Large Deflection Problems Research Articles

Improved Synchronous Characterization Theory for Surface and Interface Mechanical Properties of Thin-Film/Substrate Systems: A Theoretical Study on Shaft-Loaded Blister Test Technique

In this paper, the previously proposed shaft-loaded blister test technique for the synchronous characterization of the surface and interface mechanical properties of a thin-film/substrate system is further studied theoretically. The large deflection problem of the steady shaft-loaded blistering thin film is reformulated by surrendering the small-rotation-angle assumption of the membrane, which was previously adopted in the out-of-plane and in-plane equilibrium and radial geometric equations. A new and more accurate analytical solution to this large deflection problem is presented and is used to improve the previously presented synchronous characterization theory. The new analytical solution is numerically compared with the previous analytical solution to confirm the superiority of the new analytical solution over the previous analytical solution. An experiment is conducted to verify the beneficial effect of the improved synchronous characterization theory on improving the characterization accuracy.

Improved analytical solutions using optimized basis functions for the large deflection problem of rectangular membranes subjected to pressure load

Improved analytical solutions using optimized basis functions for the large deflection problem of rectangular membranes subjected to pressure load

A mechanics-based model for predicting flexible needle bending with large curvature in soft tissue

Percutaneous insertion is one of the most common minimally invasive procedures. Compared with traditional straight rigid needles, bevel-tipped flexible needle can generate curved trajectories to avoid obstacles and sensitive organs. However, the nonlinear large deflection problem challenges the bending prediction of the needle, which dramatically influences the surgical success rate. This paper analyzed the mechanism of needle-tissue interaction, and established a mechanics-based model of the needle bending during an insertion. And then, a discretization of the bending model was adopted to accurately predict the large bending of the needle in soft tissue. Insertion experiments were conducted to validate the bending prediction model. The results showed that the large needle bending was predicted with the mean/RMSE/maximumu error of 0.42 mm / 0.26 mm / 1.08 mm, which was clinically acceptable. This proved the rationality and accuracy of the proposed model.

Improved Power Series Solution of Transversely Loaded Hollow Annular Membranes: Simultaneous Modification of Out-of-Plane Equilibrium Equation and Radial Geometric Equation

The ability to accurately predict the shape of a transversely loaded hollow annular membrane is essential to the design of bending-free hollow annular shells of revolution, which requires a further improvement in the hollow annular membrane solution to meet the needs of this accurate prediction. In this paper, the large deflection problem of a transversely loaded hollow annular membrane is reformulated by simultaneously modifying the out-of-plane equilibrium equation and radial geometric equation, and a newer and more refined power series solution is derived. The reason why the classical radial geometry equation induces errors is revealed. The convergence and asymptotic behavior of the power series solution obtained is analyzed numerically. The newly derived solution is compared with the two previously derived solutions graphically, showing that the newly derived solution performs basically as well as expected. In addition, the anticipated use of the hollow and not-hollow annular membrane solutions for the design application of bending-free annular shells of revolution is discussed.

An Improved Mathematical Theory for Designing Membrane Deflection-Based Rain Gauges

This paper is devoted to developing a more refined mathematical theory for designing the previously proposed membrane deflection-based rain gauges. The differential-integral equations governing the large deflection behavior of the membrane are improved by modifying the geometric equations, and more accurate power-series solutions of the large deflection problem are provided, resulting in a new and more refined mathematical theory for designing such rain gauges. Examples are presented to illustrate how to analyze the convergence of the power-series solutions and how to numerically calibrate membrane deflection-based linear rain gauges. In addition, some important issues are demonstrated, analyzed, and discussed, such as the superiority of the new mathematical theory over the old one, the reason why the classical geometric equations cause errors, and the influence of changing design parameters on the input–output relationships of rain gauges.

Modeling Method for Static Large Deflection Problem of Curved Planar Beams in Compliant Mechanisms Based on a Novel Governing Equation

Abstract Slender beams serve as typical flexible components to transfer motion, force, and energy in compliant mechanisms. Therefore, the accurate and efficient kinetostatic modeling for the slender beams is highly needed in the synthesis and analysis of compliant mechanisms. Several impressive and great modeling methods have been completed by the pioneering researchers based on the governing equation of slender beams, which can solve the deflection of the beam while the beam-tip loads are known. However, parts of the beam-tip loads are unknown in some scenarios and the traditional governing equation becomes inefficient in these scenarios. The aim of this paper is to propose a novel modeling method for the compliant mechanism, which can solve the large deflection problem without iterative algorithm. In this research, a novel governing equation of slender beams has been established based on Castigliano’s principle and simplified by the differential geometry. In the proposed governing equation, the statically force equilibrium equations and geometric constraint equations between different slender beams can be established in the boundary conditions, therefore the model of compliant mechanisms can be solved directly even if parts of the beam’s tip loads and displacements are unknown. The proposed governing equation provides the deformed shapes and cross-sectional loads of the slender beams, which help the designer to check the mechanical interference and strength in the design process. The numerical examples are presented to demonstrate the feasibility of proposed method.

