Strong Geometric Nonlinearity Research Articles

A Foundational Study on Rational Optimization of Damping Ratio for Accurate Dynamic Simulation with Ultra Large Displacement

The integration of dynamic simulation analysis has become widespread in general-purpose software, providing enhanced capabilities. However, accurately tracking deformations based on complete equilibrium solutions remains a significant challenge in problems characterized by strong geometric nonlinearity. This study examines the accuracy of the combined Newmark β method and Tangent stiffness method in dynamic analysis with ultra large displacements and evaluates the utility of Rayleigh proportional damping in numerical simulations compared to experimental models. An experimental model of a slender steel plate undergoing free vibrations after being released from a deformed state was created. Video footage capturing the deformation histories was compared to computational simulations to verify accuracy. The study also examines the appropriate values of the damping ratio (ζ) and the Newmark β value in the simulations. The results indicate that adjusting various damping ratio and a β value of 1/2 yield more realistic simulations with longer conservation of mechanical energy. The findings suggest that incorporating numerical damping into actual damping settings can achieve a more realistic simulation of dynamic behaviour with ultra-large displacements. Furthermore, experiments and analyses were performed to correct natural frequency by changing Young’s modulus, which is a key factor influencing natural frequencies, with observed correlations to plate thickness, setting the stage for further research under varied conditions to develop a more rational methodology for structural analysis.

Hierarchical shape optimization for large deployable membrane reflector with spatial skirt cable

In the shape optimization analysis of electrostatically controlled deployable membrane reflector (ECDMR), the poor balance of the membrane with spatial skirt cable is the main reason for the degradation of the reflector surface precision. At the same time, while in the optimization of the pre-stress of large deployable reflectors, the membrane structure has strong geometric nonlinearity, and the lack of necessary gradient information leads to low efficiency of the calculation. Therefore, according to the structural characteristics of the cable-membrane reflector, the two optimization objectives of the highest surface precision and the most uniform stress distribution is considered, and a hierarchical shape optimization method is proposed based on membrane theory and unbalance force analysis, which realizes the solution of the multi-objective optimization problem when the three optimization objectives have different priorities. The interior effective membrane reflective surface is analyzed using membrane theory, and the external membrane with spatial skirt cable is analyzed using nonlinear finite element method. The gradient information required for optimization is also obtained by unbalance force analysis, which ensures better optimization efficiency and can solve large-scale optimization problems. Therefore, the node positions and element equilibrium stresses of the structure can be optimized simultaneously, which further expands the search space of the equilibrium state and reduces the possibility of falling into local optimal solutions. Finally, a numerical example of a 30 m aperture ECDMR is calculated and analyzed, and the results show that the proposed method can obtain a high surface precision with a fast solution speed.

Review of Wind-Induced Effects Estimation through Nonlinear Analysis of Tall Buildings, High-Rise Structures, Flexible Bridges and Transmission Lines

The nonlinear effects exhibited by structures under the action of wind loads have gradually stepped into the vision of wind-resistant researchers. By summarizing the prominent wind-induced nonlinear problems of four types of wind-sensitive structures, namely tall buildings, high-rise structures, flexible bridges, and transmission lines, the occurrence mechanism of their nonlinear effects is revealed, providing cutting-edge research progress in theoretical studies, experimental methods and vibration control. Aerodynamic admittance provides insights into the aerodynamic nonlinearity (AN) between the wind pressure spectrum and wind speed spectrum of tall building surfaces. The equivalent nonlinear equation method is used to solve nonlinear vibration equations with generalized van-der-Pol-type aerodynamic damping terms. The elastic–plastic finite element method and multiscale modeling method are widely employed to analyze the effects of geometric nonlinearity (GN) and material nonlinearity (MN) at local nodes on the wind-induced response of latticed tall structures. The AN in blunt sections of bridges arises from the amplitude dependence of the aerodynamic derivative and the higher-order term of the self-excited force. Volterra series aerodynamic models are more suitable for the nonlinear aerodynamic modeling of bridges than the polynomial models studied more in the past. The improved Lindstedt–Poincare perturbation method, which considers the strong GN in the response of ice-covered transmission lines, offers high accuracy. The complex numerical calculations and nonlinear analyses involved in wind-induced nonlinear effects continue to consume significant computational resources and time, especially for complex wind field conditions and flexible and variable structural forms. It is necessary to further develop analytical, modeling and identification tools to facilitate the modeling of nonlinear features in the future.

