Example Of Cantilever Beam Research Articles

Orbit-attitude-structure-thermal coupled modelling method for large space structures in unified meshes

Large space structures would exhibit orbit-attitude-structure-thermal coupled effects in complicated space environment. Thus, a new thermal-structure coupled modelling method is proposed using gradient-deficient absolute nodal coordinate formulation beam elements. Compared to the previous methods, both axial and circumferential heat conduction are considered. The cubic interpolation in temperature field instead of the linear interpolation is adopted to obtain a unified-mesh temperature-displacement description. The gravitational force and gravity gradient are modelled, and the effects of attitude motion and structural deformation on solar radiation intensity are considered. In addition, three kinds of quasi-static thermal-structure coupled models are proposed based on Euler-Bernoulli beam theory. Based on the quasi-static models, the theoretical formulas are derived to effectively predict the thermally induced vibrations of the beam in space. Four numerical examples are studied to validate the proposed models, including the heat conduction examples with three boundary conditions and a thermal-structure coupled cantilever beam example. Based on the proposed formulation, the orbit-attitude-structure-thermal coupled dynamics of a multibody system consisted of a rigid body and a large flexible appendage is investigated considering the gravity gradient, solar radiation, Earth's shadow, and self-shadow.

Multi-Material Optimization for Lattice Materials Based on Nash Equilibrium

Lattice materials are regarded as a new family of promising materials with high specific strength and low density. However, in the optimization of lattice materials, it is difficult in general to determine the material distribution in lattice structures due to the complex optimization formulations and overlaps between different materials. Thus, the article proposes to use the Nash equilibrium to address the multi-material optimization problem. Moreover, a suppression formula is investigated to tackle the issue of material overlapping. The proposed method is validated using a cantilever beam example, showing superior optimization results compared to single-material methods, with a maximum improvement of 20.5%. Moreover, the feasibility and stability of the approach are evaluated through L-shaped beam examples, demonstrating its capability to effectively allocate materials based on their properties and associated stress conditions within the design. Additionally, an MBB test demonstrates superior stiffness in the proposed optimized specimen compared to the unoptimized one.

Investigation of multi-input multi-output sine on random mixed vibration control

This article investigates Multi-Input Multi-Output (MIMO) mixed vibration control testing for Sinusoid on Random (SoR). The challenge of this vibration test is the mixed signal separation. The response signals at the control channels are composed of the broadband random component and the sinusoidal components. Firstly, the extracted accuracy of the Improved Correlation Integral Method (ICIM) is analyzed. The influences of the parameters used in ICIM are discussed in detail. Secondly, a new method, named Variance Minimum Method (VMM), is proposed for signal separation. Extraction accuracies of the ICIM and the proposed VMM are compared under different parameters. Finally, the control methods are proposed to iteratively update the driving signals of each component in the light of the deviations between the response signals and specified references. A cantilever beam example and biaxial vibration experiment manifest that the VMM and control method can be applied for the multi-shaker SoR vibration control tests.

Modeling the flexoelectric effect in semiconductors via a second-order collocation MFEM

In this paper, the flexoelectric effect in flexoelectric semiconductors (FSs) is studied via a second-order collocation mixed finite element method (MFEM) with the consideration of the strain gradient, electric field gradient, and drift-diffusion motion of carriers. In the present second-order collocation MFEM, the mechanical strain is assumed to be an independent quadratic polynomial function of local coordinates, and the kinematic constraint between the mechanical strain and displacement is approximately satisfied through a collocation method at 9 Gaussian quadrature points in each element. The setup for the electric field and electric potential also follows similar collocation strategy. Thus, except for the fundamental field variables, no additional DOFs are introduced. The accuracy of the proposed second-order collocation MFEM is verified by comparing numerical results with analytical solutions for a cantilever beam example. By performing the numerical experiments using the second-order collocation MFEM, the flexoelectric effect in FSs is simulated. Besides, we propose a novel mechanism for the converse flexoelectric effect in FSs which is induced by the diffusion motion of electrons. Numerical results indicate that by intensifying the diffusion motion of electrons, the converse flexoelectric effect, associated with the electric field gradient, could be enhanced.

