Fractal Contact Theory Research Articles

Effect of relative phase of loose tie rods on nonlinear dynamic behavior of rod-fastening rotor-bearing-seal system

Loosening of multiple tie rods in a circumferential rod-fastening rotor caused by thermal deformation and centrifugal loads is very common in engineering. The relative position of loose tie rods in the circumferential direction will also have an important impact on rotor dynamics. The relevant research has not been done yet. In this paper, a contact model of the rough machined surface is firstly established by combining the probability functions describing rough surfaces and fractal contact theory of asperities, then the contact stiffness model between disks is established based on the contact model. Finally, the dynamic model of the circumferential rod-fastening rotor-bearing-seal system is established and solved. The effect of relative phase of loose tie rods on the nonlinear dynamic characteristics of rod-fastening rotor-bearing-sealing system is studied. It can be concluded that relative phase of the loose tie rod has an important influence on the frequency, bifurcation, and periodic motion of the rotor system. At lower speed, the anisotropy of contact stiffness caused by loose tie rods has little effect on the rotor trajectory, while the generalized bending moment has a greater influence on dynamics of the system. At higher speed, the anisotropy of contact stiffness has a great influence on the axis rail of the rod-fastening rotor.

Vibration of bolted composite cylindrical-cylindrical flanged shells considering contact characteristics

A semi-analytical model for bolted composite cylindrical-cylindrical flanged shells (BCCFS) considering the influence of the flange and the connection characteristics of the bolted flange joint is established. In the model, flanges are treated as annular plate structures and modeled using Kirchhoff plate theory. To simulate the connectivity performance of the bolted flange joint, a surface-spring model considering the pressure distribution characteristics is developed and applied to the model of the bolted structure. A method based on fractal contact theory is proposed to identify stiffness-related parameters of the surface-spring model, which avoids additional tensile experiments of the bolted joint. Sufficient modal tests are performed to validate the accuracy of the model. Furthermore, the effects of flange thickness, length ratio of two shells, and quantity of bolts on the free vibration characteristics of the structure are studied. The frequency veering phenomenon with the variation of size parameters is found and analyzed in depth.

Dynamic Modeling and Experimental Modal Analysis for the Central Rod-Fastened Rotor With Hirth Couplings Based on Fractal Contact Theory

Abstract The central rod-fastened rotor of gas turbine exhibits pronounced noncontinuous characteristics due to the large number of contact interfaces between the compressor and turbine disks. It is necessary to establish an accurate dynamic modeling method for the central rod-fastened rotor that fully considers the contact surface effect. In this work, the contact behavior of the rough surface is characterized by the fractal theory. The normal and tangential contact stiffness models are developed, and the influence of fractal parameters is discussed. Besides, the finite element model for the central rod-fastened rotor is established by developing an improved contact element considering the equivalent stiffness segment of Hirth couplings. Finally, the proposed model is verified by conducting the modal testing and measuring the first four modes of natural frequencies and modal shapes of the central rod-fastened rotor. The results show that the numerical results are in good agreement with the experimental ones, and the fractal contact model can effectively predict the connection stiffness of Hirth couplings, which in turn improves the simulation accuracy for the modal characteristics of the central rod-fastened rotor and provides a dynamic modeling approach with high efficiency and less computational complexity.

Modeling and free vibration analysis of bolted composite flanged cylindrical-cylindrical shells under partial bolt loosening conditions

The bolted composite coupled cylindrical shell is a common substructure in modern engineering, which may encounter the issue of local loosening conditions during service. Meanwhile, the influence of flanges on the structural vibration characteristics is often overlooked in dynamic numerical analysis. In this paper, a semi-analytical dynamic model for the bolted composite flanged cylindrical-cylindrical shell structure under arbitrary connection conditions is established, and the free vibration characteristics of locally loose structures are emphatically studied. In the model, the annular plate modeling method is developed to achieve the simulation of the flange part. To fully reflect the connection characteristics of the bolted joint, a spring surface model and the corresponding parameter determination method based on fractal contact theory are developed. The equations of motion of the structure are obtained utilizing the Lagrange equations. The accuracy of the model and its predictive ability for the vibration characteristics of the bolted composite flanged cylindrical-cylindrical shell under local loosening conditions are fully demonstrated through numerous comparisons with experimental results from other literature and modal test results obtained under various working conditions. Furthermore, the influence of the number and degree of loosening bolts on the free vibration characteristics of the structure and the coupling phenomenon between different modes are discussed. The research in this paper is expected to provide references for the design of bolted composite flanged cylindrical shell structures and the monitoring of bolt connection defects.

