Stiffness Matrix For Analysis Research Articles

Dynamic modeling and optimization of mechanical drift for compliant parallel micromanipulation driven by V-shape linear ultrasonic motors

Designing micro-mechanisms that satisfy large strokes, high precision, and multiple degrees of freedom has consistently been a research focus and challenge in the field of micromanipulation systems. Linear ultrasonic motors (LUMs), with advantages such as millimeter-level travel, nanometer-level motion resolution, and power-off self-locking, significantly contribute to the advancement in the mechanism design of micromanipulation systems. However, the challenge of mechanical drift lacks a comprehensive theoretical analysis to explain its cause and limits its application. This paper presents a mechanical drift modeling method for a compliant parallel micromanipulation (CPM) driven by a LUM. The model primarily focuses on the creep characteristics of the flexure clamp element in the LUM. Based on the stiffness matrix analysis method of the compliant parallel mechanism kinematics, the mechanical drift dynamics model of the CPM is established. The influence of the main structural parameters on the CPM mechanical drift is investigated. The research indicates that the creep of flexure clamps in the CPM is the main factor leading to the mechanical drift. Optimization of mechanical drift is achieved by choosing flexure clamps with higher tangential stiffness for LUM and flexure hinge joints with greater flexibility for CPM. Finally, experimental studies reveal that under equivalent conditions, the mechanical drift of the optimized CPM was significantly reduced than before, proving the rationality of the theoretical model and optimization method.

Advanced structural health monitoring and damage detection using inverse methods and perturbation stiff-ness matrix analysis

Sensor technology advancements have the health monitoring task for modern intricate structures introduce many challenges, primarily centered around the augmentation of safety margins and operational dependabil-ity. The demand for diverse systems or methodologies is, therefore, increasing. These systems or methodolo-gies should be capable of detecting damage, monitoring remotely, and conducting continuous evaluations in real-time. As part of this study, inverse techniques rooted in wave propagation analysis within structural frameworks are utilized to identify damage response vectors. Additionally, mathematical formulations grounded in minimal rank perturbation theory facilitate determining a perturbed stiffness matrix for a com-promised structure. Various excitations are included in the analysis, which is based on intricate non-linear structural modeling. Using streamlined procedures, these findings confirm the inherent versatility of a range of methodologies capable of meeting both rapid monitoring needs and exhaustive analytical needs. In addition, the proposed approach promotes the use of intelligent structures embedded with a variety of sensors that are not restricted by limitations related to sensor type or spatial deployment. Monitoring structural health and detecting damage could be significantly improved by this strategic deployment.

Heterogeneous ultrasonic time-of-flight distribution in multidirectional CFRP corner and its implementation into total focusing method imaging

A precise time-of-flight (TOF) of phased array ultrasonic is crucial to the high-quality total focusing method (TFM) imaging in the corner of multidirectional carbon fiber reinforced plastics, but quite challenging owing to the complex material and geometric factors. With a comprehensive consideration of the curved shape, laminated structure, and elastic anisotropy, we propose a high-fidelity finite element model of the corner based on the stiffness matrix analysis. A strategy in virtue of the maximum energy of A-scan signals, is further implemented to determine the TOF and compared with the exiting minimum time method. The resulted heterogeneous TOF distributions are input into TFM imaging, respectively. Results show that of an isotropic approximation presents a larger positioning error and a higher noise level of defects. In contrast, the anisotropic algorithms, especially for the maximum energy method, obviously improve the defect echo amplitude and suppress the sub-surface noise, demonstrating a superior performance in CFRP characterization.

