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Vibration of an isotropic beam which has a variable cross-section is investigated. Governing equation is reduced to an ordinary differential equation in spatial coordinate for a family of cross-section geometries with exponentially varying width. Analytical solutions of the vibration of the beam are obtained for three different types of boundary co...
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... parameter d and the iterations were stopped when the absolute value of the differ- ence of the natural frequency obtained at two successive iterations was less than 10 À5 . Variations of the trans- verse vibration natural frequency ratios of a non-uniform beam with exponentially varying width with the non-uniformity parameter d are shown in Figs. 1-3 for the SS, CC and CF boundary conditions respectively where x 0 is the natural frequency of the uniform beam. The natural frequencies were also listed in Table 1. It may be noted that the natural frequencies ratios for the SS and CC boundary conditions are independent from the sign of d since the implicit equations for the natural ...
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Citations
... Many researchers have extended natural vibration analyses to bars with other cross-section shapes, often obtaining solutions only for special cases. For example, [3] studied the natural vibration of a bar with exponentially varying cross-sectional width, and [4] analyzed a bar in the shape of a solid of revolution with a radius changing parabolically along the length of the bar. The subject of the study of the paper [5] was a bar in the shape of a hollow truncated cone. ...
This paper presents the possibility of determining the natural frequency of flexural bars treated as Bernoulli-Euler beams using the Rayleigh method. It is assumed that the shape of the bar axis during vibration is the same as the shape of the deflection line of this bar under continuous load. By comparing the potential energy in the deflected position with the kinetic energy in the undeflected position, it is possible to determine the frequency and period of natural vibration as a function of Young's modulus and material density, as well as the parameters describing the geometric shape of the bar. Examples of solutions for truncated cone and truncated wedge-shaped bars are shown, as well as a solids of revolution with the generatrix described by linear and curvilinear (parabolic, exponential) forms. It applies to both solid and hollow bars, and different types of fixing the ends of the bar. The accuracy of the obtained results (compared with the literature and with the results of FEM calculations) is sufficient for practical applications. The authors also showed the possibility of extending the method to higher frequencies of vibration.
... Each approximate solution made assumptions to overcome the nonlinearity generated in partial differential equations due to the change in geometry and material. Some of these approximate methods are the Frobeniucs method [1][2][3][4][5][6][7], Adomian decomposition method [8,9], Galerkin method [10][11][12], finite element method , and Rayleigh-Ritz method [41][42][43][44][45][46][47][48][49][50][51]. The finite element technique was applied by Rossi and Laura [35] to simulate and investigate the dynamic behavior of the tapered homogeneous beams. ...
Non-prismatic beams, which serve crucial roles in mechanical engineering applications,
are subjected to both static and dynamic loads. Hence, their thorough analysis—
particularly under free vibration—is of significant importance. This research undertakes
an in-depth investigation into the free vibration characteristics of non-prismatic EulerBernoulli beams that exhibit linear variations in both width and height. These beams
were studied under two distinct boundary conditions: simple support and clampedclamped scenarios. The study's approach was twofold: analytical and numerical. The
analytical method entailed computing the equivalent area and the equivalent second
moment of area, thereby enabling an exploration of how variations in the second
moment of inertia and cross-sectional area affect the natural frequencies and mode
shapes of non-prismatic beams. Conversely, the numerical method employed the finite
element method through ANSYS APDL software (version 17.2). The study's findings
revealed that as the height and width of the beam decrease, natural frequencies decline
and the maximum amplitude of the mode shapes escalates. It should be noted that the
rate of decrease is more pronounced with changes in height than with alterations in
width. Furthermore, the diminishing rate of natural frequencies and the decreasing
maximum amplitude of the mode shape became more pronounced with the increase in
mode number when the beam's height and width decreased. The results derived from
the analytical procedure were validated against those from finite element analysis and
other literature sources, demonstrating the reliability of the current method. The
proposed analytical methodology, in its simplicity of use and accuracy, demonstrates
considerable promise in comparison to numerical results.
... Zhou and Cheung [4,7] investigated the bending vibrations of tapered beams using the Rayleigh-Ritz method. Mehmet et al. [5,8] used the differential quadrature method (DQM) for free vibration analysis of isotropic beam with exponentially varying width. Abdelghany et al. [6,9] studied vibration of a circular beam with variable cross sections using differential transformation method (DTM). ...
... In this paper, the governing differential equation is solved using the Frobenius method. Kim and et.al [8,11] employed State-Vector equation method to study axial vibrations of rods, and transverse vibrations of beams modeled by both Euler-Bernoulli and Timoshenko beam theories. 2nd group: Rosa and Auciello [9,12] examined the free vibration of EBB with linearly varying cross section. ...