Contact mechanics modeling of fractal surface with complex multi-stage actual loading deformation

The contact mechanical behavior of rough surfaces is related to the friction and wear of mechanical components, and also has an important impact on the dynamic performance and force transfer characteristics of assembly surfaces. This paper aims to solve the complex multi-stage mechanical deformation mode of the loading surface, and establishes all multi-stage contact mechanical models involved in the actual loading deformation process from the perspective of the actual deformation and critical deformation of the asperity. Firstly, based on the three-dimensional fractal contact theory, the force deformation relationship of a single asperity during the full deformation stage of loading is constructed. Then, six critical contact areas and four critical deformation quantities of a multi scale asperity are derived. Finally, the complex multi-stage deformation contact mechanical models of three-dimensional fractal surfaces under actual loading are established. The area power spectral density (APSD) function spectrum of rough surface under different fractal parameters, as well as the variation of contact load, contact stiffness, and contact area are analyzed numerically. Compared with several classical models, the rationality of the proposed model is verified. This study can be used as the basis for constructing the overall dynamic model of an assembly system

Time-varying Reliability Analysis of Contact Fatigue Strength for Nutation Face Gear Transmission

Background: Nutation face gear transmission is a patent about a new type of transmission based on meshing between two face gears. It replaces spur gear with internal face gear to form a “faceface” meshing gear pair and actualizes its deceleration function by combining the nutation principle, which has the advantages of both face gear and nutation drive. Objective: The purpose of this paper is to establish the time-varying reliability calculation model of contact fatigue strength based on fractal contact theory, so as to improve the reliability, safe service and further optimization design of the nutation face gear transmission. Methods: The time-varying uncertainties of the contact stress and the allowable contact stress of the face gear teeth are analyzed respectively, and then the time-varying reliability model of the contact fatigue strength is established. Finally, the drift rate and fluctuation rate of each random variable are simulated using the Monte Carlo simulation method to obtain the time-varying reliability of the contact fatigue strength for the nutation face gear drive. Results: The results show that the general trend of geometric Brownian motion locus of each random variable is consistent with the actual situation. Conclusion: For the given example of the nutation face gear transmission device, the reliability of the contact fatigue strength of the planetary face gear meshing with the fixed face gear and the rotating face gear within one year is calculated to be 0.9912 and 0.9856 respectively, and the maximum vibration noise is about 68dB, which meets the expected value.

A novel bolted joint model with stick-slip hysteresis properties considering pressure-dependent stiffness: Theoretical and experimental investigations

Considering the influence of normal pressure on tangential stiffness, in this paper, an improved Iwan model with pressure-dependent stiffness is proposed to better describe tangential stick-slip hysteresis behaviors of bolted fiber-reinforced composite joints. In the model, the expression of the force-displacement relationship is derived from the evolution of the slip area. The parameters of the model are obtained based on multi-scale fractal contact theory, which avoids the inconvenience of parameter estimation methods requiring special quasi-static experiments. Meanwhile, the tangential behavior experiment of bolted fiber-reinforced composite joints is carried out. By comparing the hysteresis curve results and the stiffness softening law, the accuracy of the model and the necessity of considering the pressure dependence of the tangential stiffness are proved.

Dynamic Friction-Slip Model Based on Contact Theory

In ultra-precision positioning equipment, the positioning accuracy is affected by the friction characteristics, especially the pre-slip stage. At present, the research on friction mainly includes contact theory and the dynamic friction process. There is no time variable in contact theory models, so they only apply to the stage of static contact, while the establishment of a friction dynamics model depends on parameter identification and cannot reflect the influence of a rough morphology and load. Therefore, neither theory can elucidate the ultra-precise slip mechanism. In this paper, maximum static friction force, tangential stiffness, and tangential damping models of contact surfaces were deduced through fractal contact theory. By substituting the contact parameters into the modified LuGre model, a dynamic slip model of the mask was established. Finally, the above model was verified by a reticle dynamic slip measurement system. The experimental results showed that in the pre-slip stage of the reticle, with an increase in the normal external load, the slippage of the reticle decreased. The theoretical value calculated by the model was basically consistent with the experimental value, and the slippage of the reticle mainly originated from the tangential deformation and relative sliding of the contact surface. With the increase in the normal external load, the proportion of the tangential deformation in the sliding of the entire contact surface was larger. For the change in slippage during different motion stages, the theoretical value was close to the experimental value after eliminating system errors such as vibration and fiber bending, which proved the correctness of the model in this paper.

