Increase In Natural Frequency Research Articles

Torsional vibration of simply supported nanoplate using nonlocal mathematical model

Present article is promoted to analyze the torsional vibration of simply supported nanoplate using nonlocal elasticity model. The equations motion of the nonlocal nanoplate are evaluated by rewriting the classical Hooke’s law through Eringen‘s model. Then, the numerical calculations are carried out to examine the reactions of the natural frequency of considered nanoplate to change of small-scale‘s coefficient, nanoplate‘s thickness, and torsional stiffness‘s parameter. The results reported that, any increase in the small-scale‘s parameter leads to a decrease in the natural frequency of nonlocal simply-supported nanoplate, while any increase of torsional stiffness‘s parameter leads to an increase of natural frequency. Additionally, the natural frequency of nonlocal nanoplates noticeably decreases as nanoplate thickness increases. Moreover, the torsional vibration analysis of nanoplates is significantly important in modelling real cases such as the nanodevices, which the nanoplates are common structural components in its designs. As a result, the findings of this investigation may be helpful in understanding torsional vibrations in nanostructures and nanodevices.

Structural and Vibro-Acoustics Optimization of a Car Body Rear Part

The perceived vibro-acoustic comfort, inside the passenger compartment, under driving conditions, is strictly related to the car body torsional behavior. The aim of this work was to identify which parts of a car body most influence the first torsional mode, in order to modify them and acquire an increase in such car body natural frequency. It was also intended to exploit the great potential of 3D printing that allows an increase in the complexity of component shapes, with an acceptable compromise with respect to production costs. A design and material (from steel to aluminum) change of a car body rear part, which was identified as the structural part of the car body with the most relevant impact on the frequency of the first torsional mode, was assessed in terms of structural and vibro-acoustic performances. In particular, with the constraint of increasing the structural and vibro-acoustic performances and, at the same time, minimize the weight of the structure itself, geometric, structural (e.g., type of connections), and material changes of the car body rear part were assessed. Working on a car model dating back to 2008, which was already compliant with structural and vibro-acoustic regulatory norms, an increase of 2 Hz on the first torsional mode frequency of the Trimmed Body model was obtained. In parallel, a weight reduction in the optimized components was also gained. It was also requested to lower the cabin sound pressure levels, optimizing the vibro-acoustic transfer functions from the accelerations at engine mounts and suspension attachment points to the cabin inside. It was shown how the combined use of advanced topological and structural optimization tools, with the capabilities of an unconventional manufacturing technology, such as 3D printing in aluminum, could guarantee an increase in the vibro-acoustics and structural car performances, also gaining a weight reduction.

Static and free vibration analysis of functionally graded annular plates using stress-driven nonlocal theory

This study presents analytical and numerical solutions of the bending and the free vibration analysis of a functionally graded annular nanoplate based on the stress-driven nonlocal theory. The nonlocal equation is obtained using the classical plate theory; the power law distribution is assumed to model changes in material properties throughout the thickness. The governing differential equation is analytically solved for different types of edge supports. The explicit analytical solution is obtained in terms of special functions, namely, hypergeometric and Meijer functions. Furthermore, based on the Galerkin technique, a finite element method is introduced to figure out the nanoplate’s flexural deformation and natural frequencies. The analytical and numerical solutions are compared with those available in the literature. The size effects in the nonlocal theory and FGM’s gradient properties have been discussed, considering clamped and simply supported boundary conditions. According to the stress-driven nonlocal model, size-dependent flexural responses of the plate demonstrate stiffening behavior with increasing nonlocal parameters. Which, in general, leads to a decrease in the value of transversal deformation and an increase in natural frequencies when compared to the plate’s local solutions.

