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Aiming at the problem that conventional digital driving methods cannot accurately track the natural frequency changes of flow tubes, we analyzed the sources of tracking errors in the natural frequency of flow tubes and proposed a new digital driving method. The method utilizes the amplitude of flow tube vibration to monitor changes in natural frequency, and tracks the natural frequency of the flow tube by estimating the frequency of the attenuation signal after stopping vibration. Then the hardware and software design and implementation of the Coriolis mass flowmeter (CMF) transmitter embedded with the proposed method was carried out, and experiments were conducted on the micro-flow differential CMF experimental platform. The experimental results illustrate that the proposed method can adjust the driving signal frequency timely when the natural frequency changes, improve the tracking accuracy of the digital driving natural frequency, and make the system have better driving performance; when the water temperature rises from $25^{\circ }\mathrm{\text{C}}$ to $65^{\circ }\mathrm{\text{C}}$, the measurement accuracy of the CMF embedded in the proposed digital driving method is improved by 37.87% compared to conventional digital drive.

This study aims to determine the natural frequencies of axially functionally graded porous material (FGPM) beams with non-uniform cross-sections under a variety of boundary conditions. Two types of pore distribution were used: even and uneven. Initially, an analytical method was used to determine the natural frequencies of FGPM beams with different cross-sections. The Fredholm integral approach was used to obtain the characteristic equations. Furthermore, the collected results are confirmed and compared to the existing literature. The artificial neural network (ANN) technique is then used to predict the fluctuations in the natural frequency of a clamped–simply supported axially FGPM beam with a non-uniform cross-section. The prediction of natural frequencies by ANN is based on a large dataset of over 1100 data points acquired from the analytical solution obtained in this study and data available in the literature. This research conducts a parametric analysis to evaluate the influence of numerous aspects, including as beam characteristics, material properties, geometric details, gradient parameters, and porosity distribution, on functionally graded (FG) beam vibration behavior. Finally, both ANN and computational analysis demonstrate that porosity distributions and non-uniform cross-sectional area have a substantial effect on the natural frequencies of axially FG beams.

Structural dynamic properties like frequencies and mode shapes have been widely adopted for bridge condition assessment and damage identification. One of the main challenges lies in environmental factors, particularly the varying temperature, which significantly influences the bridge’s natural frequencies. In some cases, the effect of temperature changes can be comparable to, or even exceed, the effect caused by damage, rendering the damage identification methods ineffective. This study investigates the influence of the cross-sectional non-uniform temperature distribution, as a result of surrounding environmental factors, on the frequency of concrete girder bridges. A theoretical analysis is conducted to derive the bridge’s frequencies considering the cross-sectional non-uniform temperature distribution. The upper and lower bounds of the frequencies are developed by using only the environmental temperature and the largest temperature gradient defined by codes. More importantly, the damage to the bridge can be detected if the frequencies exceed the bounds for a certain period. This approach is convenient and practical in real applications without embedding thermocouples in the main girder. Numerical simulations, laboratory experimental tests, and field measurements are carried out to validate the derived frequency considering the cross-sectional non-uniform temperature distribution as well as the upper and lower bounds of the frequency. In particular, the results of the Z24 bridge show that when the bridge is damaged, the measured frequency exceeds the bound continuously, verifying the capability of the developed upper and lower bounds to evaluate bridge conditions.

This study examines the impact of reinforcement type and distribution on the natural frequency, deflection, equivalent stress, and Specific Energy Absorption (SEA) of copper thick truncated cones reinforced with Graphene Platelets (GPLs) or Graphene Origami (GOri). The cone comprises $N_L$ layers with Functionally Graded (FG) distributions (X, O, V, UD patterns). Using 2D axisymmetric elasticity theory and the quadratic finite element method, free vibration and static problems are solved. The impact of the GOri and GPLs’ mass fraction and distribution pattern, folding degree of GOri, different boundary conditions and cone angle on the natural frequencies, displacement, equivalent stress and SEA have been investigated and compared. Results show that increasing GPL mass fraction raises natural frequency, while GOri exhibits a folding degree–dependent relationship, with higher folding degrees reducing frequencies. Stress distribution varies by pattern and reinforcement; GOri increases maximum stress in UD and V-patterns but decreases it in X and O-patterns at higher folding degrees. Maximum stress is 8.23 MPa (UD, 100% folding) and minimum 4.25MPa (X-pattern). SEA decreases with mass fraction; GOri enhances SEA at 100% folding and below 2.5% mass fraction but reduces it at higher fractions. GPL-reinforced cones show the lowest SEA below 2.5% mass fraction.