Highly Accurate Wavelet Solution for Bending and Free Vibration of Circular Plates Over Extra-Wide Ranges of Deflections

Abstract The wavelet multiresolution interpolation Galerkin method in which both the unknown functions and nonlinear terms are approximated by their respective projections onto the same wavelet space is utilized to implement the spatial discretization of the highly coupled and nonlinear Von Karman equation for thin circular plates with various types of boundary conditions and external loads. Newton’s method and the assumption of a single harmonic response are then used for solving the static bending and free vibration problems, respectively. Highly accurate wavelet solutions for an extremely wide range of deflections are finally obtained by the proposed method. These results for moderately large deflections are in good agreement with existing solutions. Meanwhile, the other results for larger deflections are rarely achieved by using other methods. Comparative studies also demonstrate that the present wavelet method has higher accuracy and lower computational cost than many existing methods for solving geometrically nonlinear problems of thin circular plates. Moreover, the solutions for large deflection problems with concentrated load support the satisfactory capacity for handling singularity of the proposed wavelet method. In addition, a trivial initial guess, such as zero, can always lead to a convergent solution in very few iterations, even when the deflection is as large as over 46 times thickness of plate, showing an excellent convergence and stability of the present wavelet method in solving highly nonlinear problems.

New Analytical Solution of the Large Deflection Problem of a Circular Thin Plate with Edge Clamped but Sliding Freely Under a Combined Loading

This paper intends to study the large deflection problem of a circular thin plate with an edge clamped but sliding freely under the combined action of uniformly distributed load and centrally concentrated load by the Adomian decomposition method. If the uniformly distributed load is not decomposed, the parameters obtained by the Adomian decomposition method are the same as the perturbation solution studied by scholars before. If the uniformly distributed load is decomposed into a finite series, a new analysis solution can be obtained through the Adomian decomposition method. From the error comparisons between the new solution in this paper and the perturbation solution, one can see that the result in this paper is better. Finally, the internal stress and deflection are discussed through the graphics based on the new analysis method.

Explicit Solutions to Large Deformation of Cantilever Beams by Improved Homotopy Analysis Method I: Rotation Angle

An improved homotopy analysis method (IHAM) is proposed to solve the nonlinear differential equation, especially for the case when nonlinearity is strong in this paper. As an application, the method was used to derive explicit solutions to the rotation angle of a cantilever beam under point load at the free end. Compared with the traditional homotopy method, the derivation includes two steps. A new nonlinear differential equation is firstly constructed based on the current nonlinear differential equation of the rotation angle and the auxiliary quadratic nonlinear differential equation. In the second step, a high-order non-linear iterative homotopy differential equation is established based on the new non-linear differential equation and the auxiliary linear differential equation. The analytical solution to the rotation angle is then derived by solving this high-order homotopy equation. In addition, the convergence range can be extended significantly by the homotopy–Páde approximation. Compared with the traditional homotopy analysis method, the current improved method not only speeds up the convergence of the solution, but also enlarges the convergence range. For the large deflection problem of beams, the new solution for the rotation angle is more approachable to the engineering designers than the implicit exact solution by the Euler–Bernoulli law. It should have significant practical applications in the design of long bridges or high-rise buildings to minimize the potential error due to the extreme length of the beam-like structures.

The Large Deflection Solution of Circular Elastic Membrane under Composite Loads

In this paper, the large deflection problem of axisymmetric isotropic elastic membranes subjected to the composite loads is studied. The Hencky transformation is extended and the general solution of large deflection of the membranes under the composite loads is obtained. The general solution is given in different forms in different values of loads. The obtained results of the numerical values of elastic characters can be applied in practice.

Экспериментальное и аналитическое исследование геометрически-нелинейного изгиба консоли под действием распределенной нагрузки гравитационного типа

This paper describes an approximate analytical solution for the geometrically nonlinear bending of a thin elastic cantilever beam under a uniformly distributed gravity load. The solution is based on the linearized Euler-Bernoulli equation of mechanics of materials. Traditionally, such a linear approach is used for small (geometrically linear) deflections. The authors have modified the original equation with an arc-length preservation condition. The modified solution allows one to obtain bending shapes, deflection, and axial displacement in the range of loads corresponding to geometrically nonlinear bending of a beam (large deflections). An experimental study is conducted to verify the proposed solution. A thin steel band bent by gravity is used as a sample. Changes in the length of the bent sample part allow one to obtain various dimensionless load parameters. The deflections and axial displacements averaged on experimental statistics are determined. Bending shapes are obtained by the least square method of 5th order. Experimental and theoretical data are shown to be in good agreement. This fact confirms that the approximate analytical solution can be applied to solve large deflection problems in a wider range of loads than normally considered in the original linear theory.