Experimental and theoretical studies on friction contact of bolted joint interfaces

This paper presents experimental and theoretical modeling studies on the tangential contact stiffness and friction hysteresis of bolted joint interfaces under transversal vibration. A new test rig was developed to measure the interface fretting response of bolted joints. This rig is based on a servo-hydraulic fatigue testing system in which the contact surfaces are arranged symmetrically to avoid bending of the specimen that could affect the friction measurement. Force-displacement curves were measured under different tangential loads and compared with those found in the literature. A multi-scale contact modeling method was proposed to calculate the tangential contact stiffness and reproduce the friction hysteresis of bolted joint interfaces. This method overcomes the complexity of identifying contact stiffness, which usually requires a model of the dynamic behavior of the joint and frequency responses measured with dedicated fretting test stands. Specifically, this method comprehensively considers the multi-scale features of randomly rough surfaces, the non-uniform contact pressure distribution, and the convenience of phenomenological contact models in simulating frictional hysteresis, relying on fractal contact theory and discretized Iwan model. The effectiveness of the proposed method was verified by comparing the simulation results with the experimental counterparts in different cases of tangential loading. The successful validation demonstrates the potential of the proposed modeling method for application to structures with bolted joint.

Nonlinear dynamic analysis of four-station rotary tool holder subjected to harmonic excitation

Four-station rotary tool holder on a lathe is fabricated to achieve transposition operation of turning tools and carry the violent cutting force prior to feed system. Its dynamic characteristic has a potential influence on the structural dynamics and cutting stability of a machine tool. This paper develops a mechanical model for the dynamic characteristic analysis and vibration response prediction. Using the fractal contact theory, the analytical model highlights the rough joint interfacial contact of main components including tool holder body, trapezoidal meshing gear plate and screw shaft. The elastic supports of thrust bearing unit and hollow shaft are also considered. The multi-degree-of-freedom system involving the coupled displacement and piecewise restoring force function is established using lumped mass method. The modal and harmonic excitation tests are performed to investigate the natural property and verify the proposed model. Furthermore, numerical simulation evaluates the effect of external excitation, locking force, fractal parameter, bearing preload and shaft stiffness on dynamic characteristic and stability. The proposed model is promising for the structural optimal design of four-station rotary tool holder.

A forward modeling method of assembly interface contact characteristics based on user-defined elements

The forward construction of the assembly interface model plays a fundamental role in the high-performance digital assembly model building. Aiming at the problem that it is difficult to accurately solve the interface thin-layer element parameters with dynamic inverse calculation method, a thin-layer element parameter modeling method based on multiscale morphology simulation is proposed. First, the user-defined elements (UEL) are developed, and its attribute parameters are continuously distributed in space. Then, the combination of macroscopic interface contact pressure and microscopic morphology realizes the efficient and accurate characterization of interface contact stiffness. On the macro scale, the pressure distribution characteristics of the assembly surface are extracted based on finite element theory. On the micro scale, the mapping relationship between the material properties of the thin-layer and the surface morphology is established based on fractal and Hertzian contact theory. Furthermore, the mapping relationship between the elastic modulus of the thin-layer element and the contact stiffness is established, which realizes the forward characterization of the interface contact characteristics. The equivalent model of the assembly interface is built according to the UEL and thin-layer theory. Finally, the dynamic experimental results are in good agreement with the simulation results, which verifies the effectiveness of the proposed modeling method. This study provides a basis for the forward modeling of assembly interface contact characteristics.

Dynamic Modeling of Tie-bolt Rotors via Fractal Contact Theory and Virtual Material Method

Tie-bolt rotors, comprised of multiple stages of discs fastened by tie bolts, are commonly used in heavy-duty gas turbines. The contact interfaces between adjacent discs have considerable influence on the dynamic characteristics of tie-bolt rotors. Although many studies on the modeling of tie-bolt rotors have been reported, it is still a challenge to develop a relatively simple, efficient, and accurate model for such a kind of complex rotors. In this article, a new virtual material modeling method is proposed to analyze the dynamic characteristics of tie-bolt rotors. The elastic modulus and Poisson ratio of the virtual material are determined from the interface contact stiffness, which is derived based on the three-dimensional fractal contact theory. To verify the modeling approach proposed here, modal tests are conducted on a tie-bolt rotor specimen, of which the measured natural frequencies are compared with the simulation results. Finally, by using the verified modeling approach, the dynamic behaviors of tie-bolt rotors are investigated comprehensively, where the effects of fractal dimension parameters and preload level on rotor dynamics are evaluated.

