scholarly journals Natural frequency of bending vibration for stepped beam of different geometrical characters and materials

2018 ◽  
Vol 50 (1) ◽  
pp. 3-12
Author(s):  
Shun-Cai Li ◽  
Li Liang ◽  
Qiu Yu
Keyword(s):  
Natural Frequency ◽  
Rectangular Cross Section ◽  
Zero Solution ◽  
Frequency Equation ◽  
Continuity Condition ◽  
Bending Vibration ◽  
Cross Sectional ◽  
Vibration Equation ◽  
Stepped Beam ◽  
Modal Solution

Based on the bending vibration theory of the beam with equal cross-sectional area, the free bending vibration equation of each subsection of stepped beam is given. The modal function of the stepped beam is assumed by the method of separating variable. First, substitute the modal function into free vibration equation so as to obtain the corresponding vibration shape function and then the modal solution with undetermined coefficients and natural frequency is developed. Next, substitute the modal solution into the boundary condition and the continuity condition to get the corresponding equations. Solving the equations by elimination method, the frequency equation of the step beam is obtained by the condition that the system of linear equations has non-zero solution. Taking the cantilever step beam and simplified supported step beam as examples, the first-order natural frequency of transverse bending vibration of stepped beams with different cross-sectional shapes, different materials, different areas, and different lengths for the two portions was obtained by solving the frequency equation with Mathcad software, and the corresponding curves of natural frequency for the circular and rectangular cross-section beam have been given.

The Exact One-Dimensional Theory for End-Loaded Fully Anisotropic Beams of Narrow Rectangular Cross Section

2001 ◽  
Vol 68 (6) ◽  
pp. 865-868 ◽  
Author(s):  
P. Ladeve`ze ◽  
J. G. Simmonds
Keyword(s):  
Cross Section ◽  
Plane Stress ◽  
Rectangular Cross Section ◽  
Dimensional Theory ◽  
Explicit Formulas ◽  
Cross Sectional ◽  
Rectangular Shape ◽  
One Dimensional ◽  
Anisotropic Elastic ◽  
Derivatives Of

The exact theory of linearly elastic beams developed by Ladeve`ze and Ladeve`ze and Simmonds is illustrated using the equations of plane stress for a fully anisotropic elastic body of rectangular shape. Explicit formulas are given for the cross-sectional material operators that appear in the special Saint-Venant solutions of Ladeve`ze and Simmonds and in the overall beamlike stress-strain relations between forces and a moment (the generalized stress) and derivatives of certain one-dimensional displacements and a rotation (the generalized displacement). A new definition is proposed for built-in boundary conditions in which the generalized displacement vanishes rather than pointwise displacements or geometric averages.

scholarly journals Vibrational analysis of nanoplate with surface irregularity via Kirchhoff plate theory

2021 ◽  
Vol 11 ◽  
pp. 184798042110011
Author(s):  
Mahmoud M Selim ◽  
Taher A Nofal
Keyword(s):  
Natural Frequency ◽  
Plate Theory ◽  
Vibrational Analysis ◽  
Equation Of Motion ◽  
Frequency Equation ◽  
Kirchhoff Plate ◽  
Surface Irregularity ◽  
Structural Element ◽  
Kirchhoff Plate Theory ◽  
Parabolic Form

In this work, an attempt is done to apply the Kirchhoff plate theory to find out the vibrational analyses of a nanoplate incorporating surface irregularity effects. The effects of surface irregularity on natural frequency of vibration of nanomaterials, especially for nanoplates, have not been investigated before, and most of the previous research have been carried for regular nanoplates. Therefore, it must be emphasized that the vibrations of irregular nanoplate are novel and applicable for the nanodevices, in which nanoplates act as the main structure of the nanocomposite. The surface irregularity is assumed in the parabolic form at the surface of the nanoplate. A novel equation of motion and frequency equation is derived. The obtained results provide a better representation of the vibration behavior of irregular nanoplates. It has been observed that the presence of surface irregularity affects considerably on the natural frequency of vibrational nanoplates. In addition, it has been seen that the natural frequency of nanoplate decreases with the increase of surface irregularity parameter. Finally, it can be concluded that the present results may serve as useful references for the application and design of nano-oscillators and nanodevices, in which nanoplates act as the most prevalent nanocomposites structural element.

