scholarly journals Exact Longitudinal Vibration Characteristics of Rods with Variable Cross-Sections

2011 ◽  
Vol 18 (4) ◽  
pp. 555-562 ◽  
Author(s):  
Bulent Yardimoglu ◽  
Levent Aydin
Keyword(s):  
Differential Equation ◽  
Cross Section ◽  
Cross Sections ◽  
Longitudinal Vibration ◽  
Transformation Method ◽  
Variable Cross ◽  
Cross Sectional ◽  
Cross Section Area ◽  
Section Area ◽  
Sinusoidal Functions

Longitudinal natural vibration frequencies of rods (or bars) with variable cross-sections are obtained from the exact solutions of differential equation of motion based on transformation method. For the rods having cross-section variations as power of the sinusoidal functions ofax+b, the differential equation is reduced to associated Legendre equation by using the appropriate transformations. Frequency equations of rods with certain cross-section area variations are found from the general solution of this equation for different boundary conditions. The present solutions are benchmarked by the solutions available in the literature for the special case of present cross-sectional variations. Moreover, the effects of cross-sectional area variations of rods on natural characteristics are studied with numerical examples.

scholarly journals Load bearing capacity of thin-walled rectangular and I-shaped steel sections of short both empty and concrete-filled columns

2021 ◽  
Vol 15 (58) ◽  
pp. 77-85
Author(s):  
Amor Bouaricha ◽  
Naoual Handel ◽  
Aziza Boutouta ◽  
Sarah Djouimaa
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Load Capacity ◽  
Cross Sectional ◽  
Short Columns ◽  
Cross Section Area ◽  
Section Area ◽  
Specimen Height ◽  
Steel Sections ◽  
Thin Walled

In this experimental work, strength results obtained on short columns subjected to concentric loads are presented. The specimens used in the tests have made of cold-rolled, thin-walled steel. Twenty short columns of the same cross-section area and wall thickness have been tested as follows: 8 empty and 12 filled with ordinary concrete. In the aim to determine the column section geometry with the highest resistance, three different types of cross-sections have been compared: rectangular, I-shaped unreinforced and, reinforced with 100 mm spaced transversal links. The parameters studied are the specimen height and the cross-sectional steel geometry. The registered experimental results have been compared to the ultimate loads intended by Eurocode 3 for empty columns and by Eurocode 4 for compound columns. These results showed that a concrete-filled composite column had improved strength compared to the empty case. Among the three cross-section types, it has been found that I-section reinforced is the most resistant than the other two sections. Moreover, the load capacity and mode of failure have been influenced by the height of the column. Also, it had noted that the experimental strengths of the tested columns don’t agree well with the EC3 and EC4 results.

A unified procedure for the vibration analysis of elastically restrained Timoshenko beams with variable cross sections

2020 ◽  
Vol 68 (1) ◽  
pp. 38-47
Author(s):  
Gang Wang ◽  
Wen L. Li ◽  
Wanyou Li ◽  
Zhihua Feng ◽  
Junfang Ni
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Vibration Analysis ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Timoshenko Beams ◽  
Unified Approach ◽  
Cross Section Area ◽  
Section Area ◽  
Elastically Restrained

A generalized analytical method is developed for the vibration analysis of Timoshenko beams with elastically restrained ends. For a beam with any variable cross section along the lengthwise direction, the finite element method is the only unified approach to handle those kinds of problems, since the analytical solutions could not be obtained by the governing equations when the cross section area and the second moment of area changing variably lengthwise. In this article, a unified approach is proposed to study the Timoshenko beam with any variable cross sections. The cross section area and second moment of area of the beam are both expanded into cosine series, which are mathematically capable of representing any variable cross section. The translational displacement and rotation of cross section are expressed in the Fourier series by adding some admissible functions which are used to handle the elastic boundary conditions with more accuracy and high convergence rate. By using Hamilton's principle, the eigenvalues and the coefficients of the Fourier series are both obtained. Some examples are presented to illustrate the excellent accuracy of this method. Analytical solutions of the vibration of the beam are achieved for different combinations of boundary conditions including classical and elastically restrained ones. The derived results can be used as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of Timoshenko beams with any variable cross section.

scholarly journals Klein Nishina Differential Equation for the Selection of Radiation Shielding Material (C, AL, Fe, and Zn) on the Basis of Attenuation and Cross sectional Area

