Elastic Boundary Research Articles

Nonlinear vibration analysis of functionally graded material tubes with conveying fluid resting on elastic foundation by a new tubular beam model

In this paper, nonlinear free vibration analysis of shear deformable functionally graded material (FGM) tubes conveying fluid with immovable support conditions resting on a Pasternak-type foundation is presented. Based on Hamilton’s principle, the equations of motion and boundary conditions are obtained by using a new high-order shear deformation tubular beam model with von Kármán nonlinearity and the exact expression of curvature, in which contributions of fluid velocity to the kinetic energy and body forces are also considered. The differential quadrature method (DQM) is employed to determine the nonlinear frequencies and amplitude–frequency responses of fluid-conveying FGM tubes with different boundary conditions. A detailed parametric study is conducted to analyze the influences of different types of boundary conditions, geometric and physical properties. The numerical results reveal that the geometrical and physical properties, including elastic foundation, boundary conditions and flow velocity in the fluid-conveying pipes are crucial factors on their dynamical behavior.

Vibration Characteristics of Island-Bridge Structure on Porous PDMS Substrates for Stretchable Electronics

Abstract Stretchable electronics employing island-bridge structure design can achieve controllable and reversible stretchability. The use of a porous substrate, which provides excellent breathability for wearable devices bonded to skin, not only satisfies this static superiority but also has a profound impact on the dynamic performance of the stretchable electronics. In this paper, the vibration characteristics of the island-bridge structure based on porous polydimethylsiloxane (p-PDMS) substrates are studied by utilizing an analytical model, which takes account of geometric nonlinearity due to mid-plane stretching, buckling configuration, elastic boundary conditions considering the p-PDMS substrates and the mass of the island. In numerical examples, the accuracy of the analytical model is first verified by finite element analysis (FEA). After that, we investigate the effects of some primary factors, i.e., the prestrain of the substrate, spring stiffnesses at the ends of the interconnect, porosity and thickness of the substrate, and the mass of the island, on the natural frequencies and vibration mode shapes of the island-bridge structure. Results show that the vibration characteristics of the island-bridge structure can be tuned conveniently by adjusting the porosity of the substrate and the mass of the island, which are expected to be helpful to mechanical design and optimization of stretchable electronics in complex noise environments.

Analysis of flow characteristics of the dual media shale gas reservoir with the elastic outer boundary

In order to more accurately describe the seepage characteristics of shale gas reservoirs, in this paper, an elastic outer boundary condition is introduced, and a new dual-media shale gas seepage model is established to describe the seepage characteristics of shale gas reservoirs more accurately, while considering the adsorption and desorption process. Combining Laplace transformation and Similar Structure Method, the solution of the percolation model is obtained in Laplace space. Furthermore, the solutions are inverted into real space with the Stehfest numerical inversion method. Type curves of dimensionless pressure and dimensionless pressure derivative can be used to evaluate the reservoir characteristics. The results show that the conventional three kinds of outer boundary conditions (infinite, constant pressure and closed) are actually three special cases of elastic outer boundary. The introduction of elastic outer boundary conditions not only expands the scope of data interpretation, but also closer to the actual situation of the outer boundary of the reservoir. The theory of similar structure greatly simplifies the complex process of solving the model and will have a far-reaching impact on the development of well test analysis software in the future.

Boundary control of a vertical nonlinear flexible manipulator considering disturbance observer and deflection constraint with torque and boundary force feedback signals

In this paper, boundary control (BC) laws are designed to find a BC solution for a single-link nonlinear vertical manipulator to suppress the link’s transverse vibrations and control the rigid body nonlinear large rotating motion. The governing equations of motions and boundary conditions, which all consist of a set of PDEs and ODEs have been derived based on the Hamilton principle. It is desired to regulate large angular orientation, suppress the flexible link’s transverse vibrations and compensate the boundary disturbance simultaneously. The amount of elastic boundary vibration has remained within the constraint range. By considering novel Barrier-Integral Lyapunov functional in order to prevent the amount of boundary deflection from violating the constraints and avoiding any simplifications, in the presence of external boundary disturbance, proper control feedbacks and boundary disturbance observer are adopted in order to reach control objectives and to compensate external boundary disturbance effects simultaneously. By choosing proper boundary feedback, system states are proven to be uniform ultimate bounded and converge exponentially toward a small neighbourhood of zero. In order to illustrate the performance of the proposed control approach, numerical simulation results are provided.

