Timoshenko Beam Finite Element Research Articles

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.

DISCRETE TIMOSHENKO BEAM MODEL FOR MODELING BEAMS WITH NON-UNIFORM CROSS-SECTIONAL AREA, CURVATURE, AND LARGE DEFLECTION

The Timoshenko beam theory is widely used in various fields due to its ability to accurately model the behavior of beams with specific characteristics. However, when it comes to analyzing large deflection beams, the Timoshenko beam theory becomes less precise. This paper introduces a proposed discrete Timoshenko beam model to predict the mechanical behavior of slender beams subjected to large deflections under various loadings. A mathematical analysis was developed and validated using finite element analysis in conjunction with a compliant mechanism. The modeling results demonstrate that the force-displacement behavior of a bistable mechanism can be accurately captured by the discrete Timoshenko beam model. In particular, when comparing the discrete Timoshenko beam model and finite element analysis, the maximum and minimum forces exhibited a strong agreement, with a percentage difference of less than 3% and 5%, respectively. In terms of the total elastic strain energy and maximum principal stress, the deviations between the discrete Timoshenko beam model and finite element analysis were approximately 2% and 7%, respectively. The proposed model can be integrated into an optimization algorithm for designing compliant mechanisms.

Mixed‐mode delamination of layered structures modeled as Timoshenko beams with linked interpolation

AbstractIn this work, a finite‐element formulation for modeling mixed‐mode delamination in layered structures, consisting of two‐node Timoshenko beam finite elements with quadratic linked interpolation and corresponding 4‐node interface elements is presented and compared to a more common approach where linear Lagrange interpolation is used. The principal novelty of the proposed approach is that the vertical displacements of the beam elements, as well as the transversal relative displacements of the interface elements, are interpolated using a quadratic linked interpolation that also takes into account the nodal cross‐sectional rotations of the beam elements. At the same time, the axial displacements and cross‐sectional rotations are interpolated using linear Lagrange polynomials. A bilinear cohesive zone model is embedded in the interface finite elements for delamination modes I and II. Numerical analyses based on the examples from the literature with metal joints show that the formulation with quadratic linked interpolation improves the convergence and robustness of the solution with respect to the approach with linear interpolation. On the other hand, in case of composites with stiff adhesives this formulation exhibits a peculiar behavior with spurious oscillations of the normal interface tractions that leads to a poor performance in mode‐I and mixed‐mode tests. This problem can be easily solved by canceling the quadratic term in the interpolation function and using the standard Lagrange interpolation in such cases.

Topology optimization of structural frames based on a nonlinear Timoshenko beam finite element considering full interaction

AbstractThis article presents an approach for the topology optimization of frame structures composed of nonlinear Timoshenko beam finite elements (FEs) under time‐varying excitation. Material nonlinearity is considered with a nonlinear Timoshenko beam FE model that accounts for distributed plasticity and axial–shear–moment interactions through appropriate hysteretic interpolation functions and a yield/capacity function, respectively. Hysteretic variables for curvature, shear, and axial deformations represent the nonlinearities and evolve according to first‐order nonlinear ordinary differential equations (ODEs). Owing to the first‐order representation, the governing dynamic equilibrium equations, and hysteretic evolution equations can thus be concisely presented as a combined system of first‐order nonlinear ODEs that can be solved using a general ODE solver. This avoids divergence due to an ill‐conditioned stiffness matrix that can commonly occur with Newmark–Newton solution schemes that rely upon linearization. The approach is illustrated for a volume minimization design problem, subject to dynamic excitation where an approximation for the maximum displacement at specified nodes is constrained to a given limit, that is, a drift ratio. The maximum displacement is approximated using the p‐norm, thus facilitating the derivation of the analytical sensitivities for gradient‐based optimization. The proposed approach is demonstrated through several numerical examples for the design of structural frames subjected to sinusoidal base excitation.

Rapid Prototyping of Inertial MEMS Devices through Structural Optimization.

