Geometric Nonlinearity Research Articles

Large-amplitude nonlinear free vibrational behavior of the rotating rectangular plate reinforced by functionally graded carbon nanotubes

The present study discusses the nonlinear free vibrational behavior of the rotating rectangular plate reinforced by functionally graded carbon nanotubes (FG-CNTs). Mechanical properties of the plate have been continuously changed along the thickness of the plate by considering four different distribution patterns of CNTs. Based on the first-order shear deformation theory (FSDT) and von Ḱarḿan’s geometrical nonlinearity, the governing equations of the plate are derived using Hamilton’s principle. Galerkin discretization technique is employed to convert the partial differential equations of motion to ordinary differential equations. Afterward, the nonlinear time-dependent equations of the plate were analytically solved using the multiple time scales method. Eventually, the effect of CNT parameters, the geometrical ratios and rotational speed on the nonlinear frequency of the plate have been studied. For verification purposes, the present numerical results have been compared with those available in the literature with evident agreement. The present research may give benchmark solutions and some guidelines for designing FG-CNTs rotating plates.

Effect of slug characteristics on the nonlinear dynamic response of a long flexible fluid-conveying cylinder

Hydrocarbon flows in a marine riser may appear in multiple phases with varying flow patterns, among which, slug flows are known to exhibit complex flow characteristics due to fluctuations in multiphase mass, velocities, and pressure changes. Bulk of the literature concerning internal single and two-phase flow induced vibrations have largely focused on the use of a linearized tension beam model. In this study, the fluid-structure interaction phenomena of a submerged long flexible cylinder conveying two-phase slug flows is investigated taking into account the geometric and hydrodynamic nonlinearities. A semi-empirical theoretical model which consists of nonlinear structural equations of coupled out-of-plane, in-plane and axial structural motions is presented for the analysis and prediction of two-phase slug flow induced vibrations (SIV). The model includes equations involving centrifugal and Coriolis forces to capture mass variations in the slug flow regime. A numerical space-time finite difference approach combined with a frequency domain analysis is used for analyzing the highly nonlinear three-dimensional responses. The model assumes constant geometric properties across the span of the cylinder and utilizes an idealized slug unit concept, wherein slug properties are considered to be fully developed and undisturbed by pipe oscillations. Model validations are performed through comparisons with published experimental internal flow-induced vibration results. Parametric study captures the effects of several key slug characteristics on the vibration responses of a fluid-conveying cylinder. The results demonstrate amplitude-modulation response, mean displacements, three-dimensional cylinder displacements due to geometric nonlinear couplings and the influence of internal flow velocities. Higher dominant modes and oscillation frequencies were observed when the cylinder experiences large amplitude motions at specific slug formations. A new dimensionless parameter is introduced to predict scenarios of high amplitude oscillations for long flexible cylinders carrying two-phase slug flows.

Combined vibration analysis of a top tensioned riser with the geometrical nonlinearity

A coupling model representing the interactions between the riser and vortices is proposed, of which the time-varying top tension and the internal nonlinear axial force due to the bending of the riser are considered. In addition, the van der Pol oscillator is introduced to simulate the time-varying characteristic of the vortices. The motion equations are derived and the first-order mode approximation is obtained with the Galerkin approach. The multiple scale method is applied to study the steady-state solutions of the system. Effects of system parameters (the structural damping, nonlinearity, the amplitude ratio of the varying-tension, the shedding frequency) on the responses are investigated in detail. The synchronous and non-synchronous motions between the structure and wake oscillator are studied. The bifurcation diagrams for the responses in terms of the amplitude ratio and the nonlinear parameter are obtained in the large parameter ranges. The results show that responses with different topological properties including quasi-periodic, periodic and chaotic solutions can occur under different parameter conditions. A pair of chaotic attractors can be found for the large parameter range of the amplitude ratio. The joint effects of the amplitude ratio and the nonlinear parameter on the responses in the large parameter range are studied as well, which are further verified by the phase projections and Poincaré sections. The results show that the topological properties of responses can be controlled by the amplitude ratio. These results can be helpful to the dynamic design of the riser in practice.

