In this paper, a combination solution method based on the analogy of the generalized-α (AG-α) method is proposed to solve linear rigid-flexible coupling structural dynamic problems, which is termed the G-G(ρ1∞, ρ2∞) combination solution method. In the G-G(ρ1∞, ρ2∞) method, the degrees of freedom of the system are divided into two groups. One group with larger stiffness is solved by using the AG-α method with parameters determined by the spectral radius ρ1∞ corresponding to the infinite frequency. The other, with smaller stiffness, is solved by using the AG-α method with parameters determined by ρ2∞. Taking two differential equations of second order as test equations, the amplification matrix of the G-G(ρ1∞, ρ2∞) method has two pairs of principal eigenvalues and two spurious eigenvalues. The percentage amplitude decay and period elongation are defined and investigated with the two algorithmic frequencies and two algorithmic damping ratios. In addition, the accuracy order and stability of the combination method are discussed. Numerical examples show that the G-G(ρ1∞, ρ2∞) method can filter out the high-frequency modes and simultaneously keep the low-frequency modes when solving rigid-flexible coupling structural dynamic problems, and its numerical performances are superior to the AG-α method.

For structural dynamics systems with nonlinear stiffness, a two-step method with all parameters controlled by rho infinity which is the spectral radius for infinity frequency, called the rho infinity-TSM, was constructed based on the parameter spectral analysis theory (Eur. J. Mech. A-Solid. 94: 104582 (2022)). The rho infinity-TSM has unconditional stability, but it is second-order accurate only when rho infinity is equal to 1. To address this issue, a three-step method and a four-step method are designed in this work, and their numerical properties including stability, accuracy order, and calculation accuracy are investigated. Unlike the rho infinity-TSM, when rho infinity is within a certain range less than 1, the three-step and four-step methods have second-order accuracy. Besides, they possess desirable stability and controllable high-frequency dissipation for both linear and nonlinear dynamic systems. Numerical experiments show that for nonlinear structural dynamics problems, the three-step and four-step methods have advantage in stability over the rho infinity-Bathe method and the OALTS method, and which method has higher calculation accuracy depends on the problem to be solved.

According to the Mindlin plate theory and the first-order piston theory, this work obtains accurate closed-form eigensolutions for the flutter problem of three-dimensional (3D) rectangular laminated panels. The governing differential equations are derived by the Hamilton’s variational principle, and then solved by the iterative Separation-of-Variable (iSOV) method, which are applicable to arbitrary combinations of homogeneous Boundary Conditions (BCs). However, only the simply-support, clamped and cantilever panels are considered in this work for the sake of clarity. With the closed-form eigensolutions, the flutter frequency, flutter mode and flutter boundary are presented, and the effect of shear deformation and aerodynamic damping on flutter frequencies is investigated. Besides, the relation between panel energy and the work of aerodynamic load is discussed. The numerical comparisons reveal the following. (A) The flutter eigenvalues obtained by the present method are accurate, validated by the Finite Element Method (FEM) and the Galerkin method. (B) When the span-chord ratio is larger than 3, simplifying a 3D panel to 2D (two-dimensional) panel is reasonable and the relative differences of the flutter points predicted by the two models are less than one percent. (C) The reciprocal relationship between the mechanical energy of the panel and the work done by aerodynamic load is verified by using the present flutter eigenvalues and modes, further indicating the high accuracy of the present solutions. (D) The coupling of shear deformation and aerodynamic damping prevents frequency coalescing.

This work presents an analytical model for large-amplitude free vibrations of toroidal sandwich shells with functional graded material surface layers and a honeycomb core layer. By employing Hamilton's principle, the nonlinear equations of motion are derived based on the higher order shear deformation theory and von Karman nonlinearity theory. The frequency-dependent parameters H1(omega) and H2(omega) are introduced to formally decouple the displacement components. With the concept of energy balance, the expressions of nonlinear frequency dependent on the frequency-dependent parameters are obtained by integrating the time infinitesimal with respective to amplitude. The direct iterative method is employed to solve the nonlinear frequency. For linear free vibrations, an eigenvalue method for this model is also presented as an alternative. The influences of material distribution/properties, configuration geometrical parameters, and vibration amplitude on the nonlinear free vibration behaviors are examined in parameter study. The results show that, for toroidal shells circumferentially open, the nonlinear free vibration behaviors in the positive and negative half vibration cycle can be unsymmetrical, even if the material distribution and properties are symmetrical in thickness direction.