This study investigates the transverse free vibration of general type of non-uniform beams with general non-classical boundary conditions at both ends based on the exploitation of Cauchy’s formula for iterated integrals. A review of related sources shows that despite of many studies and various methods used to investigate the transverse vibrations of non-uniform and inhomogeneous beams, no solutions applicable for free vibrations in such beams with general boundary conditions were proposed. In this article, Hamilton's principle is utilized to derive the partial differential equation governing the transverse vibration of the inhomogeneous non-uniform beam as well as the general corresponding boundary conditions. The effect of the boundary conditions, mass object, eccentricity and aspect ratios on the natural dimensionless transverse frequencies has been analyzed. The flowcharts of analyzing frequency parameters and mode shapes are also given in details to state the proposed method visually. The performance of the proposed method has been verified through numerical examples available in the published literature. All results exhibit that the proposed method is capable to analyze the free vibration of general type of non-uniform and axially FG Euler–Bernoulli beams with various general boundary conditions. Utilizing the proposed method and their related equations, it is possible to analyze the vibrations of different types of homogeneous uniform, homogeneous non-uniform, functionally graded uniform, and functionally graded non-uniform beams.
... Ece et. al [12] reported vibration response of variable cross section beam. Ganesan and Zabihollah [13][14] conducted parametric study on vibration of tapered beams. ...
... Putting (12) in (11) ...
In the present article bending, vibration and buckling analyses of a tapered beam using Eringen non-local elasticity theory is being carried out. The associated governing differential equations are solved employing Rayleigh-Ritz method. Both Euler-Bernoulli and Timoshenko beam theories are considered in the analyses. Present results are in good agreement with those reported in literature. Non-local analyses for tapered beam with simply supported - simply supported (SS) , clamped - simply supported (CS) and clamped - free (CF) boundary conditions are conducted and discussed. It is observed that the maximum deflection increases with increase in non-local parameter value for SS and CS boundary conditions. Further, vibration frequency and critical buckling load decrease with increase in non-local parameter value for SS and CS boundary conditions. Non-local parameter effect on deflection, frequency and buckling load for CF supports is found to be opposite in nature to that of SS and CS supports. In case of thick beams non-local structural response is observed to be sensitive to length to thickness ratio.
... Zhou and Cheung [4,7] investigated the bending vibrations of tapered beams using the Rayleigh-Ritz method. Mehmet et al. [5,8] used the differential quadrature method (DQM) for free vibration analysis of isotropic beam with exponentially varying width. Abdelghany et al. [6,9] studied vibration of a circular beam with variable cross sections using differential transformation method (DTM). ...
... In this paper, the governing differential equation is solved using the Frobenius method. Kim and et.al [8,11] employed State-Vector equation method to study axial vibrations of rods, and transverse vibrations of beams modeled by both Euler-Bernoulli and Timoshenko beam theories. 2nd group: Rosa and Auciello [9,12] examined the free vibration of EBB with linearly varying cross section. ...
Considering the lipid concentration and side effects regarding the stents used by surgeons, a new heart stent model is proposed. In the new stent, a few piezo plates are designed and attached to the stents by which release of the lipids can take place due to the applied alternative voltages. Due to the vibrations of small-scale piezoelectric plates, the deposition of low-density-lipoproteins (LDL) floating in the blood flow in the coronary arteries is prevented. Small-scale effects are considered using nonlocal elasticity theory, and the interaction between fluid and solid is modeled using the Navier-Stokes equation. The effect of fluid parameters as well as applied voltage and geometry structure is reported. Developing of smart stents maybe the key for prevention of short time conventional stents failure.
... Plates with a variable cross-section are also being studied. In particular, the bending characteristic of a sandwich plate with a variable cross-section was investigated in [8]. The authors of this paper identified the most adverse regional limit-bearing capacity, the stress-strain ratio, and the three-dimensional degree of stress in the ultimate state. ...
... The expressions for finding the potential energy of the deflection of section 1 and section 2 of the diamond-shaped plate at the first natural frequency of oscillation will take the form: (8) where the parameter D can be determined from the equation: To verify the accuracy of the calculation method, the plate with the specified parameters is investigated in Ansys. It is determined that the first natural frequency of oscillations of the plate is υ = 49,61Hz (figure 5). ...
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... To apply the TMM [36][37][38][39][40][41], the truncated wedge of an ABH structure should be simplified as a stepped beam with N constant-thickness segments. According to the continuities of deflection, slope, bending moment and shear force, the relationship of mode functions between two adjacent segments is established. ...
In this paper, the aeroelastic characteristic and active control of one-dimensional acoustic black hole structures in supersonic airflow are analyzed. Utilizing Hamilton's principle, the equation of motion of the one-dimensional acoustic black hole (ABH) structures in supersonic airflow with attached piezoelectric elements is established. The first-order piston theory is used to simulate the aerodynamic pressure of the composites, and the controller is designed based on speed feedback and proportional feedback algorithms. Through numerical simulations, the variation of the natural frequency concerning the aerodynamic pressure is carried out studies to determine the flutter point of the system. Besides, the influences of piezoelectric actuators and sensors on the aeroelastic flutters of one-dimensional ABH structures are analyzed. Results show that the velocity negative feedback can improve the critical pneumatic pressure of the system, thereby increasing the aerodynamic elastic stability of the composite structure. The kinetic and strain energies of the proposed system varied with the different feedback gains are also investigated. We found that the application of velocity negative feedback can effectively suppress the vibration of the proposed system, whereas, there are no apparent effects on the dynamics for varying the proportional feedback.
