Timoshenko Beam Element Research Articles

Study on the Effect of Cracks in Diaphragm Couplings on the Dynamic Characteristics of Shaft System

Diaphragm couplings are prone to developing diaphragm cracks under prolonged high-speed operating conditions, which can lead to degradation in the performance of the transmission system and affect the dynamics of the shafting system. To investigate the effects of diaphragm cracks on the dynamics of couplings and the shafting system, a finite element model of a diaphragm coupling with a crack failure is established using ANSYS finite element software to analyze the time-varying characteristics of the diaphragm coupling’s angular and radial stiffness. A shaft dynamics model of the diaphragm coupling with a crack is developed using Timoshenko beam elements to analyze the impact of different crack lengths and locations on the dynamics of the shafting system. The validity of the dynamic model for a diaphragm coupling with a crack is verified through a constant speed experiment conducted on a rotor test bench. The results indicate that diaphragm crack failure causes a change in the periodicity of the time-varying stiffness of the diaphragm coupling, leading to a distinct 2× component appearing in the frequency domain of the transmission shaft system.

A tribo-dynamics model of ball screws based on flexible body element

Dynamic actuation accuracy of ball screws is dramatically affected by screw flexible deformation and the behaviors of ball-groove interface such as friction, stiffness and damping. However, few dynamics studies have considered the screw flexible deformation and rubbing interface behaviors in ball screws. In this study, a dynamics model is developed by including both the screw flexible deformation and lubrication interface behavior. Discretization of the screw/nut is completed using the 6 degree-of-freedom Timoshenko beam element. The tribological properties of ball-groove interface are solved by the mixed lubrication model. Comparison with experimentally tested nut acceleration frequencies verifies the correctness of theoretical model. On this basis, the effects of nut motion, nut position and rough surface on the dynamic response are analyzed. With the increase of time, the screw axial displacement decreases with a certain slope and is accompanied by periodic vibrations. The centers of axis trajectories at different nut positions are far apart. From Hertz, smooth surface to rough surface, the interface stiffness coefficient decreases and the nut vibration amplitude grows. The novel tribo-dynamics model can provide a theoretical basis for fault diagnosis and performance optimization of ball screws.

Non-Linear Elastic Beam Deformations with Four-Parameter Timoshenko Beam Element Considering Through-the-Thickness Stretch Parameter and Reduced Integration

Non-Linear Elastic Beam Deformations with Four-Parameter Timoshenko Beam Element Considering Through-the-Thickness Stretch Parameter and Reduced Integration

A vibration model of a flexible multiple shaft gear system with the tip relief modification

The fixed-shaft gear system (FSGS) is one critical part in gear transmission systems (GTSs), whose vibrations can significantly influence performances of GTSs. To reduce the vibrations of rigid FSGS caused by installation and manufacture errors, the tooth profile modifications without the shaft deformation were widely introduced. This paper establishes a flexible multiple shaft gear system (FMSGS) model of an unmanned underwater vehicle, which can obtain vibrations of rotor at different axial positions. However, the previous rigid FSGS model cannot obtain those vibrations. In the FMSGS model, the Timoshenko beam elements are applied to the establishment of shaft segments considering flexibility. The meshing stiffness for helical gears with different ranges of tooth profile modification are quantitatively expressed by a gear slicing method. Based on the kinetic relationship between the gear pairs, shaft segment, and bearings, the dynamic equations of FMSGS are derived. By comparing the time-domain waveforms and spectra of FMSGS, the effects of the tooth profile modification amounts and lengths on the vibrations of FMSGS are revealed. The proposed FMSGS model is validated by an experiment. This study extends the dynamic methods of FSGSs, and some suggestions on the vibration control of FSGSs are given.

Dynamical Modeling and Dynamic Characteristics Analysis of a Coaxial Dual-Rotor System

The dual-rotor structure is the primary excitation source of aero-engine vibration. It is essential to study the dynamical model and analyse the dynamic characteristics such as critical speed and the unbalanced response for rotor system dynamics. The coaxial dual-rotor support scheme of an aero-engine was introduced in the previous work, and the physical model with a high-speed flexible inner rotor of a sizeable length-diameter ratio has been designed. Then a finite element dynamic model based on the Timoshenko beam elements and rigid body kinematics of the dual-rotor system is modelled, with the Newmark-β method and Newton-Raphson method used for the numerical calculation to study the dynamic characteristics of the system. Three different simulation models, including beam-based FE (1D finite element) model, solid-based FE (3D finite element) model, and TM (transfer matrix) model, were designed to study the characteristics of mode and the critical speed characteristic of the dual-rotor system. The unbalanced response of the dual-rotor system was analysed to study the influence of mass unbalance on the rotor system. The effect of different disc unbalance phases and different speed ratios on the dynamic characteristics of the dual-rotor system were investigated in detail. The experimental result shows that the beam-based FE model is effective and suitable for studying the dual-rotor system. Conflict of Interest Statement The authors declare no conflicts of interest.

