Kirchhoff Plate Theory Research Articles

Influences of blade crack on the coupling characteristics in a bladed disk with elastic support

In aero-engines, the blades are exposed to severe environments and subjected to complex conditions, which can cause damage and cracks. Existing analytical models focus primarily on the blade crack in a single blade. Even in the cracked-bladed disk system, most theoretical models are based on the rigid disk assumption, and few studies pay attention to the boundary conditions of the disk. A new model is proposed in this paper to compensate for deficiencies in existing theoretical models. The flexible disk and blade are deduced based on Kirchhoff plate and Timoshenko beam theories. The crack-caused stiffness loss is calculated through the released strain energy. In addition, the elastic support on the disk's inner edge is simulated by the linear spring stiffness and torsional stiffness. The accuracy of the developed model is verified by comparing it with the finite element method and experimental test. After that, the influences of the crack depth and crack location on the coupling characteristics of the system are systematically investigated. The results indicate that blade crack mainly affects the blade-predominated modes, leading to the mistuned disk mode shapes, which are symmetrically distributed along the line between the cracked blade and the disk center. Moreover, as the blade crack becomes severe or approaches the blade root, the first natural frequency of the paired orthogonal modes decreases; the first mode shape of the paired tuned high nodal diameter degenerates into mistuned low nodal diameter; the new coupling characteristics between the mistuned disk and elastic support become more stronger. The proposed methodology can serve as a theoretical basis for diagnosing faults and monitoring safety in the bladed disk coupling system.

Improving climate resilience: Reduced-order mechanical modelling of a temperature-adaptable sun protection facade system

Analytical models to describe the mechanical behaviour of an adaptive sun protection facade system (Adaptex) are presented. First, a one-dimensional (1D) rigid link model is developed. Stretching and bending deformation of textile bands are represented with the aid of extensional and rotational springs respectively. The model is described by two generalized coordinates only. The second analytical model employs a continuous description based on the Kirchhoff plate theory. The Rayleigh–Ritz method is applied to analyse the deformation behaviour of the textile bands. In comparison with the experimental results, it can be found that the continuous model simulates the real deformation of the facade system, thus its morphological behaviour, very effectively. The results of a parametric study indicate that the stiffness of the textile bands, the geometric dimensions and the position of load application have significant effects on the deformation behaviour.

Free Vibration Analysis of Rectangular Plate with Cutouts under Elastic Boundary Conditions in Independent Coordinate Coupling Method

Based on Kirchhoff plate theory and the Rayleigh-Ritz method, the model for free vibration of rectangular plate with rectangular cutouts under arbitrary elastic boundary conditions is established by using the improved Fourier series in combination with the independent coordinate coupling method (ICCM). The effect of the cutout is taken into account by subtracting the energies of the cutouts from the total energies of the whole plate. The vibration displacement function of the hole domain is based on the coordinate system of the hole domain in this method. From the continuity condition of the vibration displacement function at the cutout, the transition matrix between the two coordinate systems is constructed, and the mass and stiffness matrices are completely obtained. As a result, the calculation is simplified and the computational efficiency of the solution is improved. In this paper, numerical examples and modal experiments are presented to validate the effectiveness of the modeling methods, and parameters related to influencing factors of the rectangular plate are analyzed to study the vibration characteristics.

Axisymmetric bending vibration analysis of bidirectional functionally graded rotating micro-disk under thermo-mechanical loading

The free vibration behavior of a bidirectional functionally graded rotating micro-disk that is subjected to uniform transverse pressure and high-temperature thermal loading has been studied. The micro-disk is functionally graded along the radial and thickness directions. The problem is mathematically formulated using two distinct but interrelated steps within the framework of Kirchhoff plate theory and modified couple stress theory. The first step determines the time-invariant deformed configuration of the micro-disk under centrifugal, pressure, and thermal loading using minimum potential energy principle. The second step determines the free vibration behavior of the micro-disk in the neighborhood of the deformed configuration using Hamilton’s principle. The solutions of the governing equations for both these steps are obtained using the Ritz method. The mathematical model is successfully validated with various reduced problems. The numerical results for the first four axisymmetric bending vibration modes are presented to investigate the effects of wide range of parameters such as rotational speed, applied pressure, thermal loading, size-dependent thickness, volume fraction indices, and radius ratio. The mode shapes of vibration are illustrated through surface and contour plots.

