Archive of Applied Mechanics最新文献

Euler–Poisson equations describe the temporal evolution of a rigid body’s orientation through the rotation matrix and angular velocity components, governed by first-order differential equations. According to the Cauchy–Kovalevskaya theorem, these equations can be solved by expressing their solutions as power series in the evolution parameter. In this work, we derive the sum of these series for the case of a free symmetric rigid body. By using the integrals of motion and directly summing the terms of these series, we obtain the general solution to the Euler–Poisson equations for a free symmetric body in terms of elementary functions. This method circumvents the need for standard parametrizations like Euler angles, allowing for a direct, closed-form solution. The results are consistent with previous studies, offering a new perspective on solving the Euler–Poisson equations.

This study explores the thermomechanical 3D wave propagation behavior of a sandwich nanosensor plate with an auxetic core, leveraging nonlocal strain gradient elasticity and sinusoidal higher-order shear deformation theories. The plate comprises functionally graded ceramic (Si₃N₄) and metal (Ti₆Al₄V) face layers, with an auxetic Ti₆Al₄V core having a negative Poisson's ratio. Governing equations are derived using Hamilton's principle, leading to the Navier solution for 3D wave propagation. The results indicate that increasing the β₁ parameter enhances phase velocities and wave frequencies, while smaller β₃ values significantly impact stiffness and frequency. These findings provide a framework for optimizing the design of nanosensors, ensuring improved performance and reliability in high-temperature applications across various industries.

The classical Euler–Bernoulli beam theory, rooted in continuum mechanics and localization theories, fails to incorporate the effects of foundation deformation and atomic lattice interactions, thus inadequately capturing the true mechanical properties of nanobeams. This study aims to address these limitations by proposing a novel computational framework to accurately model the global coupling and mechanical behavior of nanobeams. First, the computational approach integrates both foundation deformation and atomic lattice interactions, constructing a physical model for the nonlocal vibration of nanobeams under arbitrary loading conditions. A validation methodology is also developed to ensure the model’s reliability. Second, using the Laplace transform, the time-domain problem is transformed into a frequency-domain analysis. The Hasselman complex modal synthesis method is employed, and for the first time, the space-state transfer function of the nanobeam vibration model, based on a modified Euler–Bernoulli beam theory incorporating nonlocal effects, is derived. Analytical solutions are presented and validated. Finally, the mechanism of global coupling in nanobeams is examined through the nonlocal intrinsic structure, using the material points of the beam to accurately reflect nanoscale mechanical behavior. Results reveal that the nonlocal factor (ranging from 0 to 0.3) significantly influences the frequency peaks of the nanobeam. As the mode order nnn increases, the frequency peak shifts along the transverse axis toward decreasing nonlocal factors and its magnitude diminishes. Conversely, with increasing beam length, the frequency peak moves toward increasing nonlocal factors on the transverse axis, and the magnitude increases. Furthermore, the influence of the nonlocal factor on the frequency and amplitude is pronounced in lower-order modes, gradually diminishing as the mode order increases. These findings not only enhance the understanding of the vibration characteristics of nanobeams but also provide a robust framework for analyzing the impact of nonlocal effects, offering new insights and avenues for future research in nanostructural mechanics.

The dynamic behavior of a polymer ball bearing system is mostly dependent on the contact characteristics among the ball and the races of the bearing. Although the well-known Hertz contact theory is widely used to model contacts in conventional bearings, it cannot be directly applied to polymer bearings due to the viscoelastic characteristics of the polymer structures. In this study, contacts in the polymer hybrid ball bearing are modeled using elastoplastic characteristics. The contact between inner/outer raceways and the ball is solved in elastic, elastoplastic, and plastic characteristics regions depending on the polymer structure and the loads. Then, two-degree of freedom rigid rotor-bearing system is simulated under different rotational speeds as well as different rotor weights. In order to investigate the nonlinear nature of the dynamic response, results are analyzed with different methods such as waterfalls, bifurcation diagrams, phase diagrams and Poincaré sections. The characteristic changes in the contact form elastic to elastoplastic regions are observed as a new peak in the time history that may lead to chaotic motion. A similar response is also seen when a single ball-race contact is in the elastoplastic region. The results are helpful to understand the cause and result of contact in a viscoelastic contact condition.

In this paper, a generalized conforming element formulated by the quadrilateral area coordinate method is developed for upper bound limit analysis of thin plates. This element can prevent loss of accuracy in severely distorted meshes since the transformation between the area and Cartesian coordinates is always linear. Once the deflection field is approximated and the upper bound theorem applied, upper bound limit analysis of thin plates can be formulated by minimizing the dissipation power subject to a set of equality constraints. In order for overcoming the difficulties caused by the nonsmoothness of the goal function, a direct iterative method is utilized to solve this optimization problem, which distinguishes the rigid zones from the plastic zones at each iteration. Numerical examples show that the proposed method for upper bound limit analysis of thin plates is reasonable and effective and possesses the advantages of high accuracy and reliability even for severely distorted meshes.