Analysis of nonlinear wave propagation within architected materials consisting of nonlinear Timoshenko beam structural elements

In the present work, a full nonlinear Timoshenko beam employing nonlinear shape functions is developed. The extended Hamilton principle is employed for deriving the differential equations of motion and the associated boundary conditions. The general form of the boundary conditions is then utilized to determine the static solution of the beam motion. Using this solution for the deformation and rotation of the beam, the nonlinear shape functions of the beam are identified, which leads to the linear and nonlinear mass and stiffness matrices of the Timoshenko beam element. The nonlinear dispersion diagram incorporating the non-linear corrections is obtained using the Linstedt–Poincaré perturbation method. An analysis of the effect of internal transverse shear and bending on the nonlinear dispersion characteristics of wave propagation in two-dimensional periodic network materials made of nonlinear Timoshenko beams is done. The formulated theory shows that the percentage of correction factor of the nonlinear kinematics versus the linear dynamical behavior is inversely proportional to the frequency amplitude. The shear and extension modes are shown to have the higher effect in the non-linear correction term in comparison to the flexural mode.

Vibration characteristics of a bolted joint rotor-bearing system with rubbing and pedestal looseness coupled fault: Simulation and experiment

Rotor-stator rub and pedestal looseness are common faults in rotating machinery, such as large gas turbines that typically use bolted joint structures to fasten adjacent disks together. It is of great significance to investigate the vibration behaviors of the bolted joint rotor-bearing system with pedestal looseness and rubbing coupled fault, which can provide proofs for the fault diagnosis. In the present work, a dynamic model of the bolted joint rotor-bearing system with coupled fault is developed by using the finite element (FE) method, and then the vibration characteristic analysis is carried out. The rotor shafts that connected by the bolted joint element (BJE) are discretized as two sets of Timoshenko beam elements. The BJE is composed with four springs (two linear translational springs and two rotational springs with piecewise-linear stiffness). The interactions of rotor-stator contact are then modeled based on the Hertz contact theory and Coulomb's friction law, and the bearing forces are also calculated based on the Hertz contact theory. Moreover, the pedestal looseness fault is modeled through the piecewise-linear stiffness spring. The equations of motions of the bolted joint rotor-bearing system with coupled fault are then developed through FE theory and solved by the Newmark method. The dynamic responses are then calculated, and the frequency spectrums, bending stiffness of bolted joint, and vibration responses under different rotation speeds under the coupled fault are analyzed. The experimental study is finally conducted based on a bolted joint rotor test rig. Simulation results are in agreement with experiment results. The research results can provide a basis for understanding the rotor dynamic features of a bolted joint rotor system with coupled faults of rubbing and pedestal looseness.

A closed-form Timoshenko beam element with multiple localized singularities for nonlinear material behavior using the strong discontinuity approach

The development of a novel closed-form beam element for nonlinear material behavior in structures is described, which is formulated according to Timoshenko beam theory and the embedded strong discontinuity approach. Multiple strong transverse and rotation discontinuities are considered to model several dissipative mechanisms due to localized singularities in the shear strain and curvature fields. Issues of dependence on numerical techniques, as occurs in displacement- and forced-based beam elements with the finite element method, are overcome since the formulated element is derived from a pure mathematical treatment. Theoretical principles are stemming from a Lagrangian variational model, from which singularities are naturally derived. The proposed formulation is solved by the conventional methods of calculus of variations, resulting in the boundary value problem. Then, closed-form expressions for any boundary conditions are obtained. By particularizing closed-form expressions for a basic system, a symmetric closed-form flexibility matrix is obtained. This matrix naturally leads to a symmetric closed-form stiffness matrix for any type of loading and unloading process, in which the nodeless degrees of freedom of the multiple embedded strong discontinuities are naturally condensed. Since Gauss integration methods are not used, the multiple localized singularities are not influenced by predefined positions of the integration points. However, the accuracy depends on the chosen number of singularities and their positions. Then, the proposed element is mesh independent when sufficient discontinuities are considered, which does not imply an increase in the degrees of freedom. Softening discontinuities are modeled at arbitrary positions in a single element, which overcomes the drawbacks of other embedded beam theories where more than one element is necessary to identify failure since strong discontinuities are generally located at the midpoint of the beam element. The obtained closed-form stiffness matrix is not always positive definite for strain-softening problems. Nevertheless, there are no major convergence issues as it is a symmetric matrix that does not require a constraint equation, so even the complex snap-back behavior in structures is adequately modeled. The capability of the formulated beam element for nonlinear material behavior in structures is validated with representative examples.