Vibration attenuation of rotating disks via acoustic black holes

Acoustic black holes (ABHs) have been found to exhibit desirable damping characteristics for vibration attenuation of thin-walled structures. However, up until now, no research has been carried out on evaluating ABH effects on vibrations of rotating structures, of which vibration attenuation issues are usually critical. To fill this knowledge gap, this work proposes a unified modeling approach for rotating thin disks with ABH indentations. The equations of motion of the system are formulated using the Rayleigh-Ritz method together with the Kirchhoff plate theory. The modified Fourier series are employed to form the displacement admissible functions. Finite element analysis is conducted to validate the proposed model. Damping performance of the ABH disk is then investigated under both non-rotating and rotating states. Finally, experiments are carried out to study the vibration attenuation effect of the ABH on non-rotating and rotating disks. The natural frequencies of the ABH disk with viscoelactic layers at different rotating speeds obtained from experiments and numerical calculations are compared to further verify the modeling approach. With all the simulation and experimental results, it is found that ABHs are capable of realizing vibration reduction for rotating structures, where the effectiveness is influenced by rotating speeds and excitation types.

Analysis of surface effect and flexoelectric effect on the electromechanical responses of bilayered transversely isotropic rectangular micro-plate

A new model of a bilayered transversely isotropic piezoelectric rectangular micro-plate with a distributed load is developed on the basis of Kirchhoff's plate theory and the extended linear piezoelectricity theory to characterize the piezoelectric micromachined ultrasonic transducer. The model takes into account both the surface effect and the flexoelectricity effect. The governing equation at the simply supported boundary condition is derived according to the variation principle. Based on the new model, the size dependent electromechanical coupling behaviors of the bilayered piezoelectric rectangular micro-plate are investigated. Considering the flexoelectric effect and surface effect synchronously, the numerical result indicates that the size dependence of the normalized central deflection decreases as the residual surface stress increases. For negative residual stress, the surface effect is the main influencing factor. While for positive residual stress, the surface effect dominates only when the ratio of thickness to length is smaller than about 25; otherwise, the flexoelectric effect will be more crucial. Moreover, if the thickness of the piezoelectric layer is less than about 40 nm, the electrical potential and polarization show a stronger size dependence. These results will be helpful to design and manufacture a piezoelectric micromachined ultrasonic transducer.

Modelling and vibration evaluations of rotational mistuning-connecting disk-drum coupling structures with partial hard-coating damping treatment

Mistuning connections between drums and disks are very common in engineering. However, this point has not been mentioned in previous studies. This work develops discontinuous arc connections with different connecting stiffness to simulate actual mistuning connections by modifying the traditional spring method with continuous distributions. The admission functions are represented by good-convergence orthogonal polynomials. Through considering the rotation-generated gyroscopic and centrifugal effects and utilizing Kirchhoff's plate theory and Sanders’ shell theory, the dynamic model of rotational mistuning-connecting disk-drum coupling structures with partial hard-coating damping treatment (PHDT) is derived through Lagrange equations and verified by utilizing FEA and existing literature, in which the partial-coating model is more suitable for practical vibration suppression than the existing whole coating model. Moreover, further mistuned vibration evaluations show that the mistuning connections induce the change of mode order, frequency division, and vibration localization. The veering phenomenon caused by mistuning is also revealed, and the modal coupling and dominant mode exchange during veering are explained by the change of mode shape and MAC. Furthermore, the mistuning will also seriously affect the key information (critical speeds, instability positions, and veering positions) in Campbell diagrams. The present model provides key technical references for the vibration prediction, connection fault diagnosis, and coating parameter optimization of rotational disk-drum coupling structures with PHDT.

The nonlocal elasticity theory for geometrically nonlinear vibrations of double-layer nanoplate systems in magnetic field

The geometrically nonlinear vibrations of simply supported double-layer graphene sheet systems under in-plane magnetic field are considered in the presented manuscript. The interaction between layers is taken into account due to van der Waals forces. The investigation is based on the nonlocal elasticity theory, Kirchhoff plate theory and von Kármán theory. The effect of the magnetic field is due to the Lorentz force based on Maxwell’s equations. The governing equations are used in mixed form by introducing the stress Airy function. The analytical presentation of the nonlinear frequency ratio for in-phase vibration and anti-phase vibration modes is presented. It is shown that the nonlocal parameter in the compatibility equation can significantly change the vibration characteristics.