Weakfish sonic muscle: influence of size, temperature and season

2002 ◽  
Vol 205 (15) ◽  
pp. 2183-2188 ◽  
Author(s):  
M. A. Connaughton ◽  
M. L. Fine ◽  
M. H. Taylor
Keyword(s):  
Natural Frequency ◽  
Pulse Duration ◽  
Dominant Frequency ◽  
Forced Response ◽  
Acoustic Energy ◽  
Honest Signal ◽  
Muscle Twitch ◽  
Cross Sectional ◽  
Fish Size ◽  
Sonic Muscle

SUMMARYThe influence of temperature, size and season on the sounds produced by the sonic muscles of the weakfish Cynoscion regalis are categorized and used to formulate a hypothesis about the mechanism of sound generation by the sonic muscle and swimbladder. Sounds produced by male weakfish occur at the time and location of spawning and have been observed in courtship in captivity. Each call includes a series of 6-10 sound pulses, and each pulse expresses a damped, 2-3 cycle acoustic waveform generated by single simultaneous twitches of the bilateral sonic muscles. The sonic muscles triple in mass during the spawning season, and this hypertrophy is initiated by rising testosterone levels that trigger increases in myofibrillar and sarcoplasmic cross-sectional area of sonic muscle fibers. In response to increasing temperature, sound pressure level (SPL), dominant frequency and repetition rate increase, and pulse duration decreases. Likewise, SPL and pulse duration increase and dominant frequency decreases with fish size. Changes in acoustic parameters with fish size suggest the possibility that drumming sounds act as an `honest' signal of male fitness during courtship. These parameters also correlate with seasonally increasing sonic muscle mass. We hypothesize that sonic muscle twitch duration rather than the resonant frequency of the swimbladder determines dominant frequency. The brief (3.5 ms), rapidly decaying acoustic pulses reflect a low-Q, broadly tuned resonator, suggesting that dominant frequency is determined by the forced response of the swimbladder to sonic muscle contractions. The changing dominant frequency with temperature in fish of the same size further suggests that frequency is not determined by the natural frequency of the bladder because temperature is unlikely to affect resonance. Finally, dominant frequency correlates with pulse duration (reflecting muscle twitch duration),and the inverse of the period of the second cycle of acoustic energy approximates the recorded frequency. This paper demonstrates for the first time that the dominant frequency of a fish sound produced by a single muscle twitch is apparently determined by the velocity of the muscle twitch rather than the natural frequency of the swimbladder.

Comparison of Static and Dynamic Stiffness for Beams Undergoing Flexural and Torsional Loading With an Intermediate Mass

2000 ◽  
Author(s):  
Arnoldo Garcia ◽  
Arnold Lumsdaine ◽  
Ying X. Yao
Keyword(s):  
Closed Form ◽  
Natural Frequency ◽  
Cross Sections ◽  
Dynamic Stiffness ◽  
Closed Form Solution ◽  
Frequency Equation ◽  
Form Solution ◽  
Torsional Loading ◽  
Intermediate Mass ◽  
Form Equation

Abstract Many studies have been performed to analyze the natural frequency of beams undergoing both flexural and torsional loading. For example, Adam (1999) analyzed a beam with open cross-sections under forced vibration. Although the exact natural frequency equation is available in literature (Lumsdaine et al), to the authors’ knowledge, a beam with an intermediate mass and support has not been considered. The models are then compared with an approximate closed form solution for the natural frequency. The closed form equation is developed using energy methods. Results show that the closed form equation is within 2% percent when compared to the transcendental natural frequency equation.

Study on Dynamic Characteristics and Parameter Sensitivity of Three Spans Prestressed Continuous Beam

2013 ◽  
Vol 351-352 ◽  
pp. 386-391
Author(s):  
Lu Ning Shi ◽  
Hao Xiang He ◽  
Wei Ming Yan ◽  
Yan Jiang Chen ◽  
Da Zhang
Keyword(s):  
Finite Element ◽  
Natural Frequency ◽  
Frequency Equation ◽  
Dynamic Parameters ◽  
Continuous Beam ◽  
Analytical Expression ◽  
Beam Dynamic ◽  
Frequency Sensitivity ◽  
The Impact ◽  
Good Agreement

Established the three spans prestressed continuous beam dynamic equation, obtained analytical expression of frequency equation. To solve the frequency equation for natural frequency, and compared with the finite element numerical analysis results, the frequency both with analytical expression and with finite element are in good agreement. The formula can be obtained accurately the dynamic parameters of three spans prestressed continuous beam such as frequency. At the same time, the paper also studied the natural frequency sensitivity analysis of three spans prestressed continuous beam, and focuses on the impact on the frequency with effective prestress and prestressed eccentricity.

