Arc-length Method Research Articles

Non-linear vibration response of functionally graded circular cylindrical shells subjected to thermo-mechanical loading

The analysis of the non-linear vibration response is carried out for functionally graded (FG) circular cylindrical shells subjected to thermal environment along with mechanical in-plane non-uniformly distributed loading along the edges and harmonic radial force. The temperature dependent material properties of the simply supported shell are assumed to vary in the radial direction according to power-law distribution. Based on the first-order shear deformation theory and von-Kármán type geometric nonlinearity, the strain-displacement relationships are established for circular cylindrical shells. The coupled governing equations of motion for functionally graded cylindrical shells are then derived using Hamilton’s principle. Employing Galerkin’s method, the coupled partial differential equations of motions are reduced to a set of non-linear ordinary differential equations. In order to obtain the free and forced vibration response of the FG shell, the incremental harmonic balance method, in conjunction with the arc-length method, is used. The non-uniform in-plane loading is converted to Fourier series and the pre-buckling analysis is performed to determine the stress distribution within the shell. The non-linear frequency-amplitude response is studied to examine the effects of volume fractions of the constituents, static partial edge loadings, thermal loads, and radial periodic loadings.

Real-time spatial intersecting seam tracking based on laser vision stereo sensor

A real-time robotic weld tracking system based on laser vision sensor is designed for intersecting seam. Firstly, the traditional image processing method is used to determine the weld feature point in the first frame when there has no arc noise. Then, an extending and adopting Kalman Filter with Spatio-Temporal Context algorithm was proposed to extract the weld feature point when the laser stripe was blocked by heavy arc and splash noise during welding. Next, in order to control torch to weld automatically, the torch frame was established through novel three points principle and an expanding circle and arc length method. In addition, a segmentation weld method and an intermediate proportional interpolation method for step-size control of mobile torch were proposed to guarantee the accurate of weld. Finally, Experiments and analyses results show that the tracking system is good in real-time, accuracy, stability and flexible, which can meet the weld requirements.

Analysis and structure optimization on buckling destabilization and wrinkling of an automobile weather-strip seal in assemblage

Buckling destabilization and wrinkling of an ethylene-propylene-diene monomer automobile weather-strip seal in assemblage and its structural optimization were studied in this paper. First, an innovative approach that traces buckling bifurcation paths was developed based on an arc-length method, and algorithmic parameters of the method were defined. A finite element analysis model of the automobile weather-strip seal in assemblage was then developed and analyzed using the arc-length method. The maximum buckling load, the deformation of the seal, and the thickness decrease of the lower tube wall in the critical region where it was prone to wrinkle were obtained by this finite element analysis method. Finally, an optimization seal structure was proposed and analyzed, and the deformations and the thickness decrease of the original and optimal structures in the critical regions were compared. The analysis conclusion implies that the optimal structure is more stable. The proposed analysis and optimization method can shorten the product design cycle, improve the structural stability, and decrease the design and trial-product cost considerably.

Numerical investigation on the nonlinear dynamics of a breathing cracked rotor supported by flexible bearings

The steady-state response and breathing mechanism of a cracked rotor supported by flexible bearings are investigated in this paper. The generalized and efficient method proposed in this paper can be used to study the dynamics of complicated cracked structures without much modification. First, a three-dimensional finite element model of the cracked rotor-bearing system is established in the rotating frame and a general contact model for modeling the breathing crack is proposed. A component mode synthesis is used to form a reduced-order model. Then, a procedure combining multi-harmonic balance method with arc-length method is used to search the response solution. To accelerate the calculation, the analytical formulations for calculating the tangent stiffness matrix are used. Finally, the gravity induced response and breathing mechanism of a cracked rotor-bearing system are obtained. Interesting result is that the rotational speed and the crack depth will influence the breathing mechanism even if the load remains unchanged.