Haar wavelet method for frequency analysis of the combined functionally graded shells with elastic boundary condition

In this paper, the Haar wavelet discretization method (HWDM), one of the effective, convenient and accurate numerical solution methods is presented for analyzing the free vibration of combined functionally graded shells (CFGSs). The combined shells consist of elliptic, hyperbolic, parabolic, and cylindrical shells, and each shells are composed of the same functionally graded material (FGM) determined by four parameters. The theoretical model of the CFGSs is formulated by the application of the first-order shear deformation theory (FSDT). The displacement components at any point of the CFGS are expressed as Haar wavelet series in the meridian direction and as a trigonometric function series in the circumferential direction. The boundary and the continuous condition are modeled by the spring stiffness technique, by adding boundary condition equations to the equation of the main system, the constant expressed in the integral form of the Haar wavelet series is satisfied. Then, by solving the characteristic equation of the whole system, the vibration characteristics of the CFGS such as the natural frequency and the corresponding mode shape can be obtained. Through comparison with previous literature and finite element method (FEM) results, it is shown that current method has high accuracy, reliability, and good convergence for free vibration analysis of CFGS. Finally, new free vibration results of the various CFGS, which can be used as benchmark data for researchers in this field, are reported along with parameter studies.

EvoStencils: a grammar-based genetic programming approach for constructing efficient geometric multigrid methods

For many systems of linear equations that arise from the discretization of partial differential equations, the construction of an efficient multigrid solver is challenging. Here we present EvoStencils, a novel approach for optimizing geometric multigrid methods with grammar-guided genetic programming, a stochastic program optimization technique inspired by the principle of natural evolution. A multigrid solver is represented as a tree of mathematical expressions that we generate based on a formal grammar. The quality of each solver is evaluated in terms of convergence and compute performance by automatically generating an optimized implementation using code generation that is then executed on the target platform to measure all relevant performance metrics. Based on this, a multi-objective optimization is performed using a non-dominated sorting-based selection. To evaluate a large number of solvers in parallel, they are distributed to multiple compute nodes. We demonstrate the effectiveness of our implementation by constructing geometric multigrid solvers that are able to outperform hand-crafted methods for Poisson’s equation and a linear elastic boundary value problem with up to 16 million unknowns on multi-core processors with Ivy Bridge and Broadwell microarchitecture.

Improving the hydrogeologic conceptualization of a remote semiarid palaeovalley groundwater system using airborne electromagnetics, seismic refraction and reflection, and downhole nuclear magnetic resonance

A hydrogeologic conceptualization is critical to understand, manage, protect, and sustain groundwater resources, particularly in regions where data are sparse and accessibility is difficult. We used airborne electromagnetics (AEM), shallow seismic reflection and refraction, and downhole nuclear magnetic resonance (NMR) logs to improve our understanding of an arid groundwater system influenced by palaeovalleys. We found that there is a limited connection between the palaeovalley and fractured bedrock aquifers because they are separated by a spatially variable layer of saprolite, which is the layer of chemically altered rock on top of the fractured bedrock. The AEM data provided an estimate of the top of the saprolite but failed to effectively image the bottom. In contrast, the seismic data provided an estimate of the bottom of the saprolite but failed to image the top. This geophysical combination of electrical and seismic data allowed us to map saprolite thickness in detail along a 1.7 km long transect that runs perpendicular the main trunk of a well-defined palaeovalley. These data indicate that the palaeovalley is lined with a heterogeneous layer of saprolite (approximately 3–120 m thick) that is thickest near its edges. Despite the observed variability, only a small percentage of the fractured bedrock aquifer (8%–17%) appears to be in contact with the palaeovalley aquifer. Furthermore, the lack of an elastic boundary at the top of saprolite suggests that the porosity of the saprolite is similar to the palaeovalley sediments — an observation that is supported by the downhole NMR-derived water contents. The electrical change at the top of saprolite is caused by a combination of a decrease in total dissolved solids of the groundwater in the saprolite and a change in pore structure associated weathering in situ versus transported weathered materials. The presence of saprolite, which commonly behaves as an aquitard, may limit the groundwater exchange between the palaeovalley and bedrock aquifers, with implications for the regional groundwater resource potential.

