) is the distributed external inplane bending moment acting on the centreline of the ring. The condition of inextensionality of the ring centreline takes the form [8,9] ∂w ¼u ∂θ Therefore, the axial force and the bending moment can be described as     EA ∂w EI 1 ∂2 u P¼ u þ ; M1 ¼ 2 u þ 2 R ∂θ ∂θ R

(4)

(5)

where E is Young's modulus of elasticity and I1 is the area moment of inertia of the ring cross-section. If the shearing force is neglected, then the slope of the deflection curve can be expressed as [8,9]   1 ∂u þw (6) ϕ¼ R ∂θ From Eqs. (1) and (2) it is easily shown that   ∂2 F ∂u ∂f ðθ; tÞ ∂2 ∂u þk R pðθ; tÞR ¼ ρAR w þ F  k R Rw þ p f ∂θ ∂θ ∂t 2 ∂θ ∂θ2

(7)

The equations of motion for the in-plane flexural vibrations of the thin ring with the modified complex Winkler foundation are given by Eqs. (3)–(7). These equations can be combined into a single equation in terms of the radial deflection u(θ,t) as    2    EI 1 ∂6 u ∂4 u ∂2 u kS ∂4 u ∂2 u ∂ u þ 2 þ þ 2 þ u þ k  k u R  p f R ∂θ4 ∂θ4 ∂θ2 ∂θ2 ∂θ2 R3 ∂θ6       ∂2 ∂2 u ∂ ∂f ðθ; tÞ ∂ ∂2 m1 ðθ; tÞ  pðθ; tÞ R   u ¼ þ m (8) þ ρAR 2 1 ∂θ ∂θ ∂θ ∂t ∂θ2 ∂θ2 The effects of rotary inertia and shear deformation are included in the equations of motion for the system under consideration. For this case, the shear force and the bending moment can be expressed in terms of the displacement components [8]   kAG ∂u EI 1 ∂ϕ þ w Rϕ ; M 1 ¼ (9) F¼ R ∂θ R ∂θ where k is the shear correction factor and G is the shear modulus. The rest of the designations are defined above. Therefore, the appropriate equation of rotational motion of a differential element shown in Fig. 1 can be described as ∂M1 ∂2 ϕ þ FR kS Rϕþ m1 ðθ; tÞR ¼ ρI 1 R 2 ∂θ ∂t

(10)

Using Eqs. (4), (7), (9) and (10), the partial differential equation of motion for the in-plane flexural vibrations of the considered system, including the effects of rotary inertia and shear deformation, can be expressed in terms of the radial deflection u(θ,t) as ! ! ∂6 u R2 R2 ∂4 u R2 R2 R4 R4 þ kf kS  2kS þ 2 kf þ 1 þkp þkf kS kAG EI 1 ∂θ4 kAG EI 1 kAGEI 1 EI 1 ∂θ6 ! ! 2 4 4 2 2 ∂2 u R R R ρR ρR ∂6 u þ  kS þkp kS þkp u EI 1 kAGEI 1 EI 1 E kG ∂θ4 ∂t 2 ∂θ2 ! ρ2 R4 ∂6 u ρR2 ρR4 ρR2 ρR4 ρAR4 ∂4 u þ kf þ þ kS þ þ 2 þ kGE ∂θ2 ∂t 4 E kAGE kG kGEI 1 EI 1 ∂θ2 ∂t 2 !   ρ2 R4 ∂4 u ρR2 ρR4 ρR4 ρAR4 ∂2 u ρR4 ∂2 ∂pðθ; tÞ ∂2 f ðθ; tÞ  þ k þ k   þ ¼  p S ∂θ kGE ∂t 4 E kAGE kGEI 1 EI 1 ∂t 2 kAGE ∂t 2 ∂θ2    2 4 2  2 2 R ∂ ∂pðθ; tÞ ∂ f ðθ; tÞ R ∂pðθ; tÞ ∂ f ðθ; tÞ   þ  kS ∂θ ∂θ kAG ∂θ2 kAGEI 1 ∂θ2 ∂θ2     R4 ∂pðθ; tÞ ∂2 f ðθ; tÞ R3 ∂m1 ðθ; tÞ ∂3 m1 ðθ; tÞ    (11)  ∂θ ∂θ EI 1 EI 1 ∂θ2 ∂θ3 Next, the centrifugal effect is included in the equation of motion of the system. The analysis is restricted to the thin ring with the proposed Winkler-type complex foundation. According to Ref. [16], Eqs. (1) and (2) take the form   ∂F P 0 ∂2 u ∂w ∂2 u ∂w þ P  kf Ru þ f ðθ; tÞRþ ¼ ρAR 2  2ρAΩ0 R (12) þ 2 ∂θ ∂θ ∂t R ∂θ ∂t

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∂P ∂2 w ∂u  F  kp Rw þpðθ; tÞR ¼ ρAR 2 þ 2ρAΩ0 R ∂θ ∂t ∂t

(13)

where P 0 is the normal force of a steady centrifugal action and Ω0 is the rotational speed. By proceeding as described above, the equation of motion is solved in terms of the radial deflection u(θ,t) as    2    EI 1 ∂6 u ∂4 u ∂2 u kS ∂4 u ∂2 u ∂ u þ2 4 þ 2  þ 2 2 þu þ kf 2  kp u R 3 6 4 R ∂θ ∂θ ∂θ ∂θ ∂θ R ∂θ     ∂2 ∂2 u ∂2 u P 0 ∂4 u ∂2 u   u  4ρARΩ0 þ 2 þ ρAR 2 2 4 ∂t∂θ R ∂θ ∂t ∂θ ∂θ     ∂ ∂f ðθ; tÞ ∂ ∂2 m1 ðθ; tÞ pðθ; tÞ R  þ m1 (14) ¼ 2 ∂θ ∂θ ∂θ ∂θ