Analytical Determination and Influence Analysis of Stiffness Matrix of Ball Bearing under Different Load Conditions

Bearing stiffness, as one of the most important service characteristics for ball bearing, plays a crucial role in the bearing design and rotor dynamic analysis. To rapidly and accurately calculate the stiffness matrix of ball bearing under the arbitrary load conditions, a 5-DOF analytical model for bearing stiffness matrix analysis has been established by the ball–raceway contact analysis, implicit/explicit differential method, and matrix operations. The model has been validated comparing with the previous methods and experimental results. Based on this, the model has been used to investigate the influences of the load and operation conditions, the structural parameters variation on stiffness of ball bearing. The results show that property increasing axial preload can inhibit the attenuation of speed-varying stiffness, and the contact states between balls and raceways also have significant influence on the change in the stiffness of ball bearings. Besides, a larger curvature coefficient of inner raceway and a small curvature coefficient of outer raceway can effectively improve the stiffness of ball bearing at high speed. Therefore, the proposed method can be a useful tool in bearing optimize design and performance analysis of ball and rotor system under various load conditions.

The stiffness characteristics analysis of the duplex angular contact ball bearings based on a comprehensive multi-degree-of-freedom mathematical model

Although the duplex angular contact ball bearings (DACBBs) are widely used in industry, the published mathematical model of DACBBs is sparse. To analyze the stiffness characteristics of DACBBs, this paper proposes a comprehensive multi-degree-of-freedom (multi-DOF) mathematical model for the DACBBs in three traditional configurations. The implicit differential method is adopted to derivate the analytical stiffness matrix formulation of DACBBs. The geometrical constraints and interactions inside the DACBBs are presented based on the vector-and-matrix method. The detailed iterative algorithm with two layers for solving the presented model is given based on the Newton-Raphson method. A systematic study for evaluating the impacts of angular misalignment on the stiffness characteristics of DACBBs has been carried out under three typical load conditions: the pure axial load condition, the pure radial load condition, and the combined-loaded condition. The results indicate that the angular misalignment remarkably affects the stiffness characteristics of DACBBs. Under the pure axial and radial load conditions, the stiffness curves of DACBBs can be divided into two segments according to the value of misalignment angle, and the angular misalignment will be the predominant factor when the angular misalignment is great. Under the combined-loaded condition, the mode (single or combined angular misalignment) and the direction of angular misalignment significantly influence the stiffness characteristics of DACBBs. Also, the angular misalignment effects depend on the radial load. The proposed model can also analyze the mechanical performance of double-row angular contact ball bearings.

Study on Environmental Effects Induced by Quasirectangle Shield Tunnelling with Analytical Stiffness Matrix

During the quasirectangle shield tunnelling, the soil around the tunnel is inevitably disturbed which results in different degrees of additional stresses and displacement of the soil. In this paper, the environmental effects caused by the construction of quasirectangle shield are analyzed by the analytical stiffness matrix method, focusing on the law of the additional stress field and the deformation of the multilayered soil induced by the construction of a quasirectangle shield. The results show that the additional stresses field has localized characteristics around the quasirectangle shield. The further away from the excavation face, the smaller the stress is. Also, shell friction and synchronous grouting pressure have a significantly greater influence on the additional stress field. Finally, the horizontal settlement groove of the quasirectangle shield is roughly symmetrical and parabolic along the tunnel axis, with a groove width range of about 20m. Synchronous grouting pressure has a greater effect on longitudinal ground deformation.

Mechanical metric for skeletal biomechanics derived from spectral analysis of stiffness matrix

A new metric for the quantitative and qualitative evaluation of bone stiffness is introduced. It is based on the spectral decomposition of stiffness matrix computed with finite element method. The here proposed metric is defined as an amplitude rescaled eigenvalues of stiffness matrix. The metric contains unique information on the principal stiffness of bone and reflects both bone shape and material properties. The metric was compared with anthropometrical measures and was tested for sex sensitivity on pelvis bone. Further, the smallest stiffness of pelvis was computed under a certain loading condition and analyzed with respect to sex and direction. The metric complements anthropometrical measures and provides a unique information about the smallest bone stiffness independent from the loading configuration and can be easily computed by state-of-the-art subject specified finite element algorithms.