Semi-Implicit Integration and Data-Driven Model Order Reduction in Structural Dynamics With Hysteresis

AbstractStructural damping is often empirically rate-independent wherein the dissipative part of the stress depends on the history of deformation but not its rate of change. Hysteresis models are popular for rate-independent dissipation; and a popular hysteresis model is the Bouc-Wen model. If such hysteretic dissipation is incorporated in a refined finite element model, then the model involves the usual structural dynamics equations along with nonlinear nonsmooth ordinary differential equations for a large number of internal hysteretic states at Gauss points used within the virtual work calculation. For such systems, numerical integration is difficult due to both the distributed nonanalytic nonlinearity of hysteresis as well as large natural frequencies in the finite element model. Here, we offer two contributions. First, we present a simple semi-implicit integration approach where the structural part is handled implicitly based on the work of Piché, while the hysteretic part is handled explicitly. A cantilever beam example is solved in detail using high mesh refinement. Convergence is good for lower damping and a smoother hysteresis loop. For a less smooth hysteresis loop and/or higher damping, convergence is noted to be roughly linear on average. Encouragingly, the time-step needed for stability is much larger than the time period of the highest natural frequency of the structural model. Subsequently, data from several simulations conducted using the above semi-implicit method are used to construct reduced order models of the system, where the structural dynamics is projected onto a few modes and the number of hysteretic states is reduced significantly as well. Convergence studies of error against the number of retained hysteretic states show very good results.

A dynamic-static coupling topology optimization method based on hybrid cellular automata

To improve the ability of the structure to resist static load (SL) and random vibration load (RVL) and prolong the fatigue life, a random vibration topology optimization method (RVTOM) and dynamic-static coupling topology optimization method (DSTOM) are proposed. RVTOM can directly use RVL for topology optimization, avoiding the huge workload caused by gradient information calculation. DSTOM combines the analytic hierarchy process (AHP) and hybrid cellular automata (HCA), and comprehensively considers SL and RVL in topology optimization. The cantilever beam example shows that the SL topology is challenging to resist random vibration load, and the equivalent load cannot replace the RVL for structural topology. Under random vibration excitation, the fatigue life of RVTOM and DSTOM models increased by 32.98% and 24.35%. The control arm example shows that both models have obtained infinite life. The RVTOM model is reasonable, and the load path obtained enhances the ability of the structure to resist random vibration loads and prolongs the fatigue life. DSTOM model can form a load path to fight SL and RVL effectively.

Identification of a cantilever beam’s spatially uncertain stiffness

This study identifies non-homogeneous stiffnesses in a non-destructive manner from simulated noisy measurements of a structural response. The finite element method serves as a discretization for the respective cantilever beam example problems: static loading and modal analysis. Karhunen–Loève expansions represent the stiffness random fields. We solve the inverse problems using Bayesian inference on the Karhunen–Loève coefficients, hereby introducing a novel resonance frequency method. The flexible descriptions of both the structural stiffness uncertainty and the measurement noise characteristics allow for straightforward adoption to measurement setups and a range of non-homogeneous materials. Evaluating the inversion performance for varying stiffness covariance functions shows that the static analysis procedure outperforms the modal analysis procedure in a mean sense. However, the solution quality depends on the position within the beam for the static analysis approach, while the confidence interval height remains constant along the beam for the modal analysis. An investigation of the effect of the signal-to-noise ratio reveals that the static loading procedure yields lower errors than the dynamic procedure for the chosen configuration with ideal boundary conditions.

Structural reliability with credibility based on the non-probabilistic set-theoretic analysis

A structural reliability analysis method based on the non-probabilistic credible set model is proposed in this paper. Compared with the probabilistic method, the developed method overcomes the high requirement for the sample size of structural uncertain parameters. Moreover, it surpasses the traditional non-probabilistic reliability analysis method by relating the reliability index to its corresponding credibility. Via introducing the non-probabilistic credible set uncertainty quantification, the distribution range of the required credibility for structural uncertain parameters can be determined by scant samples. Accordingly, the improved uncertainty propagation analysis methods based on the non-probabilistic credible set uncertainty quantification are established to obtain the bounds of structural response by transforming into local coordinate system. The non-probabilistic credible reliability index is constructed and three advanced efficient approaches are developed to calculate that. Moreover, the procedure of structural non-probabilistic credible reliability analysis method is built by combining the above content. Besides, the influence of credibility on non-probabilistic credible reliability is discussed. A cantilever beam example, an aeroengine turbine disk case and a pressure vessel case are utilized to verify the feasibility of applying the proposed approach to various engineering problems.