Free-Vibration and Buckling Analyses of Beams using Kriging-Based Timoshenko Beam Elements with the Discrete Shear Gap Technique

A family of locking-free Kriging-based Timoshenko beam elements with a new implementation of the discrete shear gap technique was recently developed (referred to as K-beam-DSG1 elements). Performance of the K-beam-DSG1 elements has been shown to be very satisfactory in the linear static analysis of beams for a wide variety of thicknesses. This paper presents further development of the K-beam-DSG1 elements to free vibration and bifurcation buckling analyses of prismatic and non-prismatic beams. Consistent Kriging-based mass matrices are used for free vibration analysis and similarly, consistent Kriging-based geometric stiffness matrices are used for buckling analysis. The results show that for most of the cases, the K-beam-DGS1 elements yield remarkably accurate natural frequencies and critical compressive loads using a reasonable number of elements to discretize the beam. For an axially functionally graded fixed-fixed supported beam, however, the elements fail to predict the critical load accurately.

A convergence‐enhanced Timoshenko beam element for the stochastic nonlinear analysis of reinforced concrete components

AbstractAn effective and precise physical model is generally required for the performance assessment of reinforced concrete (RC) components. In this study, a convergence‐enhanced Timoshenko beam element considering finite deformation is proposed for the stochastic nonlinear analysis of RC components. The unified framework of the rank 2 correction matrix is obtained by approximating the iterative matrix through the linearly independent basis vectors, and two types of Broyden–Fletcher–Goldfarb–Shanno (BFGS) operators are constructed. Then, a consistent solution scheme with superior numerical stability and efficiency is proposed based on the second kind of BFGS operator and improved quasi‐Newton method. Accurate discretization of the displacement fields and description of nonlinear sectional behavior are achieved by the high‐order Lagrangian interpolation functions, concrete damage‐plasticity model and J2 plasticity model. In an updated Lagrangian formulation, these shape functions and constitutive relationships are used to develop a complete secant stiffness with higher‐order terms in the strain tensor and residual force vector. To validate the proposed Timoshenko beam element, numerical applications involving material nonlinearity and geometrical nonlinearity are conducted. The numerical stability and efficiency of the solution scheme, the shear‐locking free technique, and the oscillation of axial force are thoroughly examined and discussed by solution of RC columns, cantilever beam and progressive collapse of planar RC frame. Finally, the parametric study considering the geometric and material randomness indicates that the RC column designed for flexural failure mode suffered considerable degradation of loading capacity and a larger scatter of residual load.

Rub-impact dynamic analysis of a dual-rotor system with bolted joint structure: Theoretical and experimental investigations

Large gas turbines often utilize dual-rotor structures to improve efficiency. To facilitate manufacture and maintenance, the large rotor systems are typically constructed by connecting components made from different materials together through the bolted joint. In this work, a dynamic model of a dual-rotor system with bolted joint included in the high-pressure (HP) rotor is established based upon the finite element theory of Timoshenko beam element, as well as taking into account the bearing forces and fixed-point rubbing fault. To evaluate the influence of rubbing faults on the dynamic behavior of a dual-rotor system, this study analyzed the response characteristics of the rotor under the cases of the LP rotor and the HP rotor subjected to the rubbing fault, respectively. In order to further reveal the effect of bolted joint structure on the system response while rubbing fault occurs, piecewise linear bending stiffness is also considered in this study. The nonlinear vibration responses of the bolted joint dual rotor-bearing system are studied through numerical simulation. Effect of rotor–stator contact stiffness and the occurrence of rubbing faults at the low-pressure (LP) rotor and the HP rotor are investigated through frequency-amplitude curves, waterfall diagrams, time-domain responses, and bending stiffness of the bolted joint. The results indicate that the presence of rubbing faults, whether occurring at the LP rotor or the HP rotor, leads to a decrease in the critical speed. Furthermore, under a rubbing fault, the bending stiffness of the bolted joint enters the stiffness softening region earlier and exhibits a wider range, which is exacerbated with increased contact stiffness. This phenomenon is considered one of the main causes of system response instability. Finally, the experimental studies are carried out on a bolted joint dual-rotor test rig to validate the numerical simulation results.