3. Free vibration analysis The objective of this section is to determine an analytical solution for free vibrations of the considered system. According to Eq. (8), the partial differential equation of motion for the free in-plane flexural vibrations of the thin circular ring with the wheel-plate as an elastic foundation can be written in terms of the radial deflection u(θ,t) in the following form:   ∂6 u ∂4 u ∂2 u ∂2 ∂2 u þ ð2  h1 Þ 4 þ ð1  2h1 þ a1 Þ 2 ðc1 þ h1 Þu þ b1 2 u ¼ 0 (15) 6 2 ∂t ∂θ ∂θ ∂θ ∂θ where a1 ¼

kf R4 ; EI 1

b1 ¼

ρAR4 ; EI 1

kp R4 ; EI 1

c1 ¼

and

h1 ¼

kS R2 EI 1

(16)

Using Eq. (11) to include the effects of rotary inertia and shear deformation the equation of motion for the free in-plane flexural vibrations of the thick ring with the wheel-plate as a foundation can be expressed in terms of u(θ,t) as     2 ∂6 u a0 ∂ 4 u a0 h0 ∂ u þ 2 b k  k þ 1 þb k  2k þ k k þ k a p 0 S 0 S S 0 f f f ρI 1 ∂θ6 ∂θ2 R2 ∂θ4 R2   6 6 a0 h0 ∂ u ∂ u ∂4 u  kS 2 þkp kS þ kp a0 u  ðc0 þ d0 Þ 4 2 þc0 d0 2 4  c0 d0 4 ρI ∂t ∂t ∂t ∂θ ∂θ 1 R   4   2 h0 A ∂ u h0 A ∂ u þ  2c0 þkf h0 þd0 þkS þ ρAa0  c þ k h þk þ ρAa ¼0 (17) p 0 0 S 0 I1 I1 ∂t 2 ∂θ2 ∂t 2 where a0 ¼

R4 ; EI 1

b0 ¼

R2 ; kAG

c0 ¼

ρR2 ; E

d0 ¼

ρR2 ; kG

and

h0 ¼

ρR4 kEAG

(18)

The general solution to Eqs. (15) and (17) takes the form [8] (19) uðθ; tÞ ¼ UðθÞeiωt pffiffiffiffiffiffiffiffi where ω is the natural frequency of vibration and i ¼  1 is the imaginary unit. After substituting Eq. (19) into Eq. (15), the formula for U(θ) becomes ! 6 4 2 2 d U d U d U d U 2 þ ð2  h1 Þ 4 þ ð1  2h1 þ a1 Þ 2  ðc1 þ h1 ÞU ω b1 U ¼ 0 (20) dθ6 dθ dθ dθ2 Substituting Eq. (19) into Eq. (17) provides the following expression:   4   2 6 d U a0 d U a0 h0 d U þ 2  b k  k þ 1 þ b k 2k þk k þk a p 0 S 0 S S 0 f f f ρI 1 dθ6 dθ2 R2 dθ4 R2   4 2 a0 h0 d U d U  kS 2 þkp kS þkp a0 U þ ðc0 þ d0 Þω2 4 þc0 d0 ω4 2  c0 d0 ω4 U ρI 1 dθ dθ R     2 h0 A h0 A 2d U 2 þρAa0 ω þ c þk h þ k þ ρAa  2c0 þ kf h0 þ d0 þ kS p 0 0 S 0 ω U¼0 I1 I1 dθ2

(21)

The general solution to Eqs. (20) and (21) can be assumed to be 3

UðθÞ ¼ ∑ C jn sin ðnθ þ φjn Þ; j¼1

n ¼ 2; 3; …

(22)

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where C jn and φjn are constants. When Eq. (22) is substituted into Eq. (20), it yields the natural frequencies of vibration for the thin ring theory ω2n ¼

n6  ð2  h1 Þn4 þ ð1  2h1 þ a1 Þn2 þ ðc1 þ h1 Þ ; b1 ðn2 þ 1Þ

n ¼ 2; 3; …

Substituting Eq. (22) into Eq. (21) gives the following frequency equation for the thick ring theory:    h0 A  c0 d0 ðn2 þ 1Þω4n þ ðc0 þd0 Þn4 þ  2c0 þ kf h0 þd0 þ kS þ ρAa0 n2 I1     h0 A a 0 þ ρAa0 ω2n  n6 þ 2 b0 kf  kS 2 n4 ; n ¼ 2; 3; … þ c0 þ kp h0 þ kS I1 R     a0 h0 a0 h0 þkf a0 n2  kS 2 þ kp kS þ kp a0 ¼ 0  1 þ b0 kp  2kS 2 þ kf kS ρI 1 ρI 1 R R

(23)

(24)

Eq. (24) is a quadratic equation with respect to ω2n , with two frequency values related to each value of n. The smaller ωn value refers to the flexural mode, and the larger value corresponds to the thickness-shear mode. It is notable that when n ¼ 1 in Eqs. (23) and (24), only pure rigid body oscillations exist, without any alteration of the ring shape [8]. For both of the ring system cases described, the general solution for the natural frequencies ωn may be written as ! 1

uðθ; tÞ ¼ ∑

n¼1

3

∑ C jn sin ðnθ þ φjn Þ eiωn t

(25)

j¼1

and the normal modes of the ring system can be expressed as un ðθ; tÞ ¼ C jn sin ðnθ þφjn Þeiωn t