Modeling and analysis of assembly variation with non-uniform stiffness condensation for large thin-walled structures

The increase of thickness for large thin-walled structure causes the nonlinear enhancement of local structural stiffness, which leads to that the variation is concentrated on the local assembled area. The strong geometrical nonlinearity exists in the assembly process, which causes the non-uniform constraint for the material in assembled area and the variation of assembled large thin-walled structure is difficultly predicted. In this paper, the large thin-walled structure is partitioned into the assembled area and non-assembled area. The nonlinear stiffness matrix of each area is constructed based on continuum mechanics theory. The large non-assembled area is condensed as the nonlinear stiffness constraint and attached on the small assembled area. The non-uniform stiffness condensation method is developed and a new assembly variation model is proposed for large thin-walled structure with high local stiffness. The initial bending and torsional variations are constructed according to the manufacturing process of thin-walled structure and the assembly variations of two large cylindrical thin-walled structures are calculated. The predicted results by using the non-uniform condensation model are compared with that by the commercial software and the classic method of influence coefficients, in which the accuracy of the proposed model is verified. The proposed model may predict the local variation characteristics of large thin-walled structures effectively and is significant for the analysis of variation propagation in the series assembly process of large thin-walled structures.

Wind-Induced Response and Equivalent Static Wind Load of Transmission Lines Considering the Location Updating Effect

Transmission lines will significantly deviate from the initial location under wind loadings due to the strong geometric nonlinearity of the conductor. Ignoring the location updating (LU) effect can greatly reduce the modeling accuracy of wind load. This paper proposes a framework to quantify the influence of the LU effect on wind-induced responses and equivalent static wind load (ESWL), and emphasizes the need to consider this effect through systematic discussions. To facilitate the actual engineering design, a simplified method based on the rigid pendulum assumption is developed to modify the ESWL. The results indicate that the mean wind load is amplified by the LU effect, and the amplification effect becomes increasingly noteworthy with increasing basic wind speed. Conversely, the gust response factor is hardly affected. After considering the LU effect, the average component of various responses increases to varying degrees, particularly the horizontal tension and lateral reaction. The simplified method has a relative error of only -0.45% under the design condition and exhibits great robustness to the height difference. Compared with the methods recommended by various codes, the proposed method provides a more accurate estimation of ESWL, which helps to balance the contradiction between structural safety and construction cost.

An investigation into model extrapolation and stability in the system identification of a nonlinear structure

Estimating a nonlinear model from experimental measurements of a vibrating structure remains a challenge, despite huge progress in recent years. A major issue is that the dynamical behaviour of a nonlinear structure strongly depends on the magnitude of the displacement response. Thus, the validity of an identified model is generally limited to a certain range of motion. Also, outside this range, the stability of the solutions predicted by the model are not guaranteed. This raises the question as to how a nonlinear model derived using data from relatively low amplitude excitation can be used to predict the dynamical behaviour for higher amplitude excitation. This paper focuses on this problem, investigating the extrapolation capabilities of data-driven nonlinear state-space models based on a subspace approach. The experimental vibrating structure consists of a cantilever beam in which magnets are used to generate strong geometric nonlinearity. The beam is driven by an electrodynamic shaker using several levels of broadband random noise. Acceleration data from the beam tip are used to derive nonlinear state-space models for the structure. It is shown that model predictions errors generally tend to increase when extrapolating towards higher excitation levels. Furthermore, the validity of the estimated nonlinear models become poor for very strong nonlinear behaviour. Linearised models are also estimated to have a complete view of the performance of each candidate model for each level of excitation.

Nonlinear bifurcations of a circular dielectric-elastomer resonator based on a modified incremental harmonic balance method

When dielectric-elastomer (DE) resonators are operated under large alternating voltage excitations, strong geometric and material nonlinearities can significantly affect the dynamic performance. Based on the virtual work principle, a dynamic model of a circular DE resonator is established in this work. Static analysis is conducted to study the impacts of the control voltage on the strain and natural frequency of the DE resonator at equilibrium. A modified Incremental Harmonic Balance and arc-length methods are used to obtain dynamic responses of the system excited by alternating voltages, and stability analysis is conducted using the Floquet theory. The effects of the mechanical parameters are studied, and period-doubling bifurcations are observed, which are produced by the parametric excitation of the voltage. The bifurcation phenomenon is comprehensively investigated through analysis of the voltage, stress, and damping coefficient, which provides design guidance for DE resonators.