Exploration of Translational Joint Design Using Corrugated Flexure Units With Bézier Curve Segments

In order to satisfy particular design specifications, shape variation for limited geometric envelopes is often employed to alter the elastic properties of flexure joints. This paper introduces an analytical stiffness matrix method to model a new type of corrugated flexure (CF) beam with cubic Bézier curve segments. The cubic Bézier curves are used to depict the segments combined to form CF beam and translational joint. Mohr's integral is applied to derive the local-frame compliance matrix of the cubic Bézier curve segment. The global-frame compliance matrices of the CF unit and the CF beam with cubic Bézier curve segments are further formed by stiffness matrix method, which are confirmed by finite element analysis (FEA). The control points of Bézier curve are chosen as optimization parameters to identify the optimal segment shape, which maximizes both high off-axis/axial stiffness ratio and large axial displacements of translational joint. The results of experimental study on the optimum translational joint design validate the proposed modeling and optimization method.

An extension of a discrete-module-beam-bending-based hydroelasticity method for a flexible structure with complex geometric features

The discrete-module-beam-bending based hydroelasticity method proposed by Lu et al. (2016) deals with the hydroelastic response of a flexible structure by discretising it into several rigid submodules which are connected by Euler-Bernoulli beam elements. A stiffness matrix is determined in terms of the geometrical and physical properties of the flexible structure for each beam element and it has an analytical form for simple geometric features such as a rectangular cross section unchanged along the longitudinal direction. However, the analytical stiffness matrix may not exist for a flexible structure with complex geometric features. In the present study, with the aid of finite element method, the discrete-module-beam-bending-based hydroelasticity method proposed by Lu et al. (2016) is extended to be applicable for a flexible structure with complex geometric features.

Analytical Stiffness Matrix for Curved Metal Wires

The paper presents an analytic stiffness matrix for curved thin metal wires, derived by the application of the second Castigliano’s Theorem. The matrix accounts both bending and axial stiffness contributions in plane. The beam geometry is described by a cubic polynomial function of the curvature radius with a monotonical attitude angle as the independent variable. The solution proposed if fully analytical although a consistent number of adding factors appear. Some test cases are discussed and compared with Finite Element solutions, formed by a plentiful assembly of straight beams.

In-plane vibration response of time and frequency domain with rigid-elastic coupled tire model with continuous sidewall

The in-plane vibration characteristic of time and frequency domain for heavy-loaded radial tire with a larger flat ratio (close to 1) is researched by utilizing the rigid-elastic coupled tire model with continuous sidewall. The sidewall bending stiffness is considered and the flexible beam on the elastic continuous beam tire model is proposed and investigated analytically to simulate the in-plane vibration of the heavy-loaded radial tire within more wider frequency band. The rigid-elastic coupled tire model is derived with finite difference method and the analytical stiffness matrix; mass matrix is formed based on the geometrical and structural parameters of heavy-loaded radial tire. Structural parameters are identified utilizing genetic algorithm based on the error between the analytical and experimental modal frequency. In-plane frequency domain transfer function and time domain dynamics response of heavy-loaded radial tire is investigated and compared with the experimental result. Experimental and theoretical results show that in-plane rigid-elastic coupled tire model with sidewall bending stiffness can be used to precisely predict the transfer function and vibration feature within the frequency band of 300 Hz, compared with the tire model with the distributed independent sidewall element. The flexible beam on the elastic continuous beam tire model and rigid-elastic coupled tire model with continuous sidewall can be extended to the dynamic analysis of the tire with larger flat ratio or the tire under the impulsive loading conditions.

Analytical Stiffness Matrix Method for Flexural Vibration of Compression Bar

According to differential equation for transverse flexural vibration of compression bar considering second-order effect and inertia force, displacement vector expression was achieved. Based on the displacement boundary condition, displacement coefficient expressed by nodal displacement vector was obtained. Internal force equations of compression bar were established and then internal force at bar ends expressed by nodal displacement vector was provided. Finally, dynamic stiffness matrix colligating mass matrix and geometry matrix was given and can be applied to accurate analysis for dynamic performance and dynamic responses of bar.

Design and Analyze a New Measuring Lift Device for Fin Stabilizers Using Stiffness Matrix of Euler-Bernoulli Beam.