(26)

where quantities Cjn and φjn are determined by the initial conditions of the considered system. The normal mode of the ring system, including the centrifugal effect is [16] un ðθ; tÞ ¼ U 0 eiðnθ  ωn tÞ

(27)

where U0 is the amplitude of the radial displacement. This leads to the frequency equation in the form of a quadratic equation ðn2 þ1Þb1 ω2n  4b1 Ω0 nωn  n6 þð2  h1  hn Þn4  ð1 2h1 þ a1  hn Þn2 ðc1 þ h1 Þ ¼ 0; n ¼ 2; 3; … 0 2

(28)

0

where hn ¼ ðP R Þ=ðEI 1 Þ and the steady centrifugal tension P takes the form [16] P0 ¼

ρAR2 Ω20 1 þ ðkf R2 =AEÞ

(29)

4. The finite element representation In this section, system finite element models are formulated to discretise the continuous model given by Eq. (17). The motion equation is first transformed into a set of independent (decoupled) differential equations cast in modal generalised coordinates through the use of the normal modes of the structure. The solutions of the decoupled modal equations are then superimposed to obtain the response of the system. Lanczos' block method is used to determine the eigenpairs (eigenvalue and eigenvector) related to the natural frequencies and the normal modes of system [19]. The elaborated FE models are treated as approximations of the analytical models given by Eq. (17). As mentioned above, systems such as rings with wheel-plates as foundations are important components of toothed gears and railway wheels. This study analyses the suitability of the proposed circular ring with a wheel-plate as the threeparameter elastic foundation, using analytical models for the vibration modelling of the proposed systems. A set of simplified models of aviation gears and railway wheels is analysed for this purpose. The next paper will study the case of a non-symmetric ring of the railway wheel. For the study, the set of toothed gears is characterised by a similar geometry, however, with different numbers of teeth. Ref. [6] contains the technical data of the selected gear set. Parameters characterising the railway wheels are based on the wheels for rail coaches, cargo wagons and locomotives. In the simplified models, the wheel hubs, the gear teeth, and the railway wheel flanges are omitted. Furthermore, the web of the wheel is modelled as the wheel-plate. The primary geometric dimensions of the adopted simplified models (diameters: d1, dw, dz; and thicknesses: lr, lw) are shown in Fig. 2a and are taken from actual wheels. For the gear wheels, the outer diameters dz of the models are equal to the pitch diameters of the corresponding wheels. The principal problem in this section is the elaboration of the FE model of the elastic layer. All of the simplified geometric models of the systems consist of six sectors (see Fig. 2b) that have cyclic symmetry features. Each sector includes the ring and portion of the foundation as the wheel-plate. The ring is modelled as the solid body and the foundation is modelled as

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lr

ødz

ødl

ødw

lw

bearing Fig. 2. (a) Geometric dimensions and (b) simplified model of the system.

Fig. 3. The FE models referred to the toothed gears at different depths of the ring.

Fig. 4. The FE models related to different railway wheels.

the massless solid body. Both segments are meshed by using standard ANSYS software procedures. For each case, the 3-D solid mesh is elaborated, and the ten-node tetrahedral element (solid187), with three degrees of freedom in each node, is used to realise the sector. During the mesh generation process, the lengths of the element sides must be equal to or less than 3.5 mm for the first five models of the toothed gears. For the railway wheel models, the lengths of the element sides must be no greater than 14 mm. The largest toothed gear FE model consists of 33,259 solid elements and corresponds to the gear with twenty-seven teeth (z¼ 27). The remaining gear FE models consist of less than 26,700 solid elements. The most complex FE model is the locomotive railway wheel, which is composed of 156,204 solid elements, whilst the remaining railway wheel FE models consist of no more than 114,600 solid elements. For all of the models subject to the analysis, the boundary conditions are applied to the nodes. For each model, the degrees of freedom associated with the node radial displacement and with the node displacement along the centreline of each model are subtracted from the nodes lying on the surface, which are identified as bearings (see Fig. 2b). The elaborated FE models are displayed in Figs. 3 and 4. 5. Numerical analysis The results from the numerical analysis of the circular rings with the wheel-plates modelled as vibration-free Winkler elastic foundations are determined using the models described earlier. This section of the study discusses and compares only the first seven natural frequencies and mode shapes for these types of vibrations. The parameters characterising the systems used in the calculations are displayed in Table 1. In this table, ν is the Poisson's ratio, and the other parameters are defined in the previous sections. The first five positions in Table 1 refer to gears, and the last four positions (6–9) in this table

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Table 1 Parameters characterising the system of gear wheels (rows 1–5) and a system of railway wheels (rows 6–9). No.

z

1 2 3 4 5 6 7 8 9

dz [m]

27 40 54 67 74 – – – –

dw [m]

0.1404 0.14 0.1404 0.1407 0.1406 0.92 0.96 1 1.25

R [m]

0.1066 0.11725 0.1235 0.1275 0.12825 0.77 0.81 0.85 1.1

d1 [m]

0.06175 0.06431 0.06598 0.06705 0.06721 0.4225 0.4425 0.4625 0.5875

lr [m]

lw [m]

ρ [kg/m3]

E [Pa] 11

0.025

0.018

0.006

2.08  10

0.185 0.185 0.2 0.2

0.135 0.135 0.135 0.14

0.02

2.1  1011

7.83  10

3

ν 0.3

7.85  103

Table 2 Ring depth of the model. No. h [m] No. h [m]

1 0.002 8 0.016

2 0.004 9 0.018

3 0.006 10 0.02

4 0.008 11 0.022

5 0.01 12 0.024

2.5

60

εn [%]

εn [%]