Convergence improvement in computation of strain-softening solids by the arc-length method

This paper deals with problems emerging in a non-linear analysis of systems modelled by the strain softening material. Apart from sharp turns on the load-displacement diagram that characterize such numerical models, snap-back, artificial unloading and possibility of bifurcation occurrence are often encountered. These aspects challenge a non-linear system solver, in a sense of finding the appropriate equilibrium path. In this paper the CDPM2 concrete material model based on damage-plasticity is employed on notched geometry models in order to analyse convergence. A number of techniques for the solution of non-linear systems are tested and special emphasis is given to the snap-back occurrence and its influence on the convergence of the solution. A relatively simple new technique is developed for achieving better convergence throughout iterations to serve as an aid to conventional methods such as arc-length and line search.

Buckling and Postbuckling Analyses of Structure using Absolute Nodal Coordinate Formulation

In this paper, the three dimensional higher order beam element based on the absolute nodal coordinate formulation (ANCF) is used to study both the classical buckling and the nonlinear postbuckling problem. The analyses are performed using the Newton-Raphson method and the arc length method. The Newton-Raphson method is used for the Euler buckling whereas the Crisfield’s arc length method is applied to track the equilibrium path of the William’s Toggle. The solutions agree well with the analytical one or that given by the commercial finite element software ANSYS. Hence, the validity of the analyses is demonstrated.

PSEUDO ELASTIC METHODS FOR DAMAGE ASSESSMENT OF COMPOSITE TUBES

Rapid urbanization globally has led to increased traffic congestion within cities which often result in low speed collisions. These low speed collisions can happen during parking of Vehicles also. Bumpers provided for the Vehicles primarily absorb the impact energy during these low speed collisions. However, a portion of this, impact could get transferred to the thin walled tubular structures attached to the bumpers whose main purpose is to effectively absorb the Impact energy and compliment occupant safety, in the event of an accident. These structures are usually provided with notches of specific sizes at select locations to improve crush (energy absorption) efficiency and result in uniform crushing pattern. These notches could experience localized damage during low speed collisions. Such a scenario could make the crush tubes ineffectual as energy absorbers, after few such collision cycles due to localized partial damage. Hence it is desirable to check the design for possibility of localized damage during low speed collisions. Computation of damage requires prediction of notch energy or associated strains. Usage of Finite Element methods for damage evolution prediction of composite elements poses several challenges, especially for anisotropic materials like Composites. Several approximate methods have been proposed and are being used for evaluation of Elasto-Plastic strains, for isotropic materials. The adaptability and relative accuracies of these methods namely The ESED rule, arc Length Method and Neuber’s rule under axial compressive loads for anisotropic composite materials have been investigated and presented.

An Efficient Strategy for Solving Structural Nonlinear Equations by Combining the Orthogonal Residual Method and Normal Flow Technique

This paper presents a new procedure for solving structural nonlinear problems by combining the orthogonal residual method (ORM) and normal flow technique (NFT). The perpendicularity condition to the Davidenko flow, introduced by the NFT, which must be satisfied during the iterative process, overcome the difficulties, i.e. the poor convergence and inefficiency of the ORM close to the limit points, particularly the displacement limit points (snap-back behavior). Basically, the idea of the proposed strategy is to adjust the load parameter, which is treated as a variable in the nonlinear incremental-iterative solution process, assuming that the unbalanced forces (residual forces) must be orthogonal to the incremental displacements. This constraint is used together with the NFT perpendicularity condition. The proposed procedure is tested, and its efficiency is corroborated through the analyses of slender shallow and nonshallow arches and an L-frame since they exhibit highly nonlinear behaviors under certain loading conditions. It is concluded that the proposed procedure can overcome the numerical instability problems in the neighborhood of critical points when using only the conventional OR process, and the procedure compares favorably with the arc-length method, minimum residual displacement method, and generalized displacement control method.