This study addresses a critical gap in the literature by investigating the static and natural frequency characteristics of functionally graded (FG) auxetic metamaterial annular plates reinforced with graphene origami (GOri), a novel area previously unexplored in the context of composite constructions, particularly for circular plates. The governing equations are derived utilizing higher-order shear deformation theory along with Hamilton’s principle, and solved using the finite element approach. For the first time, a comprehensive parametric study including the folding degree and mass fraction, and distribution pattern of GOri, is investigated on the static and natural frequency properties of annular plates. It is found that the natural frequency generally increased with higher mass fractions and decreased with greater folding degrees, though the X and V patterns at a 3% mass fraction showed an atypical increase in frequency with higher folding degrees. The impact of distribution patterns varied with weight fraction: the X-pattern caused the highest deflection at 1% weight fraction but the lowest at 3%, while the O-pattern caused the least deflection overall.

The mass inertia effects of soil and moving masses significantly alter the dynamic behavior of structures. The latest advancements in the practice of structural engineering suggest that abrupt changes in mass and sub-structural components can substantially affect structural performance, with the inertia effects of moving masses and soil being especially crucial. Currently, the dynamic investigation of beam with elastic foundation support considering transverse inertia effects is gaining increasing attention in engineering applications. A finite-depth Winkler foundation beam dynamic model was formulated using the Euler–Bernoulli beam theory, incorporating the impacts of soil–structure interaction and moving mass forces on its lateral vibration. The Galerkin and Newmark methods were applied to solve the system numerically, enabling the evaluation of the impact of soil motion on the beam’s natural frequency and dynamic behavior. Finally, through dynamic interaction analysis, a parametric investigation was conducted on the influences of foundation soil mass, moving masses, foundation stiffness, and moving speed on the beam’s dynamic behavior. The results demonstrate that the incorporation of soil–structure interaction significantly alters the beam’s vibration characteristics beam under moving masses excitation. The mass and stiffness of the foundation soil, which participate in soil motion, alter the influence of the moving masses inertia on the beam’s natural frequency and dynamic behavior. These effects depend on the underlying mechanisms of soil–structure interaction.The findings provide new insights into the beam’s dynamic characteristics under load excitation, offering significant implications for the overall design and refined study of vehicle-track systems.

This paper examines the effects of axial force and temperature changes on the dynamic characteristics of pipes. The Galerkin method, combined with the eigenvalue method, is employed to solve the fluid–structure interaction (FSI) motion equations for the pipe. The study investigates the impact of axial force, temperature changes, fluid velocity, fluid pressure, and pipe structure parameters on the pipe’s natural frequency. Numerical results reveal that different parameters have varying effects on the pipe’s natural frequency, leading to the determination of stability conditions’ varying influences. Combined with the finite element method (FEM), the impact of temperature changes on the pipe’s natural frequency is analyzed. The correctness of the numerical results of the Galerkin-eigenvalue method was verified through comparative methods. The research results confirm that this numerical method provides strong theoretical support for analyzing the nonlinear dynamic characteristics of pipelines under temperature changes in random vibration environments. This is of guiding significance to the vibration reduction design of the pipeline system and helps prevent and reduce the structural damage and accident risks caused by the change in the pipeline’s dynamic characteristics.

