scholarly journals The vibration analysis of the elastically restrained functionally graded Timoshenko beam with arbitrary cross sections

Author(s):  
Guofang Li ◽  
Gang Wang ◽  
Junfang Ni ◽  
Liang Li

In this study, an investigation on the free vibration of the beam with material properties and cross section varying arbitrarily along the axis direction is studied based on the so-called Spectro-Geometric Method. The cross-section area and second moment of area of the beam are both expanded into Fourier cosine series, which are mathematically capable of representing any variable cross section. The Young’s modulus, the mass density and the shear modulus varying along the lengthwise direction of the beam, are also expanded into Fourier cosine series. The translational displacement and rotation of cross section are expressed into the Fourier series by adding some polynomial functions which are used to handle the elastic boundary conditions with more accuracy and high convergence rate. According to Hamilton’s principle, the eigenvalues and the coefficients of the Fourier series can be obtained. Some examples are presented to validate the accuracy of this method and study the influence of the parameters on the vibration of the beam. The results show that the first four natural frequencies gradually decrease as the coefficient of the radius [Formula: see text] increases, and decreases as the gradient parameter n increases under clamped–clamped end supports. The stiffness of the functionally Timoshenko beam with arbitrary cross sections is variable compared with the uniform beam, which makes the vibration amplitude of the beam have different changes.

Materials ◽  
2020 ◽  
Vol 13 (17) ◽  
pp. 3820
Author(s):  
Chiung-Shiann Huang ◽  
S. H. Huang

This study aimed to develop series analytical solutions based on the Mindlin plate theory for the free vibrations of functionally graded material (FGM) rectangular plates. The material properties of FGM rectangular plates are assumed to vary along their thickness, and the volume fractions of the plate constituents are defined by a simple power-law function. The series solutions consist of the Fourier cosine series and auxiliary functions of polynomials. The series solutions were established by satisfying governing equations and boundary conditions in the expanded space of the Fourier cosine series. The proposed solutions were validated through comprehensive convergence studies on the first six vibration frequencies of square plates under four combinations of boundary conditions and through comparison of the obtained convergent results with those in the literature. The convergence studies indicated that the solutions obtained for different modes could converge from the upper or lower bounds to the exact values or in an oscillatory manner. The present solutions were further employed to determine the first six vibration frequencies of FGM rectangular plates with various aspect ratios, thickness-to-width ratios, distributions of material properties and combinations of boundary conditions.


Fractals ◽  
2020 ◽  
Vol 28 (04) ◽  
pp. 2050063
Author(s):  
XUEZAI PAN ◽  
MINGGANG WANG ◽  
XUDONG SHANG

The purpose of this research is to show how the complicated and irregular fractal interpolation function is represented by Fourier series. First, on the closed interval [0,1], even prolongation is operated to the fractal interpolation function generated by iterated function system constituted by affine transform and Fourier cosine series representation of fractal interpolation function is proved. Second, for fractal interpolation function, odd prolongation is done and Fourier sine series formula of fractal interpolation function is proved. Final, Fourier series expansion of fractal interpolation function on the closed interval [Formula: see text] is proved. The result shows that complex fractal interpolation function can be represented by Fourier sine series and Fourier cosine series, so relatively simple Fourier series can be used to represent relatively complicated fractal interpolation function.


2021 ◽  
Author(s):  
Shengrong Xie ◽  
Yiyi Wu ◽  
Dongdong Chen ◽  
Ruipeng Liu ◽  
Xintao Han ◽  
...  

