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This paper presents a new approach for tire dynamic analysis. By using this method, transient response for rotating tires under various loading situations can be analyzed. The well-known model of ring on elastic foundations (REF) is utilized to model tires. The general forced solution of undamped inextensible vibration is derived by the use of a mo...
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... of rotating speed and equal to the first-order natural frequency of non-rotating rings. Because the parameters of tire ring model are already specified previously, given applied force f (here f ¼ 10 000 N) we can give numerical results of the tire response including wheel center load and tire displacements, some of which are graphically shown in Figs. 2-5. It is seen that the tire displacements at the loading point contain multiple frequencies (see Fig. 5) in contrast with the wheel center load (see Fig. ...
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... Therefore, a coupled rigid-flexible ring model for tires is proposed based on two-dimensional (2D) ring models [47,50,51] and extended to the uncertainty quantification of dynamic tire response. The proposed model integrates the advantages of the rigid ring model [47,48] and the flexible ring model [49,[52][53][54][55][56]. Firstly, the deformation and physical characteristics of the tire are theoretically modeled using the flexible ring model to obtain the steady-state deformation of the tire and contact forces. ...
... The radial tires model plays an important role in the dynamic models of these transportation machineries. The REF model is currently the most widely used simplified dynamic model for radial tires, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and it is used in the dynamic characteristics analysis of flexible gear 19 and mechanical elastic wheel. 20 The REF model of radial tires is made up of two parts: spring groups and flexible ring, which represent sidewall and tread, respectively. ...
... 20 The REF model of radial tires is made up of two parts: spring groups and flexible ring, which represent sidewall and tread, respectively. The spring groups often consist of circumferential/radial springs whose stiffness can be obtained by hammering experiment, 3,4 and the flexible ring is assumed as an Euler-Bernoulli annular beam, [3][4][5][6][7][8] or Timoshenko annular beam. [9][10][11][12] In recent years, the REF model has been further developed into the 3D cylindrical shell on elastic foundation model 1,[13][14][15][16] to analyze the transverse vibration of radial tires. ...
... 20 The REF model of radial tires is made up of two parts: spring groups and flexible ring, which represent sidewall and tread, respectively. The spring groups often consist of circumferential/radial springs whose stiffness can be obtained by hammering experiment, 3,4 and the flexible ring is assumed as an Euler-Bernoulli annular beam, [3][4][5][6][7][8] or Timoshenko annular beam. [9][10][11][12] In recent years, the REF model has been further developed into the 3D cylindrical shell on elastic foundation model 1,[13][14][15][16] to analyze the transverse vibration of radial tires. ...
Radial tires are widely used in transportation machinery, such as large aircraft, new energy vehicles, and engineering vehicles. The ring on the elastic foundation (REF) model is often used for dynamic analysis of radial tires, but the influence of elastic boundary on the dynamic characteristics of radial tires is ignored in this model. In this paper, based on the absolute nodal coordinate formulation (ANCF) and plane stress theory, a rotating thick ring on the elastic foundation (RTREF) model is established to explore the influence of elastic boundary position and other factors on dynamic characteristics. Because existing rectangular and triangular elements in ANCF can’t accurately describe radial/circumferential mesh of a thick ring, a novel ANCF-annular-sector element (AASE) is proposed for the RTREF model. By analyzing the influences of elastic boundary position, thick wall, and rotational speed on the vibration characteristics and nonlinear forced transient response of the RTREF model, it is demonstrated that even if the ring wall is thin, the influence of elastic boundary positions on the dynamic characteristics of the model can’t be ignored.
... To evaluate the impacts of the tire structural parameters and the uniformity parameters, the analysis model is required to have the ability to describe the structural and material properties of tires. As mentioned in the literature review, ring models not only provide a fundamental description of the tire structure but can also ensure a concise mathematical solution [8,23,31]. The previous works have categorized the tire ring models as rigid ring models [8,41,42] and flexible ring models [11,12]. ...