Large Deflection Analysis of Peripherally Fixed Circular Membranes Subjected to Liquid Weight Loading: A Refined Design Theory of Membrane Deflection-Based Rain Gauges.

The anticipated use of elastic membranes for deflection-based rain gauges has provided an impetus for this paper to revisit the large deflection problem of a peripherally fixed circular membrane subjected to liquid weight loading, a statics problem when the fluid–structure interaction of membrane and liquid reaches static equilibrium. The closed-form solution of this statics problem of fluid–structure interaction is necessary for the design of such membrane deflection-based rain gauges, while the existing closed-form solution, due to the use of the small rotation angle assumption of the membrane, cannot meet the design requirements for computational accuracy. In this paper, the problem under consideration is reformulated by giving up the small rotation angle assumption, which gives rise to a new and somewhat intractable nonlinear integro-differential equation of the governing out-of-plane equilibrium. The power series method has played an irreplaceable role in analytically solving membrane equations involving both integral and differential operations, and a new and more refined closed-form solution without the small rotation angle assumption is finally presented. Numerical examples conducted show that the new and more refined closed-form solution presented has satisfactory convergence, and the effect of giving up the small rotation angle assumption is also investigated numerically. The application of the closed-form solution presented in designing such membrane deflection-based rain gauges is illustrated, and the reliability of the new and more refined closed-form solution presented was confirmed by conducting a confirmatory experiment.

Large Deformation Problem of Bimodular Functionally-Graded Thin Circular Plates Subjected to Transversely Uniformly-Distributed Load: Perturbation Solution without Small-Rotation-Angle Assumption

In this study, the large deformation problem of a functionally-graded thin circular plate subjected to transversely uniformly-distributed load and with different moduli in tension and compression (bimodular property) is theoretically analyzed, in which the small-rotation-angle assumption, commonly used in the classical Föppl–von Kármán equations of large deflection problems, is abandoned. First, based on the mechanical model on the neutral layer, the bimodular functionally-graded property of materials is modeled as two different exponential functions in the tensile and compressive zones. Thus, the governing equations of the large deformation problem are established and improved, in which the equation of equilibrium is derived without the common small-rotation-angle assumption. Taking the central deflection as a perturbation parameter, the perturbation method is used to solve the governing equations, thus the perturbation solutions of deflection and stress are obtained under different boundary constraints and the regression of the solution is satisfied. Results indicate that the perturbation solutions presented in this study have higher computational accuracy in comparison with the existing perturbation solutions with small-rotation-angle assumption. Specially, the computational accuracies of external load and yield stress are improved by 17.22% and 28.79% at most, respectively, by the numerical examples. In addition, the small-rotation-angle assumption has a great influence on the yield stress at the center of the bimodular functionally-graded circular plate.

Numerical solution of large deflection beams by using the Laplace Adomian decomposition method

PurposeThis paper presents the problems using Laplace Adomian decomposition method (LADM) for investigating the deformation and nonlinear behavior of the large deflection problems on Euler-Bernoulli beam.Design/methodology/approachThe governing equations will be converted to characteristic equations based on the LADM. The validity of the LADM has been confirmed by comparing the numerical results to different methods.FindingsThe results of the LADM are found to be better than the results of Adomian decomposition method (ADM), due to this method's rapid convergence and accuracy to obtain the solutions by using fewer iterative terms. LADM are presented for two examples for large deflection problems. The results obtained from example 1 shows the effects of the loading, horizontal parameters and moment parameters. Example 2 demonstrates the point loading and point angle influence on the Euler-Bernoulli beam.Originality/valueThe results of the LADM are found to be better than the results of ADM, due to this method's rapid convergence and accuracy to obtain the solutions by using fewer iterative terms.

New approximate analytical solution of the large deflection problem of an uniformly loaded thin circular plate with edge simply hinged

In this paper, the large deflection problem of an uniformly loaded circular plate with edge simply hinged is considered. First, a new algorithm for the deflection problem is proposed based on Adomian decomposition method. Then, a new approximate analytic solution of the large deflection problem is obtained by the algorithm. Final, its bending moments are analyzed corresponding to different center deflections. And the relationship between the center deflection and the external load is discussed. Through comparing the errors of the obtained approximate solution with the previous perturbation solution and homotopy perturbation solution, one can see that the obtained solution in this paper is a better one and the result in this paper has a certain practical value.