7 0.014 14 0.028

ε2

2.0 40

6 0.012 13 0.026

ε3

1.5

ε4 ε5 ε6 ε7 ε8

1.0

20 0.5 0.0

0 3

8

13

18

23

28

33

38

3

43

8

13

R/h

18

23

28

33

38

43

R/h

Fig. 5. Results of calculations related to the first variant.

refer to railway wheels. The differences between the analytical solutions and the FE solutions will be applied as they are applied in Ref. [20]: εn ¼ ðωfn  ωcn Þ=ωcn  100%

(30)

ωfn

where and ωcn are the natural frequencies of the FE and the analytical models, respectively. Eq. (30) defines a so-called frequency error [20]. In the calculations, the Kirchhoff modulus G is determined by the formula [7–9] G¼

E 2ð1 þνÞ

(31)

5.1. Comparison of the analytical models The computations for the thin ring and thick ring analytical models of the system are executed with different values of kf, kp and kS. The impact of the system's ring depth on the solutions is analysed. This study considers the analytical solution using the model based on Timoshenko's theory to be more accurate than the solution using the thin ring model. The model shown in Fig. 2a is used for the calculations. In the adopted model, the ring centreline has a radius R¼0.0875 m and the width of the ring is lr ¼0.008 m. Dynamic analysis is performed for the model cases by varying the ring depth h as shown in Table 2. The remaining necessary technical parameters are identical to those in Table 1. From the many analysed cases, two are presented and discussed in this subsection (see Figs. 5 and 6). Figs. 5 and 6 display the frequency errors, determined by Eq. (30), versus R/h for the analysed examples. For these instances in Eq. (30), ωfn is the nth natural frequency coming from the thin ring theory analytical model.

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60

3.0

ε2 2.5 40

ε3

ε4

εn [%]

εn [%]

ε7

ε6

2.0

20

ε5

ε8

1.5 1.0 0.5

0

0.0 3

8

13

18

23

28

33

38

43

3

8

13

18

R/h

23

28

33

38

43

R/h

Fig. 6. Results of calculations related to the second variant. Table 3 Value of the R/h parameter for the analysed models. No. R/h

1 3.65

2 5.65

3 7.81

4 10.16

5 10.88

6 5.63

7 5.9

8 6.17

9 7.83

Table 4 Natural frequencies of the considered system ωn [Hz] (Timoshenko's theory). n

1. 2. 3. 4. 5. 6. 7. 8. 9.

kf [N/m2]  109

kp [N/m2]  108

kS [N/m]  107

0 15 0 16.7 0 16.7 0 17 0 17.8 0 8.98 0 8.9 0 8.9 0 7.9

0 6 0 6 0 6 0 6 0 6 0 6 0 6 0 6 0 6

0 7.2 0 3 0 3 0 2.5 0 2.4 0 33.5 0 34 0 34 0 38

2

3

4

5

6

7

8

2699.5 14,442.7 1715.9 16,663.1 1221.7 18,799.1 927.5 20,957 864.2 21,991.3 263 1753.5 240.2 1729.9 220.2 1712.5 137.2 1543

7242.9 20,707.9 4736.8 21,665.1 3410 23,630.8 2602.6 25,527.1 2427.4 26,488.3 726 2317.3 664.2 2259.6 610 2208.5 383.1 1913.7

13,022.4 27,248.7 8796.2 27,006.5 6421.7 28,758.6 4935.6 30,327.2 4609.7 31,179.3 1348 2960.9 1236.3 2861.9 1137.8 2770.7 721.5 2323.5

19,633.6 33,933.3 13,698.8 32,622.2 10,159.1 34,169.2 7873.1 35,431.4 7365.3 36,180.4 2098.7 3680.5 1930.1 3534.5 1780.6 3399 1141.5 2777.3

26,795.5 40,724.9 19,275.4 38,445.7 14,530.5 39,789.1 11,364.7 40,776.1 10,651.6 41,435.8 2952.4 4468.4 2722.6 4271 2518 4087.9 1633 3271.7

34,316.9 47,610.3 25,381.5 44,436 19,448.3 45,563.7 15,359.4 46,299.9 14,424.9 46,882.9 3886.8 5317 3593.7 5064.5 3331.8 4830.9 2185.9 3803

42,069.4 54,582.1 31,898.6 50,567.9 24,830.9 51,457.5 19,806.1 51,958.5 18,640.9 52,474.8 4883.9 6218.5 4526.8 5908.3 4206.5 5622.2 2791.3 4368

The solution for the case with parameters kf ¼ 1.5  1010 N/m2, kp ¼6  108 N/m2 and kS ¼1.2  105 N/m is presented in Fig. 5. The following results (see Fig. 6) are presented for the case with parameters kf ¼7  108 N/m2, kp ¼6  106 N/m2 and kS ¼6  104 N/m. In each case, the difference between the results of the thick ring model and the thin ring model grows with the increase in the number of natural frequencies. Better agreement between the analytical solutions is observed in the first variant (see Fig. 5). Moreover, for this instance, each value of the frequency error drops below 1 percent when the value of the ratio R/h exceeds 20. 5.2. Comparison of the analytical solutions to the FE model calculations This subsection compares the results from analytical and the FE models. The analysis uses the FE models of the gears and railway wheels discussed in Section 4. The R/h ratio values for the particular models are displayed in Table 3. The R/h ratio values are below 20 in every case. The values of the natural frequencies determined by the analytical solutions are presented in Tables 4 and 5 for the models identified above. The natural frequencies are determined for two cases of each wheel model. In the first case, it is assumed that the elastic layer does not exist (i.e., kf ¼kp ¼kS ¼0). In the second case, the computational results are generated for different parameter values characterising the elastic layers (see Tables 4 and 5) of all of the wheel models. The results are compared to the FE model solutions shown in Table 6.