Linear parameter-varying output-feedback for active vibration control of an elastic kinetic roof structure with experimental validation

Elastic kinetics are an approach to design transformable lightweight structures with a stable transformation process. The transformation is realized through elastic bending of structural members by exploiting the compliant material behavior. This lightweight and flexible design comes at the cost of increased sensitivity to static and dynamic disturbances. However, most of the current research focuses on the principles of elastic kinetic transformation instead of effective disturbance mitigation. This work focuses on dynamic disturbance mitigation for such transformable lightweight structures using active control. Modeling and controller synthesis are performed in the linear parameter-varying (LPV) framework, since the dynamics of elastic kinetic structures are transformation-state dependent due to geometric nonlinearities. Based on an LPV model in a grid-based representation, an LPV output-feedback control can be designed and synthesized via a gridding approach. This methodology is experimentally tested and validated for the example of an active hybrid roof structure prototype.

A Nonlinear Beam Finite Element with Bending–Torsion Coupling Formulation for Dynamic Analysis with Geometric Nonlinearities

Vibration analysis of wing-box structures is a crucial aspect of the aeronautic design to avoid aeroelastic effects during normal flight operations. The deformation of a wing structure can induce nonlinear couplings, causing a different dynamic behavior from the linear counterpart, and nonlinear effects should be considered for more realistic simulations. Moreover, composite materials and aeroelastic tailoring require new simulation tools to include bending–torsion coupling effects. In this research, a beam finite element with bending–torsion coupling formulation is used to investigate the effects of the deflection of beam structures with different aspect ratios. The nonlinear effects are included in the finite element formulation. The geometrical effect is considered, applying a deformation dependent transformation matrix. Stiffness effects are introduced in the stiffness matrix with Hamilton’s Principle and a perturbation approach. The results obtained with the beam finite element model are compared with numerical and experimental evidence.

A Comprehensive Investigation of Linear and Nonlinear Beam Models on Flexible Wind Turbine Blade Load Calculations

This study was performed to investigate the effects of structural nonlinearity and large deformations on the aeroelastic loads of flexible wind turbine blades. First, a blade structural analysis model was established using the geometrically exact beam (GEB) theory. Subsequently, the blade element momentum (BEM) theory was corrected using the geometrically exact method leading to the development of a geometrically exact blade element momentum (GE-BEM) model. The results from the GE-BEM model indicated that flapwise deformations always reduce blade fatigue loads, while torsional deformations decrease fatigue loads under low wind speeds but increase them under high wind speeds. Finally, the linear Euler–Bernoulli beam and the GEB were compared to explore the influence of geometric nonlinearity on the blade aeroelastic loads, which revealed that the Euler beam model underestimates the blade loads. The simulations that used the GEB model produced torsional root twist fatigue loads that were 57.49% greater than those generated when the Euler beam model was used. Furthermore, the flapwise bending moment fatigue loads at the root were 8.24% greater than those obtained by the Euler beam model. The smallest discrepancy between the results of the two models was 7.26%, and it corresponded to the edgewise fatigue load.

Study on equivalent mechanical properties of U-shaped bellows based on novel implementation of asymptotic homogenization method

As typical metal thin-walled structure, bellows are used widely in various engineering fields, especially in storage and transportation of floating liquefied natural gas (FLNG) system on the sea. While the shapes of the bellows are relatively complex with the characteristics of geometric nonlinearity in the structure, it is quite challenging to calculate basic mechanical properties of the bellows accurately through theoretical analysis. Concurrently, both numerical simulation and experiments also require high computational and economic cost. Given the typical one-dimensional periodicity of the structure of U-shaped bellows, novel implementation of asymptotic homogenization (NIAH) method was secondary developed in finite element software and unit-cell model of the whole structure with periodic boundary conditions was established, realizing equivalent analysis of the overall mechanical properties of the bellows accurately and efficiently. By comparing the NIAH equivalent results with the fine finite element model results, it was found that the relative error was within 3.00 % and the calculation cost was reduced by 40 times. Compared with the experimental results, the error of NIAH equivalent results was also less than 6.00 %, which verified the accuracy and high efficiency of the NIAH equivalent method. Furthermore, the influence of unit-cell model with different structural sizes on the prediction accuracy of the NIAH equivalent stiffness results of the bellows was also discussed. This study provides a new effective method for the design and analysis of the structure of bellows.