PID Parameter Tuning for Buck Controllers Based on an Improved Differential Evolution Algorithm

With the wide application of Buck converters, it is of great significance to select the optimal PID controller parameters to ensure the dynamic and steady-state performance of the Buck converters. On this basis, a PID parameter tuning method based on an improved differential evolution algorithm is proposed. Also, a crossover approach with population generation across orders of magnitude, elastic boundary absorption, and retention of some combined features is put forward. In addition, an adaptive mutation factor, and a cost function that suppresses overshoot and is sensitive to the tuning time are designed. The results of 10 tests show that the differential evolution algorithm fails in the case of parameters in a range cross orders of magnitude, while the algorithm proposed in this paper enables a better combination of PID parameters selected in terms of speed and stability, with a wide search space and stable convergence results.

Design of Laminated Composite Plates with Carbon Nanotube Inclusions against Buckling: Waviness and Agglomeration Effects.

In the present study, a buckling analysis of laminated composite rectangular plates reinforced with multiwalled carbon nanotube (MWCNT) inclusions is carried out using the finite element method (FEM). The rule of mixtures and the Halpin–Tsai model are employed to calculate the elastic modulus of the nanocomposite matrix. The effects of three critical factors, including random dispersion, waviness, and agglomeration of MWCNTs in the polymer matrix, on the material properties of the nanocomposite are analyzed. Then, the critical buckling loads of the composite plates are numerically determined for different design parameters, such as plate side-to-thickness ratio, elastic modulus ratio, boundary conditions, layup schemes, and fiber orientation angles. The influence of carbon nanotube fillers on the critical buckling load of a nanocomposite rectangular plate, considering the modified Halpin–Tsai micromechanical model, is demonstrated. The results are in good agreement with experimental and other theoretical data available in the open literature.

Mixed-triggered communication with limited elastic slot boundaries

Time-triggered systems provide dependable and deterministic communication based on strict time boundaries for tasks and messages. On the other hand, event-triggered communication is less strict on requirements and more flexible, but it is difficult to provide fault isolation. Thus, the combination of time-triggered and event-triggered communication is desirably extending a time-triggered communication system by a dynamic property. This paper improves a novel mixed-triggered communication approach based on ESBs which adds limited delay tolerance to an otherwise strict real-time communication. The presented approach overcomes the problem of delayed respectively missing messages by adding fail-operational behavior to the ESB-based communication. This is achieved by extending the basic mechanism using additional fault-tolerant features. The presented mechanism allows the system to stay operable even when messages occasionally violate their planned slot boundaries and provides fault isolation for timing violations beyond a predefined tolerance budget. The introduced fault-tolerant features are analyzed via Monte-Carlo simulations where the resulting data throughput is compared to two straightforward hard real-time communication approaches. As our results show, the proposed guards enable the system to handle extended message reception delays while, compared to strictly bounded communication, having a better performance of successfully transmitted messages.