In this paper, we propose a new geometrically exact shell model integrating higher order shear deformation theory (HSDT). Departing from the classical geometrically exact shell model, the proposed shell model fundamentally reconstructs the basic kinematic assumption through synergistic use of the unit normal vector field and director field, enabling higher order deformation pattern of transverse fibers. This reconstruction naturally leads to newly defined generalized strain components and a novel constitutive relation. Compared to the five-DOF classical geometrically exact shell model, seven DOFs are allocated to each node at the element level due to employing the unit normal vector field in constructing kinematic relations. The additional DOFs are eliminated in the global equations by enforcing the constraint conditions for the unit normal vector field according to kinematic relations. Since the unit normal vector is employed as an intermediate quantity in this model, distinct from the updates of director field, a specialized update procedure is developed to maintain orthonormality of the unit normal vector field during large rotations. The MITC scheme is borrowed in this model to address membrane and shear locking phenomena. A comprehensive set of numerical examples is presented to illustrate the effectiveness of the present formulation in predicting static behaviors of shells undergoing large deformations and rotations.

Fiber-reinforced composite laminated plates are widely used in engineering fields such as aerospace, vehicles and ships. Accurately and rapidly solving natural modes of laminates is significant for their designs and dynamic property evaluation. Considering both symmetric and antisymmetric cross-ply and angle-ply laminates, this paper presents an analytical solution method for the natural modes of rectangular thin laminates based on the iterative separation-of-variable (iSOV) method. For symmetric cross-ply laminates, high-precision closed-form analytical natural modes are obtained by the iSOV method. For symmetric angle-ply laminates with bending-torsion coupling, a semi-analytical solution method based on the Rayleigh-Ritz principle is proposed. In this method, the closed-form mode functions obtained by the iSOV method serve as basis functions, and the frequency equation and mode functions are then achieved according to the Rayleigh quotient. For the free vibration of antisymmetric laminates, this paper establishes a theoretical model that involves only one independent displacement, i.e., the deflection, by assuming in-plane resultant forces to be zero, then uses the iSOV method to solve for closed-form natural modes for some boundary conditions and semi-analytical natural modes for other boundary conditions, greatly simplifying the solution complexity. Numerical results indicate that the proposed methods can effectively produce natural modes with high accuracy.

A theoretical model for large amplitude free vibration analysis of a geometrically imperfect sandwich shallow arch is developed in this paper. The sandwich arch is composed of functionally graded surface layers and an auxetic honeycomb core layer. The Hamilton's principle in conjunction with the higher-order shear deformation theory and von Karman nonlinearity theory is employed to derive the nonlinear differential governing equations. The implicit coupling relation of displacement components is converted into an explicit form by introducing a frequency-dependent parameter. Aided by the explicit coupling relation in displacement field, the nonlinear frequency expression is derived in the context of energy balance method. As the nonlinear frequency expressions are dependent on the frequency-dependent parameter, the nonlinear frequencies are determined through the fixed-point iterative method. Based on the proposed theoretical model, a comprehensive parameter study is conducted to inspect the influences of the material distribution, arch curvature, and initial geometric imperfection on nonlinear vibrations. According to the results of parameter study, the unsymmetrical nonlinear vibration behaviors between the positive and negative half vibration cycle are captured for sandwich arches with nonzero curvature. Some other new results revealing the peculiar nonlinear vibration behaviors of the geometrically imperfect sandwich arches are also provided.