Nonlinear vibrations of fractional viscoelastic PET membranes subjected to tangential follower force

Nonlinear forced vibrations of simply supported viscoelastic Polyethylene Terephthalate (PET) membranes subjected to a harmonic excitation and tangential follower force are investigated in this paper. The consideration of vibration systems under follower force requires special attention to the modeling of non-conservative force and its influence on nonlinear analysis. Herein, a fractional Kelvin-Voigt Hamilton-based framework accounting for the viscoelastic PET membrane is developed. Based on the Hamilton principle for the non-conservative system, the mathematical modeling of viscoelastic PET membrane taking into account non-conservative force and geometric nonlinearity is formulated, and the derived equations are discretized via the Galerkin schemes. A technique of multiple scales is employed to numerically acquire the frequency–response curves with different system configurations. Numerical investigations are conducted to analyze the effects of tangential follower force, viscous coefficient, as well as fractional order on the dynamic stability of the considered structure. Further ramifications of the present study point to the paramount importance of tangential follower force and an accurate viscoelasticity model on the dynamic behaviors of the viscoelastic structures.

Comparative analysis of nonlinear dynamic behaviors of hyperelastic curved structure modelled by different constitutive laws

This work focuses on conducting numerical studies and comparisons on the nonlinear dynamic behaviors of hyperelastic curved structures subjected to external dynamic loadings. A numerical model of the hyperelastic structure is developed using the nonlinear finite element method, which considers both the geometric nonlinearity resulting from large deformations and the material nonlinearity due to hyperelasticity. Four commonly used hyperelastic constitutive laws, namely the Mooney-Rivlin model, the neo-Hookean model, the Yeoh model, and the Ogden model, are employed to describe the nonlinear stress-strain relationship of the hyperelastic material. The convergence performance of the numerical model with respect to grid element and time step is thoroughly examined through several numerical examples. Subsequently, based on the developed numerical model, the nonlinear dynamic behaviors of the hyperelastic curved structures excited by uniform pressure loadings with single and dual frequencies are studied and compared using various constitutive laws. The material constants used in these constitutive laws are estimated based on the well-known Treloar's test data on vulcanized rubber material. Additionally, comprehensive parametric analyses are conducted on the nonlinear dynamic behaviors of the hyperelastic structure simulated by different constitutive laws. The deformation evolution process, resonance curves, time-frequency analysis, bifurcation diagrams, and phase portraits are employed to demonstrate the similarities and differences in the nonlinear dynamic behaviors (i.e., super- and sub-harmonic resonance, internal resonance, chaotic motion and bifurcations) of the hyperelastic structure modeled by different constitutive laws.

Nonlinear dynamic collapse analysis of space semi-rigid frames using finite particle method

Based on the finite particle method, this work presents a general numerical method to perform the nonlinear dynamic collapse analysis of space semi-rigid structures. A space connection element composed of two particles and six zero-length springs is proposed to simulate the space semi-rigid connection. Geometrical nonlinearity is addressed through fictitious motion, while material and connection behavior nonlinearities are captured by updating the element stiffness matrix and the connection stiffness matrix, respectively. Member and connection fracture models are convenient for simulating fracture behaviors during collapse processes. To verify the effectiveness of the proposed method, the numerical results are compared with those from previous studies, and energy conversion analyses are adopted. The analysis results indicate that only member fractures occur at beam ends in the rigid and linear semi-rigid frames, while only connection fractures occur at beam ends in the nonlinear semi-rigid frame.