The aim of this article is to explore the motion of an infinitesimal body (third body) in the vicinity of the out-of-plane equilibrium points of the restricted three-body problem under the effect of continuation fraction and radiation pressure perturbations. We investigate the effect of the radiation factor of the bigger primary and continuation fraction parameter of the smaller primary on the existence, position, zero-velocity curves and stability of the out-of-plane equilibrium points. It is discovered that the presence of these points is made possible by the bigger primary’s radiation. These equilibria arise in symmetric pair, and their number may be zero or two based on the values of the radiation and mass ratio parameters. Our results reveal that all the involved parameters have strong influence on the position of the out-of-plane equilibrium points. A numerical investigation found that the perturbing parameters have impact on the geometry of the zero-velocity curves. The stability of these points is studied in the linear sense. A detailed numerical investigation found that the equilibrium points are unstable in general.

Thermoelastic dissipation (TED) is a primary source of energy loss in extremely small structures, making the precise determination of its magnitude vital for the optimal design and performance of such components. The inclusion of two-dimensional (2D) heat conduction alongside size effects in both the structural and thermal domains plays a key role in enhancing TED analysis for small-scale beam resonators. The modified couple stress theory (MCST) and Moore–Gibson–Thompson (MGT) heat equation, within the context of the energy approach, are employed in this paper to create a novel size-dependent framework for TED in small-scale beams subjected to 2D heat conduction. After comparing the developed framework with existing research, numerical simulations are carried out to reveal the differences between 2 and 1D models, as well as the impact of employing size-dependent mechanical and thermal formulations. For beams with large thickness-to-length ratios, especially under clamped–clamped (CC) boundary conditions, the proposed model shows significant differences when compared to 1D model. Based on the findings, the ratio of 2D TED to 1D TED in CC beams with an aspect ratio of 10 can be up to 1.6 times. The integration of size effects and 2D heat transfer in the established framework is expected to provide benchmark results for accurate TED simulations and facilitate the optimal design of ultra-small beam resonators.

There exist complex deformation characteristics and torque transmission mechanisms of the structure whose boundary is constrained locally. This paper proposes the power series polynomial constraining method to establish the dynamical modeling of a rectangular thin plate with a localized constraint. The boundary condition can be constructed by applying the power series polynomial with undetermined coefficients to the free boundary directly. This means that the derivation of the admissible function no longer relies on the first term associated with a specific constraint. The undetermined coefficient of the power series polynomial can be obtained while calculating the weight coefficient of the admissible function by using the Rayleigh–Ritz method. Then natural frequencies are calculated and polynomial coefficients can be further obtained. The influence of variations in boundary length, constraint length, constraint position and multiple discontinuous localized constraints on natural frequencies of the plate is studied. Convergence verification is performed for the truncated number of orthogonal polynomials and power series multipliers. Then the appropriate number of the term for the power series multiplier is determined. Natural frequencies of the cantilever plate and the opposite sides simply supported plate obtained by using the proposed method are compared with those obtained using the traditional method. Then natural frequencies of the plate with a local boundary constrained are compared with those obtained from the finite element software MSC.Patran. The fairly low relative error demonstrates the validity of the proposed method. The dynamical response analysis shows the superiority of the proposed method for the boundary locally constrained boundaries, which cannot be adequately handled by the traditional method.. The power series polynomial overcomes the limitation that the traditional Lagrange multiplier method can only construct point constraints.

The transient response of porous media is an important aspect of dynamic research. However, existing studies seldom provide solutions to the transient response problem of layered unsaturated porous media. Based on the Biot-type unsaturated wave equations, dimensionless one-dimensional wave equations are established. An appropriate displacement function is introduced to homogenize the boundary conditions. Subsequently, the transfer matrix method is used to obtain the eigenvalues and eigenfunctions of the homogeneous governing equations. Leveraging the orthogonality of the eigenfunctions, the original problem is transformed into solving a series of initial value problems of ordinary differential equations. The temporal solution within the time domain is then obtained through an improved precise time integration method. The validity of the solution presented in this paper is verified by comparing it with existing solutions in the literature. Analysis of numerical examples shows that reflection waves of opposite phases will be generated at the hard-soft and hard-harder interface, which helps in the accurate identification of weak interlayers in practical engineering applications. With increasing saturation, there is a noticeable increase in the velocities of the (P_{1}) and (P_{3}) waves, whereas the velocity of the (P_{2}) waves tends to decrease, which can be used to assess the mechanical property of medium. The peak value of pore pressure in unsaturated can be 1.64 times higher than those in saturated condition.

In this study, we define Hilfer fractional Dirac system. Our main object is to analyze the main spectral structure of the Hilfer fractional Dirac system. To this end, the self-adjointness of the Hilfer fractional Dirac operator, orthogonality of the eigen-vector-functions, and reality of the eigenvalues are displayed. Also, we obtain the representation of the solution of the system by using Laplace transforms with analytical estimations. We investigate eigen-vector functions and eigenvalues for the Hilfer fractional Dirac boundary value problem and illustrate the results in detail with tables and figures.