Identification method on stiffness weak part of cantilever structure on machine tool based on flexibility matrix

Identifying the weak link of structural stiffness is one of the important issues in improving machine tool stiffness. Aiming at the cantilever structure in machine tool, this paper proposes a new weak index to identify the weak link of stiffness. The weak index is based on the modal parameters obtained by vibration test method. In this method, the weak parameter matrix is obtained by multiplying the difference between the reciprocal of the flexibility matrix of the overall structure and the stiffness matrix of the overall structure. The diagonal elements of the weak parameter matrix are extracted to obtain the weak index. And the change of weak index is adopted to identify the weak link of structural stiffness. The method of element stiffness reduction is used to simulate the weak link of stiffness, so as to verify the proposed method. Finite element simulation is used to verify different noises and single and multiple stiffness weak links. The method is verified by numerical examples of cantilever beam and elastic support beam structure. It has higher recognition accuracy under noise interference, which is more advantageous than the flexibility difference method. Finally, an experimental study on the elastic supported cantilever beam structure and the tailstock of machine tool are carried out to verify the effectiveness of the method.

A comprehensive review of educational articles on structural and multidisciplinary optimization

Ever since the publication of the 99-line topology optimization MATLAB code (top99) by Sigmund in 2001, educational articles have emerged as a popular category of contributions within the structural and multidisciplinary optimization (SMO) community. The number of educational papers in the field of SMO has been growing rapidly in recent years. Some educational contributions have made a tremendous impact on both research and education. For example, top99 (Sigmund in Struct Multidisc Optim 21(2):120–127, 2001) has been downloaded over 13,000 times and cited over 2000 times in Google Scholar. In this paper, we attempt to provide a systematic and comprehensive review of educational articles and codes in SMO, including topology, sizing, and shape optimization and building blocks. We first assess the papers according to the adopted methods, which include density-based, level-set, ground structure, and more. We then provide comparisons and evaluations on the codes from several key aspects, including techniques, efficiency, usability, readability, environment, and compatibility. In addition, we conduct numerical experiments on the reviewed codes using the benchmark cantilever beam example to provide feedback on the overall user experience. With a systematic review and comparison, this paper aims to offer insights on the educational values and practicality for employing these codes. We try to provide not only guidance for beginners to approach various optimization methods, but also a dictionary to direct readers to effectively target the relevant codes and building blocks based on their demands. Finally, based on the findings in this review paper, we provide some perspectives and recommendations for future educational contributions.

Development of a four-node quadrilateral element-based high order numerical manifold method without linear dependency

A high-order numerical manifold method (HONMM) is developed using four-node quadrilateral (QUAD4) element to enhance accuracy and computational efficiency in solid mechanical problems. In the current study, the high-order global approximations are built by increasing the order of local approximations without facing the linear dependence (LD) problem, which is one of the main issues in the partition of unity (PU)-based methods with high-order approximations. To remove the LD problem, a new and simple scheme is proposed. Four problems are utilized to evaluate the efficiency of the QUAD4-based HONMM (QHONMM). A cantilever beam example is analyzed to compare different orders of QHONMM. Also, a block example with different loads and boundary conditions is used to investigate the sensitivity of the QHONMM results. Moreover, to demonstrate the capabilities of the QHONMM in dynamic analysis, free and forced vibrations of a simple beam under distributed and moving loads are analyzed. The results showed that using QUAD4 elements, the proposed QHONMM performs better than the conventional NMM in terms of accuracy and computational costs.