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Table 5 Natural frequencies of the considered system ωn [Hz] (the thin ring solution). n 2

kf [N/m ]  10 1. 2. 3. 4. 5. 6. 7. 8. 9.

9

0 25 0 21 0 25 0 23 0 23 0 15 0 12 0 14 0 9.9

2

kp [N/m ]  10

8

kS [N/m]  10

0 6 0 6 0 6 0 6 0 6 0 6 0 6 0 6 0 6

2

3

4

5

6

7

8

2816.2 15,097.1 1747.5 17,299.3 1233.5 21,657.2 932.8 23,542.5 868.5 24,330.4 267.9 1983.8 244.2 1788.4 223.6 1922.3 138.6 1623

7965.3 18,021.7 4942.6 20,656.2 3488.9 25,154.2 2638.4 27,326.2 2456.6 28,210 757.7 2213.5 690.8 2020.9 632.3 2137.3 391.9 1850.1

15,272.8 22,981.6 9476.9 24,799.2 6689.7 28,930.6 5058.9 31,253.4 4710.2 32,202.1 1452.9 2574.8 1324.5 2380.5 1212.5 2440.6 751.4 2121.2

24,699.5 30,660.9 15,326.2 30,161.8 10,818.6 33,405.8 8181.3 35,687.9 7617.5 36,669.6 2349.7 3184 2142.1 2962.7 1960.8 2933.3 1215.2 2486.8

36,233.6 41,018.7 22,483.3 36,875.1 15,870.7 38,698.7 12,001.8 40,697.6 11,174.7 41,671.9 3446.9 4069.1 3142.4 3788.8 2876.5 3648.4 1782.7 2965.8

49,870.9 53,870.2 30,945.3 44,969.7 21,844 44,852.6 16,518.9 46,292.4 15,380.5 47,211.2 4744.2 5217.7 4325.1 4848.5 3959.1 4583.9 2453.6 3563.4

65,609.5 69,071.3 40,711.3 54,441.9 28,737.7 51,887.1 21,732 52,474.4 20,234.4 53,283.5 6241.5 6610.8 5690 6126.7 5208.5 5727 3227.9 4279.2

6

0 1.2 0 4.2 0 4.5 0 4.5 0 4.5 0 0.335 0 14 0 14 0 60

Table 6 Natural frequencies of the considered system ωn [Hz] (the FE model). n

1. 2. 3. 4. 5. 6. 7. 8. 9.

E [Pa]  1011

ν

0 2.08 0 2.08 0 2.08 0 2.08 0 2.08 0 2.1 0 2.1 0 2.1 0 2.1

0 0.3 0 0.3 0 0.3 0 0.3 0 0.3 0 0.3 0 0.3 0 0.3 0 0.3

2

3

4

5

6

7

8

2724 14,553 1724 16,785.5 1225 18,877 929.9 20,896.5 866.3 21,482 264.3 1750 241.2 1697 221 1663 137.6 1404

7326.5 20,340.5 4765 22,677.5 3424 25,175.5 2613 27,641.5 2436 28,330.5 730.7 2381 668.1 2310 613.3 2248.5 384.4 1910

13,198 26,193 8863 27,883.5 6458 30,280 4963.5 32,791.5 4636 33,471 1359 3025 1246 2920 1146 2826 724.9 2356

19,933 32,311.5 13,825 32,881.5 10,233 34,803.5 7933 37,092.5 7417 37,694 2120 3717 1948 3566 1796 3431 1148 2791

27,245 38,795 19,485 38,006 14,661.5 39,166.5 11,474 41,017.5 10,746 41,492 2987 4483 2752 4277 2544 4094 1645 3247

34,936 45,611.5 25,698.5 43,401 19,656 43,597.5 15,539.5 44,832 14,580 45,137 3939 5327 3639 5060 3371 4823 2205 3742

42,870 52,688 32,345.5 49,110.5 25,137 48,222.5 20,080 48,701.5 18,877 48,803 4956 6240 4590 5910 4262 5615 2819 4280

Tables 7 and 8 show the frequency errors that were derived by comparing the FE models to the corresponding analytical solutions. There is generally a considerable agreement between the results based on the thick ring analytical model and the results from the FE representations. The frequency ω2 associated with the number nine wheel model is the worst fit observed. For the case without the foundation (i.e., kf ¼kp ¼kS ¼0), the frequency error grows with the increase in the number of natural frequencies for each model. A worse fit is visible between the thin ring solutions and the results of the corresponding FE models (see Table 8). This fit is worse because the value of the ratio R/h falls below 20 (see Table 3) for each model. For many cases, the frequency error exceeds 10 percent. The worst result is observed for the frequency ω8 associated with the number one wheel model. As in the previously discussed cases, the frequency error increases with the growth of the natural frequencies number when the foundation of the ring model is removed. The results associated with the thin ring theory are better than the comparable results related to the cases of the presented analytical models of the ring with the wheel-plate as the elastic foundation (the factor kS is omitted). As examples, the frequency errors are displayed in Tables 9 and 10 for cases comparing the analytical results to the corresponding FE solutions. The values of the stiffness moduli kf and kp presented in the tables mentioned above are selected during the numerical simulation to minimise the frequency error (30). In every case, the tables show that frequency errors below 10 percent can be achieved for one or two natural frequencies.

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Table 7 Frequency error εn [percent] (comparison of the FE solution with Timoshenko's theory, for the three-parameter layer). n 2

kf [N/m ]  10 1. 2. 3. 4. 5. 6. 7. 8. 9.