The dynamic response of tires directly affects the handling stability and ride comfort of a vehicle, which in turn determines the vehicle ride, handling, and driver perception. Modeling and simulation of tire dynamics typically require a reasonable computational cost while ensuring proper accuracy. In this study, a novel theoretical model, the coupled rigid-flexible ring model, is presented to analyze the characteristics of the in-plane dynamic responses of tires on uneven road surfaces. The proposed new method consists of three primary sub-models: an elastic contact algorithm featured by a flexible ring model, rolling/vibration dynamics represented by a rigid ring model, and an internal-force transmission algorithm linking the rigid and flexible ring models. In this way, the proposed new method has the merits of both high accuracy up to 150 Hz and low computing cost. A contact algorithm based on the two-dimensional flexible ring provides a pressure distribution on the tire-road contact patch and the length of the footprint under different vertical loads. The transient dynamic response is then estimated by combining the rigid ring model with the flexible ring model. The accuracy of the contact algorithm and the transient responses are validated against experimental radial stiffness and the over-cleat tests respectively. The results show that the in-plane dynamics of the tire can be predicted well. In addition, this model is extended to the analysis of the low-speed uniformity of tires with geometric defects and is validated experimentally. It indicates that the novel proposed model offers many application scenarios and extension possibilities.
... The circular beam represents the belt region, and the radial spring represents the stiffness of the sidewall and air pressure. As established in prior studies[10],[11], a single point contact model is utilized to model tire-road contact instead of a distributed model. The model fits the current drum loading scenario. ...
The deformation of part of a tire in direct contact with the road is of great interest in tire sensor development. The paper focuses on the analysis of deformation of tire carcass and its potential for estimation of the tire vertical force. Detailed empirical and physical tire models are available for simulating tire dynamics. However, these models require extensive tire parameterization. A simple model to predict vertical force is helpful to the vehicle industry during the initial design phase, where a tangible tire is not available. This paper uses the tire modeled as a beam on an elastic foundation to predict the tire radial deformation profile for a given displacement input. The predicted tire radial deformation is then used to estimate the vertical force. The predicted radial deformation and vertical force are verified by comparing them to experimental results. The study finds its use in tire sensing applications which relies on direct measurement of the radial deformation of the tire carcass for the estimation of tire vertical force. This research proposes a cost-effective tire vertical force parameterization approach based solely on measured tire geometry and air pressure for applications where high-fidelity tire models are not required. The proposed method is also helpful in cases where load test equipment is not available. In the course of the analysis, it is found that the vertical force depends significantly on the sidewall length of the tire sidewalls.
... In this study, the relationship between the strain information of the inner liner and tire dynamic parameters was established according to the flexible ring tire model [19], and the details of the model symbols are provided in the appendices. The flexible ring model provides theoretical support for the development of intelligent tire algorithms. ...
... Eq. (19) indicates that the deformation differences in the front and rear areas have the same magnitude but in the opposite direction. To analyse the effect of the longitudinal force on the radial deformation difference Δω, the longitudinal dynamic model of the thin ring is analysed under a vertical load of 4000 N, a friction coefficient of 0.7, roll velocity of 70 km/h, and slip rate range of -20 % to 20 %. ...
Tires are essential components of vehicles and the only vehicle components in contact with the road. The tire longitudinal force originating from the contact between the tire and road can enhance traction and braking and contribute to the directional stability of vehicles. If the longitudinal tire force can be accurately estimated, vehicle safety can be improved. This study established a longitudinal tire physics-based model combining the brush model with the flexible ring model to develop a strain-based intelligent tire system for estimating the longitudinal tire force. The developed longitudinal dynamic model was used to study tire strain characteristics under pure longitudinal slip conditions. An algorithm was developed for estimating the longitudinal tire force through feature extraction and data fitting of the tire strain. A finite-element tire model was established to simulate the longitudinal force. Comparing the simulated and estimated forces indicated that the proposed algorithm can accurately predict the longitudinal force of intelligent tires and thus provide useful information to vehicle stability control systems.
... kinetics analysis method for analyzing the transient response of rotating tires under various loads. 10 However, the transient response in the application of the REF model to the NPT was relatively late. Gasmi proposed a two-dimensional NPT analysis model based on the REF model. ...