Chatter Stability Prediction for Deep-Cavity Turning of a Bent-Blade Cutter.

The bent-blade cutter is widely used in machining typical deep-cavity parts such as turbine discs and disc shafts, but few scholars have studied the dynamics of the turning process. The existing mechanism of regenerative chatter in the metal-cutting process does not consider the influence of bending and torsional vibration, the change of tool profile and the complex machining geometry, so it cannot be directly used to reveal the underlying cause of the chatter phenomena in the deep inner cavity part turning process. This paper attempts to investigate the dynamic problem of the bent-blade cutter turning process. The dynamic model of a bent-blade cutter is proposed by considering the regenerative chatter effect. Based on the extended Timoshenko beam element (E-TBM) theory and finite element method (FEM), the coupling between the bending vibrations and the torsional vibrations, as well as the dynamic cutting forces, are modeled along the turning path. The vibration characteristics of the bending-torsion combination of cutter board and cutter bar, together with the dynamical governing equation, were analyzed theoretically. The chatter stability of a bent-blade cutter with a bending and torsion combination effect is predicted in the turning process. A series of turning experiments are carried out to verify the accuracy and efficiency of the presented model. Furthermore, the influence of cutting parameters on the cutting process is analyzed, and the results can be used to optimize the cutting parameters for suppressing machining vibration and improving machining process stability.

Stochastic stiffness identification and response estimation of Timoshenko beams via physics-informed Gaussian processes

Machine learning models trained with structural health monitoring data have become a powerful tool for system identification. This paper presents a physics-informed Gaussian process (GP) model for Timoshenko beam elements. The model is constructed as a multi-output GP with covariance and cross-covariance kernels analytically derived based on the differential equations for deflections, rotations, strains, bending moments, shear forces and applied loads. Stiffness identification is performed in a Bayesian format by maximising a posterior model through a Markov chain Monte Carlo method, yielding a stochastic model for the structural parameters. The optimised GP model is further employed for probabilistic predictions of unobserved responses. Additionally, an entropy-based method for physics-informed sensor placement optimisation is presented, exploiting heterogeneous sensor position information and structural boundary conditions built into the GP model. Results demonstrate that the proposed approach is effective at identifying structural parameters and is capable of fusing data from heterogeneous and multi-fidelity sensors. Probabilistic predictions of structural responses and internal forces are in closer agreement with measured data. We validate our model with an experimental setup and discuss the quality and uncertainty of the obtained results. The proposed approach has potential applications in the field of structural health monitoring (SHM) for both mechanical and structural systems.

Mechanical Behavior of 2D FGP Beam with Uneven Porosity Distribution

This study gives the mechanical behavior of 2D functionally graded porous (FGP) beams using the finite element method. The Matlab code with simple Timoshenko beam elements is written to solve 2D FGP beam problems under distributed load. The transverse deflections are plotted along the length to provide mechanical views about this structure in reality.

Influence of Bearing Wear on the Stability and Modal Characteristics of a Flexible Rotor Supported on Powder-Lubricated Journal Bearings

This study has investigated the influence of journal bearing wear on the dynamic behaviour of a flexible rotor with a central disc. Rotors supported on journal bearings are susceptible to self-excited whirling, leading to unstable conditions. Prior knowledge of the stability limit speed is important to avoid the excessive vibration of rotating machines. For the study in this paper, journal bearings were lubricated with powder owing to high-temperature applications where conventional oil lubricants would fail to perform. The governing equations for lubrication were derived using a simplified grain theory based on the theory of dense gases. The rotor shaft was discretized considering Timoshenko beam elements. Modal analysis was conducted to obtain the system’s natural frequencies, mode shapes, damping factors, stability limit speed, and unbalance response. This study has also evaluated the influence of wear depth on the dynamic behaviour of the rotor shaft system and found that bearing wear significantly affects the stiffness and damping characteristics of lubricating film. Consequently, free and forced vibration behaviour is also affected. It has been found that increased wear depth improves stability limit speed but has little influence on the unbalance response.

Piezoelectric Actuators Placement Using Genetic Algorithm and Active Vibration Control of Smart Plane Frame Structure

A Genetic Algorithm-based optimisation scheme is presented to find the optimum actuator elements infinite element analysis for active vibration control applications. The dynamic strain data is used to reduce the size of search domain for the actuators placement problem. A modal velocity, feedback control is employed to estimate the active damping of piezoelectric actuators. The structural damping index is taken as a fitness function; which directly reflects the status of active damping effect for a given piezoelectric actuator set. Modal preference is attempted by assigning a different weighting factor in the fitness function for desired modes to have better controllability. As an illustration, a 20-member smart plane frame is modeled with d3I based extension-bending piezoelectric actuators using 2-D piezoelectric Timoshenko beam element. It is shown that the vibration of first four elastic modes is actively controlled with optimally, selected four piezoelectric actuators.

