Timoshenko Beam Element Research Articles

Effect of rotating unbalance and engine excitations on the nonlinear dynamic response of turbocharger flexible rotor system supported on floating ring bearings

In practical conditions, turbochargers are supported by floating ring bearings and mounted on engines. In this paper, the effect of rotating unbalance and engine excitations on turbocharger is studied. The finite element model of turbocharger system is developed considering flexible rotor, using Timoshenko beam elements. The nonlinear fluid film forces generated in floating ring bearings are derived analytically in dimensional form using short bearing approximation. A new $$\hbox {MATLAB}^{{\circledR }}$$ code has been constructed to solve the governing differential equations of motion of system using implicit Newmark-$$\upbeta $$ numerical integration scheme along with Newton–Raphson convergence method, and dynamic response of the system is computed. The orbital plots, Poincare maps, and frequency spectrum are developed to show the nonlinear behaviour of the turbocharger system. At low rotor speeds, the system exhibits chaotic behaviour and has a wide range of sub-synchronous vibrations. As the speed of turbocharger increases, the forces due to unbalance dominate over engine excitations and nonlinear bearing forces and frequency spectrum become narrow. The behaviour of compressor and turbine disc centre is governed by their respective bearing nodes.

Improved Displacement-Based Timoshenko Beam Element with Enhanced Strains

AbstractThis paper presents an enhanced strain formulation for conventional displacement-based (DB) Timoshenko beam elements accounting for shear deformations. The proposed beam element is an exten...

Modeling of rotation accuracy of multi-support rotating machinery considering geometric errors and part deformation

PurposeThis paper aims to develop a theoretical method to analyze the rotation accuracy of rotating machinery with multi-support structures. The method effectively considers the geometric errors and assembly deformation of parts.Design/methodology/approachA method composed of matrix and FEA methods is proposed to do the analysis. The deviation propagation analysis results and external loads are set as boundary conditions of the model which is built with Timoshenko beam elements to calculate the spatial pose of the rotor. The calculation is performed repeatedly as the rotation angle increased to get the rotation trajectories of concerned nodes, and further evaluation is done to get the rotation accuracy. Additionally, to get more reliable results, the bearing motion errors and stiffness are analyzed by a static model considering manufacturing errors of parts.FindingsThe feasibility of the proposed method is verified through a case study of a high-precision spindle. The method reasonably predicts the rotation accuracy of the spindle.Originality/valueFor rotating machinery with multi-support structures, the paper proposes a modeling method to predict the rotation accuracy, simultaneously considering geometric errors and assembly deformation of parts. This would improve the accuracy of tolerance analysis.

Effects of Rub-Impact on Vibration Response of a Dual-Rotor System-Theoretical and Experimental Investigation

In order to understand the mechanism of fixed-point rub-impact in aero-engine, a dual-rotor system with inter-shaft bearing able to describe the mechanical vibration caused by unbalance and fixed-point rub-impact is developed in this paper. Firstly, a finite element (FE) model of the dual-rotor based on Timoshenko beam element and disk element is proposed, in which the effects of gyroscopic moments, rotary inertias, bending and shear deformations are taken into account. A rub-impact model is developed to derive the rubbing force between the rotor and fixed limiter, in which the softening characteristics of coating painted in casing is fully taken into account. Moreover, the Coulomb model is applied to describe the frictional behaviors. Secondly, the governing equations to describe the motion of the dual-rotor system with rub-impact are solved numerically by the Runge-Kutta method, and the dynamic characteristics are investigated by 3D water-fall plots and frequency spectrum. Combined harmonic frequency, whirling frequency and rotational frequency components of dual rotors are observed. Finally, the experiments of fixed-point rub-impact are performed on a dual-rotor test bench. Good agreement between the theoretical and experimental results shows the accuracy of the dual-rotor FE model and rub-impact model. The results indicate that (1) first backward whirling and forward whirling are excited under the influence of unbalance force; (2) several combination frequency components and second backward whirling and forward whirling frequency components are presented under fixed-point rubbing condition.