Structural modeling and investigation of cantilever load-bearing mechanism in telescopic wing

Telescopic wing requires changing the wingspan under aerodynamic load. Structural deformation caused by forces and moments will result in increased driving forces or even stagnation. Deformation and distributed contact force of linear motion pair of the load-bearing mechanism which considers the geometric nonlinearity of guide rail is necessary to predict the driving force, the mechanical efficiency, or the stiffness matching in the optimal design process. Aiming at the cantilever load-bearing mechanism, the structural model was established in terms of the flexibility of the components, contact stiffness and other parameters on the friction of moving pairs are studied based on the modified chained-beam constraint model (CBCM). The model prediction accuracy is verified through simulation and experiment. The result shows that the friction force is related to the stiffness of guide rail, slider connector, ball bearing contact as well as the geometric parameters. Improper parameter selection will increase the friction force to more than twice the optimal solution.

Effect of interstitial fluid pressure on shear wave elastography: an experimental and computational study

Objective. An elevated interstitial fluid pressure (IFP) can lead to strain-induced stiffening of poroelastic biological tissues. As shear wave elastography (SWE) measures functional tissue stiffness based on the propagation speed of acoustically induced shear waves, the shear wave velocity (SWV) can be used as an indirect measurement of the IFP. The underlying biomechanical principle for this stiffening behavior with pressurization is however not well understood, and we therefore studied how IFP affects SWV through SWE experiments and numerical modeling. Approach. For model set-up and verification, SWE experiments were performed while dynamically modulating IFP in a chicken breast. To identify the confounding factors of the SWV-IFP relationship, we manipulated the material model (linear poroelastic versus porohyperelastic), deformation assumptions (geometric linearity versus nonlinearity), and boundary conditions (constrained versus unconstrained) in a finite element model mimicking the SWE experiments. Main results. The experiments demonstrated a statistically significant positive correlation between the SWV and IFP. The model was able to reproduce a similar SWV-IFP relationship by considering an unconstrained porohyperelastic tissue. Material nonlinearity was identified as the primary factor contributing to this relationship, whereas geometric nonlinearity played a smaller role. The experiments also highlighted the importance of the dynamic nature of the pressurization procedure, as indicated by a different observed SWV-IFP for pressure buildup and relaxation, but its clinical relevance needs to be further investigated. Significance. The developed model provides an adaptable framework for SWE of poroelastic tissues and paves the way towards non-invasive measurements of IFP.

The design, fabrication and analysis of a cantilever-based tensile-mode nonlinear piezoelectric energy harvester

In this paper, a cantilever-based tensile-mode piezoelectric energy harvester is proposed. In contrast to a bending mode, the tensile mode enables a uniform strain distribution within the piezoelectric material. This stands in contrast to the strain concentration at the fixed end of a typical cantilevered design. The proposed design comprises a two-segment cantilever beam and an elastic PVDF film that are interconnected. The PVDF film undergoes pre-stretching to guarantee that the harvester maintains a state of tension rather than compression during excitations, as the film is not capable of withstanding compressive forces. The theoretical model of the proposed harvester is derived and validated via experiments. Due to the geometric nonlinearity of this mechanism, a highly nonlinear hardening effect is observed in the simulation and experiments. The effects of different parameters, including preload, dimensions of the elastic beam, tip mass, and excitation accelerations, are explored. The optimal configuration can provide a maximum peak power output of 1827 μW.