Experimental and numerical study of the shielding effect of two tandem rough cylinders in flow-induce oscillation

The shielding effect of the downstream cylinder in flow induced oscillation (FIO) of two cylinders arranged in tandem is studied experimentally and numerically at Reynolds number 30,000 to 120,000. Both cylinders are in one degree-of-freedom, transverse-oscillations, and have turbulence stimulation in the form of selective surface roughness to expand FIO beyond vortex-induced vibration (VIV) into galloping. Shielding of the downstream cylinder has a negative effect on harnessing hydrokinetic energy. To study its effect and mechanics, selective cases are studied both numerically and experimentally and discussed to demonstrate the shielding effect on the downstream cylinder and understand its cause. The main conclusions are: (1) The shielding effect for the downstream cylinder shows a strong relation to the damping ratio. As the damping ratio increases, the shielding effect is mitigated. Additionally, the oscillation of the rear cylinder becomes stable and shows stable frequency. (2) In the VIV region, as the stiffness and natural frequency increase, the shielding effect decreases substantially. (3) In the VIV region, the vorticity of the vortices shedding from both the upper and the lower sides of the downstream cylinder does not accumulate enough due to the attraction by the vortices shed from the upstream cylinder, thus resulting in partial suppression of the oscillation on the downstream cylinder. (4) In the galloping region, the shielding effect for the downstream cylinder depends on whether the vorticity near the downstream cylinder is strengthened by the vortices generated by the shear layers of the upstream cylinder or weakened.

Modeling additively-manufactured particle dampers as a 2DOF frictional system

In this paper, a two-mass model for the dynamics of additively manufactured structures with embedded particle dampers (EPDs) is proposed and verified experimentally. It is proposed to model the structure’s dynamics as a linear spring-mass-dashpot system, with an added lumped mass, representing the particles, that slides relative to the structure’s mass. The friction caused by the sliding motion introduces nonlinear effects into the model that are similar to those observed in particle dampers. The two-mass model is thoroughly studied numerically and analyzed with Hilbert-transform-based tools, used to identify the instantaneous parameters of an equivalent single-degree-of-freedom system. The analysis reveals that the system’s dynamics is linear up to a certain acceleration response amplitude, above which the system exhibits nonlinear behavior, where its natural frequency and damping ratio vary with the response amplitude. This transition from linear to nonlinear behavior is attributed to the onset of sliding motion due to the inertial forces exceeding the friction force. In the nonlinear regime, the natural frequency increases and approaches a limiting value with increasing excitation frequency, while the damping ratio increases, reaches a peak and then decreases back and approaches its original value. To verify the model, a beam structure with EPDs was fabricated and excited to characterize its dynamic behavior. The same Hilbert-transform-based analysis tools were used for the investigation of the beam, providing a comprehensive description of its dynamics. The investigation revealed that the beam exhibits similar behavior to the two-mass model: transition from a linear to a nonlinear regime at an inertial threshold, and the same variations pattern of the instantaneous natural frequency and damping ratio with increasing excitation amplitude. The similarities drawn from particle dampers to the simple two-mass system shed light on the underlying mechanisms that affect the behavior of these dampers: the transition from linear to nonlinear behavior above a certain acceleration response amplitude is explained by the inertial forces acting on the particles exceeding the friction force between the particles and the structure. Above this amplitude, the particles move more freely, collide and rub more with one another, dissipating more energy overall. When the response amplitude is increased even further, the relative energy dissipated by the particles decreases, similar to what is observed in the two-mass system model. Since the model successfully reproduces the nonlinear phenomena observed in the beam with EPDs, it can be calibrated and used to predict the beam’s response to a given excitation.

Analysis of Size-Dependent Linear Static Bending, Buckling, and Free Vibration Based on a Modified Couple Stress Theory

The purposes of this paper are to study bending, buckling, and vibration by considering micro-scale effects using the Kirchhoff thin-plate theory and to consider small deflections, neglecting higher-order nonlinear terms. The governing equations for the bending, buckling, and vibration of the system are obtained using the equilibrium method coupled with the Kirchhoff thin-plate theory and a modified couple stress theory (MCST). The concept of the equivalent bending stiffness (EBS) of micro-thin plates is proposed to describe the scale effect. The Navier method is used to obtain analytical solutions for the bending, buckling, and free vibration of thin plates under simply supported boundary conditions with scale effects. The numerical results are presented to investigate the influence of scale effects on deflection, critical buckling load, buckling topography, and thin-plate natural frequency. The results show that the scale effect increases the equivalent stiffness of the thin plate, which leads to a decrease in deflection, a larger critical buckling load, and an increase in natural frequency, but does not affect the buckling topography. The MSCT is invalid when the thickness is greater than 10 times the scale effect parameter, thus defining the scope of application of the scale effect. This research study may contribute to the design of micro-scale devices such as MEMSs/NEMSs.

