Analytical Modeling and Vibration Analysis of Partially Cracked Rectangular Plates With Different Boundary Conditions and Loading

Vibration-based damage detection methods, leveraging structural response data to identify structural damage, play an important role in structural health monitoring, particularly for critical infrastructure like bridges and high-rise buildings. In recent years, deep learning, which can autonomously learn damage-sensitive features and classify damage from complex datasets, has been widely used in structural damage detection. This paper reviews deep learning architectures, including Convolutional Neural Networks (CNNs), Recurrent Neural Networks (RNNs), and autoencoders, applied in vibration-based structural damage detection for civil structures over the past ten years, highlighting their advantages and disadvantages. The challenges and future trends for vibration-based damage detection by using deep learning are summarized.

This paper investigates the parametric resonance of electrostatically actuated MicroElectroMechanicalSystems (MEMS) cantilever resonators. The electrostatic force is modeled to include fringe effect. The MEMS consists of a cantilever over a parallel ground plate and an AC voltage between them. The actuation frequency is near first natural frequency of the cantilever beam. This leads to parametric resonance. It is of interest to investigate the amplitude frequency response of MEMS cantilever resonators. This paper uses the Homotopy Analysis Method (HAM), which is able to capture nonlinear behaviors for higher amplitudes, large parameters, and strong nonlinearities. The base method used for comparison in this work is the method of multiple scales (MMS). MMS is a perturbation method. It requires a relatively short computational time for simulations. Although the CPU time is advantageous, MMS is only accurate for weak nonlinearities and low amplitudes. It is in the interest to compare how well HAM captures the softening behavior of this system as opposed to MMS. In this paper the influences of Casimir forces and Van der Waals effects are included. Electrostatic, Van der Waals and Casimir forces are nonlinear. HAM is a deformation technique that continuously deforms the initial guess, provided to the procedure, to the exact solution. In this work the first and second order deformation equations are constructed for the equation of motion governing the behavior of the MEMS cantilever beam. In the first order deformation, HAM deviates from the solution obtained by MMS. This deviation demonstrates the power of the method to capture the softening behavior more accurately than MMS even at the 1st order deformation HAM. In the second order deformation construction, the HAM’s solution softens more than the previous, demonstrating that higher order deformation approximations result in higher accuracy. In the second order deformation, HAM contains the convergence control parameter. This parameter is chosen via the c0 curve approach. Up to 2nd order HAM deformations are evaluated for this paper. These higher order homotopy deformation solutions were developed and automated symbolically in the software Mathematica and tested numerically using Matlab software.

Based on a non classical plate theory, an analytical model is proposed for the first time to analyze free vibration problem of partially cracked thin isotropic submerged plate in the presence of thermal environment. The governing equation for the cracked plate is derived using the Kirchhoff’s thin plate theory and the modified couple stress theory. The crack terms are formulated using simplified line spring model whereas the effect of thermal environment is introduced using thermal moments and in-plane forces. The influence of fluidic medium is incorporated in governing equation in form fluids forces associated with inertial effects of its surrounding fluids. Applying the Galerkin’s method, the derived governing equation of motion is reformulated into well known Duffing equation. The governing equation for cracked isotropic plate has also been solved to get central deflection which shows an important phenomenon of shift in primary resonance due to crack, temperature rise and internal material length scale parameter. To demonstrate the accuracy of the present model, few comparison studies are carried out with the published literature. The variation in natural frequency of the cracked plate with uniform rise in temperature is studied considering various parameters such as crack length, fluid level and internal material length scale parameter. Furthermore the variation of the natural frequency with plate thickness is also established.

Non-linear vibration analysis of an elastic plate subjected to heavy fluid loading in magnetic field

A systematic way of generating sets of local boundary conditions on the gauge\nfields in a path integral is presented. These boundary conditions are suitable\nfor one--loop effective action calculations on manifolds with boundary and for\nquantum cosmology. For linearised gravity, the general proceedure described\nhere leads to new sets of boundary conditions.\n

Vibration and buckling analysis of partially cracked thin orthotropic rectangular plates in thermal environment

In this paper, the nonlinear free vibrations of thin symmetric and non-symmetric cross-ply composite plates subjected to biaxial initial stresses are investigated. Because of their excellent properties such as specific strength and specific stiffness, composite plates have wide applications in aerospace and mechanical structures. Based on Von-Karman's strain-displacement relations and using Galerkin method, the nonlinear differential equation of free vibrations of initially stressed composite plate is obtained. This nonlinear equation is solved using two different analytical perturbation methods, namely method of multiple scales (MTS) and homotopy perturbation method (HPM), to analyze the nonlinear vibrations of initially stressed cross-ply composite plates. Effects of tensile and compressive biaxial initial stresses, initial vibration amplitude, thickness, and aspect ratios of the composite plates on the frequency behavior are investigated. The validity of the results is confirmed by making a comparison with those reported in the literature. According to the results, both analytical solutions show increasing trends for natural frequency parameters by increasing normal initial stresses. Regardless of the value of initial biaxial stresses, for both symmetric and non-symmetric plates, the results of MTS and HPM are in close agreement for the smallest initial amplitude. However, for compressive initial stresses, by increasing initial amplitude ratios, the discrepancies between the results of HPM and MTS increase for symmetric and non-symmetric plates. Although HPM includes less computational effort (smaller length of formulation) than MTS, the linear-to-nonlinear frequency ratios obtained using MTS method become closer to those obtained by HPM as initial vibration amplitude is decreased and initial stress is increased.