In this paper, we propose a novel design and optimization environment for inertial MEMS devices based on a computationally efficient schematization of the structure at the a device level. This allows us to obtain a flexible and efficient design optimization tool, particularly useful for rapid device prototyping. The presented design environment—feMEMSlite—handles the parametric generation of the structure geometry, the simulation of its dynamic behavior, and a gradient-based layout optimization. The methodology addresses the design of general inertial MEMS devices employing suspended proof masses, in which the focus is typically on the dynamics associated with the first vibration modes. In particular, the proposed design tool is tested on a triaxial beating-heart MEMS gyroscope, an industrially relevant and adequately complex example. The sensor layout is schematized by treating the proof masses as rigid bodies, discretizing flexural springs by Timoshenko beam finite elements, and accounting for electrostatic softening effects by additional negative spring constants. The MEMS device is then optimized according to two possible formulations of the optimization problem, including typical design requirements from the MEMS industry, with particular focus on the tuning of the structural eigenfrequencies and on the maximization of the response to external angular rates. The validity of the proposed approach is then assessed through a comparison with full FEM schematizations: rapidly prototyped layouts at the device level show a good performance when simulated with more complex models and therefore require only minor adjustments to accomplish the subsequent physical-level design.

Hyperbolic‐Like Structure with Negative Poisson's Ratio: Deformation Mechanism and Structural Design

Mechanical metamaterials with negative Poisson's ratio (NPR) effect possess important practical applications in aerospace, nanotechnology, and other fields. The microstructure and its deformation mode directly determine the NPR ability and mechanical performance of auxetic metamaterials. Herein, inspired by the double‐negative‐index ceramic aerogels, a novel NPR structural model is proposed in which symmetric reentrant oblique ligaments and short transversal ligaments constitute the main framework inside a square, which is different from the traditional reentrant structure due to the hyperbolic‐like skeleton and deformation modes. The influence of structural parameters on the NPR effect is investigated using the Timoshenko beam model and finite element simulations. The structural model is then extended to design multiple reentrant structures and assembled structures, respectively. The multiple reentrant structures include multiple pairs of symmetrically hyperbolic‐like skeletons, and the distribution of horizontal ligaments can be regulated to achieve a better NPR effect. For assembled structures under compression, the local rotational deformation and buckling result in structural instability, demonstrated by inflection points in the NPR curves. Moreover, one special assembled structure is found for which there is a linear relationship between the compressive strain and NPR value. The insights should be significant for designing mechanical metamaterials with the NPR effect.

Modeling multi-fracturing fibers in fiber networks using elastoplastic Timoshenko beam finite elements with embedded strong discontinuities — Formulation and staggered algorithm

To model fiber failures in random fiber networks, we have developed an elastoplastic Timoshenko beam finite element with embedded discontinuities. The method is based on the theory of strong discontinuities where the generalized displacement field is enhanced by a jump. The continuum mechanics formulation accounts for a fracture process zone and a bulk material while retaining traction continuity across the discontinuity. The additional degrees of freedom that are associated with the discontinuity are represented by a midpoint node, which is statically condensed to enable the implementation in commercial software through the user element interface. We propose a quasi-brittle fracture model, where the failure-related deformation is uncoupled from the plastic deformation in the bulk material. To retain the positive definite finite element stiffness matrix of the bulk material, we neglect the fracture-related softening of the discontinuity and employ a modified Newton iteration in the strain softening domain. Our implementation facilitates the integration into commercial finite element software and examples illustrate the robustness of the method. The FORTRAN source code is freely available to benchmark our model. We show that fiber failures contribute to the nonlinear stress–strain response of paper. Together with fiber–fiber bond failures, they can potentially explain the nonlinear stress–strain response of paper and nanopaper.