The role of graphene oxide in the strength and vibration characteristics of standard and high-grade cement concrete

This study presents the results of an experimental study of the influence of graphene oxide (GO) on the static and dynamic mechanical characteristics of cement concrete. Four different dosages of GO, 0.05%, 0.10%, 0.15% and 0.20% by weight of cement and two different grades of concrete, standard concrete (M30) and high strength concrete (M60) were studied. The static properties such as compressive strength and chord modulus of elasticity were determined. The impact hammer technique was used to obtain dynamic characteristics such as fundamental natural frequency, damping ratios, and mode shapes of concrete beams under free-free condition. In addition, fundamental resonant frequencies under different modes of vibration were determined. The dynamic elasticity modulus, dynamic shear modulus and dynamic poisons ratio of all concrete mixes were also determined. Experimental results revealed that the compressive strength of the concrete with the incorporation of GO was remarkably enhanced. The highest increase in compressive strength was found at GO dosage of 0.15%, and the improvement at 7 and 28 days was 58.2% and 47.7% for standard concrete and 51.2% and 24.3% for high strength concrete respectively compared to control concrete. The impact hammer test revealed that the maximum increase in fundamental natural frequency was 8.0% and 6.6%, and the maximum decreased damping ratio was 29.1% and 26.3% for standard concrete and high strength concrete respectively when compared to the control concrete. Ultrasonic pulse velocity results indicated that the homogeneity of the concrete with GO addition was significantly improved. SEM and EDX microstructure investigations found that the addition of GO to concrete promoted the development of improved hydration phases leads to the dense microstructure. The findings of this study demonstrated that GO nanomaterial had a potential future as a reinforcing material in cement concrete for improved microstructure, strength, and free vibration characteristics.

Effect of Stiffening the Printed Circuit Board in the Fatigue Life of the Solder Joint.

Predictive analysis of the life of an electronic package requires a sequence of processes involving: (i) development of a finite element (FE) model, (ii) correlation of the FE model using experimental data, and (iii) development of a local model using the correlated FE model. The life of the critical components is obtained from the local model and is usually compared to the experimental results. Although the specifics of such analyses are available in the literature, a comparison among them and against the same electronic package with different user printed circuit board (PCB) thicknesses does not exist. This study addresses the issues raised during the design phase/life analysis, by considering a particular package with a variable geometric thickness of the user PCB. In this paper, the effect of stiffening the user PCB on the fatigue life of a ball grid array (BGA), SAC305 solder joint is studied. The board stiffness was varied by changing the thickness of the PCB, while the size of the substrate, chips, and solder balls were kept constant. The test vehicle consisted of BGA chips soldered to a user PCB. The thickness of the user PCB was varied, but the surface area of the BGA chip remained identical. The test vehicle was then modeled using a finite element analysis tool (ANSYS). Using a global/local modeling approach, the modal parameters in the simulations were correlated with experimental data. The first resonance frequency dwell test was carried out in ANSYS, and the high-cycle fatigue life was estimated using the stress-life approach. Following the simulation, the test vehicle was subjected to resonance fatigue testing by exciting at the first mode resonance frequency, the mode with the most severe solder joint failure. The resistance of the solder joint during the experiment was monitored using a daisy-chain circuit, and the point of failure was further confirmed using the destructive evaluation technique. Both the experimental and simulation results showed that stiffening the board will significantly increase the fatigue life of the solder joint. Although the amplitude of the acceleration response of the test vehicle will be higher due to board stiffening, the increase in natural frequencies will significantly reduce the amplitude of relative displacement between the PCB and the substrate.