Chatter suppression in the milling process of the weakly-rigid workpiece through a moving fixture

This article presents a new method to suppress the chatter vibrations in the milling process of the weakly-rigid workpiece through a moving device. The basic idea is to design a moving fixture to instantaneously and continuously provide additional stiffness and damping to the instantaneous contact position between the moving fixture and the workpiece, and as the device moves, the chatter vibrations of the workpiece during the whole process are suppressed without the need of any other additional hardware. Two theoretical means, i.e. the structure split-assembly technology and the fractal contact theory, are combined to mathematically model the underlying dynamic behavior of the workpiece under the support provided by the moving fixture. The workpiece is divided into three parts, i.e. the machined part, the machining part and the part to be machined. The dynamic response of each part is separately analyzed and then theoretically assembled to calculate the whole workpiece’s varying dynamic characteristics, which involve the support influence induced by the moving fixture. The effect of material removal on the workpiece’s dynamic characteristics is also integrated into the analyzing procedure. Based on the above analysis, the dynamic governing equation of the thin-wall milling process with the moving fixture is finally established and solved to predict the stability lobe diagrams (SLDs). The validity and reliability of the proposed method are verified by conducting a series of actual milling tests with and without the support of the designed moving fixture.

Nonlinear behavior of disk spring with complex contact state.

Disk springs are widely used as preload and isolation due to their unique mechanical properties. In the prior research, the effect of linear friction on the disk spring was considered, but contact stiffness, another nonlinear contact factor, is ignored. Accordingly, in this paper, the asymmetric displacements of the contact edges are first derived, and the accurate friction dissipations are obtained, as a way to evaluate the effect of friction on the system. Then the velocity of the edges was obtained to establish a dynamic friction model. Meanwhile, the contact displacement and contact stiffness of the edge are obtained by fractal contact theory. Then the nonlinear static and dynamic models of disk spring with friction and contact stiffness are established by the energy method. The load-deflection relationship, stiffness, and hysteresis of disk spring are studied with different contact states. The results show that the model considering contact stiffness and asymmetric friction dissipation can effectively evaluate the static properties of the disk spring. Friction reinforces the nonlinear behavior of the system, while contact stiffness weakens the nonlinearity of the system. And due to the influence of nonlinear contact factors, the transmissibility curves produce multiple resonance peaks.

Dynamical behavior of coal shearer under the influence of multiple factors in slant-cutting conditions

Aiming at the problem of severe vibration and abnormal wear and tear of various components in coal shearer under slant-cutting conditions, a non-linear dynamics model with 13 degrees of freedom for a coal shearer under slant-cutting conditions is developed using vibration mechanics and multi-body dynamics theory, and the characteristics of the slide shoes-middle groove contact, the ranging arm-haulage unit connection with gaps and the guidance sliding boots-pin rail multi-surface contact with gaps are described based on three-dimensional fractal theory and Hertz contact theory. Based on Huco's law, the ranging arm and the hydraulic rod are assumed to be flexible beams, the rigidity characteristics of the ranging arm itself, the connection characteristics of the haulage unit and the fuselage are described, a drum correction load with a traction speed correction factor is proposed as the external excitation of the system, and the model is solved and analyzed. The research results show that the change of traction speed has a greater influence on the vibration swing angle and displacement of the front drum, front ranging arm and front walking unit, and the vibration swing angle and displacement of the three increase with the increase of traction speed, while the change of coalface hardness coefficient has less influence on the vibration displacement of the key components of the coal shearer. Under the working parameters of v = 3 m/min and f = 3, the swing angle and displacement of the front ranging arm and front drum fluctuate in the ranges of − 0.4–0.1 rad and – 15–15 mm respectively; the vibration acceleration is – 300–300 rad/s2 and – 200–200 mm/s2 respectively, the main vibration frequencies are 16.63 Hz and 12.14 Hz respectively, and finally the results are verified by experimental methods.

Elastic Rough Surface Contact and the Root Mean Square Slope of Measured Surfaces over Multiple Scales

This study investigates the predictions of the real contact area for perfectly elastic rough surfaces using a boundary element method (BEM). Sample surface measurements were used in the BEM to predict the real contact area as a function of load. The surfaces were normalized by the root-mean-square (RMS) slope to evaluate if contact area measurements would collapse onto one master curve. If so, this would confirm that the contact areas of manufactured, real measured surfaces are directly proportional to the root mean square slope and the applied load, which is predicted by fractal diffusion-based rough surface contact theory. The data predicts a complex response that deviates from this behavior. The variation in the RMS slope and the spectrum of the system related to the features in contact are further evaluated to illuminate why this property is seen in some types of surfaces and not others.