This paper deals with the voltage response of electrostatically actuated NEMS resonators at superharmonic resonance. In this work a comparison between Boundary Value Problem (BVP) model, and Reduced Order Model (ROM) is conducted for this type of resonance. BVP model is developed from the partial differential equation by replacing the time derivatives with finite differences. So, the partial differential equation is replaced by a sequence of boundary value problems, one for each step in time. Matlab’s function bvp4c is used to numerically integrate the BVPs. ROMs are based on Galerkin procedure and use the mode shapes of the resonator as a basis of functions. Therefore, the partial differential equation is replaced by a system of differential equations in time. The number of the equations in the system is equal to the number of mode shapes (or modes of vibration) used in the ROM. One mode of vibration ROM is solved using the method of multiple scales. Two modes of vibration ROM is numerically integrated using Matlab’s function ode15s in order to obtain time responses, and a continuation and bifurcation analysis is conducted using AUTO 07P. The effects of different nonlinearities in the system on the voltage response are reported. This work shows that BVP model is a valid method to predict the voltage response of a micro/nano cantilevers.

This paper thoroughly examines the singularity of stress resultants of the form r−ξFθ for 0<ξ⩽1 as r→0 (Williams-type singularity) at the vertex of an isotropic thick plate; the singularity is caused by homogeneous boundary conditions around the vertex. An eigenfunction expansion is applied to derive the first known asymptotic solution for displacement components, from the equilibrium equations of Reddy’s third-order shear deformation plate theory. The characteristic equations for determining the singularities of stress resultants are presented for ten sets of boundary conditions. These characteristic equations are independent of the thickness of the plate, Young’s modulus, and shear modulus, but some do depend on Poisson’s ratio. The singularity orders of stress resultants for various boundary conditions are expressed in graphic form as a function of the vertex angle. The characteristic equations obtained herein are compared with those from classic plate theory and first-order shear deformation plate theory. Comparison results indicate that different plate theories yield different singular behavior for stress resultants. Only the vertex with simply supported radial edges (S(I)_S(I) boundary condition) exhibits the same singular behavior according to all these three plate theories.

Dynamics of a mass–spring–beam with 0:1:1 internal resonance using the analytical and continuation method

Combination and principal parametric resonances of axially accelerating viscoelastic beams: Recognition of longitudinally varying tensions

Nuevas perspectivas para el médico internista en la insuficiencia cardiaca

In this paper, principal parametric resonance of rotating Euler–Bernoulli beams with varying rotating speed is investigated. The model contains the geometric nonlinearity due to the von Karman strain–displacement relationship and the centrifugal forces due to the rotation. The rotating speed of the beam is considered as a mean value which is perturbed by a small harmonic variation. In this case, when the frequency of the periodically perturbed value is twice the one of the axial mode frequencies, the principal parametric resonance occurs. The direct method of multiple scales is implemented to study on the dynamic instability produced by the principal parametric resonance phenomenon. A closed-form relation which determines the stability region boundary under the condition of the principal parametric resonance is derived. Numerical simulation based on the fourth-order Runge–Kutta method is established to validate the results obtained by the method of multiple scales. The numerical analysis is applied on the discretized equations of motion obtained by the Galerkin approach. After validation of the results, a comprehensive study is adjusted for demonstrating the damping coefficient and the mode number influences on the critical parametric excitation amplitude and the parametric stability region boundary. A discussion is also provided to illustrate the advantages and disadvantages of the fourth-order Runge–Kutta method in comparison with the method of multiple scales.

The paper considers an extension of the Multiple Scales method to strongly non-linear vibration systems. It is well known that similarly to other perturbation methods, the method of Multiple Scales produces good accuracy results when the perturbation parameter is small which is not the case in many practical applications. A two-step hybrid technique aimed at dealing with strong nonlinearities has originally been developed by Noor [9]. It is based on combining a perturbation method with a direct variational procedure. The full potential of such a hybrid technique has not yet been fully realized in solving non-linear vibrational problems. This paper examines the hybrid technique of combining the Multiple Scale method with the Galerkin procedure. Firstly, a perturbation solution is generated assuming a perturbation parameter is small. The next step involves using the computed perturbation functions as the coordinate (or approximation) modes whose amplitudes are computed by applying Bubnov-Galerkin conditions. The effectiveness of the hybrid Multiple Scales-Galerkin procedure is demonstrated on a 2-degree-of-freedom autoparametric vibration absorber by comparing the solutions computed by the method of Multiple Scales and by the hybrid method with the numerical results.

In this paper, we consider the nonlinear boundary value problems for ordinary differential equations containing a small parameter by means of a modified method of multiple scales in weaker conditions than those used by other authors. The existence of solutions is proved, the asymptotic solutions are constructed, and the remainder terms are estimated by using a simpler method. The results in [1]—[6] are thus improved and extended.