This paper presents a statistical framework for identifying circular flaws in structures using natural frequency data and Bayesian inference, explicitly addressing uncertainties arising from modeling errors and measurement noise. In this approach, the circular flaw is characterized by parameters such as the center coordinates and radius. The natural frequencies of the structure, measured under known boundary conditions, serve as the input data for the identification process. The smoothed finite element method (SFEM) forward model predicts the natural frequency shifts due to the presence of flaws and is integrated into the analysis. By combining observed frequency data with prior knowledge, Bayes’ theorem is employed to refine the probability distributions of the flaw parameters. The Markov chain Monte Carlo (MCMC) algorithm is utilized to sample from the posterior distributions of the parameters, ensuring robust uncertainty quantification. A numerical case study validates the proposed method, highlighting its accuracy and effectiveness in detecting and characterizing circular flaws.

Generalized synchronization (GS) of a chaotic oscillator driven by two chaotic signals is investigated in this paper. Both receiver and drivers are the same kind of oscillators with mismatched parameter values. Partial and global GS may appear depending on coupling strengths. An approach based on the conditional entropy analysis is presented to test the partial GS, which is difficult to determine with conventional methods. A trough in conditional entropy spectrum indicates partial GS between the receiver and one of the drivers.

The stepped microbeams are typical low stiffness structures widely used in MEMS devices. Approximated solutions of the natural frequency and the pull-in voltage of stepped microbeams, including clamped-free (CF) and clamped–clamped (CC) boundary conditions are developed, and the unique pull-in behavior of the stepped microbeams is investigated. The stepped microbeam is viewed as a beam with a rectangular electrode pad at its tip for CF beam and at its center for CC beam. The motion equations are deduced based on the Euler–Bernoulli beam and the modified couple stress theory. The natural frequency and the pull-in voltage are extracted with a one-degree-of-freedom model. The present model correctness is validated by comparing with the finite element results, and the effect of the length ratio of the electrode pad to the beam and the width ratio of the beam to the electrode pad are discussed. The results show that both the natural frequency and the pull-in voltage monotonously increase with the increase of the width ratio, and first decrease and then increase with the increase of the length ratio. The minimum value of them does not appear at the same time, but is determined by the width ratio. The results can be used to design and improve the performance of MEMS devices.

In this study, the CFRP shafts made up of T700-SC multilayered composites have been designed to replace the steel shaft of a ship. An important design variable to be considered when designing composite material intermediate shafts is the natural frequency for resonance avoidance at critical rotational speed and torsional strength for axial load. In order to satisfy these, strength and modal analysis were performed. In order to minimize the deformation of the shape due to the residual stress after mandrel removal, it was laminated by axial symmetry. The fibers orientation angle has a great influence on the natural frequency of the drive shaft. The carbon fiber should be closely oriented at $30^{\circ }$ to improve the modulus of elasticity in the direction of length of the intermediate shaft and to increase the natural frequency. Also, the optimum fiber orientation for maximum torsional strength should be close to $45^{\circ }$. The stacking pattern and the stacking order were finally decided considering the results of the finite element analysis (FEA).

In this work the exact axisymmetric vibration frequencies of circular and annular variable thickness plates are found. The solution is obtained using the exact element method developed earlier. It allows for the exact solution of problems with general polynomial variation in thickness using infinite power series. The solution is exact up to the accuracy of the computer. The natural frequencies of vibration are found as the solutions of the frequency equation. Normalized values for the natural frequencies are given for linear, parabolic and cubic variations of the plate thickness, for circular and annular plates, with four types of boundary conditions on the inner and outer boundaries.

Numerical methods for calculating both the natural frequencies and buckling loads of columns with intermediate multiple elastic springs are developed. In formulating the governing equations of the column, each elastic spring is modeled as a discrete Winkler foundation of the finite longitudinal length. By using this model, the differential equations governing both the free vibration and buckled shapes of the column are derived, which are solved numerically. The Runge–Kutta method is used to integrate the differential equations, and the determinant search method combined with Regula–Falsi method is used to determine the eingenvalues, namely, the natural frequencies and buckling loads. In the numerical examples, fixed–fixed, fixed-hinged, hinged-fixed and hinged–hinged end constraints are considered. The numerical results including the frequency parameters, mode shapes of free vibrations and buckling loads are presented in non-dimensional forms.