Numerical Investigation on Diffusion Slot Hole With Various Cross-Sectional End Shapes

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
Bai-Tao An ◽  
Jian-Jun Liu
Keyword(s):  
Cross Section ◽  
Film Cooling ◽  
Separation Bubble ◽  
Rectangular Cross Section ◽  
Pressure Ratio ◽  
Smooth Transition ◽  
Shape Variation ◽  
Cross Sectional ◽  
Cooling Effectiveness ◽  
Shaped Hole

The diffusion hole constructed on a slot-type cross section has the potential to obtain high film cooling performance. However, the end shape of the cross section can greatly affect film cooling characteristics. This study examined eight cases of diffusion slot holes with various cross-sectional end shapes. The comparison of the eight diffusion slot holes and a typical fan-shaped hole was performed with a flat plate model using a three-dimensional (3D) steady computational fluid dynamics (CFD) method. The rectangular cross section had an aspect ratio of about 3.4. The end shape variation can be described based on sidewall contraction location, size, and form. The simulations were performed under an engine-representative condition of mainstream inlet Mach number 0.3 and turbulence intensity 5.2%. The simulated results showed that a strip separation bubble caused by inlet “jetting effect” occurs near the downstream wall of the diffusion slot hole and interacts with the diffusion flow. The different end shape of the rectangular cross section leads to different sidewall static pressure and exit velocity profiles, thereby produces three cooling effectiveness patterns, single-peak, bipeak, and tripeak patterns. The tripeak pattern produces higher cooling effectiveness and relatively uniform film coverage. The structure with moderate contraction and smooth transition on two sides of the downstream wall favors creation of a tripeak pattern. Compared with the fan-shaped hole, the discharge coefficient of diffusion slot hole is slightly small in low pressure ratio range, the pressure loss ratio has little difference.

Vibration of Bending-Torsion Coupled Resonance in a Rotor

2013 ◽  
Author(s):  
Hiroyuki Fujiwara ◽  
Tadashi Tsuji ◽  
Osami Matsushita
Keyword(s):  
Numerical Simulations ◽  
Natural Frequency ◽  
Torsional Vibration ◽  
Rotational Speed ◽  
Coupling Effect ◽  
The Other ◽  
Resonance Phenomenon ◽  
Horizontal Direction ◽  
Bending Vibration ◽  
Rotor Systems

In certain rotor systems, bending-torsion coupled resonance occurs when the rotational speed Ω (= 2π Ωrps) is equal to the sum/difference of the bending natural frequency ωb (= 2π fb) and torsional natural frequency ωθ(= 2πfθ). This coupling effect is due to an unbalance in the rotor. In order to clarify this phenomenon, an equation was derived for the motion of the bending-torsion coupled 2 DOF system, and this coupled resonance was verified by numerical simulations. In stability analyses of an undamped model, unstable rotational speed ranges were found to exist at about Ωrps = fb + fθ. The conditions for stability were also derived from an analysis of a damped model. In rotational simulations, bending-torsion coupled resonance vibration was found to occur at Ωrps = fb − fθ and fb + fθ. In addition, confirmation of this resonance phenomenon was shown by an experiment. When the rotor was excited in the horizontal direction at bending natural frequency, large torsional vibration appeared. On the other hand, when the rotor was excited by torsion at torsional natural frequency, large bending vibration appeared. Therefore, bending-torsion coupled resonance was confirmed.

Edgewise Bending Vibration Analysis of a Rotating Sandwich Beam with Magnetorheological Elastomer Core

2018 ◽  
Vol 18 (11) ◽  
pp. 1850134 ◽  
Author(s):  
S. Bornassi ◽  
H. M. Navazi ◽  
H. Haddadpour
Keyword(s):  
Magnetic Field ◽  
Layer Thickness ◽  
Rectangular Cross Section ◽  
Ritz Method ◽  
Sandwich Beam ◽  
Beam Theory ◽  
Core Layer ◽  
Magnetorheological Elastomer ◽  
Bending Vibration ◽  
Elastic Layers

The vibration of a rotating sandwich beam with magnetorheological elastomer (MRE) as a core between two elastic layers is theoretically analyzed in this paper. This study is focused on the bending vibration along the edgewise direction of a sandwich beam of rectangular cross-section, which, to the best of our knowledge, has not been addressed yet. The classical Euler–Bernoulli beam theory is used to model the dynamic behavior of the elastic layers. In the modeling, the effect of the MRE layer is considered by incorporating its shear strains and the inertia due to shear deformation and bending motion. The governing equations of motion of the rotating sandwich beam are derived by using the Ritz method and the Lagrange’s equations. The effects of the applied magnetic field, core layer thickness, rotational speed, setting angle and hub radius on the natural frequencies and the corresponding loss factors are investigated parametrically. The results show the significant effect of the magnetic field intensity and the MRE layer thickness on the modal characteristics of the MRE sandwich beam.