2021 ◽  
Vol 3 (1) ◽  
pp. 30-35
Author(s):  
Saddam Husain Dhobi ◽  
Santosh Kumar Das ◽  
Kishori Yadav
Keyword(s):  
Differential Equation ◽  
Cross Section ◽  
Radiation Shielding ◽  
Cross Sectional Area ◽  
Sectional Area ◽  
Protective Material ◽  
Cross Sectional ◽  
Cross Section Area ◽  
Section Area ◽  
Aluminum Iron

On studying the Electronic and Atomic Cross sectional area for low atomic masses (Carbon, Aluminum, Iron and Zinc) using Klien-Nishina differential equation. The atomic cross section among these elements for same energy of incidence photon the atomic cross section area found on order of Carbon Aluminum Iron Zinc. This show with increasing atomic number and mass the cross section area of material goes increase. But the mass attenuation goes decrease with increasing in mass and number of materials made up of high atomic weight and number. This is clearly seen in Fig. 2 and Fig. 3. Therefore, among these elements protective material is made up of Carbon has more safety than other (Al, Fe, Zn).

scholarly journals An accurate and efficient series solution for the longitudinal vibration of elastically restrained rods with arbitrarily variable cross sections

2019 ◽  
Vol 38 (2) ◽  
pp. 403-414 ◽  
Author(s):  
Deshui Xu ◽  
Jingtao Du ◽  
Zhigang Liu
Keyword(s):  
Boundary Condition ◽  
Fourier Series ◽  
Boundary Conditions ◽  
Cross Section ◽  
Longitudinal Vibration ◽  
Variable Cross ◽  
Arbitrary Boundary ◽  
Cross Section Area ◽  
Section Area ◽  
Area Variation

Longitudinal vibration of non-uniform rod has been of great significance in various engineering occasions. The existing works are usually limited to the certain area variation and/or classical boundary condition. Motivated by this limitation, an efficient accurate solution is developed for the longitudinal vibration of a general variable cross-section rod with arbitrary boundary condition. Displacement function is invariantly expressed as the summation of standard Fourier series and supplementary polynomials, with an aim to make the calculation of all derivatives more straightforwardly in the whole solving region [0, L]. Energy principle is employed for system dynamics formulation, with the elastic boundaries considered as potential energy stored in the restraining spring. Arbitrary cross-section area variation is uniformly expanded into Fourier series. Numerical examples are presented for the natural frequency and mode shapes of non-uniform rod of free and clamped boundary conditions and compared with literature data. Results show good agreement with the previous analytical solutions. Influence of cross section area variation on vibration characteristics of non-uniform rods is then studied and discussed. One of the most advantages of the proposed model is that there is no need to reformulate the problem or rewrite the codes when the cross-section area distribution and/or boundary conditions change arbitrarily.

An Analytical Method of Analyzing the Heat Conduction for the Unequal Cross-Section Straight Fin

2013 ◽  
Vol 365-366 ◽  
pp. 1211-1216
Author(s):  
Fan Zhang ◽  
Peng Yun Song
Keyword(s):  
Heat Conduction ◽  
Cross Section ◽  
Analytical Method ◽  
Thermal Analyses ◽  
Cross Sectional ◽  
Cross Section Area ◽  
Section Area ◽  
Calculation Results ◽  
The Cross ◽  
Analytical Expressions

The cross-section area of straight fin is often considered to be equal in the thermal analyses of straight fin, but sometimes it is unequalin actual situation. Taking a straight fin with two unequal cross-sectional areas as an example,an analytical method of heat conduction for unequal section straight fin is presented. The analytical expressions of temperature field and heat dissipating capacity about the fin,which has a smaller cross-section area near the fin base and a larger one, is obtained respectively. The calculation results of the unequal cross-section are fully consistent with the equal area one, so the method is proved right. The results show that the larger the cross section areanear the base,the better is the heat transfer, and the temperature at the base with larger cross-section area is lower than that with smaller cross-section area when the amount of heat is fixed.

Simulation and Analysis on Operating Characteristics of Central Exhaust System in High-Rise Residential

2012 ◽  
Vol 614-615 ◽  
pp. 656-664 ◽  
Author(s):  
Gang Li ◽  
Yue Ren Wang ◽  
Guo Hui Feng ◽  
Ming Zhi Jiang ◽  
Qian Liu
Keyword(s):  
Cross Section ◽  
Exhaust System ◽  
Variable Cross Section ◽  
Maximum Pressure ◽  
Variable Cross ◽  
Main Duct ◽  
Operating Characteristics ◽  
Cross Section Area ◽  
Section Area ◽  
Operation Rate