FE approach for dynamic response of a functionally graded spinning shaft system containing a transverse fully open crack

Free vibration and stability analysis are studied for a rotor-disc-bearing system having a radially functionally graded (FG) shaft with a transversely fully open crack, based on finite element (FE) approach. Both internal viscous and hysteretic damping are incorporated in the FE model of FG cracked shaft using two nodded Timoshenko beam element having four degrees of freedom at each node. Material properties of the FG shaft are assumed temperature dependent and graded with different material law of gradation. Aluminium Oxide (Al2O3) and stainless steel (SS) are composed as FG material. Local flexibility coefficients (LFCs) are derived analytically as a function of crack size, power-law gradient index and temperature using Paris’s equation and Castigliano’s theorem to compute the stiffness matrix at each instant in the FE analysis. Using the developed MATLAB code, the FE formulation and the cracked model are verified with the published results. Parametric studies are conducted to study the influences of different material gradient index, temperature gradient, crack size, internal damping, slenderness ratio and boundary conditions on the vibration responses of the FG cracked shaft system.

Chaotic vibration reduction of vertically suspended centrifugal pumps by the effect of the mechanical design parameter on hydraulic forces

The chaotic vibration generated by slender rotor–stator contact in vertically suspended centrifugal pumps shows itself in wear and fatigue failure of shafts, bearings, wear rings, seals and impellers. In common design methods of rotary equipment, for validating designed products, virtual representations in Computer Aided Design tools are imported to simulation tools (e.g. ANSYS software) on the last step of the design process. In this research, the ability of a non-linear finite element model is examined to support decision-making, synthesis and interaction of design choices during earlier stages of design process. So, the back wear ring diameter as the adjustable mechanical design parameter is used to reduce chaotic behavior of the pump rotor without any difficulty in manufacturing or problematic effect on the major hydraulic properties. The non-linear finite element model of the rotor is discretized into an appropriate number of Timoshenko beam elements. The major non-linear terms, the geometric stiffening effect, and the gyroscopic effect are considered. Rotor–stator contact, considering the rotor–stator clearance and the rotor–stator friction coefficient are taken into account. The full-order equations of motion are obtained using Lagrangian approach and the finite element method, and then are integrated into a computational plan. This study presents an allowable domain of the mechanical design parameter in which chaotic vibrations of the slender rotor can be reduced effectively. Furthermore, this research shows a basic method in which some design stages can be run in parallel rather than in series can be used to reduce redesigning and optimization times.

Dynamic characteristics of aero-engine’s rotor under large maneuvering flight

In view of the large maneuvering overload of the rotor system in Unmanned Aerial Vehicle (UAV), the dynamic characteristics of rotor system under large maneuvering overload conditions were analyzed in this paper. The finite element model of rotor system under large maneuvering overload was first established by Timoshenko beam element theory and finite element method. The motion differential equations of the rotor system were derived by Lagrange equation, and the additional damping matrix, stiffness matrix and excitation force were obtained. Critical speeds and unbalance response were calculated by characteristic equation method and Newmark-β numerical integration method respectively. The calculation results showed that the additional damping and stiffness leaded the critical speeds of the rotor system to change under large maneuvering flight, and the additional excitation force resulted in a certain static displacement of the whirling orbits.

Analytical elastostatic stiffness modeling of parallel manipulators considering the compliance of the link and joint

This paper discusses the analytical elastostatic stiffness modeling of parallel manipulators (PMs) considering the compliance of the link and joint. The proposed modeling is implemented in three steps: (1) the limb constraint wrenches are formulated based on screw theory; (2) the strain energy of the link and the joint is formulated using material mechanics and a mapping matrix, respectively, and the concentrated limb stiffness matrix corresponding to the constraint wrenches is obtained by summing the strain energy of the links and joints in the limb; and (3) the overall stiffness matrix is assembled based on the deformation compatibility equations. The strain energy factor index (SEFI) is adopted to describe the influence of the elastic components on the stiffness performance of the mechanism. Matrix structural analysis (MSA) using Timoshenko beam elements is applied to obtain analytical expressions for the compliance matrices of different joints through a three-step process: (1) formulate the element stiffness equation for each element; (2) extend the element stiffness equation to obtain the element contribution matrix, allowing the extended overall stiffness matrix to be obtained by summing the element contribution matrices; and (3) determine the stiffness matrices of joints by extracting the node stiffness matrix from the extended overall stiffness matrix and then releasing the degrees of freedom of twist. A comparison with MSA using Euler–Bernoulli beam elements demonstrates the superiority of using Timoshenko beam elements. The 2PRU-UPR PM is presented to illustrate the effectiveness of the proposed approach. Finally, the global SEFI and scatter matrix are used to identify the elastic component with the weakest stiffness performance, providing a new approach for effectively improving the stiffness performance of the mechanism.