A static grounding analysis model of non-pneumatic tire (NPT) was built and presented in this paper. The proposed NPT analysis model considers the non-linearity of the spoke stiffness and is suitable for the performance exploration of various structures of the NPT. First, the shear band, rigid rim, and spoke structure of the NPT were simplified, the main structural parameters and mechanical parameters were extracted, and an analysis model was established. The model can describe the deformation of the NPT when it is subjected to external forces. On this basis, the different stiffness of the spokes during tension and compression was considered, an iterative method was used to compensate for the difference in the deformation caused by the difference in radial stiffness of the spokes, and an analysis model of NPT with nonlinear spokes was established. Then, the contact between the tire and the road surface was introduced to iteratively compensate for the reaction force of the road surface, and the deformation of the NPT with nonlinear spokes on the road surface was obtained. Finally, the finite element software ABAQUS was used to verify the accuracy of the model. This model contains more comprehensive structural parameters and material parameters, which can more realistically simulate the structural characteristics and static grounding behavior of the NPT.
... The tangential stiffness, radial stiffness, equivalent flexural rigidity, and density of the tire are denoted by k v , k w , EI, and ρ, respectively. The governing equation of motion for a rotating inextensible tire with radius r, width b, thickness t, and internal pressure p is derived using the Hamilton's principle 22 . The equation of motion for a nonrotating tire to determine the natural frequencies is obtained by ignoring the tire rotation and tire loading as: ...
... Based on the experimentally observed frequencies, the flexural rigidity, radial and tangential stiffness were estimated to be 63.6 Nm 2 , 1562 kNm −2 , and 2212 kNm −2 . A damping ratio of 0.1 was assumed for all the modes of vibration 22,25 . The strain in the tire was estimated at different speeds, loads and tire pressures using Eq. ...
The transition of autonomous vehicles into fleets requires an advanced control system design that relies on continuous feedback from the tires. Smart tires enable continuous monitoring of dynamic parameters by combining strain sensing with traditional tire functions. Here, we provide breakthrough in this direction by demonstrating tire-integrated system that combines direct mask-less 3D printed strain gauges, flexible piezoelectric energy harvester for powering the sensors and secure wireless data transfer electronics, and machine learning for predictive data analysis. Ink of graphene based material was designed to directly print strain sensor for measuring tire-road interactions under varying driving speeds, normal load, and tire pressure. A secure wireless data transfer hardware powered by a piezoelectric patch is implemented to demonstrate self-powered sensing and wireless communication capability. Combined, this study significantly advances the design and fabrication of cost-effective smart tires by demonstrating practical self-powered wireless strain sensing capability.
... Furthermore, viscoelastic structures such as tires have high damping characteristics, so that resonance peaks do not appear clearly and they are difficult to identify the existence of modes in the high frequency region [4]. Therefore, tire mode identification is generally targeted at the low frequency region [5][6] [7]. From such a background, a method of identifying after reducing the number of modes from FRF has been proposed. ...
The viscoelastic structures such as tires have high damping characteristics, so that resonance peaks do not appear clearly and they are difficult to identify the existence of modes in the high frequency region. Previous studies have proposed methods that make it easier to understand modes. In that method, the number of modes are reduced by using the fact that the basis function of the mode shape of the cylindrical structure can be expressed by a periodic function. However, a large number of measurement points are placed on the structure and data is collected, so the workload of the experiment is large. In this study, we conducted a numerical experiment of the mode reduction method using some measurement point data for the purpose of reducing the workload, and evaluate the influence on the identification accuracy. As a result, it was confirmed that the identification results of the natural frequency and the structural damping coefficient had sufficient accuracy even if some measurement points were used.
... This simplified method of transforming a pneumatic tire into a ring on an elastic foundation can date back to the 1960s. 1,2 The previous works of the ring model have summarized the vibrations of tires in the condition of in-plane vibration [2][3][4][5][6] and out-of-plane vibration. 7,8 The in-plane vibration includes circumferential and radial bending. ...
