Rotation–vibration analysis by finite difference perturbation technique. Application to 1Σ+ state of NH

This paper presents an approach to the Timoshenko beam theory (TBT) using the finite difference method (FDM). The TBT covers cases associated with small deflections based on shear deformation considerations, whereas the Euler–Bernoulli beam theory neglects shear deformations. The FDM is an approximate method for solving problems described with differential or partial differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating partial differential equations with finite differences and introducing new points (additional or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness). The introduction of additional points allows satisfying boundary and continuity conditions. First-order, second-order, and vibration analyses of structures were conducted with this model. Efforts, displacements, stiffness matrices, buckling loads, and vibration frequencies were determined. In addition, tapered beams were analyzed (e.g., element stiffness matrix, second-order analysis, and vibration analysis). Finally, the direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, considering the damping. The efforts and displacements could be determined at any time.

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Nonlinear vibration analysis of the nanobeams subjected to magneto-electro-thermo loading based on a novel HSDT

10.21203/rs.3.rs-364673/v1 ◽

2021 ◽

Author(s):

Reza Mohammadi

Keyword(s):

Nonlinear Vibration ◽

Frequency Ratio ◽

Vibration Analysis ◽

Multiple Scales ◽

Numerical Technique ◽

Perturbation Technique ◽

Equations Of Motion ◽

Nonlocal Elasticity Theory ◽

Small Scale ◽

Nonlinear Frequency

Abstract In this paper, the nonlinear vibration analysis of the nanobeams subjected to magneto-electro-thermo loading based on a novel HSDT is studied. Nonlocal elasticity theory is applied to consider the small scale effect. The nonlinear equations of motion are derived using Hamilton’s principle. First, a Galerkin-based numerical technique is applied to reduce the nonlinear governing equation into a set of Duffing-type time-dependent differential equations. Afterward, the analytical solutions are derived based on the method of multiple scales (MMS) and perturbation technique. All of the mechanical properties of the beam are temperature dependent. The impacts of the several variables are investigated on the nonlinear frequency ratio of the nanobeams. The results illustrate that when maximum deflection is smaller/ greater than 0.2, its impact on the nonlinear frequency ratio will decrease/increase.

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Vibration Analysis of Axially Functionally Graded Non-Prismatic Euler-Bernoulli Beams Using the Finite Difference Method

10.20944/preprints202105.0660.v2 ◽

2021 ◽

Author(s):

Valentin Fogang

Keyword(s):

Finite Difference ◽

Finite Difference Method ◽

Differential Equations ◽

Difference Method ◽

Vibration Analysis ◽

Mass Density ◽

Functionally Graded ◽

Governing Equations ◽

Mass System ◽

The Finite Difference Method

This paper presents an approach to the vibration analysis of axially functionally graded (AFG) non-prismatic Euler-Bernoulli beams using the finite difference method (FDM). The characteristics (cross-sectional area, moment of inertia, elastic moduli, and mass density) of AFG beams vary along the longitudinal axis. The FDM is an approximate method for solving problems described with differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. In this paper, differential equations were formulated with finite differences, and additional points were introduced at the beam’s ends and at positions of discontinuity (supports, hinges, springs, concentrated mass, spring-mass system, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. Vibration analysis of AFG non-prismatic Euler-Bernoulli beams was conducted with this model, and natural frequencies were determined. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of AFG non-prismatic Euler-Bernoulli beams, considering the damping. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was always increased through a grid refinement.

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Kirchhoff-Love Plate Theory: First-Order Analysis, Second-Order Analysis, Plate Buckling Analysis, and Vibration Analysis Using the Finite Difference Method

10.20944/preprints202111.0327.v1 ◽

2021 ◽

Author(s):

Valentin Fogang

Keyword(s):

Finite Difference ◽

Vibration Analysis ◽

Plate Theory ◽

Second Order ◽

Governing Equations ◽

Order Analysis ◽

First Order ◽

Order Polynomial ◽

The Finite Difference Method ◽

Point Stencil

This paper presents an approach to the Kirchhoff-Love plate theory (KLPT) using the finite difference method (FDM). The KLPT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations. The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. Generally in the case of KLPT, the finite difference approximations are derived based on the fourth-order polynomial hypothesis (FOPH) and second-order polynomial hypothesis (SOPH) for the deflection surface. The FOPH is made for the fourth and third derivative of the deflection surface while the SOPH is made for its second and first derivative; this leads to a 13-point stencil for the governing equation. In addition, the boundary conditions and not the governing equations are applied at the plate edges. In this paper, the FOPH was made for all of the derivatives of the deflection surface; this led to a 25-point stencil for the governing equation. Furthermore, additional nodes were introduced at plate edges and at positions of discontinuity (continuous supports/hinges, incorporated beams, stiffeners, brutal change of stiffness, etc.), the number of additional nodes corresponding to the number of boundary conditions at the node of interest. The introduction of additional nodes allowed us to apply the governing equations at the plate edges and to satisfy the boundary and continuity conditions. First-order analysis, second-order analysis, buckling analysis, and vibration analysis of plates were conducted with this model. Moreover, plates of varying thickness and plates with stiffeners were analyzed. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, with damping taken into account. In first-order, second-order, buckling, and vibration analyses of rectangular plates, the results obtained in this paper were in good agreement with those of well-established methods, and the accuracy was increased through a grid refinement.

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An Approach to the Vibration Analysis of Confocal Elliptic Annular Plates

13th Biennial Conference on Mechanical Vibration and Noise: Machinery Dynamics and Element Vibrations ◽

10.1115/detc1991-0295 ◽

1991 ◽

Author(s):

Shyh-Chin Huang ◽

Jaw-Lin Wang

Keyword(s):

Finite Difference ◽

Vibration Analysis ◽

Separation Of Variables ◽

Complex Geometries ◽

Elliptic Plate ◽

Center Crack ◽

Annular Plates ◽

Receptance Method ◽

Characteristic Values ◽

Square Plate

Abstract An approach is presented for the vibration analysis of annular, confocal, elliptic plate. The approach employs the separation of variables, finite difference, and receptance concept. This method is then expanded to square plate with center crack, or with more complex geometries. The basic idea of this paper is: split the plate into several strips and get the strips’ characteristic values, then compose these strips by the receptance method to get the characteristics value of the plate. Numerical results for certain cases are compared to known data, and precise agreement is met.

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