Numerical Analysis of Natural Frequency for U-Shaped Expansion Bellows in Fixed-Fixed and Fixed-Free Conditions

In this study, MATLAB code was used to analyze the natural frequency in two types of U-shape metal expansion bellows with various supporting conditions (fixed-fixed and fixed- free). The first bellow has a 10 mm inner diameter., and a length of (100, 200, 300) mm, has (36, 75, and 111) convolutions. While the second bellow, with an inner diameter of 20mm and (100,200,300) mm, has a number of convolutions (25,54 and 82). The result shows that by lowering the bellow length and increasing the bellow diameter, the natural frequency was raised. Were. Where, the maximum frequency was recorded at a maximum value of 20 mm in diameter and a length of 100 mm, about (13549.167 Hz) in the case of fixed-fixed and (13097.528 Hz) in the case of fixed-free. In general, the natural frequency depends directly on the stiffness and total mass of the pipe conveying fluid. The stiffness of the material is determined by the moment of inertia. As a result, increasing the diameter will increase the moment of inertia, which will result in an increase in stiffness, which will result in an increase in natural frequency.

An Investigation of the Influence of Various Shaped Cutouts on the Free Vibration Behavior of Sandwich Structures

The current study deals with free vibration analysis of a sandwich beam in presence of different cutouts. PVC foam and glass fiber were used for the core and surface layers of the sandwich structures, respectively. First 6 frequencies of the sandwich structures were determined for different cutout shapes to see the effect of cutout shapes on the free vibration behavior of sandwich structures. For this, sandwich beams with different cutout shapes but with the same area were modeled. To see the effect of cutouts-shape only the shape of cutouts was changed and all other parameters were kept unchanged. Analyses were performed by using ANSYS apdl commercially available software. The analysis gives information about the influence of the cutout shape on the first six natural frequencies of the sandwich structures. From the results, it was observed that the changes of cutout shapes caused a significant decrease or increase in natural frequencies of sandwich structures beam depending on the cutout shape.

Natural frequency discontinuity of vertical liquid sheet flows at transcritical threshold

The natural and forced dynamic response of a gravitational plane liquid sheet (curtain) of finite length interacting with an unconfined gaseous ambient is numerically and experimentally investigated. The global eigenvalue spectrum obtained by means of a linear inviscid one-dimensional model, accounting for the coupling between the curtain motion and the ambient pressure disturbances, clearly shows an abrupt increase (jump) in the characteristic natural frequency of the flow when the supercritical ( $We>1$ ) to subcritical ( $We<1$ ) transition occurs, with the Weber number $We$ defined as the ratio between inertia and capillary forces. On the other hand, the numerical simulation of the forced sheet response does not show any discontinuity between supercritical and subcritical conditions, as recently found by Torsey et al. (J. Fluid Mech., vol. 910, 2021, pp. 1–14) in the case of an infinite liquid sheet subjected to imposed ambient pressure disturbances not coupled with the curtain motion. It is argued that the forced liquid sheet behaviour varies continuously in shape and amplitude between the two regimes, not depending on the specific liquid–gas interaction model considered, whilst the natural frequency of the finite flow system does undergo a discontinuity, which can be theoretically predicted by the model of sheet–ambient interaction employed here. As a major result, the experimental evidence of the natural frequency jump is for the first time provided as well.

Kinematic Chain Equivalent Method for Tube Model and Elastodynamic Optimization for Parallel Mechanism Based on Matrix Structural Analysis

The platforms of parallel mechanisms usually suffer vibration loads. In these cases, structure elastodynamic analysis and elastodynamic optimization of parallel mechanisms are important. A tube structure is very common for parallel mechanisms. This work establishes the model of a tube structure based on matrix structural analysis. The kinematic pair equivalent method is used to simulate the surface contact between the inner and outer tubes. The corresponding mass and stiffness matrices are derived through the strain energy minimization method. The reconfigurable legged lunar lander has been used as an example to verify the effectiveness of this method. By adding the mechanism configuration to the optimization process, the equivalent static load method and the desirability approach are combined and modified. A procedure for the multi-objective elastodynamic optimization of parallel mechanisms is proposed. The optimization procedure is implemented on the lander and the results show a reduction in mass and an increase in natural frequency.