Modeling and dynamic analysis of integral vertical transport system for deep-sea mining in three-dimensional space

This paper proposed a new three-dimensional numerical model of the integral vertical transport system (VTS) in deep-sea mining using the vector form intrinsic finite element (VFIFE) method. The method shows great advantages in solving complex dynamic behavior problems of VTS with large differences in mechanical properties and strong geometric nonlinearity. The governing equations are established according to Newton's second law and solved in the time domain by the central difference method. The simulation program for predicting the nonlinear dynamic behavior of VTS is developed with MATLAB and verified with the results of relevant literature. Then, a parameter sensitivity analysis is conducted to investigate the influences of the amplitude and period of the top vessel motion, current velocity, internal flow velocity and buffer mass on the VTS dynamic characteristics. The results show that the heave motions of the vessel are the most adverse motion state to the rigid pipe stress. The influence of the internal flow velocity on the system tension is significantly greater than the bending moment. The rigid pipe, which exhibits a higher sensitivity to each parameter than the flexible pipe, is the weakest area that needs more attention in the design.

Spatial weak form quadrature beam elements based on absolute nodal coordinate formulation

A new spatial nonlinear beam element allowing for arbitrary number of nodes is proposed. The element which embodies the idea of weak form quadrature is established on the basis of the absolute nodal coordinate formulation (ANCF). The order of Taylor expansion over the cross section can be chosen as desired, enabling the element to cope with complex cross-sectional behavior such as high-order shear and warping deformation. Four typical examples are presented and satisfactorily results are obtained, demonstrating the effectiveness of the formulation for the analysis of spatial beams with strong geometrical nonlinearity.

Tension and Deformation Analysis of Suspension Cable of Flexible Photovoltaic Support under Concentrated Load with Small Rise-span Ratio

The suspension cable structure with a small rise-span ratio (less than 1/30) is adopted in the flexible photovoltaic support, and it has strong geometric nonlinearity. Based on the principle of energy, the increment of cable force and the change of cable displacement under concentrated force are derived for the suspension cable in an equilibrium state under uniform distributed load, and the equivalent bending stiffness of simply supported beam under bending is given. Considering the strain energy generated by cable force variation, the method presented in the paper has higher calculation accuracy for suspension cable structures with a small rise-span ratio, and includes the special case of a large rise-span ratio. An engineering example of flexible photovoltaic support with a span of 15m is calculated and analyzed, and then compared with the finite element calculation results. The results show that the calculation error is less than 1% when the rise is 0.4 m, and the calculation error is less than 3% when the rise is 1.5m. The calculation formula in the paper is simple and accurate, which can provide a reference for static analysis and structural design of flexible photovoltaic support.

Enhancing suspension vibration reduction by diagonal inerter

The diagonal inerter is integrated into a suspension vibration reduction system (SVRS). The dynamic model of the SVRS with diagonal inerter and damping is established. The dynamic model is of strong geometric nonlinearity. The retaining non-linearity up to cubic terms is validated under impact excitation. The conditions omitting the static deformation are determined. The effects of the diagonal inerter on the vibration reduction performance of the SVRS are explored under impact and random excitations. The vibration reduction performance of the proposed SVRS with both diagonal inerter and damping is better than that of either the SVRS without them or the SVRS with the diagonal damping only.

Stress concentration effect on deflection and stress fields of a master leaf spring through domain decomposition and geometry updation technique

Abstract Theoretical and experimental large deflection and stress analysis of a master leaf spring considering stress concentration effect of clamping is reported. The non-uniformly curved master leaf spring under three point bending subjected to moving boundaries is modeled. Geometrically nonlinear strain-displacement relations, as necessary for the theoretical analysis, are derived through visualization of physics behind the large deformation problem. An embedded curvilinear coordinate system is considered, to study the combined effects of non-uniform curvature, bending, stretching and shear deformation including cross-sectional warping. Governing equation of the non-uniformly curved beam system is derived in variational form using energy method, based on linear material constitutive relations and the derived nonlinear kinematic relations. An iterative solution scheme through successive geometry updation is developed and executed in MATLAB® software to solve the governing equation involving strong geometric nonlinearity together with complicating moving boundary effect. Experimental deflection profiles under static loading are obtained through manual image processing technique using AutoCAD® software. Whereas, strain measurements are performed using strain gauges with data acquisition system (HBM-MX840B). Comparison between the theoretical and experimental results lead towards observation on stress concentration effect due to presence of geometric discontinuity in form of a small hole in the physical system. A modified formulation is proposed using domain decomposition method incorporating effect of geometric discontinuity through an equivalent curved beam geometry of variable cross-section. The modified theoretical model is validated successfully with the experimental results, and observations on stress characteristics and effect of hole diameter to beam width ratio are made.