Abstract In deep underground mining, achieving stable support for roadways along with long service life is critical and the complex geological environment at such depths frequently presents a major challenge. Owing to the coupling action of multiple factors such as deep high stress, adjacent faults, cross-layer design, weak lithology, broken surrounding rock, variable cross-sections, wide sections up to 9.9 m, and clusters of nearby chambers, there was severe deformation and breakdown in the No. 10 intersection of the roadway of large-scale variable cross-section at the − 760 m level in the Nanfeng working area of the Wuyang Coal Mine. As there are insufficient examples in engineering methods pertaining to the geological environment described above, the numerical calculation model was oversimplified and support theory underdeveloped; therefore, it is imperative to develop an effective support system for the stability and sustenance of deep roadways. In this study, a quantitative analysis of the geological environment of the roadway through field observations, borehole peeking, and ground stress testing is carried out to establish the FLAC 3D variable cross-section crossing roadway model. This model is combined with the strain softening constitutive (surrounding rock) and Mohr-Coulomb constitutive (other deep rock formations) models to construct a compression arch mechanical model for deep soft rock, based on the quadratic parabolic Mohr criterion. An integrated control technology of bolting and grouting that is mainly composed of a high-strength hollow grouting cable bolt equipped with modified cement grouting materials and a high-elongation cable bolt is developed by analyzing the strengthening properties of the surrounding rock before and after bolting, based on the Heok-Brown criterion. As a result of on-site practice, the following conclusions are drawn: (1) The plastic zone of the roof of the cross roadway is approximately 6 m deep in this environment, the tectonic stress is nearly 30 MPa, and the surrounding rock is severely fractured. (2) The deformation of the roadway progressively increases from small to large cross-sections, almost doubling at the largest cross-section. The plastic zone is concentrated at the top plate and shoulder and decreases progressively from the two sides to the bottom corner. The range of stress concentration at the sides of the intersection roadway close to the passageway is wider and higher. (3) The 7 m-thick reinforced compression arch constructed under the strengthening support scheme has a bearing capacity enhanced by 1.8 to 2.3 times and increase in thickness of the bearing structure by 1.76 times as compared to the original scheme. (4) The increase in the mechanical parameters c and φ of the surrounding rock after anchoring causes a significant increase in σc and σt; the pulling force of the cable bolt beneath the new grouting material is more than twice that of ordinary cement grout, and according to the test, the supporting stress field shows that the 7.24 m surrounding rock is compacted and strengthened in addition to providing a strong foundation for the bolt (cable). On-site monitoring shows that the 60-day convergence is less than 30 mm, indicating that the stability control of the roadway is successful.


2020 ◽  
Vol 20 (3) ◽  
pp. 159-163 ◽  
Author(s):  
Mehdi Bozorgi

In this paper, the simple yet effective mode-matching technique is utilized to compute TE-backscattering from a 2D filled rectangular groove in an infinite perfect electric conductor (PEC). The tangential magnetic fields inside and outside of the groove are represented as the sums of infinite series of cosine harmonics (half-range Fourier cosine series). By applying the continuity of the tangential magnetic field, these modes are matched on the groove to obtain the series coefficients by solving a system of linear equations. For this purpose, some oscillatory logarithmic singular integrals involving Hankel and trigonometric functions are solved numerically, starting by removing the logarithmic singularity via integration by parts. In the following, the new well-behaved highly oscillatory integrals are computed using efficient methods, and several comparisons are made to demonstrate the validity and ability of the presented procedure.


2020 ◽  
Vol 20 (05) ◽  
pp. 2050068 ◽  
Author(s):  
Zhengmin Hu ◽  
Kai Zhou ◽  
Yong Chen

In this paper, the sound radiation behaviors of the functionally graded porous (FGP) plate with arbitrary boundary conditions and resting on elastic foundation are studied by means of the modified Fourier series method. It is assumed that a total of three types of porosity distributions are considered in the present study. The material parameters are determined according to the porosity coefficient used to denote the size of pores in the plate. The governing equations of the FGP plate are derived by utilizing the Hamilton’s principle on the basis of the first-order deformation theory (FSDT). Each displacement component of the FGP plate is expanded as the Fourier cosine series combined with auxiliary polynomial functions introduced to enhance the convergence rate of the series expansions. The acoustic response of the FGP plate due to a concentrated harmonic load is calculated by evaluating the Rayleigh integral. Good agreements are attained by comparing the present results with those in available literatures, which show the accuracy and versatility of the developed method in this paper. Finally, the influences of the porosity distribution type, porosity coefficient, boundary condition and elastic foundation on the sound radiation of the FGP plate are analyzed in detail.


Micromachines ◽  
2020 ◽  
Vol 11 (11) ◽  
pp. 970
Author(s):  
J. Israel Martínez-López ◽  
Héctor Andrés Betancourt Cervantes ◽  
Luis Donaldo Cuevas Iturbe ◽  
Elisa Vázquez ◽  
Edisson A. Naula ◽  
...  

In this paper, we characterized an assortment of photopolymers and stereolithography processes to produce 3D-printed molds and polydimethylsiloxane (PDMS) castings of micromixing devices. Once materials and processes were screened, the validation of the soft tooling approach in microfluidic devices was carried out through a case study. An asymmetric split-and-recombine device with different cross-sections was manufactured and tested under different regime conditions (10 < Re < 70). Mixing performances between 3% and 96% were obtained depending on the flow regime and the pitch-to-depth ratio. The study shows that 3D-printed soft tooling can provide other benefits such as multiple cross-sections and other potential layouts on a single mold.


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