9

2

kp [N/m ]  10

0 15 0 16.7 0 16.7 0 17 0 17.8 0 8.98 0 8.9 0 8.9 0 7.9

8

kS [N/m]  10

0 6 0 6 0 6 0 6 0 6 0 6 0 6 0 6 0 6

2

3

4

5

6

7

8

0.91 0.76 0.47 0.74 0.27 0.41 0.26  0.29 0.24  2.32 0.49  0.2 0.42  1.9 0.36  2.89 0.29  9.01

1.15  1.77 0.6 4.67 0.41 6.54 0.4 8.28 0.35 6.95 0.65 2.75 0.59 2.23 0.54 1.81 0.34  0.19

1.35  3.87 0.76 3.25 0.57 5.29 0.57 8.13 0.57 7.35 0.82 2.16 0.78 2.03 0.72 2 0.47 1.4

1.52  4.78 0.92 0.79 0.73 1.86 0.76 4.69 0.7 4.18 1.02 0.99 0.93 0.89 0.86 0.94 0.57 0.49

1.68  4.74 1.09  1.14 0.9  1.56 0.96 0.59 0.88 0.14 1.17 0.33 1.08 0.14 1.03 0.15 0.73  0.75

1.8  4.2 1.25  2.33 1.07  4.32 1.18  3.17 1.07  3.72 1.34 0.19 1.26  0.09 1.18  0.16 0.87  1.6

1.9  3.47 1.4  2.88 1.23  6.29 1.38  6.26 1.27  6.99 1.48 0.35 1.4 0.03 1.32  0.13 0.99  2.01

7

0 7.2 0 3 0 3 0 2.5 0 2.4 0 33.5 0 34 0 34 0 38

Table 8 Frequency error εn [percent] (comparison of the FE solution with the thin ring theory, for the three-parameter layer). n

1. 2. 3. 4. 5. 6. 7. 8. 9.

kf [N/m2]  1010

kp [N/m2]  108

kS [N/m]  106

0 2.5 0 2.1 0 2.5 0 2.3 0 2.3 0 1.5 0 1.2 0 1.4 0 0.99

0 6 0 6 0 6 0 6 0 6 0 6 0 6 0 6 0 6

0 1.2 0 4.2 0 4.5 0 4.5 0 4.5 0 0.335 0 14 0 14 0 60

2

3

4

5

6

7

8

 3.27  3.6  1.34  2.97  0.69  12.84  0.31  11.24  0.25  11.71  1.34  11.79  1.23  5.11  1.16  13.49  0.72  13.49

 8.01 12.87  3.59 9.79  1.86 0.08  0.96 1.15  0.84 0.43  3.56 7.57  3.29 14.31 3 5.2  1.91 3.24

 13.58 13.97  6.48 12.44  3.46 4.66  1.89 4.92  1.58 3.94  6.46 17.48  5.93 22.66  5.48 15.79  3.53 11.07

 19.3 5.38  9.79 9.01  5.41 4.18  3.03 3.94  2.63 2.79  9.78 16.74  9.06 20.36  8.4 16.97  5.53 12.23

 24.81  5.42  13.34 3.07  7.62 1.21  4.4 0.79  3.84  0.43  13.34 10.17  12.42 12.89  11.56 12.21  7.72 9.48

 29.95  15.33  16.96  3.49  10.02  2.8  5.93  3.15  5.2  4.39  16.97 2.09  15.86 4.36  14.85 5.22  10.13 5.01

 34.66  23.71  20.55  9.79  12.53  7.06  7.6  7.19  6.71  8.4  20.6  5.61  19.33  3.54  18.17  1.96  12.67 0.02

Table 9 Frequency error εn [percent] (comparison of the FE solution with Timoshenko's theory, for the two-parameter layer). n 2

kf [N/m ]  10 1. 2. 3. 4. 5. 6. 7. 8. 9.

4.1 4.1 4.1 4.1 4.1 2.8 2.8 2.8 2.3

10

2

kp [N/m ]  10 6 6 6 6 6 6 6 6 6

2

3

4

5

6

7

8

 22.64  26.44  28.65  30.19  30.58  34.84  36.78  38.02  41.18

 2.36  7.25  10.5  12.95  13.67  18.14  20.22  22.05  24.94

10.65 7.64 3.88 0.45  0.67  4.34  6.37  8.34  11.45

15.83 17.28 14.65 10.88 9.44 4.5 2.97 1.36  0.95

16.41 22.06 21.97 13.84 17.27 8.86 8.02 7.05 6.37

15.1 23.25 26.04 24.42 22.98 10.31 10.03 9.62 10.75

13.29 22.36 27.46 27.81 26.68 10.26 10.34 10.3 12.78

8

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Table 10 Frequency error εn [percent] (comparison of the FE solution with the thin ring theory, for the two-parameter layer). n 2

kf [N/m ]  10 1. 2. 3. 4. 5. 6. 7. 8. 9.

2.5 2.9 3 4.1 4.1 1.5 1.2 1.4 2.3

10

2

kp [N/m ]  10 6 6 6 6 6 6 6 6 6

2

3

4

5

6

7

8

 2.3  12.91  16.78  24.53  30.63  11.77  4.5  13.05  41.18

16.78 8.39 3.85  6.09  13.86 7.61 16.4 6.78  24.94

19.08 22.09 19.13 7.9  1.14 17.56 26.06 18.57  11.45

9.8 26.48 28.72 18.25 8.54 16.81 23.98 20.24  0.95

 2.14 23.21 32.55 25.29 15.68 10.24 15.97 15.24 6.37

 12.99 15.71 31.29 29.03 20.36 2.15 6.77 7.71 10.75

 22.03 6.95 26.45 29.65 22.6  5.57  1.7 0.01 12.78

8

Fig. 7. Mode shapes related to the following frequencies: (a) ω2, (b) ω3, and (c) ω4.