Load-Bearing Capacity Investigation of Friction Pairs in Hydro-Viscous Drive Based on Fractal Contact Theory

Abstract Hydro-viscous drive has gained extensive application in fans, pumps, belt conveyors, and scraper conveyors for step-less speed regulating, soft starting, and overload protection. In the mixed friction stage, the state of contact and load-bearing capacity are always changing with the pressure of control oil. In order to predict the load-bearing capacity of friction pairs, a description of the frictional and hydrodynamic lubrication states was first made. Oil film’s pressure distribution and load-bearing capacity were analyzed with the average flow model. The fractal contact model of the asperity surface was proposed for simulating mutual contact state and contact load between the asperities. The influences of fractal parameters on the contact performance of friction pairs were conducted. Eventually, test rig of load-bearing capacity was established. As indicated by the experiment, asperity contact area and load-bearing capacity of the asperity contact increase slowly at first and then rapidly with reduced film thickness ratio. Correspondingly, oil film load-bearing capacity progressively reduces. Due to the reduction of film thickness ratio, friction pairs’ load-bearing capacity increases gradually. The load-bearing capacity of friction pairs is affected by the fractal dimension and scale factor. Results of the experiment well conform to the theoretical values, which illustrates that friction pairs’ load-bearing capacity can be accurately predicted and computed using the average flow model and M-B fractal contact model. The calculation offers a theoretical reference to the survey about friction and torque properties of the hydro-viscous drive.

Influence of creep on preload relaxation of bolted composite joints: Modeling and numerical simulation

Preload relaxation can greatly influence the durability of bolted composite joints. This paper focuses on the effects of material creep and contact creep between rough surfaces on preload relaxation. An efficient finite element model is proposed to predict bolt preload relaxation in ABAQUS. First, a creep constitutive model of the composite based on elastic-viscoplastic theory was established and the contact stiffness of uneven interfaces of the joints, featuring multi-asperity contact at the micro perspective, were derived using fractal contact theory. Then, fractal contact theory was extended to address the time-dependent elastic-viscoplastic contact problem. The models were implemented within ABAQUS software using the UMAT and UINTER subroutines. Finally, a comparison of the numerical and experimental results showed that calculation error decreased from 1.4–2.0% to 0.1–0.5% after 35 h when effects of rough surface contact were considered, thereby significantly improving model accuracy. Furthermore, the results suggest that thicker jointing materials could effectively reduce preload relaxation. In addition, according to surface fractal parameters D and G, rougher surfaces facilitate contact creep and preload relaxation. Thus, controlling surface topography could be an effective way to improve surface contact properties in practical applications.

Monitoring of bolt looseness using piezoelectric transducers: Three-dimensional numerical modeling with experimental verification

Across multiple industries, bolted connections have been widely used to hold different parts and components together, due to their advantages such as convenient use and low cost. However, bolt looseness may lead to disastrous consequences if not promptly detected. In this article, a novel numerical modeling is proposed to simulate the piezoelectrically enabled active sensing method that can monitor the looseness of bolted connection, and this investigation can help us better understand the mechanism of the active sensing of a bolted connection. Compared to prior investigations, the main contribution of this article is that we develop a new three-dimensional modeling method on microscopic roughness of bolted interface by using the fractal contact theory. The numerical results, backed by experimental verification from the testing on a bolted joint, reveal that the received signal peak amplitude is proportional to the bolt preload. This phenomenon can be attributed to the enlarging actual contact area with the increasing preload, since a larger contact surface can enhance stress wave propagation. Moreover, the proposed numerical modeling has better performance than similar research in the past, which can provide guidance for future investigations on bolt looseness monitoring.

Research on the micro and dynamic characteristics of combination surface based on fractal theory

Abstract. The dynamic characteristics of the mechanical joint surface are important aspects of the dynamic theoretical analysis and optimization design of the machine tool. In this paper, the typical mechanical joint surface is taken as the research object. Through the combination of theoretical analysis and experimental analysis, the dynamic characteristics of typical joint surface parameters with different surface textures and the influence of texture parameters on the dynamic characteristics of the joint surface are studied. Based on the Hertz elastic contact theory and the contact fractal theory, the normal and tangential contact fractal models of the joint surface are derived, and then a mathematical model of the joint surface normal and tangential contact stiffness considering the domain expansion factor is established. The influence of surface texture parameters on the dynamic characteristics of the surface is further studied according to the model. In addition, the design of the experimental device and experimental scheme design are completed by the contact resonance method and the ERA algorithm, and the joint surface parameter identification experiment with texture is conducted. The normal and tangential frequency response functions of the joint surface, the dynamic characteristic parameters of the joint surface and the influence law of the joint surface parameters on the contact characteristics are obtained through the dynamic test analysis technology.