Analytical approach to the stepped multi-span rotor-bearing system with isotropic elastic boundary conditions

This paper strives to derive the analytical solutions for the dynamic analysis of stepped multi-span rotor system, where the rotating shaft's internal damping effect is also considered. Specifically, the steady-state whirl analysis and free vibration analysis is given herein. For the steady-state whirl analysis, firstly, the stepped shaft is spilt to several uniform segments and their governing equations are established by Hamilton principle. Subsequently, by applying variable separation method and Laplace transform method to each segment's governing equation, their steady-state responses in terms of determinate interpolation functions multiplied by unknown boundary constants are derived. Then based on the transfer matrix method, each segment's boundary constants are determined with consideration of the compatibility conditions of each two adjacent segments and the end boundary conditions. For the free vibration analysis, one only need to remove the forced term in the steady-state form of transfer matrix and the rotor's characteristic equation will be obtained. Through the theoretical derivation, it is found that this analytical approach can be generalized to any isotropic elastic boundary conditions. To validate the proposed method, two case studies are proposed, where the finite element method is used as benchmark method. For the first one, the influence of internal damping on a uniform rotor's damped whirl characteristics and its stability is analyzed. The damped whirl characteristics analysis reveals that viscous internal damping result in the destabilization of forward modes as long as the spinning speed becomes higher than the critical speed. This phenomenon is also demonstrated by the time responses analysis. For the second one, the dynamic responses of a two-span three steps rotor-bearing system with isotropic viscoelastic boundary conditions are determined. All of simulated results show a great agreement between analytical results and FEM results.

Boundary Reflections of Rolling Waves in Cubic Anisotropic Material

Rolling waves have unconventional circular polarizations enabled by the equal-speed propagation of longitudinal and transverse waves in elastic solids. They can transport non-paraxial intrinsic (i.e. spin) mechanical angular momentum in the media. In this work, we analyze the rolling wave reflections and their effects on the non-paraxial spins in a cubic elastic half-space with an elastically supported boundary. Reflected waves from both normal and general oblique incidences are investigated. We show that, by adjusting the stiffness of the elastic boundary, we can precisely control the spin properties of the reflected waves, paving the way towards a broad category of spin manipulation techniques for bulk elastic waves.

3D Free Vibration Analysis of Functionally Graded Plates with Arbitrary Boundary Conditions in Thermal Environment

An analytical method for free vibration of functionally graded (FG) plates with arbitrary boundary conditions in thermal environments is developed based on 3D elasticity theory. Material properties of the FG plate are assumed to be temperature dependent but graded along with its thickness. At the edges of the plate, three different sets of linear springs are introduced, so that its boundary condition can be altered by changing the spring stiffness. Each displacement component of the plate is expanded as a standard Fourier cosine series, supplemented with closed‐form auxiliary functions. With thermal environment effects duly accounted for, the energy method is used to derive eigenvalue equations, whereas the Rayleigh−Ritz procedure is utilized to calculate the natural frequencies of the FG plate. Numerical examples show that the present method converges quickly and gives satisfactory results. The method is then utilized to calculate the natural frequencies of FG plates having different aspect ratios, volume fractions, and elastic boundaries.

A meshfree local weak-form method for free vibration analysis of an open laminated cylindrical shell with elliptical section

In this paper, for the free vibration analysis of an open laminated cylindrical shell with elliptical section (OLCSES), under different elastic boundary conditions (BCs), a new meshfree Tchebychev-radial point interpolation method (TRPIM) shape function using the basis function consisted of radial basis and Tchebychev polynomial basis, is proposed. With the help of the first-order shear deformation theory, the local weak-forms of the governing equation and BC of OLCSES are derived. Its BC is modelled as a general BC via an artificial elastic spring technique, and the displacement functions are all approximated by TRPIM shape function in two directions (x, θ) regardless of the BC. The natural frequency of OLCSES is obtained by solving its governing equation and BC equation discretized by TRPIM shape function. The numerical results are in good agreement with those of preceding literatures and finite element method (FEM). The natural frequencies of OLCSES under different BCs including classical and elastic BCs are newly obtained and the effect of some parameters such as its material property and geometry on them is studied. The results can be used as reference data for researchers studying dynamic characteristics of shell structures with elliptical section.

Free vibration analysis of rectangular plates with arbitrary elastic boundary conditions

boundary conditions are In this paper, an improved Fourier series method is presented for the free vibration analysis of rectangular plates with arbitrary elastic conditions. The stiffness value of the restraining springs is determined as required to simulate the arbitrary elastic boundary conditions. The exact solution of plates with arbitrary elastic boundary conditions is solved by the introduced supplementary func-tions. The matrix eigenvalue equation of plates is derived by using boundary conditions and the governing equations. Compared with exist methods, the presented method can be easily applied to most of plate vibration problems with different boundary conditions. To validate the accuracy of the presented method, numerical simulations with different boundary conditions are presented.presented.