Effect of ribs on vibration characteristics of cantilever plate

Analytical solution for the vibration response results of the ribbed rectangular cantilever plate using the Kirchhoff plate theory is presented in this paper. The validity of the analytical solution is verified by comparison with the results of finite element analysis. The analytical model is then used to study the effect of the single rib, periodic ribs and orthogonal ribs on the modal vibration of the cantilever plate. According to the minimum inertia axis and torsion center line of the cantilever plate, the modal vibration shapes generated by the cantilever plate are divided into flexural and torsional vibration mode. The effect of the insertion of single rib on the modal vibration of the cantilever plate and the curvature of the insertion position are analyzed. It can be found that the effect of single rib on modal vibration of cantilever plate depends on the vibration mode and the location of the rib insertion, where a rib can be used as a mass loading, a stiffness attachment, or a combination of both. And the effect of periodic ribs on the vibration characteristics of the cantilever plate is further investigated. When periodically ribbed, the effect of the periodic ribs on the modal vibration of the cantilever plate depends on the dominant of the total of the mass loading or the stiffness enhancement played by each of these ribs. However, when another rib is added to the torsion center line of the periodic ribbed cantilever plate, it shows stiffness enhancement in both flexural and torsional vibration.

Bioinspired periodic panels optimized for acoustic insulation.

The design of structures that can yield efficient sound insulation performance is a recurring topic in the acoustic engineering field. Special attention is given to panels, which can be designed using several approaches to achieve considerable sound attenuation. Previously, we have presented the concept of thickness-varying periodic plates with optimized profiles to inhibit flexural wave energy propagation. In this work, motivated by biological structures that present multiple locally resonant elements able to cause acoustic cloaking, we extend our shape optimization approach to design panels that achieve improved acoustic insulation performance using either thickness-varying profiles or locally resonant attachments. The optimization is performed using numerical models that combine the Kirchhoff plate theory and the plane wave expansion method. Our results indicate that panels based on locally resonant mechanisms have the advantage of being robust against variation in the incidence angle of acoustic excitation and, therefore, are preferred for single-leaf applications. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)'.

Interdependent effects of surface and flexoelectricity on the electromechanical behavior of BNRC nanoplate

The present work deals with the static and dynamic behavior of boron-nitride (BN) reinforced piezoelectric nanocomposite subjected to mechanical load while considering flexoelectric and surface effects. Based on the modified Kirchhoff plate theory and Navier's solution, an analytical model is derived to investigate the electromechanical response of simply supported (SS) boron-nitride reinforced nanocomposite (BNRC) plates. In addition, a micromechanical model by employing the Mori-Tanaka (MT) approach is derived to investigate the effective elastic and piezoelectric properties of the BNRC lamina. The BNRC lamina is composed of BN nanofiber and polyimide matrix, such that the BN nanofibers are oriented along the 3-direction. Our outcomes reveal that the incorporation of the BN nanofiber shows a substantial enhancement in the effective longitudinal and transverse piezo-elastic coefficients of the BNRC lamina. Further, the outcomes of the analytical model by adopting the Kirchhoff plate theory illustrate that the effects of flexoelectricity and surface show significant enhancement in the stiffness of the composite at the nanoscale, although the surface effect is more pronounced when compared with the flexoelectric effect. Based on the present analysis, we observed that the effects of flexoelectricity and surface influence the static and vibrational properties of BNRC. Thus, at the nanoscale, these size-dependent properties cannot be neglected. This work presents an opportunity for the development of high-performance and efficient BNRC nanoplates.

Asymptotic analysis of 3D dynamic equations in linear elasticity for a thin layer resting on a Winkler foundation

Abstract A 3D dynamic problem for a thin elastic layer resting on a Winkler foundation is considered. The stiffness of the layer is assumed to be much greater than that of the foundation in order to allow low-frequency bending motion. The leading long-wave approximation valid outside the vicinity of the cut-off frequency is derived. It is identical to the classical Kirchhoff plate theory. A novel near cut-off 2D approximation is also established. It involves both bending and extension motions which are not separated from each other due to the effect of the foundation. The associated dispersion relation appears to be non-uniform over the small wavenumber domain.

Performance optimization and broadband design of piezoelectric energy harvesters based on isogeometric topology optimization framework