Natural Frequency Analysis of Horizontal Piping System Conveying Low Viscosity Oil–Gas–Water Slug Flow

The water cut (WC) has a significant effect on the flow parameters, such as liquid holdup, liquid phase velocity, and flow regimes of the low-viscosity oil–gas–water slug flow, and it can change the vibration characteristics of piping systems. To study the effect of water cut on the vibration characteristics of piping systems conveying such internal flow, a new dynamic model is developed. Galerkin’s method is used to discretize the equation and determine the natural frequencies by solving for the eigenvalues of the equation coefficient matrix. The results show that in the range of 10–90% WC, the natural frequency increases and then decreases, and the turning point occurs near the phase inversion region (WC = 40–60%). The main reason is the highly effective viscosity in the phase inversion region, which leads to an increase in the liquid holdup. The natural frequency increases and then decreases with superficial gas velocity, and the inflection point decreases with the increase in water cut in the oil-based flow regime and increases with the increase in water cut in the water-based flow regime. The critical gas velocity is lowest near the phase inversion region, but it should be noted that the presence of the critical gas velocity is related to the pipe length and superficial liquid velocity. The results of the study provide a reference for the design of safe pipeline operation.

Magnetoelastic nonlinear free vibration of an annular plate under a nonuniform magnetic field of the long straight current‐carrying wire

AbstractIn this article, the magnetoelastic nonlinear free vibration of a thin conductive annular plate under the nonuniform toroidal magnetic field is investigated, where the magnetic field is generated by a long straight current‐carrying wire. Based on electromagnetic theory, expressions of the nonuniform magnetic field, electromagnetic forces are deduced. The nonlinear free vibration equation of plate is derived by using the Hamilton principle. Considering different boundary conditions, the axisymmetric vibration differential equation is achieved by the Galerkin method. The method of multiple scales is employed to obtain the second‐order approximate analytical solutions and natural frequency. In numerical calculation, the characteristics of nonlinear natural vibration and singular points varying with different parameters, for example, current intensity and plate size, are presented, respectively. The results indicate that, with the increase of current, natural frequency increases and then stabilizes in the inner‐clamped and out‐free (C‐F) boundary, but decreases and stabilizes in the cases of the inner‐clamped and out‐clamped (C‐C) boundary or inner‐simply and out‐simply (S‐S) boundary. Moreover, when wire current is off, natural frequency maintains a constant, where the equilibrium solution of the system is a center. In addition, for all boundary conditions, natural frequency increases significantly with increase of internal radius and thickness, but decreases with the increase of outer radius.

Free vibration analysis of multi-layer rectangular plate containing magnetorheological fluid and flexible core layers rested on Winkler-Pasternak foundation

Magneto-rheological (MR) fluids viscosity can be varied by changing the magnetic field intensity. Therefore, they can improve structural rigidity and damping property. The current study presents a free vibration analysis of a multilayer rectangular plate with two layers of MR fluid and a flexible core layer, rested on a Winkler-Pasternak foundation based on exponential shear deformation theory (ESDT). In this theory, the exponential functions are applied in terms of thickness coordinates to include the effect of transverse and inertial rotational shear deformation. The flexible core displacement is modeled via the second-order Frostig model. The Hamilton’s principle is used to derive the equations of motion. These equations are solved using the Navier method to obtain the natural frequencies of the plate. The accuracy of the derived equations is validated, and the obtained results are compared with a specific case. Finally, the results show that by applying and increasing the magnetic field intensity, the natural frequencies and loss factor increase. With the increase in the mode number for each specific magnetic field intensity, the natural frequency increases and the loss factor decreases. The natural frequencies and loss factor decrease with the increase of the MR layer thickness to overall thickness ratio and the flexible core layer thickness to overall thickness ratio. The natural frequencies increase when the parameters of the foundation increase.

Natural characteristics for transverse vibration of Euler Bernoulli beams with variable end constraints

Abstract The transverse vibration of Euler Bernoulli beam with mass of ends and springs is studied. The exact frequency equation is derived and natural frequencies and the corresponding mode shapes are calculated. With the linearly increasing mass of ends, natural frequencies and the rate of frequency change of the beam system initially decrease sharply and then level out, which demonstrates that the beam system is transforming from the free beam to the pinned beam. When the springs are added at two tips, the natural characteristics of the beam are affected by mass of ends and spring stiffness. If the added mass has much lower magnitude than that of the beam, the stiffness of springs exerts major impact on the increase of natural frequencies. While the added mass of ends is increased to the same magnitude of the beam, the natural characteristics of the beam are determined by both the mass of ends and spring stiffness. As the growing magnitude of added mass, mass of ends performs a dominant role in decreasing the natural frequencies. Therefore, spring stiffness and mass of ends should be first considered to establish different dynamic models accurately.