Fig. 8. Mode shapes related to the following frequencies: (a) ω5, (b) ω6, (c) ω7, and (d) ω8.

Fig. 9. Mode shapes related to the following frequencies: (a) ω2, (b) ω3, and (c) ω4 (analytical solutions).

The results show that introducing the additional coefficient kS to the analytical models to model the distortion of the foundation by the in-plane rotation of a ring element reduces the differences between the analytical solutions (particularly the thick ring solutions) and the corresponding FE solutions for broader range of the natural frequencies.

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Fig. 10. Mode shapes related to the following frequencies: (a) ω5, (b) ω6, (c) ω7, and (d) ω8 (analytical solutions).

Table 11 Results of the simulation case where the centrifugal effect is included (first velocity of the train, FE solution). n

3

4

5

6

7

8

Natural frequencies of the considered railway wheels ωn [Hz] 6 98 1750 7 94 1697 8 90 1663 9 72 1404

2381 2310 2249 1910

3025 2920 2826 2356

3718 3566 3431 2791

4484 4277 4094 3248

5327 5060 4823 3742

6241 5910 5616 4280

Frequency error εn [%] 6 98 7 94 8 90 9 72

0 0 0.022 0

0 0 0 0

0.027 0 0 0

0.022 0 0 0.031

0 0 0 0

0.016 0 0.018 0

No. of wheel

2 rotational speed Ω0 [rad/s]

0 0 0 0

Table 12 Results of the simulation case where the centrifugal effect is included (second velocity of the train, FE solution). n

3

4

5

6

7

8

Natural frequencies of the considered railway wheels ωn [Hz] 6 213 1750 7 204 1697 8 196 1663 9 157 1404

2382 2311 2249 1910

3026 2921 2826 2357

3719 3567 3432 2791

4485 4278 4095 3248

5329 5062 4824 3743

6242 5912 5617 4281

Frequency error εn [%] 6 213 7 204 8 196 9 157

0.042 0.043 0.022 0

0.033 0.034 0 0.042

0.054 0.028 0.029 0

0.045 0.023 0.024 0.031

0.038 0.04 0.021 0.027

0.032 0.034 0.036 0.023

No. of wheel

2 rotational speed Ω0 [rad/s]

0 0 0 0

Figs. 7 and 8 show the mode shapes of the vibrations coming from the FE solution for the number eight simplified model. For the purpose of comparison, Figs. 9 and 10 show the mode shapes obtained from the analytical solution (26) for the same model. Figs. 9 and 10 show the normal nodes of the system's ring centreline. The impact of the rotational speed of the wheels on the natural frequencies and the mode shapes of the systems under consideration is analysed. For the sake of brevity, this analysis is limited to the set of railway wheels. Two cases of train movements are studied. In the first case, the train is moving at a speed of 160 km/h. The indicative values of the angular velocities of the railway wheels are determined for these conditions. The wheel angular velocity values range from 72 rad/s to 98 rad/s (see Table 11). The second case involves a train moving at a speed of 350 km/h. The angular velocity values of these railway wheels range from 157 rad/s to 213 rad/s (see Table 12). The discussion is limited to the analysis of the results obtained from the FE models (described in Section 4) of the railway wheels. The centrifugal effect is accounted for by determining the stress distributions resulting from rotation for each FE model during the static analysis computational step (the so-called pre-stress effect). This stress distribution is included in the computational step associated with the modal analysis. Tables 11 and 12 show the values of the natural frequencies for the cases discussed. These results are compared to

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the values of the natural frequencies of the cases with no centrifugal effect (see Table 6, positions 6–9). In each case, the values of the frequency error (30) are below 0.06 percent or equal to zero. Additionally, the changes in the shape of the normal modes as a result of rotation are difficult to notice. This indicates that the centrifugal effect has a small influence on the values of the natural frequencies and the shape of the corresponding normal modes of the ring in plane flexural vibration. Therefore, the mode shapes of the rotational systems are not displayed in this paper. 6. Experimental verification The issues with the experimental verification of the considered analytical and numerical models are discussed in this section. The LMS measurement environment was used in the experimental investigation. The measuring set consisted of the PCB model 086C03 modal hammer equipped with a steel gauging point, the PCB model 353B18 accelerometer, the LMS SCADA data acquisition system, and the SCM-V4E measuring module supported by the LMS Test.Lab software. The analysis of the experimental results was conducted on a portable computer using the actual measured values. The measurement experiment was scheduled and conducted to identify the natural frequencies and the corresponding mode shapes of the inplane flexural vibrations of the considered objects. Three objects of the geometry shown in Fig. 2a were made for the measurement experiment. Material data and geometric dimensions of the systems are shown in Table 13. These dimensions correspond to model numbers two, eight, and fourteen in Table 2. These models have a significant difference in ring depths (see Table 13). It is notable that the objects prepared for the test have the same values for the radii of the ring centrelines. Each of the tested systems was bolted to a fixed object (see Fig. 11). Thirty-two measurement points were marked on the periphery of each tested object (see Figs. 11 and 12). The accelerometer was placed at each marked measuring point to measure the system response in the radial direction. The excitation point was located between points 1 and 2 on each of the tested systems. An impulse response was registered, which caused modal hammer vibrations at the measurement points for all of the discussed cases. Table 14 shows the values of the excited natural frequencies. The mode shapes of the excited frequencies for the number fourteen test model are displayed in Figs. 12 and 13. The shapes of the identified normal modes are not as smooth as those in Figs. 9 and 10, but are satisfactory. The FE models and the analytical models were prepared for every tested object. The FE models were elaborated in the same manner as the models in Section 4, except that the masses of the elastic layers were included. This permitted the selection of the physical properties of the models that were similar to the physical properties of the tested objects. The solutions obtained from the FE models were compared to the results from the experimental investigation. The values of the natural frequencies from the FE models of the tested objects and the frequency errors (30) are shown in Table 14. The frequency error was above 4 percent for only one of the natural frequencies of each case. In the next step, the analytical models of the tested objects were prepared and verified. As mentioned earlier, the proposed analytical models consisted of a thin (or thick) ring and a massless elastic foundation. To obtain the physical properties of the analytical models that represented the tested objects, the values of the equivalent mass density ρz of the Table 13 Parameters characterising the objects subjected to the experimental investigation. No. of models 2 8 14