Identifying the transient milling force coefficient of a slender end-milling cutter with vibrations

Milling is a complex process during the coupling effect for multiple physical fields, especially for slender end-milling cutters. Because of the weak rigidity of its process system, it is easily affected by the physical behavior such as the milling force. In addition, it produces large vibrations and deformations, which affects the machining quality. To accurately predict the milling force of a slender end-milling cutter, a method to identify the milling force coefficients while considering vibrations is proposed. When considering the small diameter of the milling cutter edge, it is difficult to install sensors to measure the vibration. By combining the measured frequency response function of the cutter handle end and the cutter’s Timoshenko beam finite element model, the interface dynamics parameters between the cutter and the cutter handle are identified. This is based on the receptance coupling substructure analysis, and the cutter edge’s frequency response function is established. When considering the measured milling force data and the state equation of the cutter edge vibration, the real-time vibration of the cutter edge during the milling process is predicted. In addition, the undeformed cutting thickness during the superposition of the cutter edge vibration is obtained. The milling force coefficients are identified based on the undeformed cutting thickness of the coupled vibration, and the influence of the process parameters on the variation of the milling force coefficient is examined. The prediction accuracy and the average milling force coefficient without the vibrations are compared to verify the accuracy.

An ultrasonic vibration-assisted system development for inner-diameter sawing hard and brittle material

The inner-diameter (ID) sawing technology is dominant in small-lot and multi-specification sawing process of hard and brittle materials. However, the surface quality obtained in the conventional ID sawing cannot meet practical requirements. In this paper, a novel processing technology combining ultrasonic machining and ID sawing is proposed to improve the ID sawing process and surface quality. A one-dimensional ultrasonic vibration system is developed by combining Timoshenko beam theory and finite element optimisation technology. An experimental setup is established to implement the ultrasonic vibration-assisted ID sawing (UVAIDS). With single-crystal silicon as the sawn workpiece, a series of ID sawing experiments with and without ultrasonic vibration are carried out. The surface morphology and surface roughness of sawn wafers under different cutting parameters are analyzed and discussed. The experimental results show that the application of ultrasonic vibration assistance can improve the surface quality by nearly 30 % in ID sawing process, proving the ultrasonic vibration system is available and UVAIDS technology is feasible and effective. The developed UVAIDS technology can be widely applied to the slicing process of hard and brittle material.

Aeroelastic Optimization of Aircraft Wings Using a Coupled Three-Dimensional Panel-Beam Model

This work introduces an aeroelastic optimization framework with a coupled three-dimensional panel method and Timoshenko beam finite element model. The method allows for optimization of both the exterior surface of the wing and interior structural properties. We investigate the effects of curved wall spars on the aeroelastic performance of converged designs, which have been shown to provide an improved performance due to the ability to create tradeoffs between bending and torsional stiffnesses. Studies also highlight the importance of calculating aerodynamic loads in the deformed configuration and solving the coupled aeroelastic problem to convergence.

Accurate Timoshenko Beam Elements For Linear Elastostatics and LPB Stability

Several methods to derive accurate Timoshenko beam finite elements are presented and compared. Two application problems are examined: linear elastostatics and linearized prebuckling (LPB) stability analysis. Accurate elements can be derived for both problems using a well known technique that long preceeds the Finite Element Method: using homogeneous solutions of the governing equations as shape functions. An interesting question is: can accurate elements be derived with simpler assumptions? In particular, can linear-linear interpolation of displacements and rotations with one-point integration reproduce those elements? The answers are: no if standard variational tools based on classical functionals are used, but yes if modified functionals are introduced. The connection of modified functionals to newer methods, in particular templates, modified differential equations and Finite Increment Calculus (FIC) are examined. The results brings closure to a 50-year conumdrum centered on this particular finite element model. In addition, the discovery of modified functionals provides motivation for extending these methods to full geometrically nonlinear analysis while still using inexpensive numerical integration.

Multi-fidelity analysis and uncertainty quantification of beam vibration using co-kriging interpolation method

In this paper, a multi-fidelity surrogate model is created using co-kriging methodology to determine the natural frequencies of beams by combining the fidelities of Euler-Bernoulli (low-fidelity) and Timoshenko (high-fidelity) beam finite element models. This study of free vibration of beams involves uncertainties in material properties. The sampling space for the co-kriging surrogate model is created using an optimal latin hypercube sampling technique. It is shown that the co-kriging surrogate model predicts the natural frequencies with high computational efficiency and accuracy, in the entire design space. The computational efficiency and utility of the multi-fidelity model is demonstrated through its application to a problem of quantifying uncertainties in the natural frequencies of a tapered beam using Monte Carlo Simulation.