Planar Weak Form Quadrature Beam Elements Based on Absolute Nodal Coordinate Formulation: A Structural Mechanics Approach

A planar nonlinear weak form quadrature beam element of arbitrary number of axial nodes is proposed on the basis of the absolute nodal coordinate formulation (ANCF). Elastic forces of the element are established through geometrically exact beam theory, resulting in good consistency with classical beam theory. Two examples with strong geometrical nonlinearity are presented to verify the effec-tiveness of the formulation.

Low frequency sound absorption of adjustable membrane-type acoustic metamaterials

The membrane-type acoustic metamaterials with negative pressure cavity are designed to achieve near perfect sound absorption and sound absorption adjustment of low frequency spectrum line control. The soft membrane is replaced by PET membrane as the substrate to enhance tolerance. The membrane material has achieved strong geometric nonlinearity by the static negative pressure of the back cavity. The change in geometric stiffness caused by geometric nonlinearity is used to adjust structural sound absorption peak for low frequency spectrum line. The combined method of finite element and modal superposition is used to predict the sound absorption of membrane-type acoustic metamaterials. The experimental results agree well with the one from the analysis method. The advantages of this structure are that the location of the sound absorption peak can be adjusted. And a near perfect sound absorption can be achieved without damping adjustment. This structure has positive significance for low frequencies spectrum line noise control.

Vibration control of stress ribbon bridges subjected to moving vehicles

Stress ribbon bridges experience excessive vibration when subjected to moving vehicles due to flexibility and low damping, affecting the running safety and ride comfort of the vehicles. It is of great challenge to suppress the vibration of a vehicle-stress ribbon bridge (VSRB) system because of the strong geometric nonlinearity of the stress ribbon bridge, which might also lead to the inapplicability of the conventional optimal design of tuned mass dampers (TMD). This paper proposes an eddy-current tuned mass damper (ECTMD) and its parameter optimization method for suppressing the excessive vibration of stress ribbon bridges subjected to moving vehicles. A dynamic analysis method of the VSRB system is first presented for evaluating the control performance of the ECTMD. Then, a response surface method is proposed to optimize the parameters of the ECTMD for improving control performance. The control performance of the ECTMD optimized by the proposed procedure is validated using a numerical study of a real stress ribbon bridge. Numerical results demonstrate that the conventional design methodology of the TMD based on the linear theory is not the optimal design for the VSRB system due to its nonlinear vibration feature, whereas the proposed response surface method well optimizes the ECTMD's parameters, thus achieving the optimal control performance. This study provides an effective passive control approach for reducing the dynamic responses of stress ribbon bridges excited by moving vehicles.

Mathematical modeling of single action pressing of powder materials under dry friction conditions

The paper presents a theoretical analysis of the single action pressing of powder materials featuring plasticity and compressibility. It takes into account dry external friction between the die material and side walls, which determines the strong nonlinearity of the problem considered. This problem has a number of features that complicate its numerical solution: the presence of external friction, the elastic-plastic law of material behavior description, as well as the calculation of large displacements and, as a consequence, strong geometric nonlinearity. To consider these features, a combination of Fleck–Kuhn–McMeeking and Gurson– Tvergaard–Needleman models was used to consider a wide range of changes in the porosity of materials. The numerical solution of the problem was carried out using finite element analysis with isoparametric elements. The increment of plastic deformations at each step was determined from nonlinear equations of plastic flow. Stresses at the Gaussian points were updated according to the specified increments of deformations to calculate the material behavior during deformation. Unknown density and strain values as functions of coordinate and time were calculated. The influence of the different height-to-diameter ratio of the blank and the value of external friction of the material stress-strain state and compaction kinetics were considered. The distribution of equivalent stresses and the value of volumetric plastic deformations in the material, as well as the nonuniformity of relative density at the end of the pressing period were studied. The theoretical analysis made it possible to study the basic compaction kinetics laws for powder materials with nonuniform density under conditions of dry friction on side walls. The results obtained are relevant for predicting possible negative changes in the blank geometry when implementing the single action pressing scheme for powder materials.