Material and geometrical parameters of tires involve some degree of uncertainty mainly related to production processes. Accordingly, the associated structural responses are affected by these uncertainties. In this study, a novel theoretical ring model is presented to describe the in-plane and out-of-plane vibrations as well as the steady-state response of tires, and then to evaluate the influence of the uncertainties in structural parameters on the natural frequencies and the sound radiation characteristics under uncertain excitations. The Hamilton principle is applied here to derive the governing equations. The modal superposition method is used to calculate the steady-state response of the tire. In the sound radiation analysis, the in-plane and out-of-plane bending and torsional vibrations under a set of harmonic unit forces and moments are treated as the source of noise generation. On this basis, the generalized polynomial chaos expansion method is then adopted to evaluate the influence of the uncertainty on the natural frequencies and the sound power. To obtain the unknown coefficients of the expansions, the nonintrusive probabilistic collocation method is employed. Moreover, considering the concept of linear independence of vectors, the number of collocation points is reduced. It is applied to investigate the impacts of the elastic and structural uncertainties on the natural frequencies of the tire. This yields an efficient simulation in terms of computational costs. Finally, the distributions of the sound power due to the forced vibration under the random concentrated line forces are given.
... Meanwhile, other techniques, such as the wave method [32] and spectral element method [33] are also used to investigate free vibrations. The forced vibrations are emphasized in [11,12,14,15,20,35,36] using classical mode superposition, except [37], in which the authors employed a new modal analysis method proposed by Meirovitch in [38,39] for linear gyroscopic systems to obtain the transient response of a rotating ring subjected to a point excitation. ...
Rotating ring-like structures are very commonly used in civil, mechanical and aerospace engineering. Typical examples of such structures are components in turbo-machinery, compliant gears, rolling tyres and flexible train wheels. At the micro-scale, rotating ring models find their applications in the field of ring gyroscopes, in which high accuracy of modelling is required. The in-plane vibrations of rotating rings are of particular interest since such structural components are usually subject to in-plane loads. The focus in this thesis is therefore placed on the in-plane dynamics of rotating rings. While the radial and circumferential motions of a stationary ring are coupled due to curvature, a steadily rotating ring, as any gyroscopic system, is subject to two additional fictitious forces induced by the gyroscopic coupling due to rotation, i.e. the Coriolis and centrifugal forces. Among them the centrifugal force associated with the steady rotation of the ring (quasi-static force) introduces an axi-symmetric radial expansion and a hoop stress; the latter has the tendency to stiffen the ring. In contrast, the dynamic part of the centrifugal force has the tendency to soften the system. Next to that, the Coriolis force bifurcates the natural frequencies of the ring. The proper consideration of the rotation effects is essential to determine the dynamic behaviour of rotating rings, such as stability of free vibrations and resonance of rotating rings under stationary loads. Although various models exist, the considerations of rotation effects are not always in agreement, resulting in distinct theoretical predictions of critical speeds associated with instability and resonance of rotating rings. In addition, in all the existing rotating ring models, the equations of motion were derived assuming the inner and outer surfaces of the ring to be traction-free. However, when one considers a ring whose inner surface is elastically restrained by distributed springs, this assumption is violated. The traction at the inner surface can significantly influence the stress distribution along the thickness of the ring and this effect has to be properly accounted for since the internal stresses may show a strong gradient from the inner surface to the outer surface, especially in the case of rings rotating at high speeds or when the latter are supported by stiff foundation. The primary aim of this thesis is to develop a highly accurate rotating ring model that properly accounts for both the rotation and boundary effects with rigorous mathematical derivation to fill the gap regarding the modelling and prediction of the dynamic behaviour of high-speed rotating rings. To achieve this aim, the following four objectives are set: (i) identify the reasons of disagreements between various existing rotating ring models and clarify the mathematically sound derivations of governing equations; (ii) develop a high-order rotating ring model which properly accounts for the rotation effects, as well as the non-zero tractions at boundaries; (iii) close the debate on the prediction of critical speeds associated with instability of free vibrations and resonance of forced vibrations; and (iv) apply the developed high-order model to predict the steady-state response of rotating rings under stationary loads and the stability of rotating ring-stationary oscillator system.