This paper presents a novel 7-DOF geometrically exact shell formulation that describes through-thickness stretching using a five-parameter extensible director field. Unlike conventional resultant-based shell models, the director is defined throughout the shell domain rather than solely on the mid-surface, which brings the kinematics conceptually closer to a three-dimensional solid element while retaining the rigor in rotation treatment from geometrically exact shell theory. The model directly incorporates fully three-dimensional hyperelastic constitutive relations without any stress-state assumptions, thereby enables its applications with complicated nonlinear material models. Membrane, shear, and bending locking are mitigated using a higher order Mixed Interpolation of Tensorial Components (MITC) scheme, while thickness locking is inherently avoided through the asymmetric through-thickness deformation afforded by the five-parameter extensible director. Three formulation variants—full, reduced, and uniform thickness-extension—are systematically compared via a suite of numerical examples. Results show that the full formulation, which fully accounts for the influence of the fifth director parameter on all strain components, delivers superior accuracy, robustness, and convergence, especially for incompressible or nearly incompressible materials under significant Poisson effects. The proposed model thus offers a versatile and physically transparent framework for the finite element analysis of shells undergoing large deformations and complex material behavior.

Using the closed-form natural mode functions of rectangular plates as Ritz basis functions, this work presents a semi-analytical solution method for the free vibration analysis of arbitrarily shaped plates with stiffeners and cutouts. In the present method, the plates are approximated by the Kirchhoff plate theory, the stiffeners are modeled by a curvilinear beam theory, and the cutouts are considered as removing energy stored in cutout domain. The arbitrarily shaped plate (physical plate) is built in a rectangular plate (background plate) that covers the physical plate, retaining the energy of the physical plate and removing the residual energy of the background plate. The closed-form natural mode functions of the background plate are used as basis functions. The closed-form natural modes are obtained by the iterative Separation-Of-Variable (iSOV) method, and an effective procedure to make use of the iSOV method is presented to obtain the natural modes as many as acquired. The present method decomposes the arbitrarily shaped plate into several curvilinear triangular and quadrilateral patches, and transforms each patches into a standard square domain using the blending function method, where the integration of energy functional is performed by Gauss quadrature. To validate the present method, comprehensive comparisons of the present results with those of commercial software and available literature are presented, revealing that the present method using fewer basis functions can achieve accurate natural modes, even higher modes, attributing to the superiority of the present novel basis functions. The present basis function derived from rectangular plates can be viewed as a natural extension and improvement of the beam functions widely used in Ritz method.

The separation-of-variable (SOV) methods, such as the improved SOV method, the variational SOV method, and the extended SOV method, have been proposed by the present authors and coworkers to obtain the closed-form analytical solutions for free vibration and eigenbuckling of rectangular plates and circular cylindrical shells. By taking the free vibration of rectangular thin plates as an example, this work presents the theoretical framework of the SOV methods in an instructive way, and the bisection-based solution procedures for a group of nonlinear eigenvalue equations. Besides, the explicit equations of nodal lines of the SOV methods are presented, and the relations of nodal line patterns and frequency orders are investigated. It is concluded that the highly accurate SOV methods have the same accuracy for all frequencies, the mode shapes about repeated frequencies can also be precisely captured, and the SOV methods do not have the problem of missing roots as well.

This work presents a semi-analytical solution method for free vibration and supersonic flutter of rectangular stiffened thin plates. In this semi-analytical method, the displacements of stiffened plates are expressed as the superposition of the closed-form natural modes of unstiffened plates; the Kirchhoff plate theory and the linear piston theory are employed; and the stiffeners are modeled using a curvilinear beam theory. The present method can deal with rectangular stiffened thin plates with arbitrary homogenous boundary conditions. Numerical results demonstrate that the present semi-analytical solutions using fewer base functions can yield more accurate higher modes, which is a key advantage compared to other superposition methods. In addition, considering a rectangular plate with two opposite edges simply supported and stiffened by a single straight stiffener, exact solutions are achieved by an analytical method, where the governing differential equations of the stiffened plate are derived according to the generalized Hamilton variational principle and solved by the semi-inverse method. The present results are validated by detailed comparisons with those of the MSC/NASTRAN Software System and available literature.