A numerical method of lower bound dynamic shakedown analysis for 3D structures

PurposeThe safety assessment of engineering structures under repeated variable dynamic loads such as seismic and wind loads can be considered as a dynamic shakedown problem. This paper aims to extend the stress compensation method (SCM) to perform lower bound dynamic shakedown analysis of engineering structures and a double-closed-loop iterative algorithm is proposed to solve the shakedown load.Design/methodology/approachThe construction of the dynamic load vertexes is carried out to represent the loading domain of a structure under both dynamic and quasi-static load. The SCM is extended to perform lower bound dynamic shakedown analysis of engineering structures, which constructs the self-equilibrium stress field by a series of direct iteration computations. The self-equilibrium stress field is not only related to the amplitude of the repeated variable load but also related to its frequency. A novel double-closed-loop iterative algorithm is presented to calculate the dynamic shakedown load multiplier. The inner-loop iteration is to construct the self-equilibrated residual stress field based on the certain shakedown load multiplier. The outer-loop iteration is to update the dynamic shakedown load multiplier. With different combinations of dynamic load vertexes, a dynamic shakedown load domain could be obtained.FindingsThree-dimensional examples are presented to verify the applicability and accuracy of the SCM in dynamic shakedown analysis. The example of cantilever beam under harmonic dynamic load with different frequency shows the validity of the dynamic load vertex construction method. The shakedown domain of the elbow structure varies with the frequency under the dynamic approach. When the frequency is around the resonance frequency of the structure, the area of shakedown domain would be significantly reduced.Research limitations/implicationsIn this study, the dynamical response of structure is treated as perfect elastoplastic. The current analysis does not account for effects such as large deformation, stochastic external load and nonlinear vibration conditions which will inevitably be encountered and affect the load capacity.Originality/valueThis study provides a direct method for the dynamical shakedown analysis of engineering structures under repeated variable dynamic load.

A model validation framework based on parameter calibration under aleatory and epistemic uncertainty

Model validation methods have been widely used in engineering design to evaluate the accuracy and reliability of simulation models with uncertain inputs. Most of the existing validation methods for aleatory and epistemic uncertainty are based on the Bayesian theorem, which needs a vast number of data to update the posterior distribution of the model parameter. However, when a single simulation is time-consuming, the required simulation cost for the validation of a simulation model may be unaffordable. To overcome this difficulty, a new model validation framework based on parameter calibration under aleatory and epistemic uncertainty is proposed. In the proposed method, a stochastic kriging model is constructed to predict the validity of the candidate simulation model under different uncertainty input parameters. Then, an optimization problem is defined to calibrate the epistemic uncertainty parameters to minimize the discrepancy between the simulation model and the experimental model. K–S test finally decides whether to accept or reject the calibrated simulation model. The performance of the proposed approach is illustrated through a cantilever beam example and a turbine blade validation problem. Results show that the proposed framework can identify the most appropriate parameters to calibrate the simulation model and provide a correct judgment about the validity of the candidate model, which is useful for the validation of simulation models in practical engineering design.

Reduced order models for wind turbine blades with large deflections

Non-intrusive nonlinear reduced order modeling (ROM) techniques have been applied by researchers to obtain computationally cheap and yet accurate structural responses of aircraft panels. However, its application to wind turbine blades is new and challenging due to much larger deflections of wind turbine blades. This study improves a non-intrusive nonlinear ROM method for wind turbine blades going through large deflections. In the nonlinear ROM, the nonlinear stiffness is described by the quadratic and cubic functions, and the secondary motions induced by the primary large deflections are described by the modal derivative vectors in the reduction basis. The non-intrusive nature of the method requires a geometrically nonlinear solver, and HAWC2 is chosen in this study for the computation of nonlinear stiffness terms. Two examples, including a cantilever beam example and the NREL 5MW wind turbine blade model, are used to evaluate the accuracy and computational effectiveness of the nonlinear ROMs. The cantilever beam example shows that the nonlinear ROM can accurately capture the axial displacements due to large deflections reaching 20 % of span length as well as the torsion coupled with flapwise and edgewise motions. The NREL blade example shows that the nonlinear ROM is accurate for the tip displacements more than 5.9 m. Because the size of the nonlinear ROM is much smaller than that of HAWC2 model, a speedup factor of 8.5 for computational time is observed for the NREL blade example.

New hybrid reliability-based topology optimization method combining fuzzy and probabilistic models for handling epistemic and aleatory uncertainties

This study presents a hybrid reliability-based topology optimization (RBTO) method for handling epistemic and aleatory uncertainties. First, we establish a new triple-nested RBTO model based on fuzzy and probabilistic theory for describing the multi-source uncertainties. Subsequently, an efficient single-loop optimization method is proposed to degrade the triple-nested optimization problem into a deterministic optimization problem using the Karush–Kuhn–Tucker optimality condition. Furthermore, the sensitivities of the hybrid reliability constraint with respect to the random probabilistic variables, fuzzy variables, and deterministic design variables are derived using the adjoint variable method. Finally, a cantilever beam example, an L-shape beam design and a 3D example are tested to verify the validity of the proposed single-loop method.