Analysis of the transient flow of non-Newtonian power-law fluids in homogeneous reservoirs with the elastic outer boundary

Analysis of the transient flow of non-Newtonian power-law fluids in homogeneous reservoirs with the elastic outer boundary

Analytic Static Deflection Solutions of Uniform Cantilever Beams Resting on Nonlinear Elastic Rotational Boundary

Analytic Static Deflection Solutions of Uniform Cantilever Beams Resting on Nonlinear Elastic Rotational Boundary

Identification of Young’s modulus and equivalent spring constraint boundary conditions of the soft tissue with locally observed displacements for endoscopic liver surgery

In endoscopic surgery, the surgical navigation system needs to calculate the deformation of soft tissue by biomechanical model which requires elastic properties and boundary conditions. However, patient-specific elastic parameters and boundary conditions of soft tissue are hard to measure accurately from the preoperative images, especially the boundary conditions will change during the operation due to the ligament cutting. In addition, simple boundary conditions such as fixed constraints and free-force constraints are not physically adequate to simulate the elastic effect of ligaments attached to the liver. In this paper, we present a novel method to identify the Young’s modulus and equivalent spring constraint boundary conditions of a locally observed soft tissue. Based on the spring constraint boundary condition, a two-step inverse algorithm is developed based on the finite element method (FEM) with integration of energy regularized Gauss-Newton (GN) method and -regularized method, which takes external forces and displacements of observable nodes as inputs. A series of numerical simulations and physical hydrogel phantom experiments were conducted. The results of simulation and physical experiments show that the Young’s modulus and equivalent spring constraint boundary conditions identified by the proposed method agree well with their setup true values.

Buckling behavior analysis of prestressed CFRP-reinforced steel columns via FEM and ANN

Prestressed (PS) carbon fiber reinforced polymer (CFRP)-reinforced steel columns are novel multiparameter systems exhibiting complex nonlinear buckling behavior. In this study, this behavior was investigated with the finite element method (FEM) and an artificial neural network (ANN). First, FEM models of the columns under axial and eccentric compression were built. The numerical and experimental force–displacement curves, failure modes, and CFRP stress–displacement curves were in good agreement. Moreover, the influencing rules of 9 key parameters (i.e., CFRP initial prestressing force, supporting length, eccentricity, steel yield strength, slenderness, CFRP elastic modulus, initial imperfection and boundary conditions) on the buckling capacity and reinforcing efficiency of the reinforced columns were determined. Afterward, 312 datasets from the validated finite element model covering 8 input parameters were generated via the ANSYS parametric design language (APDL). Finally, as ANNs can manage highly complex and computationally intensive nonlinear problems, a practical ANN tool was developed to predict the buckling capacity of PS CFRP-reinforced steel columns.

Dynamics Sensitivity Analyses of Functionally Graded Porous (FGP) Curved Beams with Variable Curvatures and General Boundary Conditions

A method is proposed to obtain the exact solution for the dynamic analysis of functionally graded porous (FGP) curved beams with general boundary conditions and variable curvatures. First, the model of a curved beam of variable curvature is constructed, and then the beam is divided into a number of free beam segments via a multi-segment segmentation (MSS) strategy. Second, the first-order shear deformation theory (FSDT) is adopted to obtain the displacement fields of each segment, and then the kinetic energy and potential energy of the structure are expressed by the displacement field. Finally, the exact solution is obtained by the Hamilton principle. Using the springs to simulate various boundary conditions, the frequency parameters, modal shapes and forced vibration responses of the structure with elastic boundary conditions are calculated, with the convergence and correctness verified. Finally, effects of the FGP curved beams, such as porosity distribution types, porosity ratios, boundary condition types, geometry parameters and load types, are investigated in detail.