Nonlinear deformations of size-dependent porous functionally graded plates in a temperature field

In this paper, the Variational Iteration Method (VIM) or Extended Kantorovich Method (EKM) is formulated for the first time for flexible porous functionally graded (PFGM) plates with different boundary conditions and geometric nonlinearity according to the theory of Theodore von Karman. Its accuracy and efficiency are demonstrated. The plate is subjected to a uniform transversal load and temperature field. The displacement field of the plate is approximated based on the classical plate theory (CTP) or Kirchhoff's plate theory. The governing equations are derived using Hamilton's principle. Modified Coupled Stress Theory (MCST) is used to account for size-dependent effects. The material properties vary with thickness and are temperature dependent. Four porosity distribution patterns are considered in this study. Several examples are solved to demonstrate the proposed algorithm. The results obtained are compared with solutions obtained by the Bubnov-Galerkin Method (BGM) in higher approximations, the Finite Difference Method (FDM) of second order accuracy, as well as with results obtained by the Finite Element Method (FEM) of other authors. The results include an analysis of the effect of size dependent parameters, porosity type pattern, porosity index, functionally graded index, temperature field and different types of boundary conditions on the stress–strain state and bending deflection of plates.

Phase‐field model for ductile fracture in the stress resultant geometrically exact shell

SummaryWe present a phase‐field fracture model for a stress resultant geometrically exact shell in finite deformation regime where the configuration manifold evolves according to deformation and fracture. The Reissner–Mindlin shell problem is first solved via the finite element method, where the independent unknown fields are the displacement and director. The phase‐field ductile fracture model is then coupled with the verified geometrically exact shell by enriching the three‐field elasto‐plastic free energy with a regularized crack surface energy. To capture the geometrical nonlinearity due to the large plastic deformation exhibited in the processing zone, the corresponding finite‐strain stress resultant elasto‐plasticity model is coupled with the phase field model. This coupling between the stress resultant elasto‐plasticity model and the phase‐field fracture model enables us to predict the different amounts of energy dissipation attributed to plastic deformation and fracture and hence simulate the crack propagation in the ductile regime properly. We introduce a mixed finite element model with displacement, director, and phase‐field order parameters as the nodal degree of freedoms formulated on the mid‐surface. An energy‐based arc‐length method is generalized to track the equilibrium path and mitigate instabilities arising from material nonlinearity. Four numerical examples are presented to validate the implementation of the model and demonstrate its capability to simulate ductile fracture in shell structures. These examples include a plane‐stress tension/shear fracture model, the Muscat‐Fenech and Atkins plate, axial tension of a notched cylindrical shell, and ductile fracture of a simply supported plate under uniform pressure.

A novel geometric predictive algorithm for assessing compressive elastic modulus in MEX additive processes, based on part nonlinearities and layers stiffness, validated with PETG and PLA materials

MEX (Material Extrusion) is an intrusive technological process that inherently induces alterations in the elastic and mechanical parameters of plastic materials. Manufacturers provide initial mechanical parameters for plastic filaments, which undergo modifications during MEX manufacturing, influenced by intrinsic manufacturing factors such as temperature and pressure changes, as well as geometric and technological parameters of the 3D additive process. These factors, compounded by the inherent geometric nonlinearities in plastic components, directly impact the post-manufacture mechanical and elastic properties of the material. Presently, material characterization in MEX manufacturing relies on manual experimental testing, necessitating new tests for any variation in manufacturing parameters. In this scenario of mechanical uncertainty, rigorously validating component behavior involves costly experimental trials. Intending to solve the problems of MEX components manufacturing, the paper presents an innovative methodology based on the use of a new predictive algorithm created by the researchers capable of obtaining the elastic modulus of a plastic material manufactured with MEX and its mechanical behavior in the elastic zone under compressive loads. The predictive algorithm only needs as input the compressive elastic modulus of the isotropic plastic material filament and the manufacturing parameters of the MEX process. The smart developed algorithm calculates the stiffness of each layer considering the number of holes in the projected area. The innovative predictive algorithm has been experimentally and numerically validated using PETG (Polyethylene Terephthalate Glycol) material and PLA (Polylactic Acid) on test specimens and on a case study of variable topology. The results show deviations from [0.2%–4.3%] for PETG and [0.4%] for PLA concerning the experimental tests and [1.1%–13.5%], to the numerical analyses. In this line, the presented algorithm greatly improves the results obtained by the simulation software since this software currently can not consider the geometric and technological parameters associated with the 3D manufacturing process of the component. The predictive algorithm is valid for each print pattern and manufacturing direction. The new algorithm improves the existing state of the art significantly since this algorithm extends its utility to most plastic polymer materials suitable for MEX 3D printing, provided that the mechanical and elastic properties of the filament are known. Its versatility extends to complex component geometries subjected to uniaxial compression loads, eliminating the need for mechanical analysis software or expensive experimental validations.