Fin-angle feedback control is usually used in conventional fin stabilizers, and its actual anti-rolling effect is difficult to reach theoretical design requirements. Primarily, lift of control torque is a theoretical value calculated by static hydrodynamic characteristics of fin. However, hydrodynamic characteristics of fin are dynamic while fin is moving in waves. As a result, there is a large deviation between actual value and theoretical value of lift. Firstly, the reasons of deviation are analyzed theoretically, which could avoid a variety of interference factors and complex theoretical derivations. Secondly, a new device is designed for direct measurement of actual lift, which is composed of fin-shaft combined mechanism and sensors. This new device can make fin-shaft not only be the basic function of rotating fin, but also detect actual lift. Through analysis using stiffness matrix of Euler-Bernoulli beam, displacement of shaft-core end is measured instead of lift which is difficult to measure. Then quantitative relationship between lift and displacement is defined. Three main factors are analyzed with quantitative relationship. What is more, two installation modes of sensors and a removable shaft-end cover are proposed according to hydrodynamic characteristics of fin. Thus the new device contributes to maintenance and measurement. Lastly, the effectiveness and accuracy of device are verified by contrasting calculation and simulation on the basis of actual design parameters. And the new measuring lift method can be proved to be effective through experiments. The new device is achieved from conventional fin stabilizers. Accordingly, the reliability of original equipment is inherited. The alteration of fin stabilizers is minor, which is suitable for engineering application. In addition, the flexural properties of fin-shaft are digitized with analysis of stiffness matrix. This method provides theoretical support for engineering application by carrying out finite element analysis with computers.

Optimal Bicubic Finite Volume Methods on Quadrilateral Meshes

AbstractIn this paper, an optimal bicubic finite volume method is established and analyzed for elliptic equations on quadrilateral meshes. Base on the so-called elementwise stiffness matrix analysis technique, we proceed the stability analysis. It is proved that the new scheme has optimal convergence rate in H1 norm. Additionally, we apply this analysis technique to bilinear finite volume method. Finally, numerical examples are provided to confirm the theoretical analysis of bicubic finite volume method.

Compliance modeling and analysis of a 3-RPS parallel kinematic machine module

The compliance modeling and rigidity performance evaluation for the lower mobility parallel manipulators are still to be remained as two overwhelming challenges in the stage of conceptual design due to their geometric complexities. By using the screw theory, this paper explores the compliance modeling and eigencompliance evaluation of a newly patented 1T2R spindle head whose topological architecture is a 3-RPS parallel mechanism. The kinematic definitions and inverse position analysis are briefly addressed in the first place to provide necessary information for compliance modeling. By considering the 3-RPS parallel kinematic machine(PKM) as a typical compliant parallel device, whose three limb assemblages have bending, extending and torsional deflections, an analytical compliance model for the spindle head is established with screw theory and the analytical stiffness matrix of the platform is formulated. Based on the eigenscrew decomposition, the eigencompliance and corresponding eigenscrews are analyzed and the platform’s compliance properties are physically interpreted as the suspension of six screw springs. The distributions of stiffness constants of the six screw springs throughout the workspace are predicted in a quick manner with a piece-by-piece calculation algorithm. The numerical simulation reveals a strong dependency of platform’s compliance on its configuration in that they are axially symmetric due to structural features. At the last stage, the effects of some design variables such as structural, configurational and dimensional parameters on system rigidity characteristics are investigated with the purpose of providing useful information for the structural design and performance improvement of the PKM. Compared with previous efforts in compliance analysis of PKMs, the present methodology is more intuitive and universal thus can be easily applied to evaluate the overall rigidity performance of other PKMs with high efficiency.

Wave interpretation of numerical results for the vibration in thin conical shells

The dynamic behaviour of thin conical shells can be analysed using a number of numerical methods. Although the overall vibration response of shells has been thoroughly studied using such methods, their physical insight is limited. The purpose of this paper is to interpret some of these numerical results in terms of waves, using the wave finite element, WFE, method. The forced response of a thin conical shell at different frequencies is first calculated using the dynamic stiffness matrix method. Then, a wave finite element analysis is used to calculate the wave properties of the shell, in terms of wave type and wavenumber, as a function of position along it. By decomposing the overall results from the dynamic stiffness matrix analysis, the responses of the shell can then be interpreted in terms of wave propagation. A simplified theoretical analysis of the waves in the thin conical shell is also presented in terms of the spatially-varying ring frequency, which provides a straightforward interpretation of the wave approach. The WFE method provides a way to study the types of wave that travel in thin conical shell structures and to decompose the response of the numerical models into the components due to each of these waves. In this way the insight provided by the wave approach allows us to analyse the significance of different waves in the overall response and study how they interact, in particular illustrating the conversion of one wave type into another along the length of the conical shell.