An efficient multibody dynamic model of three-dimensional meshing contacts in helical gear-shaft system and its solution

The dynamics of helical gear-shaft systems are characterized by three-dimensional (3D) meshing contacts that have significant variations in the location and size of the contact area, resulting in noise that is unavoidably transmitted to the gearbox through the shaft. Accurate and efficient predictions of the dynamic behaviors of helical gear and shaft are indispensable in reliable and cost-effective gearbox design. Available analytical methods, though computationally feasible, cannot consider multi-point contacts and uneven tooth-load distribution. In contrast, the finite element (FE) method provides a high-fidelity approach to compute the dynamic behaviors of a general gear-shaft system at high expenses of computation. This paper aims to establish a high-efficiency multibody dynamic model for 3D contacts in helical gear-shaft systems, in which the helical gear is pertinently represented under the framework of Arbitrary Lagrangian Eulerian formulation and the shaft is discretized by 3D Timoshenko beam elements. The computational efficiency is greatly improved through the following four steps. First, the low-frequency approximation technique is adopted to reduce the degrees of freedom (DOFs) resulting from the fixed boundary normal modes. Second, under the framework of ALE formulation, only the FE nodes of three meshing tooth-faces are defined as boundary nodes. Then, the dynamic equations and Jacobian matrix are simplified by ignoring the inertial forces associated with deformation. Finally, a two-step algorithm is adopted to accelerate the contact detection process. The accuracy and efficiency of the proposed method are demonstrated through five numerical tests with correlation to commercial nonlinear finite element software.

Rubbing response comparisons between single blade and flexible ring using different rubbing force models

In order to simulate the rubbing characteristics of the single blade-casing system with flexible supports, two types of the self-programmed elements (i.e., two-node Timoshenko beam and spring elements) are adopted to establish corresponding finite element models (FEMs) in sequence. Then a node to surface contact algorithm is programmed to build the interface contact between the blade tip and the casing. Next, the forward increment Lagrange multiplier and penalty methods in combination with the model order-reduction technique are separately utilized to solve the rubbing dynamic responses of the studied system under the effects of eccentricity and friction coefficient, and corresponding result comparisons are also made with each other. Finally, some main conclusions are summarized as follows: (1) Relative to the penalty method (PM), the Lagrange multiplier method (LMM) can eliminate the penetration between the blade tip and the casing, but results in the larger vibration amplitudes of rubbing force and displacement especially for the large penetration induced by the large eccentricity and the high rotating speed, generally speaking, those vibration amplitudes obtained from both methods are relatively consistent with each other under the small penetration; (2) Serious rubbing cause the appearance of resonance bands in the amplitude-frequency responses and sidebands in the spectrum; (3) Increasing friction coefficient makes the soft nonlinearity of the system more distinct, meanwhile, strain energy combining with the spectrum is very suitable for identifying the dominant mode of the system.

Modelling and Dynamic Analysis of the Spiral Bevel Gear-Shaft-Bearing-Gearbox Coupling System

To accurately study the dynamic characteristics of the spiral bevel gear transmission system in a helicopter tail transmission system, the finite element model of the gear shaft was established by a Timoshenko beam element, and the mechanical model of the spiral bevel gear was created by the lumped mass method. The substructure method is employed to extract the dynamic parameters from the gearbox’s finite element model, and the dynamic model of the spiral bevel gear-shaft-bearing-gearbox coupling system was built according to the interface coordination conditions. In the model, the influences of time-varying stiffness, a time-varying transmission error, gearbox flexibility, unbalance excitation, and a flexible shaft and bearing support on the system vibration were taken into account simultaneously. On this basis, the dynamic differential equations of the full coupling system of the spiral bevel gear were derived, and the effects of the gearbox flexibility, the shaft angle, and the unbalance on the dynamic properties of the system were analysed. The results show that the gearbox flexibility can reduce the gear meshing force and bearing force, in which there is a more significant impact on the bearing force. The shaft angle affects the position, size, and direction of the system’s axis trajectory. Meanwhile, the meshing force and the bearing force of the system are also varied because of the various pitch angles of the driving and driven gears under different shaft angles. The unbalance of the gear shaft has an effect on the vibration of the spiral bevel gear transmission system in all directions, wherein the influence on the torsional vibration is the most significant, and the influence increases as the unbalance rises. The unbalance of the gear shaft also affects the meshing force and bearing force, which increases as the rotational speed rises. This research provides a theoretical basis to optimize dynamic performance and reduce the vibration and noise of a spiral bevel gear full coupling system.