An Optimized Approach for Tracing Pre- and Post-Buckling Equilibrium Paths of Space Trusses

In this paper, a novel optimization-based method is proposed to analyze steel space truss structures undergoing large deformations. The geometric nonlinearity is considered using the total Lagrangian formulation. The nonlinear solution is obtained by introducing and minimizing an objective function subjected to the displacement-type constraints. The proposed approach can fully follow the equilibrium path of the geometrically nonlinear space truss structures not only before the limit point, but also after it, namely, including both the pre- and post-buckling paths. Moreover, a direct estimation of the buckling loads and their corresponding displacements is possible by using the method. Particularly, it has been shown that the equilibrium path of a structure with highly nonlinear behavior, multiple limit points, snap-through, and snap-back phenomena can be traced via the proposed algorithm. To demonstrate the accuracy, validity, and robustness of the proposed procedure, four benchmark truss examples are analyzed and the results compared with those by the modified arc-length method and those reported in the literature.

The buckling strength of plexiglass protective shield under static water pressure

The protective shield of PMT (photon multiplier tubes) is a typical spherical-cylindrical thin-walled shell structure which works in the environment of deep water pressure with the material of plexiglass and character of low Young's Modulus. Therefore, it is significant to analyze the stability of the structure. The validity of eigenvalue buckling prediction and arc length method was verified by classic buckling theory of rotation shell firstly. Then the stability of plexiglass shield was researched by these two methods and a general calculation formula of shallow shell was fitted. The results show that the critical stress of plexiglass shield decreases with the reduction of section roundness; if the height of cylindrical shell is less than 0.3R, there is no influence on sphere-cylindrical shell structure, but when the height is more than 0.3R, the structure stability decreases with the increasing of cylindrical shell height. If the influence of exhaust orifice is considered, the geometry of shell is the main factor for the stability of shell when the orifice radius is less than 0.3 times of orifice spacing. Otherwise, orifice is the main factor for the stability of shell. There is a significant influence of initial imperfection on the critical stress of the protective shield and the critical stress decreases with the increasing of initial deformation. The research of this paper can provide useful guidance for the design of PMT plexiglass shield.

Higher-order finite element models for the static linear and nonlinear behaviour of functionally graded material plate-shell structures

In this work, finite element formulations based on higher order shear deformation theories are used for the nonlinear static analysis of Functionally Graded Material plate-shell type structures. Linear and geometric nonlinear behaviour of the plate-shell type structures are considered. For the nonlinear analysis, the incremental equilibrium path is obtained using the updated Lagrangian procedure and Newton-Raphson incremental-iterative method, incorporating the automatic arc-length method for the cases of snap-through occurrence. The finite element models are based on a non-conforming triangular flat plate/shell element with 3 nodes and 8 or 11 degrees of freedom per node. The solutions of some illustrative plate-shell examples are performed, and the results are presented and discussed with numerical alternative models.

Effects of Stator Stiffness, Gap Size, Unbalance, and Shaft’s Asymmetry on the Steady‐State Response and Stability Range of an Asymmetric Rotor with Rub‐Impact

The asymmetric rotor and the rub‐impact behavior are important sources of instability and may cause severe vibrations. However, the dynamics of the rotor‐bearing system simultaneously considering the two factors has not gained sufficient attention in available investigations. In this paper, the steady‐state response and stability of an asymmetric rotor with rub‐impact were evaluated. The asymmetric rotor was modeled by beam elements with asymmetric cross section, and the nonlinear equations of motion were established in the rotating frame. The multiharmonic balance (MHB) method was employed to obtain the linearized form of the nonlinear equations of motion. Either the asymmetry of rotor or rub‐impact can result in instability and make the problem difficult to solve. Thus, the arc‐length method was utilized to trace the branch of the solutions. In order to improve the calculation speed and accurately predict the solution, the alternating frequency/time domain (AFT) was adopted to calculate the iteration of the arc‐length method. Based on the proposed method, the effects of stator stiffness, gap size, unbalance, and asymmetric in shaft on the steady‐state response and stability were obtained.