Numerical study of tornado-induced unsteady crosswind response of railway vehicle using multibody dynamic simulations

The tornado-induced unsteady crosswind responses of railway vehicles are investigated by using multibody dynamic simulations. Firstly, a tornado-induced aerodynamic force model is proposed by using the equivalent wind force method and the quasi-steady theory and validated by the experimental data. The Uetsu line railway accident caused by tornado winds on December 25, 2005 is then investigated by the proposed tornado-induced aerodynamic force model and the multi-body dynamic simulation. The predicted accident scenes show favorably agreement with those obtained from the accident survey when the maximum tangential velocity of tornado is around 41 m/s and the core radius is 30m. Finally, the dynamic amplification factor (DAF) for railway vehicles in tornado winds is systematically studied and it increases as the passing time decreases. It is found that the DAF can be effectively suppressed as the damping parameters increase while it decreases slightly as the natural frequency increases. A simple method to predict the DAF is also proposed based on simulation results.

Effect of magnetorheological elastomer thickness and magnetic flux on vibrational characteristics

The maximum vibration is achieved when the natural frequency of any vibrating device reaches its resonance value. In a conventional suspension system, springs are used as vibration absorber. Since the stiffness of the spring is constant and the natural frequency of the road surface could vary at some point, the natural frequency of the spring can become the same as the natural frequency of the road. This would lead to resonance, and the vibration would be maximum and hence discomfort for the passengers. There are many active vibration absorbers under testing and development to overcome this disadvantage. In this paper, one such alternate vibration absorber is studied. The vibration absorber analyzed here is a magnetorheological elastomer (MRE). Magnetorheological materials fall under the smart material category as they could exhibit different properties under different conditions. The MRE laminate containing silicone and carbonyl iron was prepared for 6, 8, and 10 mm in thickness and was cut to the required dimension. The vibration characteristics of the vibration absorber were analyzed with and without magnetic flux. Acceleration of the MRE laminate fixed in cantilever mode as a function of natural frequency for 6, 8, and 10 mm thick with and without magnetic flux. The natural frequency increases with increasing MRF composite thickness due to the increasing stiffness of the composite. With applying magnetic flux, the rise in natural frequency has increased for all thicknesses. It shows that the increase in natural frequency achieved by applying magnetic flux to a 10 mm thick laminate is greater. The natural frequency increases as the thickness increases for modes 1 and 2. The rise in natural frequency for mode 2 compared to mode 1 has increased for all thicknesses.

Stiffness and natural frequency effects on a hollow truncated conical shaft

Larger tool length-diameter ratios generate undesired mechanical vibrations during machining processes. Previous studies show that natural frequency increases by adding a central hole and reducing the mass. Using FEA, the static stiffness and natural frequency response for different hole geometries, applied to conical shafts of various slenderness ratios were determined. The results show a convergence between an increase of natural frequency and decrease of static stiffness. In some cases, it was found that the working frequency range increases without considerably decreasing the rigidity, enhancing the response against vibrations. These findings could be implemented to develop reliable machining tools.

Free vibration analysis of a cylindrical micro-shell based on MCST

Dynamic analysis of a functionally-graded (FG) cylindrical micro shell is investigated in this work. Effect of small parameter in micro scale is included in fundamental relations based on modified couple stress theory (MCST). For more precise analyzing the structure, third-order shear deformation theory (TSDT) is employed as kinematic relations. The shell is made from FG materials, using power function law. A solution procedure based on an analytical method is developed for special boundary condition. The numerical results are computed in graphical form to seek how changes of main parameters of the study such as micro parameter, in-homogeneity index and some geometric values change the free vibration responses. As an important output of this paper, it is concluded that an increase in micro parameter leads to increase of natural frequency. Furthermore, increase of in-homogeneous index yields more soft structure and then decrease of natural frequency.