1/2 and 1/3 Subharmonic Resonance Behaviors of a Rotating FG-CNTRC Beam with Geometric Imperfections

This paper aims at investigating 1/2 and 1/3 subharmonic resonances of a rotating functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beam in the presence of geometric imperfections. A general function is introduced to denote three types of imperfections containing sine, global, and local modes. At first, based on the von-Kármán geometric nonlinearity assumption, the forced vibration equation is set up. Then, the Galerkin method and multiple scale method are utilized to study subharmonic resonance behaviors. Finally, coupled effects of FG-CNTRCs, rotating motion, and geometric imperfections on subharmonic resonance responses are investigated. It is found that a specific range of the excitation amplitude and frequency motivates the subharmonic resonance behavior. The coupling of geometric imperfections and geometric nonlinearities generates the 1/2 resonance phenomena, and geometric nonlinearities result in the 1/3 resonance behavior. Both the imperfection mode and amplitude can affect the resonance response and resonance regions. Moreover, in contrast with the 1/3 subharmonic resonance, geometric imperfections produce greater influence on the 1/2 subharmonic resonance.

Assessment of Axisymmetric Dynamic Snap-Through and Thermally Induced Vibrations in FGM Cylindrical Shells Under Instantaneous Heating

This paper addresses the investigation of axisymmetric thermally induced vibrations in Functionally Graded Material (FGM) cylindrical shells. It considers temperature-dependent (TD) properties and geometric non-linearity (the von Karman effect). The study systematically solves a transient heat conduction equation using finite differences method and the Crank–Nicolson method. During the heating stages, the evaluation of thermal forces and moments takes place. Equations of motion are derived through the application of Hamilton’s principle. Spatial dependencies are discretized using the generalized Ritz method, while temporal dependencies are approximated using the [Formula: see text]-Newmark method with Newton–Raphson linearization. A comparative analysis validates the procedure’s efficiency and precision. Parametric studies explore the influence of parameters, including the temperature-dependency material properties, geometric nonlinearity, and shell’s power-law index, providing valuable insights into FGM shell behavior under thermal shock.

Nonlinear Finite Element Analysis of Tubular Steel Wind Turbine Towers near Man Door and Ventilation Openings to Optimize Design against Buckling

The safe and cost-effective design of wind turbine towers is a critical and challenging aspect of the future development of the wind energy sector. This process should consider the continuous growth of towers in height and blades in length. Among potential failure modes of tubular steel towers, shell local buckling due to static axial compressive stresses from the rotor, blades, and tower weight, as well as dynamic flexural compressive stresses from wind actions on the rotating blades and the tower itself, are dominant as thickness is optimized to reduce weight. As man door and ventilation openings are necessary for the towers’ operation, the local weakening of the tower shell in those areas leads to increased buckling danger. This is compensated for by tower manufacturers by the provision of stiffening frames around the openings. However, the cold-forming and welding of these frames are among the most time-consuming aspects of tower fabrication. Working towards the optimization of this design aspect, the buckling response of tubular steel towers near such openings is investigated by means of nonlinear finite element analysis, accounting for geometrical and material nonlinearity and imperfections (GMNIA), and also considering several wind directions with respect to the openings. The alternatives of stiffened and unstiffened openings are investigated, revealing that a thicker shell section around the opening may be sufficient to restore lost stiffness and strength, while the stiffener frame may also be eliminated, offering substantial benefits in terms of manufacturing effort, time and cost.

A nonlinear numerical scheme to investigate the influence of geometric nonlinearity on post-flutter responses of bridges

A nonlinear numerical scheme to investigate the influence of geometric nonlinearity on post-flutter responses of bridges