Nonlinear multimodal electromagnetic device for vibration energy harvesting

A multimodal vibration energy harvesting in a periodic system is proposed. The multimodal approach and the nonlinearity are implemented in order to improve the performances of the studied device. The periodic system, based on electromagnetic transduction, consists of two weakly coupled magnets mechanically guided by two elastic beams. The quasi-periodic system is obtained by varying the mass of one of the moving magnets which leads to the vibration energy localization in regions close to the imperfections. This phenomenon is exploited to maximize the harvested energy. The mechanical nonlinearity is introduced by considering large displacements of the beams. The system is modeled by two coupled forced Duffing equations. The governing equations are solved using finite difference method combined with arc-length method. It is shown that the introduction of the nonlinearity leads to the enlargement of the bandwidth and the increase of the amplitude of the vibration.

A Minimum Arclength Method for Removing Spikes in Empirical Mode Decomposition

Empirical mode decomposition (EMD) is an extensively utilized tool in a time–frequency analysis. However, disturbances, such as impulse noise, can result in a mode-splitting effect, in which one physically meaningful component is split into two or more intrinsic mode functions (IMFs). In this paper, we propose a novel method, minimum arclength EMD (MA-EMD), to robustly decompose time series data with impulse-like noises. The idea is to apply a minimum arclength criterion to adjust the knot positions of impulses during the sifting process in EMD. In this way, the impulse-like artifact is extracted with the first IMF, and the mode splitting effect of the latter decomposition is alleviated. Furthermore, when the first IMF contains the desired information, we separate the spikes and the first IMF by adding a pair of masking signals. For using this masking-aided MA-EMD (MAMA-EMD) method, we also mathematically derived the appropriate ranges of the frequency and the amplitude of the masking signal. The MAMA-EMD is utilized to deal with the simulated Duffing wave and four real-world data, including electrical current, vibration signals, the cyclic alternating pattern in sleep EEG (electroencephalography), and circadian of core body temperature. The results show that the MA-EMD and MAMA-EMD have a sound improvement when encountering impulse noises.

퇴화 쉘요소를 사용한 원통형 복합재 패널의 비선형 좌굴후 현상해석

In this study, the post - buckling analysis of cylindrical panel of isotropic material and composites was investigated. The arc-length nonlinear analysis method has been used for the analysis of post-buckling phenomena where snap-through and snap-back phenomenon occur. In this study, a method to find bifurcation point in buckling analysis using arc-length method is presented. The bifurcation point of the buckling was found in the cylindrical shell structure of isotropic material to ensure the reliability of the current approach. The bifurcation buckling point was successfully calculated with composites having extremely high structural instability. The formulations, mathematical model and analysis procedures and boundary conditions proposed in this study can be used as important data for the unstable buckling analysis of other structures.

Reinforcement design of concrete sections based on the arc-length method

Abstract The reinforcement design of concrete cross-sections with the parabola-rectangle diagram is a well-established model. A global limit analysis, considering geometrical and material nonlinear behavior, demands a constitutive relationship that better describes concrete behavior. The Sargin curve from the CEB-FIP model code, which is defined from the modulus of elasticity at the origin and the peak point, represents the descending branch of the stress-strain relationship. This research presents a numerical method for the reinforcement design of concrete cross-sections based on the arc length process. This method is numerically efficient in the descending branch of the Sargin curve, where other processes present convergence problems. The examples discuss the reinforcement design of concrete sections based on the parabola-rectangle diagram and the Sargin curve using the design parameters of the local and global models, respectively.