Nonlinear dynamics of directional drilling with fluid and borehole interactions

In rotary drilling, a drillstring is an assembly of slender pipes. It is used to transmit the driving torque of a motor at the drilling surface to the drill bit at the bottom hole of a 3D well. Numerous vibratory phenomena are induced during the drilling: whirling, stick-slip, bit-bouncing, lateral instability, inducing in particular reduction of the rate of penetration and mean time between failures. For the rotordynamics prediction of such a structure, the drillpipes are modelled with Timoshenko beam elements, containing 12° of freedom, equipped with distributed radial stop-ends. The rotary motion is assumed to have a constant speed of rotation imposed at the top of the drillstring. The drilling mud is taken into account by using a fluid-structure interaction model. The numerical simulations concern a real 3D-borehole and a parametric analysis is carried out for determining the role of the mud density and of the flows rate on the drillstring dynamics. It is shown that increasing the flow rate and densifying the drilling fluid reduce the fluid damping effect that increases drillstring lateral vibrations.

Effect of Joint Stiffness on Deformation of a Novel Hybrid FRP–Aluminum Space Truss System

To achieve light weight, high load-carrying capacity, and corrosion resistance, a novel hybrid FRP-aluminum space truss system is proposed in detail. An advanced pretightened teeth connection (PTTC) and its combined joint–aluminum bolt–ball connecting system (ABCS) were designed to interconnect pultruded FRP elements. In view of the significant effect of the ABCS on structural deformation, the axial stiffness properties of the ABCS were first investigated through experimental tests and numerical simulations. Then, in establishing a structural calculation model for the hybrid space truss, a new combined-line-elements solution was proposed to simulate the complex axial stiffness variation in the ABCS, to which the traditional Timoshenko beam element cannot be applied. Beam elements with different cross sections were adopted to simulate the FRP truss member and the PTTC. The structural calculation model was verified with a previously published case. The design of a hangar structure using this hybrid space truss system is discussed here. The preliminary design was made using the traditional simple-link-system model, and a comparative calculation was made with the new combined-line-elements model. The results indicated that the structural design of this unique hybrid system under a flexural moment is stiffness-driven rather than strength-driven owing to the low elastic modulus of the materials. The conventional simple-link-system model ignores actual differences in component stiffness and thus is dangerous for structural design. Based on the discussion, a design strategy for the hybrid space truss system is proposed.

Tribo-dynamics model of a spur gear pair with gyroscopic effect and flexible shaft

This study proposes a tribo-dynamic model of a spur gear by coupling an elastic dynamic model and an isothermal elastohydrodynamic lubrication (EHL) model. In this model, the gear shafts are modelled as flexible bodies by the resultant elements of Timoshenko beam and bar elements, and the gyroscopic effect of gear rotors and shafts is also included. The transient and non-Newtonian effects are considered in the EHL model to predict tooth friction forces and moments. Then, an example is provided to analyze the interaction between gear tribology and dynamic behavior. Results indicate that the fluctuations of mesh force and surface tangential velocity caused by gear vibration considerably affect the pressure and thickness of the lubricant film. The gyroscopic effect exerts a significant influence on the dynamic and tribology properties of the gear drive at high speed. The proposed model has important guidance for the design of lubrication systems and dynamic properties in gear drives.

The modal analysis of a double disc flexible rotor-bearing system with isotropic stiffness and damping properties

This paper shows the analysis of the modal characteristics of a flexible rotor system having two non-identical discs mounted on it using the finite element method. It uses the Timoshenko beam elements theory including shear deformation, rotary bending effects and gyroscopic effects. The bearing is considered to be isotropic having translational stiffness and damping values constant in x and y directions. There are no cross-coupling terms. The conduct of the framework is investigated by reference to natural frequency maps and mode shape diagrams. A comparative study has been conducted between this system having isotropic bearings of stiffness and damping properties and the one in which the bearings have only stiffness keeping other parameters same. The gyroscopic effect is seen to be prominent at higher rotating speeds for both the cases which cause the splitting of frequency pairs into forward and backward whirls. The Campbell diagram is computed which helps in finding the critical speeds associated with each pair of diverging natural frequencies. The bearing system represented by stiffness and damping properties shows a comparatively better control over the critical speed range.