Qualitative research on the relationship between spindle vibration characteristics and bearing thermal load

The dynamic performance of machine tool is one of the most important factors affecting the machining accuracy. As the core part of machine tool, the dynamic performance of spindle directly determines the performance of machine tool. In this paper, the spindle of machine tool is taken as the research object, and the change of fit clearance between bearing inner and outer rings, rotating shaft and bearing seat caused by spindle bearing heat, and its influence on the stiffness of spindle system are analyzed. Based on Timoshenko beam theory and finite element method, this paper qualitatively analyzes the variation characteristics of spindle stiffness caused by bearing overheating and grease lubrication failure during high-speed machining.

Dynamic Analysis of Bolted Cantilever Beam with Finite Element Method Considering Thread Relation

There are complex nonlinear behaviors and mechanisms in the bolted joint interface. Thus, the bolted joint is crucial to the complex nonlinear dynamic response of the structure. However, in the traditional structural dynamic analysis, the screw connection is usually neglected, which makes it challenging to analyze and study the nonlinear dynamic behavior of bolted structures. Hence, based on the Timoshenko beam theory and finite element method, this paper introduces a model considering thread connection to analyze the dynamic response under different excitation. Eventually, the results indicate that owing to the local nonlinearity of bolts, the whole bolted cantilever beam shows hardening-type characteristics. In addition, the frequency response curve also depicts the typical nonlinear phenomenon of instability and uncertainty, namely bifurcation, which preliminarily verifies the correctness and accuracy of the bolted cantilever beam model.

Modeling, Analysis, and Identification of Parallel and Angular Misalignments in a Coupled Rotor-Bearing-Active Magnetic Bearing System

Abstract The standard techniques used to detect misalignment in rotor systems are loopy orbits, multiple harmonics with predominant 2X component, and high axial vibration. This paper develops a new approach for the identification of misalignment in coupled rotor systems modeled using two-node Timoshenko beam finite elements. The coupling connecting the turbine and generator rotor systems is modeled by a stiffness matrix, which has both static and additive components. While the magnitude of static stiffness component is fixed during operation, the time-varying additive stiffness component displays a multiharmonic behavior and exists only in the presence of misalignment. To numerically simulate the multiharmonic nature coupling force/moment as observed in experiments, a pulse wave is used as the steering function in the mathematical model of the additive coupling stiffness (ACS). The representative turbogenerator (TG) system has two rotor systems, each having two disks and supported on two flexible bearings—connected by coupling. An active magnetic bearing (AMB) is used as an auxiliary bearing on each rotor for the purposes of vibration suppression and fault identification. The formulation of mathematical model is followed by the development of an identification algorithm based on the model developed, which is an inverse problem. Least-squares linear regression technique is used to identify the unbalances, bearing dynamic parameters, AMB constants, and importantly the coupling static and additive stiffness coefficients. The sensitivity of the identification algorithm to signal noise and bias errors in modeling parameters has been tested. The novelty of paper is the representation and identification of misalignment using the ACS matrix coefficients, which are direct indicators of both type and severity of the misalignment.