dz [m] 0.179 0.191 0.203

dw [m] 0.171 0.159 0.147

h [m] 0.004 0.016 0.028

R [m] 0.0875

d1 [m] 0.02

lr [m] 0.008

Fig. 11. The measuring test.

lw [m] 0.002

ρ [kg/m3]

E [Pa] 2.1  10

11

7.85  10

3

ν 0.28

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Fig. 12. Mode shapes related to the following frequencies: (a) ω2, (b) ω3, and (c) ω4 (experimental results).

Table 14 Results of verifications of the FE models of the objects subjected to the experimental investigation. n No. of models

2

5

6

7

8

Natural frequencies of the considered objects ωn [Hz] (experimental data) 2 11,731.9 17,215 22,933.8 8 8660 12,943.8 16,802.5 14 7207.5 11,537.5 16,058.8

27,451.3 20,618.1 20,933.8

31,481.9 25,211.9 26,278.8

35,205.6 29,550.6 31,647.5

39,487.5 34,155.6 37,106.3

Natural frequencies of the considered models ωn [Hz] (the FE solutions) 2 11,135 17,855 22,887 8 8393.2 13,199 17,141 14 7276.9 12,015 16,549

27,357 21,085 21,387

31,491 25,219 26,498

35,389 29,595 31,799

39,112 34,209 37,214

Frequency error εn [%] 2 8 14

 0.34 2.27 2.17

0.03 0.03 0.83

0.52 0.15 0.48

 0.95 0.16 0.29

 5.09  3.08 0.96

3

3.72 1.97 4.14

4

 0.2 2.02 3.05

Fig. 13. Mode shapes related to the following frequencies: (a) ω5, (b) ω6, (c) ω7, and (d) ω8 (experimental results).

ring models were determined under conditions where the mass of the model and the mass of the object under test were comparable. In the analysed cases, the following values of the equivalent mass density were used: for the number two model, ρz ¼ 2.8  104 kg/m3; for the number eight model, ρz ¼1.22  104 kg/m3; and for the number fourteen model, ρz ¼ 9.97  103 kg/m3. The values of the natural frequencies of the selected analytical models of the tested objects and the frequency errors are presented in Table 15. The values of the stiffness moduli kf, kp and kS presented in this table were selected from the numerical simulations to minimise the frequency errors (30). The results that correspond to the number two model were derived by using the thin ring theory. The thick ring theory was used for the number eight and number fourteen model cases to determine the results presented. The analytical model of the number eight tested object obtained for best fit. In this case, the errors were below 3.6 percent for every natural frequency. The values of the frequency errors were above 6 percent but below 7.9 percent for three natural frequencies of the two remaining model cases. The measured results were satisfactory and confirmed the decision to introduce the additional factor kS.

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Table 15 Results of verifications of the analytical models of the objects subjected to the experimental investigation. No.

n

2 2

kf [N/m ]  10 Natural 2 8 14

8

2

kp [N/m ]  10

8

kS [N/m]  10

frequencies of the considered models ωn [Hz] (analytical solutions) 46.4 6 0.695 12,432 34 6 43 8875 9.98 6 38.5 7049

Frequency error εn [%] 2 46.4 8 34 14 9.98

6 6 6

0.695 43 38.5

3

4

5

6

7

8

16,860 12,919 12,269

21,564 17,116 17,319

26,414 21,352 22,244

31,351 25,603 27,097

36,348 29,859 31,904

41,392 34,117 36,684

 2.06  0.2 6.34

 5.97 1.87 7.85

 3.78 3.56 6.26

 0.42 1.55 3.11

3.25 1.04 0.81

4.82  0.11  1.14

7

5.97 2.49  2.2

7. Conclusions This paper presents the studies of the free in-plane vibrations of circular rings with wheel-plates as three-parameter elastic foundations. A generalised Winkler elastic foundation model is proposed to represent this system. The effects of rotary inertia and shear deformation are included in the analytical models of the systems studied. Both analytical solutions of the free vibrations of the systems under study are determined by using the separation of variables method. It is clear that Timoshenko's theory leads to a more complicated solution. The analytical solution associated with the thick ring theory is treated as the most satisfactory solution. The effectiveness of the proposed approach is tested by using the simplified models of aviation gears and railway wheels. The analytical solutions are compared to the corresponding FE results. Theoretical and numerical studies are presented and verified successfully during the experimental investigation. The proposed analytical models give satisfactory results for a wide range of frequencies. It is notable that the analytical approach reduces computational time and computer memory requirements significantly when compared to the FE solution. At this stage of investigation, it is clear that using the proposed generalised Winkler model of the elastic foundation (particularly in the case of Timoshenko's theory) yields satisfactory results. The proposed method may be useful for dynamic analysis of systems such as aviation gears and railway wheels.

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