Efficient analysis of shear wall-frame structural systems

PurposeThis paper aims to propose an efficient method to conduct the preliminary analyses of medium or high-rise wall-frame structural systems with vertically varying properties. To this end, a finite element is formulated to take the shear deformation of the shear wall and the constrained moment of the link beam.Design/methodology/approachThe differential equation of the structure is derived from the total potential energy. Its homogenous solutions are functions of initial parameters (deflections and inner forces). To solve the structure with vertically non-uniform properties, the authors first use the classical Timoshenko beam element and then heuristically propose a finite element that uses the initial parameter solutions as shape functions and is easier to implement. A post-processing method to compute the shear force in the frame and shear wall is developed. Modal analysis using the consistent mass matrix is also incorporated. Numerical examples demonstrate the accuracy and mesh independency of the proposed element.FindingsThe shear deformation of the shear wall and the constrained moment of the link beam significantly influence the static response of the structure. Taking into account the shear deformation can eliminate the misleading result of zero-base shear force of the frame and give much better predictions of the system natural frequencies.Originality/valueThe proposed method achieves higher accuracy than the classical approach most often used. The finite element formulation derived from transformations of the initial parameter solutions is simple and has superior numerical performance. The post-processing method allows for a fast determination of the shear force distributions in the shear wall and frame.

Simulations of the dynamic response of planetary gears in the presence of localised tooth faults

This paper is aimed at analysing the influence of local tooth faults such as pitting on the dynamic behaviour of planetary gears. A model of one-stage planetary gear combining lumped parameters and Timoshenko beam elements is presented, which accounts for deformable shafts and ring gears. Local tooth fault are simulated by material removal from tooth flanks, which can be positioned on the sun-gear, the planets and the ring-gear. The corresponding state equations are solved by combining a Newmark time-step integration scheme combined with a unilateral normal contact algorithm, which verifies that all contact forces on gear teeth are compressive and that no contact can occur outside the contact areas. A number of results are presented, which show the influence of tooth fault positions, depths and extents on displacement and acceleration signals. The contribution of a deformable ring-gear is analysed and the possibility to detect such localised tooth faults from vibration monitoring is discussed.

Finite element based stability analysis of a rotor-bearing system having a functionally graded shaft with transverse breathing cracks

In spite of gaining practical importance of functionally graded (FG) shafts, stability analysis of such FG rotor-bearing system has not been reported yet. This paper presents the development of a finite element (FE) procedure and code for stability analysis of cracked FG rotor-bearing system under thermal environment. Two-noded Timoshenko beam elements are used to model the FG shaft considering the effects of gyroscopic moments, translational and rotary inertia, bending and shear deformation and material damping. Zirconia (ZrO2) and stainless steel (SS) are considered as constituents of the radially graded FG shaft with temperature dependent material properties. Considering breathing crack behaviour, local flexibility coefficients (LFCs) are derived using the Paris's equation and the Castigliano's theorem. Results show that while the depth, orientation and locations of cracks, thermal gradient and material damping affect the stability threshold speed, it is important to choose the material gradient index judiciously. Thus, even when surface cracks appear on the FG shaft, threshold speed could still be in the desired range, which is important for damage tolerant design in high temperature applications.

Ballistic response of hexagonal boron nitride monolayer under impact of a nano-projectile

The response of a hexagonal boron nitride (h-BN) monolayer subjected to ballistic impact is studied using an explicit finite element (FE) model based on molecular structural mechanics where BN bonds are represented by the nonlinear Timoshenko beam elements. The elastic properties and bond strength of h-BN monolayer and graphene are verified by nano-indentation using FE and molecular dynamics simulations. It shows that the ballistic response of h-BN monolayer is dependent of the impact speed of the projectile. The stress and deformation wave propagations in the h-BN monolayer interact with the movement of projectile and the material bond failure, which, in turn, influence the ballistic performance of the h-BN monolayer target. It shows that h-BN material, in comparison with conventional armor materials, has potential to increase the ballistic performance significantly if it could be practically made for armor application.

Elastic Wave Propagation in Open-Cell Foams

This work examines elastic wave propagation phenomena in open-cell foams with the use of the Bloch wave method and finite element analysis. Random foam topologies are generated with the Surface Evolver and subsequently meshed with Timoshenko beam elements, creating open-cell foam models. Convergence studies on band diagrams of different domain sizes indicate that a representative volume element (RVE) consists of at least 83 cells. Wave directionality and energy flow features are investigated by extracting phase and group velocity plots. Explicit dynamic simulations are performed on finite size domains of the considered foam structure to validate the RVE results. The effect of topological disorder is studied in detail, and excellent agreement is found between the wave behavior of the random foam and that of both the regular and perturbed Kelvin foams in the low-frequency regime. In higher modes and frequencies, however, as the wavelengths become smaller, disorder has a significant effect and the deviation between regular and random foam increases significantly.