Energy harvesting is attracting considerable attention in devices that cannot replace exhausted batteries, where improving the energy harvesting efficiency of piezoelectric energy harvesters (PEHs) through different optimization methods is always a research hotspot. This paper proposes an isogeometric topology optimization framework to implement the simultaneous optimization of fiber angles and piezoelectric material and reveal the feasibility of broadband design for PEHs. We deduce the isogeometric formulation of PEHs based on Kirchhoff plate theory and Hamilton's principle. The optimization formulations are established with minimum energy factor or maximum output power considered, and the distributions of fiber angles and piezoelectric material are respectively defined by the Heaviside Penalization of Discrete Material Optimization (HPDMO) model and the Piezoelectric Material with Penalization (PEMAP) model. The numerical results show that there is only a slight difference in the optimized designs of fiber angles between the static load and harmonic load when the objective function is the minimum energy factor. Furthermore, taking maximum output power as the objective function, resistance has a substantial effect on the optimized distributions of fiber angles and piezoelectric material, and there is an optimal resistance making the output power of the optimized design increase 9 times compared with the initial model. More important, by combining the designability of composite material, the proposed method is extended to the broadband design of PEHs. It is concluded that the optimized design can not only generate two resonance peaks in the predetermined frequency range but also the second-order output power amplitude is 14 times higher than that of the benchmark model.

Free Vibration Analysis of Rectangular Thin Plates Resting on Nonhomogeneous Elastic Foundations by Using Matched Interface and Boundary Algorithms and their Multi-Domain Formulations

Matched interface and boundary (MIB) algorithms and their multi-domain formulations are developed to solve free vibration problems of thin plates resting on unidirectionally and bidirectionally nonhomogeneous elastic foundations. Winkler and Pasternak types of elastic foundations are considered. Three types of MIB algorithms are introduced to handle unidirectional interfaces caused by nonhomogeneous foundations. On the basis of in-depth understanding of MIB method and Kirchhoff plate theory, new multi-domain MIB (MD-MIB) algorithms are proposed to handle bidirectional interfaces caused by nonhomogeneous foundations that cannot be handled by using the existing MIB method. Three types of MD-MIB algorithms are developed by using three domain decomposition strategies. A number of examples are chosen to validate the applicabilities of different MIB and MD-MIB algorithms to the title problems. Numerical results obtained by using MIB and MD-MIB algorithms are compared with the existing solutions or finite element solutions. Numerical studies show that MIB and MD-MIB algorithms are efficient approaches to deal with nonhomogeneous elastic foundations.

Properly-tuned continuum and atomistic models for vibrational analysis of the silicon nanoplates

The high computational costs of atomistic simulations for the investigation of nanostructures, despite their accuracy, necessitate efforts to develop efficient continuum models. Since the classical continuum mechanics assume the matter as continuous and ignore the size dependency of material properties, these theories fail to capture the behavior of materials at the nanoscale. Size-dependencies of nanostructures could result in the emergence of surface effects. This paper, for the first time, aims to develop and tune surface-enhanced continuum models to predict the vibrational properties of nanoplates which are inherently discrete at the nanoscale. Regarding this aim, we use a composite core-shell modeling scheme for integrating Kirchhoff and Mindlin plate theories with the surface effects. We have performed Molecular Dynamics (MD) simulations to determine the accurate vibrational behavior of silicon nanoplates. We present a systematic procedure for obtaining continuum model parameters, including the surface elastic parameters and surface-induced stress, from our MD simulation results. In doing so, this simulation method benefits from the merits of both continuum- and atomistic-based modeling. We further examine the validity of the developed continuum models by comparing the results achieved using calibrated continuum models and the validation set of MD results. It is demonstrated that properly tuned continuum models, especially the Mindlin plate theory with surface effects, can accurately predict the vibrational behavior of ultrathin Si nanoplates. Furthermore, our calibrated continuum models based on the Mindlin plate theory offer an effective tool for capturing higher natural frequencies of Si nanoplates, introducing new paradigms in nanomechanical resonators.

Modeling method for analyzing veering and nonlinear vibration of rotating hard-coated drum-disk structures considering the strain-amplitude dependency

This paper proposes a modeling method of rotating hard-coated drum-disk structures considering the strain-amplitude dependency of coatings and studies the veering of frequency curves and nonlinear coupled vibration. Firstly, Sanders’ shell theory and Kirchhoff's plate theory are used to derive the energy expressions of the hard-coated drum and hard-coated disk, respectively. The arbitrary multi-arc joints between disk and drum and arbitrary boundaries are presented based on the artificial spring. Furthermore, the displacements of the drum and disk are expanded by using the orthogonal polynomials, and Lagrange's equations are employed to derive the linear semi-analytical model. Secondly, the multi-domain dividing approach is used to develop a general method considering the strain-amplitude dependency in the rotation state, and the nonlinear semi-analytical model of rotating hard-coated drum-disk structures is created and further solved by an efficient algorithm proposed in the present work. The convergence and validation of the semi-analytical model are verified. Finally, the veering of frequency curves caused by the variations in rotation speeds, connection and boundary conditions, and thickness of coatings and substrates is revealed. The variation law and mechanism of nonlinear coupled vibration during veering are also investigated. The proposed modeling method can provide helpful references for the modeling and vibration analysis of multi-stage hard-coated drum-disk structures in aero-engine and other rotating hard-coated structures.