According to the Donnell–Mushtari shell theory, this work presents a closed-form solution procedure for free vibration of open laminated circular cylindrical shells with arbitrary homogeneous boundary conditions (BCs). The governing differential equations of free vibration are derived from the Rayleigh quotient and solved by the iterative separation-of-variable (iSOV) method. In addition, considering axial aerodynamic pressure, simulated by the linear piston theory, the exact eigensolutions for the flutter of open laminated cylindrical shells with simply supported circumferential edges and closed laminated cylindrical shells are also achieved. The governing differential equations of cylindrical shell flutter are derived from the Hamilton variational principle and solved by the separation-of-variable (SOV) method. The influence of circumferential dimension on flutter speed is investigated for open cylindrical shells, which reveals that the number of circumferential waves in critical flutter mode increases with circumferential length, and there exists an infimum for flutter speed that is an invariant independent of circumferential length. The present results agree well with those obtained by the Galerkin method, the finite element method, and other analytical methods.

Thin-walled circular cylindrical shells are important components in diverse engineering fields. There have been extensive researches on the buckling of open circular cylindrical shells, but still few researches on analytical solutions for non-Levy boundary conditions (BCs). This work develops an extended separation-of-variable (eSOV) method for eigenbuckling analysis of open circular cylindrical shells with arbitrary homogenous BCs. In this eSOV method, buckling mode functions are the product of eigenfunctions of two coordinate directions, and critical buckling loads corresponding to two-direction eigenfunctions are independent of each other. By employing Rayleigh principle, two eighth-order characteristic differential equations are derived, and the two-direction eigenfunctions are expressed in terms of the eigenvalues of the corresponding characteristic differential equations, then closed-form mode functions and explicit equations for critical load are achieved. The results in this work are validated by numerical methods, and several interesting phenomena are found.

Stiffened plates are widely used in various engineering fields as main load-carrying components. Semi-analytical methods are effective for studying the eigenbuckling problems of stiffened rectangular plates, but there are no semi-analytical solutions available for stiffened plates with arbitrary homogeneous boundary conditions. This work aims to develop a semi-analytical method for solving the eigenbuckling problems of orthotropic stiffened thin plates. In this method, base functions for constructing the mode functions of stiffened plate are the mode functions of a simple plate (a plate without stiffener), and the base functions are obtained with the extended separation-of-variable method, which is a closed-form solution method for the eigenvalue problems of plates. The critical buckling load is achieved by substituting the mode functions into the Rayleigh's principle. The present method can deal with arbitrary homogeneous boundary conditions without anticipating the forms of the base functions for different boundary conditions. The accuracy can be improved by using more superposition terms. Besides, an exact closed-form solution of simply supported stiffened plates is achieved, serving as benchmark solutions. Numerical experiments validate the accuracy of the present solutions, and the study on the minimum stiffener stiffness is conducted for different boundary conditions.

The panel flutter analysis considering the shear deformation is performed with a particular focus on the exact eigensolutions and the energy transfer. According to Mindlin plate theory and the first-order piston theory, this work obtains the exact eigensolutions of two-dimensional (2D) panel flutter at supersonic speeds. The boundary conditions (BCs) considered include: two edges are simply supported (SS), two edges are clamped (CC), and two edges are clamped and free (CF) respectively. Using the obtained exact eigensolutions, the relations of the kinetic energy, the potential energy and the work done by aerodynamic force are examined for the states of divergent vibration and periodic vibration. Besides, the coupling effect of shear deformation and aerodynamic damping on flutter frequencies is also revealed, and the present results are compared with those of the Galerkin method and the 2D Kirchhoff panel.

The Wilson-u03C1u221E technique is formed by establishing a relationship between u03C1u221E and u03B8, as the high-frequency dissipation of the Wilson-u03B8 method cannot be properly controlled by u03C1u221E (spectral radius at infinite frequency). The numerical performances of this approach are compared with those of the Generalized-u03B1 method. In the Wilson-u03C1u221E method, there are two different u03B8 for a given u03C1u221E. The characteristic roots of the Jacobi matrix corresponding to both u03B8 are different, and the corresponding Wilson-u03C1u221E method has different numerical performances. A better u03B8 is recommended according to the properties of the spectral radius. In addition, an analog system of a single degree-of-freedom forced vibration system is constructed with the dissipation and frequency of the Wilson-u03C1u221E method, and the initial conditions of which and the forces acting on the analog system are the same with those of the original system. It is evident that the steady state responses have no cumulative amplitude errors and phase errors, and the results of the Wilson-u03C1u221E method match the analytical solutions of the analog system.