Stability Analysis of a Simple Strut with Elastic Springs Through Sensitivity Surfaces and Gradients

The global elastic stability of a strut supported by three inclined springs is studied using the exact displacement geometry. The discrete phenomenological model in the theory of stability, described by Thompson and Gaspar, exhibits different postbuckling behaviors, depending on the initial geometry for large imperfections, and this has various applications in engineering. Using the total potential energy method, a system of nonlinear algebraic equations is derived for the nondissipative system. The arc length method is adopted to solve the system of nonlinear equations, considering the stability of equilibrium points at each position, while detecting and traversing the critical points with the possibility of intervention. The characteristic equilibrium paths and their corresponding trajectories are shown and compared with the asymptotic solutions at the first critical point. A parametric analysis is performed, and sensitivity surfaces are constructed for several initial positions of the springs, represented by the initial angle in the horizontal plane. A wide range of independent imperfections with large amplitudes in two directions is considered. The concept of load capacity gradient functions of sensitivity surfaces is introduced and used to qualitatively and quantitatively analyze the stability behavior of the system located far away from the initial position. A few interesting observations and conclusions are obtained from the sensitivity analysis of a simple strut in the postcritical region. The critical combinations of imperfections for each static system are determined through analysis of the gradients. Further, the “stable” and “unstable” regions of the sensitivity surfaces are identified, with some observations made. Finally, the applications of load capacity gradient functions in structural optimization and form findings are reviewed.

Evaluation of geometrically nonlinear effects due to large cross-sectional deformations of compact and shell-like structures

This paper investigates geometrically nonlinear effects due to large deformations over the cross sections of beam-like and shell-like structures. Finite elements are used to provide numerical solutions along with the Newton–Raphson technique and the arc-length method. Refined theories able to capture cross-sectional deformation are constructed by referring to the Carrera unified formulation. Full nonlinear Green-Lagrange strains and second Piola-Kirchoff stresses are employed in a total Lagrangian scenario. The numerical results demonstrate that geometrical nonlinearities play a fundamental role when cross-sectional deformations become significant and theories of structures with nonlinear kinematics are utilized. In other words, this means that the use of refined beam models may be ineffective if geometrical nonlinear relations are not employed. These phenomena become particularly evident in thin-walled/shell-like type structures.

Analyzing primary resonant dynamics of functionally graded nanoplates based on a surface third-order shear deformation model

In the context of Gurtin-Murdoch (GM) surface elasticity theory, a size-dependent third-order shear deformable plate model is developed herein in order to study the nonlinear forced vibration behavior of rectangular nanoplates with considering surface stress effect. Nanoplates are assumed to be made of functionally graded materials (FGMs) whose properties are graded in the thickness direction based on a power-law distribution. First, the constitutive relations of GM model are matricized. Then, Hamilton’s principle is used to derive the governing equations. The variational differential quadrature, a numerical Galerkin, time periodic discretization, and pseudo arc-length methods are also employed for numerical solution of the geometrically nonlinear forced vibration problem. The frequency-response curves of rectangular nanoplates with different boundary conditions are investigated for different values of thickness, power-law index, surface constants and side length-to-thickness ratio. The results reveal that the surface stress has an important influence on the frequency-response curve of nanoplates at very small scales.

Ultra-Precision Machining of a Compound Sinusoidal Grid Surface Based on Slow Tool Servo.

Compound sinusoidal grid surface with nanometric finish plays a significant role in modern systems and precision calibrator, which can make the systems smaller, the system structure more simple, reduce the cost, and promote the performance of the systems, but it is difficult to design and fabricate by traditional methods. In this paper, a compound freeform surface constructed by a paraboloidal base surface and sinusoidal grid feature surface is designed and machined by slow tool servo (STS) assisted with single point diamond turning (SPDT). A novel combination of the constant angle and constant arc-length method is presented to optimize the cutting tool path. The machining error prediction model is analyzed for fabricating the compound sinusoidal grid surface. A compound sinusoidal grid surface with 0.03 mm amplitude and period of 4 is designed and cutting process is simulated by use of MATLAB software, machining experiment is done on ultra-precision machine tool, the surface profile and topography are measured by Taylor Hobson and Keyence VR-3200, respectively. After dealing with the measurement data of compound freeform surface, form accuracy 4.25 μm in Peak Village value (PV), and surface roughness 89 nm in Ra are obtained for the machined surface. From the theoretical analysis and experimental results, it can be seen that the proposed method is a reasonable choice for fabricating the compound sinusoidal grid surface.