Experimental and Analytical Resin Impregnation Characterization in Carbon Fiber Reinforced Thermoplastic Composites

2020 ◽  
Author(s):  
Satoshi Kobayashi ◽  
Toshiko Osada
Keyword(s):  
Carbon Fiber ◽  
Optical Microscope ◽  
Textile Composites ◽  
Continuity Condition ◽  
Thermoplastic Composites ◽  
Molding Pressure ◽  
Cross Sectional ◽  
Molding Temperature ◽  
Resin Impregnation ◽  
Molding Condition

Abstract Effect of molding condition on resin impregnation behavior and the associated mechanical properties were investigated for carbon fabric reinforced thermoplastic composites. Carbon fiber yarn (TORAYCA, Toray) was used as a reinforcement, and thermoplastic PI (AURUM PL 450 C, Mitsui Chemicals) was used as the matrix. CFRTP textile composites were compression-molded with a hot press system under the molding temperature, 390 °C, 410 °C and 430 °C, molding pressure 2 MPa and 4 MPa and molding time 0∼300 s. In order to evaluate the impregnated state, cross sectional observation was performed with an optical microscope. Specimen cross-section was polished and finished with alumina slurry for a clear observation. The images observed were processed through image processing software to obtained impregnation ratio which defined as the resin impregnation area to the cross-sectional area of a fiber yarn. Resin impregnation was accelerated with molding temperature and pressure. At molding temperature more than 410 °C, resin impregnation was similar irrespective of temperature. Tensile test results indicated that modulus and strength increased with resin impregnation. Resin impregnation during molding was predicted using the analytical model based on Darcy’s law and continuity condition. The analysis could successfully predict the impregnation behavior despite the difference in molding pressure and temperature.

Experimental Analysis of Vortex Induced Vibration in the Bladeless Small Wind Turbine

2019 ◽  
Author(s):  
Micha Premkumar Thomai ◽  
Lasoodawanki Kharsati ◽  
Nakandhrakumar Rama Samy ◽  
Seralathan Sivamani ◽  
Hariram Venkatesan
Keyword(s):  
Energy Conversion ◽  
Wind Turbine ◽  
Natural Frequency ◽  
Cross Section ◽  
Vortex Shedding ◽  
Rectangular Cross Section ◽  
The Body ◽  
Test Facility ◽  
Vortex Induced Vibration ◽  
Small Wind Turbine

Abstract Vortex-induced vibration is one of the predominant fundamental concepts for forced oscillation which attracts considerable practical engineering application for energy conversion. In this work, an oscillation of a mast arising as a result of wind force is utilized for energy conversion. The paradigm for energy conversion from vortex-induced vibration in the mast is the bladeless wind turbine. It consists of a rigid mass known as a mast, fixed in the spring of stiffness (k) and allowed to oscillate along the direction of the flow. In this work, four different types of mast have been fabricated and tested. The first is uniform tapered hollow conical mast (MAST1), the cross-section of the second is uniform tapered plus symbol (MAST2), the third is uniform tapered inversed plus symbol (MAST3) and the fourth is uniform tapered simple rectangular cross-section (MAST4). All the masts were fabricated using fiber carbon. The experiments were conducted in a versatile small wind turbine testing facility of Hindustan Institute of Technology and Science, Chennai. This test facility contained an open jet wind tunnel with variable frequency drive and other measuring instruments. The vibration sensor was located in the mast where it experienced a large oscillation in a free stream. In this experiment, an increase in wind velocity led to a terrible change in the amplitude of vibration. A vigorous oscillation was experienced in this mast at this critical frequency, when the natural frequency of the mast was synchronized with the frequency of the vortex shedding and the frequency of the oscillation of the mast. The total force in this oscillation was a summation of the body force due to the mass of the mast and vorticity force that is mainly which was the result of the shedding of the vortices. In this work, extensive studies have been carried out for Reynolds number ranging from 2.5 × 105 to 5.0 × 105. The mast length to diameter ratio of 13 was exposed to various speeds of wind and response was measured. The occurrence of the maximum oscillation in a simple rectangular mast was seen where vortex shedding due to the bluff body was large for constant mass and spring stiffness. The frequency of the oscillation at maximum amplitude of the rectangular cross-section mast was equal to the natural frequency, due to vortices shedding at critical velocity. This demonstrated the appropriateness of the simple rectangular cross-section for harnessing the low rated wind energy and its suitability for renewable energy conversion in the small bladeless wind turbine.

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