An Accelerating Convergence Rate Method for Moving Morphable Components

In the structural topology optimization approaches, the Moving Morphable Components (MMC) is a new method to obtain the optimized structural topologies by optimizing shapes, sizes, and locations of components. However, the size of the mesh has a strong influence on the rate of which the component builds the initial topological configuration by moving. The influence may slow down the convergence rate. In this paper, a hierarchical mesh subdivision solution method that can accelerate the convergence rate for the MMC is developed. First, the coarse mesh is used as the starting point for the optimization problem, and the construction process of the initial topology structure is increased speed by accelerating the movement of components. Second, the optimized solution obtained by the coarse mesh is equivalently mapped to the same problem with a finer mesh and used to construct a good starting point for the next optimization. Finally, two-dimensional (2D) MBB beam example and three-dimensional (3D) short cantilever beam example are provided so as to validate that with the use of the proposed approach, demonstrating that this method can improve the convergence rate and the stability of the MMC method.

Numerical shape optimization based on meshless method and stochastic optimization technique

This paper puts forward a newer approach for structural shape optimization by combining a meshless method (MM), i.e. element-free Galerkin (EFG) method, with swarm intelligence (SI)-based stochastic ‘zero-order’ search technique, i.e. artificial bee colony (ABC), for 2D linear elastic problems. The proposed combination is extremely beneficial in structural shape optimization because MM, when used for structural analysis in shape optimization, eliminates inherent issues of well-known grid-based numerical techniques (i.e. FEM) such as mesh distortion and subsequent remeshing while handling large shape changes, poor accuracy due to discontinuous secondary field variables across element boundaries needing costly post-processing techniques and grid optimization to minimize computational errors. Population-based stochastic optimization technique such as ABC eliminates computational burden, complexity and errors associated with design sensitivity analysis. For design boundary representation, Akima spline interpolation has been used in the present work owing to its enhanced stability and smoothness over cubic spline. The effectiveness, validity and performance of the proposed technique are established through numerical examples of cantilever beam and fillet geometry in 2D linear elasticity for shape optimization with behavior constraints on displacement and von Mises stress. For both these problems, influence of a number of design variables in shape optimization has also been investigated.

Research of vibration control based on current mode piezoelectric shunt damping circuit

The piezoelectric shunt damping circuit using current mode approach is imposed to control the vibration of a cantilever beam. Firstly, the simulated inductance with large values are designed for the corresponding RL series shunt circuits. Moreover, with an example of cantilever beam, the second natural frequency of the beam is targeted to control for experiment. By adjusting the values of the equivalent inductance and equivalent resistance of the shunt circuit, the optimal damping of the shunt circuit is obtained. Meanwhile, the designed piezoelectric shunt damping circuit stability is experimental verified. Experimental results show that the proposed piezoelectric shunt damping circuit based on current mode circuit has good vibration control performance. However, the control performance will be reduced if equivalent inductance and equivalent resistance values deviate from optimal values.

Sensitivity-Based Parameter Calibration and Model Validation Under Model Error

In calibrating model parameters, it is important to include the model discrepancy term in order to capture missing physics in simulation, which can result from numerical, measurement, and modeling errors. Ignoring the discrepancy may lead to biased calibration parameters and predictions, even with an increasing number of observations. In this paper, a simple yet efficient calibration method is proposed based on sensitivity information when the simulation model has a model error and/or numerical error but only a small number of observations are available. The sensitivity-based calibration method captures the trend of observation data by matching the slope of simulation predictions and observations at different designs and then utilizing a constant value to compensate for the model discrepancy. The sensitivity-based calibration is compared with the conventional least squares calibration method and Bayesian calibration method in terms of parameter estimation and model prediction accuracies. A cantilever beam example, as well as a honeycomb tube crush example, is used to illustrate the calibration process of these three methods. It turned out that the sensitivity-based method has a similar performance with the Bayesian calibration method and performs much better than the conventional method in parameter estimation and prediction accuracy.

Determination of the Optimum Reinforcement Layout in RC Structural Members: Cantilever Beam Example

Determination of the Optimum Reinforcement Layout in RC Structural Members: Cantilever Beam Example