Operating characteristics of variable cross section area of main duct in the main-branch central exhaust system was simulated based on various stories, operation rate and cross section area of main duct by FLUENT. The results show that there are certain obvious induced flow in the lower part of the system when the users in the upper floor work in the central exhaust system,and the internal pressure of main duct appears low- high-low trend in the system from the bottom to the top floor. The maximum pressure position is not in the bottom, but in the middle and lower floor of the system, where is the most disadvantageous floors of the exhaust system. It indicates that 400 mm x 500 mm is the necessary cross section area of main exhaust duct for 10 layers buildings,and 400 mm x 750 mm is the necessary one for 20 layers buildings.

scholarly journals Numerical simulation of cutting layer in internal corners milling

Mechanik ◽  
2019 ◽  
Vol 92 (7) ◽  
pp. 412-414
Author(s):  
Jan Burek ◽  
Rafał Flejszar ◽  
Barbara Jamuła
Keyword(s):  
Numerical Simulation ◽  
Numerical Model ◽  
Cross Section ◽  
Cross Sectional Area ◽  
Sectional Area ◽  
Cross Sectional ◽  
Cross Section Area ◽  
Section Area ◽  
The Cross ◽  
The Stability

The analytical and numerical model of the cross-section of the machined layer in the process of milling of concave rounding is presented. Simulation tests were carried out to determine the cross-sectional area of the cutting layer. A strategy has been developed that allows to increase the stability of the cross-section area of the cutting layer when the mill enters the inner corner area.

scholarly journals AN EXPERIMENTAL INVESTIGATION OF SOIL LAYER COUPLING FAILURE CHARACTERISTICS ON NATURAL GRASSLAND BY PASSIVE SUBSOILER-TYPE OPENERS

2020 ◽  
pp. 49-58 ◽  
Author(s):  
Changbin He ◽  
Yong You ◽  
Decheng Wang ◽  
Hongjian Wu ◽  
Bingnan Ye
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Layer Structure ◽  
Soil Layer ◽  
Failure Characteristics ◽  
Design And Optimization ◽  
Cross Section Area ◽  
Section Area ◽  
Draft Force ◽  
Natural Grassland

Unclear soil layer coupling failure characteristics on natural grassland impeded the design and optimization of appropriate tillage tools. The coupling failure characteristics including surface disturbance and profile, disturbed cross-section area, soil over-turning rate, and coupling forces between the soil layer of natural grassland and selected passive subsoiler-type openers were investigated in this paper. Three single-shoot openers (i.e. CO, AO and WAO) and a test unit were designed, and furrow opening experiments under different working depths were conducted. Results showed that, along the passages, U-shaped disturbed cross-sections were usually created with soil-root clods overturned along the furrows. The roots were usually broken in a pulling or dragging way underground. Both disturbed cross-section area and draft force values increased with the working depth increasing linearly (R2≥0.93), contrary to the tendency of the specific draft force with the depth. Winged opener (i.e. WAO) had larger draft forces and disturbed soil layer cross-section areas than no-winged openers (i.e. AO and CO). The soil layer failure processes of the natural grassland were affected by its composite soil layer structure and the geometry parameters of the openers. The results provide original references for designing novel furrow openers applicable to improve degraded natural grassland.

Heat Transfer Characteristic Inside Two-Pass Variable Cross-Section Channels With Ribs and Bleed Holes

2004 ◽  
Author(s):  
Xiangyun Liu ◽  
Shuiting Ding ◽  
Zhi Tao ◽  
Guoqiang Xu
Keyword(s):  
Heat Transfer ◽  
Cross Section ◽  
Transfer Characteristic ◽  
Reynolds Numbers ◽  
Variable Cross Section ◽  
Heat Transfer Process ◽  
Variable Cross ◽  
Relative Roughness ◽  
Cross Section Area ◽  
Section Area

Heat transfer characteristics were experimentally studied for two-pass channels with variable cross-section with and without ribs. Rectangular ribs were placed symmetrically on two opposite walls with attack angles of 90° and 60° to the main flow. The bleed holes were only distributed on the rib roughened surface along the second pass. The cross-section area of the channel varied in the way that it increased along the first pass and decreased along the second. The relative roughness heights were 0.092 and 0.123; the relative roughness pitches were 10 and 20. The experiments were carried out at Reynolds numbers from 7100 to 60000. Analysis of the experimental results may lead to the following conclusions: (1) The heat transfer was enhanced by the presence of variable cross section. (2) The 60° ribs was not necessarily the optimistic option as published in the literature, test showed that 90° ribs produced a hither heat transfer for the relative roughness height of 0.092. (3) For up to 30% of bleeding were outlet, the existence of bleed holes enhanced the heat transfer process.

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