Closed-Form Solution for Circular Membranes under In-Plane Radial Stretching or Compressing and Out-of-Plane Gas Pressure Loading

The large deflection phenomenon of an initially flat circular membrane under out-of-plane gas pressure loading is usually involved in many technical applications, such as the pressure blister or bulge tests, where a uniform in-plane stress is often present in the initially flat circular membrane before deflection. However, there is still a lack of an effective closed-form solution for the large deflection problem with initial uniform in-plane stress. In this study, the problem is formulated and is solved analytically. The initial uniform in-plane stress is first modelled by stretching or compressing an initially flat, stress-free circular membrane radially in the plane in which the initially flat circular membrane is located, and based on this, the boundary conditions, under which the large deflection problem of an initially flat circular membrane under in-plane radial stretching or compressing and out-of-plane gas pressure loading can be solved, are determined. Therefore, the closed-form solution presented in this paper can be applied to the case where the initially flat circular membrane may, or may not, have a uniform in-plane stress before deflection, and the in-plane stress can be either tensile or compressive. The numerical example conducted shows that the closed-form solution presented has satisfactory convergence.

Kinetostatics modeling and analysis of parallel continuum manipulators

Parallel continuum manipulators (PCMs) have increasingly attracted attentions in the field of mechanism and robotics, owing to their inherent merits, such as compliance in structure, reduction of backlash, and easiness in implementation. This paper presents a general approach for the kinetostatics modeling and analysis of spatial PCMs. A discretization-based technique is employed for the nonlinear large-deflection problems of the coupled slender flexible links in these flexible parallel manipulators. Benefiting from the mechanism approximation to the force-deflection characteristics of the flexible links, the kinetostatics models of the PCMs can be established analytically in the realm of robot kinematics/statics, rather than continuum mechanics. Moreover, a closed-form solution to the gradient of these nonlinear algebraic equations can be derived, such that gradient-based searching algorithms can be implemented to efficiently determine the equilibrium configurations in a variety of given conditions. A prototype of a three-limb 6-DOF PCM is developed, on which validation experiments are conducted. And the results verify the effectiveness of the proposed method for the kinetostatics modeling and analysis of parallel continuum manipulators.

Numerical analysis of large deflection of the cantilever beam subjected to a force pointing at a fixed point

In many cable-driven compliant mechanisms, the need for controlled motion in the flexible structure often demands the actuation cables to pass through a fixed point (e.g. a cable routing channel in a segment disk or a fixed pulley) to constrain the force angle on the cable. This condition can be modeled as the large deflection problem of a cantilever beam with two parameters, namely the distal slope and the force angle, which is more complex than the regular one-parameter deflection problem of cantilever beams. In order to solve this mechanics problem, a numerical method based on variable transformation and heuristic cuckoo search (CS) algorithm is presented to analyze the large deflection shape of a beam subjected to a tip force pointing at a fixed point. Firstly, a two-parameter second-order differential equation governing the cantilever beam with large deflection and tip force constrained to a particular point is established. Secondly, a new variable is defined to replace the two parameters based on the variable transformation, allowing the governing equation to be simplified to a one-parameter second-order differential equation. Thirdly, this mathematical problem is written as a one-parameter optimization problem that can be solved by the CS algorithm. The proposed approach is then verified by commercial finite element analysis solution. Comprehensive simulation results and analysis are presented under different combinations of critical parameters, leading to potentially more accurate model-based design for compliant structures under the studied force constraint.

A Stiffness Variable Passive Compliance Device with Reconfigurable Elastic Inner Skeleton and Origami Shell

Devices with variable stiffness are drawing more and more attention with the growing interests of human-robot interaction, wearable robotics, rehabilitation robotics, etc. In this paper, the authors report on the design, analysis and experiments of a stiffness variable passive compliant device whose structure is a combination of a reconfigurable elastic inner skeleton and an origami shell. The main concept of the reconfigurable skeleton is to have two elastic trapezoid four-bar linkages arranged in orthogonal. The stiffness variation generates from the passive deflection of the elastic limbs and is realized by actively switching the arrangement of the leaf springs and the passive joints in a fast, simple and straightforward manner. The kinetostatics and the compliance of the device are analyzed based on an efficient approach to the large deflection problem of the elastic links. A prototype is fabricated to conduct experiments for the assessment of the proposed concept. The results show that the prototype possesses relatively low stiffness under the compliant status and high stiffness under the stiff status with a status switching speed around 80 ms.

Analytical and experimental investigations on large deflection analysis of composite cantilever beams

An analytical study for the problem of large deflection in the structural composite components was applied in this work due to the lack of similar investigations. The large deflection response of the composite cantilever beam was investigated using analytical and experimental studies. The analytical formulation of this work was intended to be validated for composite laminated beams by comparing the results with those obtained in the literature. Several bending experiments were performed on isotropic Plexiglass (PG) and random short fiber with unsaturated polyester resin (RSM) cantilever beams, where, the obtained results were well-founded with the proposed analytical model.