A theoretical study of weight-balanced mechanisms for design of spring assistive mobile arm support (MAS)

This paper studies the underlying theory of weight-balanced mechanism for the design of a class of spatial mobile arm support (MAS), a spring assistive MAS. Conventional designs of spring assistive MAS and their associated spring balancing techniques are analyzed based on the stiffness matrix analysis in order to highlight the structural novelty of the proposed MAS design concept. This MAS comprises two ideal zero-free-length springs directly installed to the arm mechanism without using any auxiliary link. Through the passive assistance provided by the springs, the MAS can facilitate the arm movement in space by the complete weight compensation of the upper limb at any possible posture. The design is believed to have benefited from its simple structure and the easiness of adjustment compared to other conventional designs. The conceptual design of the MAS is proposed and followed with a simulation model. The gravity balancing is verified with an example of a quasi-static motion. The results show that the MAS is capable of fully balancing the weights of user's upper limb and the device during the full range of motion.

Proximal-point method for finite element model updating problem

The problem of finding the optimal approximation to analytical stiffness matrix modeled by the finite element method is considered in this paper. Desired matrix properties, including satisfaction of the dynamic equation, symmetry, positive semidefiniteness and physical connectivity, are imposed as side constraints of the minimization problem. To the best of the author's knowledge, the finite element model updating problem containing all these constraints simultaneously has not been proposed in the literature earlier. By partial Lagrangian multipliers technique, the optimization problem is transformed into a matrix linear variational inequality and the proximal-point method is first used to solve the equivalent problem. The results of numerical examples show that the proposed method works well even for incomplete measured data.

Matrix linear variational inequality approach for finite element model updating

The problem of finding the least change adjustment to analytical stiffness matrix modeled by finite element method is considered in this paper. Desired matrix properties, including satisfaction of the dynamic equation, symmetry, positive semidefiniteness and sparse pattern, are imposed as side constraints of the nonlinear optimization problem. To the best of the author's knowledge, the matrix updating problem containing all these constraints simultaneously has not been proposed in the literature earlier. By partial Lagrangian multipliers technique, the optimization problem is first reformulated as an equivalent matrix linear variational inequality (MLVI) and solved by extended projection and contraction method. The results of numerical examples show that the proposed method works well even for incomplete measured data.

Fast dynamic macro element and finite element 2D coupled model for an electromagnetic launcher study

PurposeThe modelling of the electromagnetic devices with moving objects such as launchers, electrical machines and actuators, necessitates considering the motion. This paper aims to examine this subject.Design/methodology/approachThis task can be performed by introducing the velocity term in the electromagnetic equation. However, the application of this method leads to a non‐symmetrical finite element matrix. This numerical problem can be avoided either by the finite element meshing domain every displacement step or by using special techniques coupled to the finite element method like the moving bound, sliding surface and macro element (ME). The ME solution, based on an analytical model in the air‐gap of the devices, is more solicited for its low cost and accuracy by comparison with the other one. This technique keeps unchanged the finite element topology during the simulation, where the motion is taken into account by modification of the ME formula's every displacement.FindingsThis paper sought to present a new formula of the ME which is called dynamic ME. This new formula keeps unchanged the finite element topology and the terms of the analytical stiffness matrix too during the movement simulation.Research limitations/implicationsThe developed model is limited to analyzing the 2D devices with moving objects in linear or non‐linear case with saturation of the magnetic circuits. Extending the model to consider the 3D effects is the perspective of this work.Originality/valueThe developed formula is more economical than the classical one.