Effect of Geometric Nonlinearity on Membrane Roof Stability in Air Flow

Membrane materials are widely used in construction engineering with small mass and high flexibility, which presents strong geometric nonlinearity in vibration. In this paper, an improved multiscale perturbation method is used to solve the aerostatics stability of membrane roofs on closed and open structures by quantifying the effect of geometric nonlinearity on the single-mode aeroelastic instability wind velocity. Results show that the critical wind velocities of two models are smaller when the geometrical nonlinearity of the membrane material is neglected. In addition, under normal wind load, the influence of geometrical nonlinearity of the membrane on the aerodynamic stability of the roof can be neglected. However, under strong wind load, when the roof deformation reaches 3% of the span, the influence of geometric nonlinearity should be considered and the influence increases with the decrease of transverse and downwind span of the membrane roof. The results obtained in this paper have an important theoretical reference value for the design membrane structures.

Nonlinear vibration control effects of membrane structures with in-plane PVDF actuators: A parametric study

In recent years, with the increasing use of membrane structures in space applications, the problems of large-amplitude (compared to the thickness) vibrations arise; leading to the degradation of their working performance. It is challenging to control the large-amplitude vibrations of membranes due to the high flexibility, low damping, and strong geometrical nonlinearity. In this study, polyvinylidene fluoride (PVDF) actuators are considered to develop the active vibration control of a pre-tensioned Kapton membrane. The governing equations are established based on the geometrically nonlinear theory of plates, taking into account the modal control force induced by the actuators. To analyze the nonlinear dynamic characteristics of the membrane, the theoretical and approximate solutions of nonlinear frequencies of the membrane are obtained, and the discrepancies between the two solutions under different vibration amplitudes are discussed. Control performance for both free vibration and harmonic forced vibration of the membrane is numerically studied. A comprehensive parametric study is carried out to investigate the influences of different system parameters, including the pretension and the size of the membrane, and the position and number of the actuators, on the control performance. Analytical results indicate that the vibration of the membrane can be effectively suppressed with the appropriately laminated PVDF actuators. The results of this research can be extended to the active control of different gossamer space structures.

An improved multidimensional modal pushover analysis procedure for seismic evaluation of latticed arch‐type structures under lateral and vertical earthquakes

SummaryPushover methods for seismic assessment of buildings under multidimensional earthquakes have been studied and retrofitted. However, these current methods are not suitable when applied to widely adopted arch‐type structures characterized by strong geometrical nonlinearity and coupling effects. An improved multidimensional modal pushover procedure with two‐stage analyses is proposed for seismic evaluation of latticed arches. Taking overall multidimensional response into consideration, modal stiffness of the equivalent single‐degree‐of‐freedom system is derived, and its capacity curve is determined during the first‐stage analysis. To provide a deformation profile with algebraic signs of response retained, the second‐stage analysis is conducted using the pushover load pattern derived from modal displacement superposition. The objective of the improved procedure is to overcome the drawback of the conventional modal pushover method, which describes the capacity curve resorting to base shear and roof displacement, and that of quadratic combination rules which eliminate the sign reversals of response. To validate its serviceability, nodal displacements and element stresses, as well as the yielding members, of two typical latticed arches are calculated. Through comparative analysis, the results by the improved procedure exhibit good agreement with those by response history analysis. Additionally, this procedure demonstrates great superiority over the conventional method for its satisfying accuracy.

Nonlinear dynamics of an origami wheel with shape memory alloy actuators

Abstract Origami generates three-dimensional structures from two-dimensional sources and is inspiring engineers to design systems related to different applications as robotics, biomedical and aerospace engineering. The use of smart materials increases the range of applicability of origami systems exploiting adaptive behavior. In this regard, shape memory alloy (SMA) actuators are being used to promote geometrical changes in origami structures. This paper deals with the nonlinear dynamics of an origami wheel with SMA actuators. The origami wheel has strong geometric and constitutive nonlinearities presenting a complex dynamical response. Symmetry assumptions related to waterbomb folding pattern allow one to develop a single degree of freedom reduced-order model system that describes origami dynamics. Based on that, numerical simulations are carried out representing different operational conditions presenting a general comprehension of the origami dynamical response. Origami complex behavior is of concern showing chaotic motions and strong parameter sensitivity.