Study on theoretical modeling and vibration performance of an assembled cylindrical shell-plate structure with whirl motion

Hub-blade rotor structures are widely applied in many spinning machines, such as aero-engine, gas-turbine and so on. In theory, the hub and blade can be modeled by elastic cylindrical shell and plate, respectively. This paper firstly presented an investigation on the coupled modeling and vibration behaviors of a spinning assembled cylindrical shell-plate structure. According to the Donnell shell theory and the Kirchhoff plate theory, the coupled governing equations of the assembled cylindrical shell-plate structure are derived by employing the Lagrange's equation and considering the displacement continuity condition. Then, the method of hypothesis modes is adopted to obtain the free vibration results of the assembled cylindrical shell-plate structure. Moreover, the modeling and vibration analysis in this paper are verified by the finite element method. Special attention is paid to the effects of structural parameters, including plate length-to-thickness ratio, plate width-to-thickness ratio, plate position, cylindrical shell length-to-thickness ratio and cylindrical shell radius-to-thickness ratio on the traveling wave frequencies of the assembled cylindrical shell-plate structure.

Elastodynamic analyses of transversely isotropic unsaturated subgrade–pavement system under moving loads

AbstractIn this paper, elastodynamic responses of the layered transversely isotropic unsaturated subgrade–pavement system subjected to the moving load are investigated. The unsaturated subgrade soil is described via a triphasic Biot‐type model introducing the effect of matric suction, immiscible flows of water and air, and transverse isotropy. The rigid pavement is modeled by the Kirchhoff plate theory, while the flexible pavement is modeled by the stratification medium theory. First, the governing equations of the unsaturated subgrade soil are obtained after algebraical manipulations. The precise integration method (PIM) is further combined with the double Fourier transform to solve the governing equations. The subgrade soil solution is utilized as the flexible coefficient to describe the plate–soil interaction, and the solution of the pavement is further acquired via the double inverse Fourier transform. Four verification examples regarding literatures and COMSOL are presented to verify the correctness of the theory. Finally, numerical examples are conducted to investigate the effect of various parameters on the dynamic behavior of the unsaturated subgrade–pavement system.

Nonlinear Bending and Vibration Analysis of Imperfect Functionally Graded Microplate with Porosities Resting on Elastic Foundation Via the Modified Couple Stress Theory

This paper represents the nonlinear bending and free vibration analysis of a simply supported imperfect functionally graded (FG) microplate resting on an elastic foundation based on the modified couple stress theory and the Kirchhoff plate theory (KPT) together with the von-Kármán’s geometrical nonlinearity. The FG microplates with even and uneven distributions of porosities are considered. Analytical solutions for the nonlinear bending and free vibration are obtained. Comparing the obtained results with the published one in the literature shows the accuracy of the current solutions. Numerical examples are further presented to investigate the effects of the material length scale parameter to thickness ratio, the length to thickness ratio, the power-law index and the elastic foundation on the nonlinear bending and free vibration responses of the FG microplate.

Vibration suppression and energy absorption of plates in subsonic airflow using an energy harvester enhanced nonlinear energy sink

In this paper, a nonlinear energy sink (NES) and a giant magnetostrictive-piezoelectric (GMP) energy harvester (NES-GMP) are investigated to suppress the nonlinear aeroelastic responses and to absorb the mechanical energy of an embedded plate interacting with external subsonic airflow. The analytical model of the embedded plate attaching the NES-GMP is established by using the Hamilton’s principle based on the Kirchhoff plate theory and incompressible subsonic aerodynamic model. The natural frequencies of the plate with the NES-GMP and with the NES are analyzed by solving the generalized eigenvalue problems. The global amplitude-frequency responses of the pure plate, the plate with NES, and the plate with NES-GMP are compared to show the vibration suppression effects of the proposed method. Based on the energy analysis, the energy transfer mechanisms of the plate with the NES-GMP are studied. The results reveal that the input energy of the NES-GMP converts into the strain energy by the Terfenol-D layer and then transforms into the electric energy by the piezoelectric layer. Furthermore, numerical simulations also indicate that the maximum harvested energy is obtained when the airflow velocity approaches the critical divergence flow velocity of the plate, and the most effective harvesting installation position is in a specific annular region near the edge of the plate.