... The constants C i , D i , E i , and F i for the ith mode depend on the boundary conditions. For beams with varying cross-sectional area, such general and closed-form expression of mode shape can be found in the literature [44][45][46][47]. ...
In civil, mechanical, and aerospace structures, full-field measurement has become necessary to estimate the precise location of precise damage and controlling purposes. Conventional full-field sensing requires dense installation of contact-based sensors, which is uneconomical and mostly impractical in a real-life scenario. Recent developments in computer vision-based measurement instruments have the ability to measure full-field responses, but implementation for long-term sensing could be impractical and sometimes uneconomical. To circumvent this issue, in this paper, we propose a technique to accurately estimate the full-field responses of the structural system from a few contact/non-contact sensors randomly placed on the system. We adopt the Compressive Sensing technique in the spatial domain to estimate the full-field spatial vibration profile from the few actual sensors placed on the structure for a particular time instant, and executing this procedure repeatedly for all the temporal instances will result in real-time estimation of full-field response. The basis function in the Compressive Sensing framework is obtained from the closed-form solution of the generalized partial differential equation of the system; hence, partial knowledge of the system/model dynamics is needed, which makes this framework physics-guided. The accuracy of reconstruction in the proposed full-field sensing method demonstrates significant potential in the domain of health monitoring and control of civil, mechanical, and aerospace engineering systems.
... Due to the ABH effect, the bending waves in the structure are concentrated in the edge region, resulting in a violent deformation of the edge, which is difficult to accurately describe by traditional shape functions. The transfer matrix method [47][48][49][50] adopts the idea of a piecewise function and divides the variable-thickness structure into a combination of multiple sections of uniform beams. When the number of sections is sufficient, the original structure can be replaced. ...
In the view of the potential for vibration control and energy harvesting of the acoustic black hole
(ABH), the transfermatrix scheme combinedwith the finite elementmethod is utilized to establish the governing
equation of a one-dimensional ABH beams attached with a damping layer. According to the continuous
condition of generalized forces and displacements between the two adjacent uniform sections, the transfer
relationship is derived. The energy ratio is defined as the ratio of the edge part to the entire wedge, which
illustrates the energy concentration effect. A damping layer is introduced for controlling the fluctuation.
Numerical simulation is presented to illustrate the effectiveness of presented control method. The influences
of physical parameters such as excitation frequency, power exponent and thickness of the damping layer on
energy concentration are discussed
... Thus, the hydrophobic diaphragm vibrates with almost no increase in linear density ρA, even though the area covered by the narrow water column increases with the increase in the incident power of the transducer. [58][59][60] According to Eq. (1), the eigenfrequency f of the diaphragm remains constant when the linear density ρA is constant. Owing to the persistence of the eigenfrequency f of the hydrophobic diaphragm, the peak shift of the FFT signal is so small that a constant metallic sound can be clearly heard. ...
... As the incident power of the transducer increases, the area covered by the water film increases; thus, the diaphragm covered with the water film vibrates with an increasing linear density ρA. [58][59][60] According to Eq. (1), as the linear density ρA increases, the eigenfrequency f of the diaphragm decreases by a factor of 1/(ρA) 1/2 . This decrease in the eigenfrequency f for the diaphragm covered with the water film was detected as a peak shift to the lower-frequency side of the FFT signal and could be heard as the changes in the metallic sound tone. ...
In this study, the mechanisms of jet atomization were analyzed based on a frequency analysis of atomization sounds in the audible range (∼20 kHz). Jet atomization is a two-dimensional, high-speed atomization using a diaphragm, and interesting acoustic signals and atomization phenomena were detected on hydrophobic and hydrophilic diaphragms. The hydrophilic diaphragm strongly interacted and resonated with the surface wave, resulting in symmetrical jet atomization relative to the diaphragm. The resonance between the diaphragm and the surface wave was supported by a calculation of the eigenfrequency of the diaphragm and the coincidence of the droplet diameters as calculated from Lang's equation. Notably, the diaphragm excited by the ultrasonic transducer acted as a new transducer vibrating perpendicular to the transducer. As a result, when the diaphragm and the surface wave were in resonance at 2.4 MHz, a symmetrical two-dimensional high-speed jet atomization was generated in the direction perpendicular to the transducer’s vibration direction. This study also revealed that the atomization state can be determined based on the acoustic analysis. This acoustic analysis of atomization sounds can be applied in more advanced atomization control, such as for providing uniform dispersions of droplets containing DNA, drugs, or microplastics.