On the structural behaviour of existing RC bridges subjected to corrosion effects: Numerical insight

The evaluation of the structural behaviour of existing reinforced concrete (RC) bridges represents one of the most current structural engineering research topics due to their strategic importance, especially if they are subjected to corrosion effects which can lead to a significant reduction of load-bearing capacity of the main structural elements (e.g., the piers). In the last decades, different types of numerical approaches have been proposed for the evaluation of the structural behaviour of these strategic infrastructures, especially after the recent collapses that have affected this type of structures during last years. In this paper, the structural behaviour of an existing RC bridge subjected to corrosion effects due to carbonation is analysed by means of an efficient procedure based on the implementation of a Finite Element Model (FEM) where the main structural elements are implemented using only Timoshenko beam elements. The safety level of the bridge has been evaluated considering different load conditions (e.g. traffic load, seismic action, etc.) calculated according to the Italian Design Code (NTC2018). Finally, a retrofitting intervention is proposed in order to guarantee and adequate safety level of the bridge under the considered different load combinations.

A locking free virtual element formulation for Timoshenko beams

The virtual element method (VEM) provides new ways of deriving discretization for problems in structural and solid mechanics, starting with the contribution by Beirão da Veiga et al. (2013) for elastic solids. Interestingly, the virtual element method allows also to revisit the construction of different elements which have the same shape as finite elements. This is even true for one-dimensional structures like trusses and beams. Here we study a virtual element development of the Timoshenko beam which surprisingly leads to a straight forward formulation of locking free and in the linear range even exact Timoshenko beam elements. These elements can be easily incorporated into classical finite element codes since they have the same number of unknowns as finite elements for beams. The formulation allows to compute nonlinear structural problems undergoing large deflections and rotations using the formulation provided in Reissner (1972).

Modeling strategy and dynamic analysis of a dual-rotor-bearing-casing system in aero-engine

In this study, a novel modeling method for a dual-rotor-bearing-casing system is proposed based on the finite element method (FEM) and the model reduction method. The dual-rotor system is modeled using Timoshenko beam and rigid disc elements, while the casing is modeled using shell elements. To reduce the number of degrees of freedom (DOFs), the modified Craig–Bampton method is used to downscale the finite element model of the casing. In addition, the Hertzian contact force and the DOFs of the bearing outer race and pedestal are considered in the bearing model. Through a specific assembly method, a dynamic model of the dual-rotor-bearing-casing is established at the expense of slight increase in the number of DOFs. Based on this dynamic model with fewer DOFs, the effects of two disc imbalances and the rub-impact fault between the low-pressure (LP) turbine and casing are then introduced as an application of the proposed model. The calculation results indicate that the critical speeds of each order of the whole aero-engine model are significantly lower than those of the casing-free model. In the amplitude-frequency diagram, the peak of the critical speed of the whole aero-engine model is found to appear earlier, and the extent of rightward bending is larger. The study also includes parametric analysis to examine the effects of parameters such as the coefficient of friction, the casing-disc contact stiffness, and the casing-foundation connection stiffness. After introducing the nonlinear bearing model, a large number of fractional frequency components are found in the frequency-domain diagram, the dynamic characteristics of the model become more complicated. The coupled bending-torsional effect has also been studied in detail, the calculation results show that it is reasonable to ignore the torsional deformation.

Vibration attenuation characteristics of squeeze film dampers in spiral bevel gear systems

A central drive bevel gear system of aero-engines extracts power from a high-pressure rotor. The high speed of the high-pressure rotor aggravates the gear system's vibration. This work adopts squeeze film dampers (SFD) as damping devices to suppress excessive vibration of the spiral bevel gear rotor system. This paper establishes the general Reynolds equation of SFD and calculates the nonlinear oil-film force of SFD in real-time based on the finite element method. Besides, the finite element method and Timoshenko beam element are employed to model the flexible gear shaft. Then, the dynamic model of the spiral bevel gear system supported by SFDs is established, including the nonlinear oil-film force of SFD and time-varying meshing stiffness of the gear system. The dynamic response of the spiral bevel gear system is analyzed theoretically and experimentally to study the vibration attenuation characteristics of SFD. The findings indicate that SFD can effectively suppress the vibration amplitude of the system at high speeds. The theoretical analysis and experimental amplitude are within an order of magnitude, and the changing trend is consistent.