All dielectric materials including ceramics, semiconductors, biomaterials, and polymers have the property of flexoelectricity, which opens a fertile avenue to sensing, actuation, and energy harvesting by a broad range of materials. However, the flexoelectricity of solids is weak at the macroscale. Here, we achieve an ultrahigh flexoelectric effect via a composite foam based on PDMS and CCTO nanoparticles. The mass- and deformability-specific flexoelectricity of the foam exceeds 10,000 times that of the solid matrix under compression, yielding a density-specific equivalent piezoelectric coefficient 120 times that of PZT. The flexoelectricity output remains stable in 1,000,000 deformation cycles, and a portable sample can power LEDs and charge mobile phones and Bluetooth headsets. Our work provides a route to exploiting flexible and light-weight materials with highly sensitive omnidirectional electromechanical coupling that have applications in sensing, actuation, and scalable energy harvesting.

This paper proposes a composite method for the analysis of rigid body rotations based on Euler parameters. The proposed method contains three sub-steps, wherein for keeping as much low-frequency information as possible the first two sub-steps adopt the trapezoidal rule, and the four-point backward interpolation formula is used in the last sub-step to flexibly control the amount of high-frequency dissipation. On this basis, in terms of the relation between Euler parameters and angular velocity, the stepping formulations of the proposed method are further modified for maximizing the accuracy of the angular velocity. For the analysis of rigid body rotations, the accuracy of the proposed method can converge to second-order, and the amount of its high-frequency dissipation can smoothly range from one (conservative scheme) to zero (annihilating scheme). Additionally, in the proposed method, the constraints at the displacement and velocity levels are strictly satisfied, and the numerical drifts at the acceleration level can be effectively eliminated. Furthermore, the proposed method is generalized to the field of rigid-flexible multibody systems described by Euler parameters, and in this work, its implementation procedure is provided. Several benchmark rigid body rotations and rigid-flexible multibody problems show the advantages of the proposed method in stability, accuracy, dissipation, efficiency, and energy conservation.

Skilled human resource becomes an essential resource for implementing intelligent manufacturing in the new era, prompting high demands on Intelligent Manufacturing Training (IMT). Empowering the effective IMT, the new mode of IMT based on Industrial Metaverse is proposed as well as detailed comparison with traditional training modes. The layered technical architecture is discussed as a guidance for training system construction, as well as specific solutions for the six key technologies based on primary research, including rapid modeling, natural interaction, real-time communication, industrial avatar/agent, industrial tools access, industrial AIGC, etc. Verifying the effectiveness of Industrial Metaverse based IMT, a prototype system “TrAiN” for industrial internet skill training is built, constructing a private Industrial Metaverse based on specific industrial equipment and fields in certain factory, facilitating the virtual training. Future research hotspots on Industrial Metaverse based IMT are prospected at the end based on the primary research and application.

This paper develops a new homogenization method for free vibration problems of periodic composite plates. In this new method, three-dimensional (3D) periodic plates are equivalent to Reissner–Mindlin plates with both effective stiffnesses and effective inertia coefficients. The effective stiffnesses for the dynamic problems are the same as those for the static problems, and they can be achieved by the equivalence principle of macro- and microscopic internal virtual work. To fully take the inertia effects into account, the effective inertia coefficients including the effective translational, translational–rotational and rotational inertias are determined by the two-scale equivalence principle of kinetic energies under three rigid modes. In addition, cell size effects in the thickness direction of composite plates are investigated by using the proposed method and the asymptotic homogenization method (AHM). Numerical experiments validate the effectiveness of the proposed equivalent method for different scale factors, and show that the rotational inertia cannot be ignored for out-of-plane deformations, especially for higher-order modes. Besides, numerical comparisons show that the cell size effects are not negligible when using the AHM to analyze the out-of-plane deformations, and three or more repeated unit cells in the thickness direction are required to ensure accuracy.