Dynamic analysis of a double-helical geared rotor system with oil-film bearing

PurposeThis paper aims to study a dynamic analysis of a double-helical geared rotor system with oil-film bearing.Design/methodology/approachA finite element model of a double-helical geared rotor system with oil-film bearing is developed, in which a rigid mass is used to represent the gear and the Timoshenko beam finite element represents the shaft; the equations of motion are obtained by applying Lagrange’s equation. Natural frequencies, Campbell diagram, lateral responses, axial responses, bearing stiffness coefficients, bearing damping coefficients and bearing force are investigated.FindingsNatural frequencies and Campbell diagram of a double-helical geared rotor system with oil-film bearing are investigated. An increased helical angle enhanced the axial response of the system and reduced its lateral response. The distance between the node and bearing affected the lateral response magnitude on the node. The farther away the gear pair was from the central part of the shaft, the higher the system’s resonance frequency became. The different gear pair position has a significant influence on the bearing stiffness coefficient and bearing force, but it just has a little effect on the bearing damping coefficient.Practical implicationsThe model of a double-helical geared rotor system with oil-film bearing is established in this paper. The dynamic characteristics of a double-helical geared rotor system with oil-film bearing are investigated. The numerical results of this study can be used as a reference for subsequent personnel research.Originality/valueAlthough the dynamics characteristics of geared rotor bearing system have been reported in some literature, the dynamic analysis of a double-helical geared rotor-bearing system is still rarely investigated. This paper showed some novel results that lateral and axial response results are obtained by the different helical angle and different gear positions. In the future, it makes valuable contributions for further development of dynamic analysis of a double-helical geared rotor-bearing system.

3D thermo-hydro-mechanical coupled discrete beam lattice model of saturated poro-plastic medium

In this paper, we present a 3D thermo-hydro-mechanical coupled discrete beam lattice model of structure built of the nonisothermal saturated poro-plastic medium subjected to mechanical loads and nonstationary heat transfer conditions. The proposed model is based on Voronoi cell representation of the domain with cohesive links represented as inelastic Timoshenko beam finite elements enhanced with additional kinematics in terms of embedded strong discontinuities in axial and both transverse directions. The enhanced Timoshenko beam finite element is capable of modeling crack formation in mode I, mode II and mode III. Mode I relates to crack opening, mode II relates to in-plane crack sliding, and mode III relates to the out-of-plane shear sliding. The pore fluid flow and heat flow in the proposed model are governed by Darcy\'s law and Fourier\'s law for heat conduction, respectively. The pore pressure field and temperature field are approximated with linear tetrahedral finite elements. By exploiting nodal point quadrature rule for numerical integration on tetrahedral finite elements and duality property between Voronoi diagram and Delaunay tetrahedralization, the numerical implementation of the coupling results with additional pore pressure and temperature degrees of freedom placed at each node of a Timoshenko beam finite element. The results of several numerical simulations are presented and discussed.

Multi-fidelity analysis and uncertainty quantification of beam vibration using correction response surfaces

A multi-fidelity model for beam vibration is developed by coupling a low-fidelity Euler-Bernoulli beam finite element model with a high-fidelity Timoshenko beam finite element model. Natural frequencies are used as the response measure of the physical system. A second order response surface is created for the low-fidelity Euler-Bernoulli model using the face centered design. Correction response surfaces for multi-fidelity analysis are created by utilizing the high-fidelity finite element predictions and the low-fidelity finite element predictions. It is shown that the multi-fidelity model gives accurate results with high computational efficiency when compared to the high-fidelity finite element model.

Effect of multi-frequency parametric excitations on the dynamics of on-board rotor-bearing systems

In the transportation domain such as automotive turbochargers and aircraft turbines, the vibrations of on-board rotors are induced not only by the mass unbalance excitation but also by various movements of their support. The dynamics of an on-board rotor mounted on hydrodynamic finite-length bearings is investigated in the presence of support motions which create multi-frequency parametric excitations. The developed on-board rotor model is based on the gyroscopic Timoshenko beam finite element with two nodes and six degrees of freedom per node for 3D motions (transverse and axial displacements as well as rotations due to the bending and to the torsion). The equations of motion highlight time-varying parametric terms due to the mass unbalance, the support rotations, the coupling between both phenomena and the combination of mass unbalance and support translations. These parametric terms can yield a dynamic instability because they contribute as generators of internal excitation. In the presented applications, single-frequency and multi-frequency parametric excitations are used. Namely, the rotor is excited either by simple and combined sinusoidal support rotations or by a rotating mass unbalance combined with sinusoidal support translations to examine the stability of the static equilibrium point through the Floquet theory.

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.