The differential equations governing free vibrations of the elastic, parabolic arches with unsymmetric axes are derived in Cartesian coordinates rather than in polar coordinates. The formulation includes the effects of axial extension, shear deformation and rotatory inertia. Frequencies and mode shapes are computed numerically for arches with clamped-clamped, clamped-hinged, hinged-clamped and hinged-hinged ends. The convergent efficiency is highly improved under the newly derived differential equations in Cartesian coordinates. The lowest four natural frequency parameters are reported as functions of four non-dimensional system parameters: the rise to chord length ratio, the span length to chord length ratio, the slenderness ratio and the shear parameter. Typical mode shapes of vibrating arches are also presented.

Previously, it was found that the analytical deflections computed for towers using computer software are less than those from test results. The present study is aimed at deriving a relationship between the ratio of the test to theoretical deflection, and a nondimensional parameter to serve as an index for monitoring the structural displacements during testing. Currently, structural dynamic evaluation plays little or no role in the design of towers, partly due to the difficulties involved in the analysis and the relatively high cost of field testing. Using the fundamental frequency of a tower, the peak response of the tower to gusty wind and the impact force caused by conductor breakage can be evaluated. Both theoretical and experimental studies have been carried out to evaluate the natural frequencies of the towers tested at TTRS, SERC, Chennai, India. Based on these data, an equation was derived in this paper using the tower geometry and test/theoretical deflection ratios, which allows us to predict the natural frequency of the tower in a way closer to its actual value.

In this paper, an electromechanical coupled dynamic equation of a micro beam under an electrostatic force as well as under an electromechanical coupled force is presented. The linearization of above dynamic equation is made, allowing the equation to be divided into a linear dynamic equation for dynamic displacement and a static balance equation for static displacement. Using the balance equation, the changes of the voltage along with displacement are studied. It is shown that there is a critical voltage at which the micro beam will buckle. From the linear dynamic equation, natural frequencies and vibration modes of the micro beam, and its forced responses to voltage excitation are derived. The results show that the natural frequencies and vibrating magnitudes of the micro beam are affected by mechanical and electric parameters. Smaller beam length and voltage as well as larger beam thickness and clearance should be selected in order to obtain smaller vibrating magnitudes. It is also shown that for higher vibration modes, more positions of the peak dynamic displacements occur.

A high-order finite element model is presented to perform the vibration analysis of beams. The equations of motion are formulated by applying the principle of total potential energy in elastic dynamic system and the "set-in-right-position" rule for the construction of system matrices first proposed by the author. The primary advantage of the principle and rule lies in its simplicity and efficiency in solving the modeling problem of complex dynamic system. The requirement of strain continuity has certainly not being met at element interfaces with the use of conventional cubic Hermitian formulation. Hence, it is difficult to predict the dynamic responses of beams accurately. In order to overcome this problem, a beam element with simple higher-order interpolation function is chosen as the analysis model. Although the number of nodal degrees of freedom is increased herein, usually a coarse mesh will suffice. The present formulation is able to provide results of high accuracy with low computational effort. For the purpose of illustration, the dynamic characteristics analysis and dynamic response analysis are carried out on beam models. The solutions obtained for all the examples are in good agreement with the exact solutions found by fundamental theory of vibration.

The initial stresses due to dead loads have an influence on the natural frequencies of bridges. In this paper, a dynamic stiffness-based method is proposed for determining the natural frequencies of uniform elastic beams with allowance for the dead load effect. Firstly, the governing differential equation including the effect of dead loads is derived. Next, the analytical dynamic stiffness matrix is obtained by applying the displacements and forces boundary conditions at the ends of the beam. In order to solve analytically the governing differential equation, the modified dynamic stiffness matrix is defined by converting the governing quasi-static boundary value problem into an equivalent set of initial value problems. Finally, the Wittrick–Williams algorithm is implemented to extract the natural frequencies from the modified dynamic stiffness matrix. Numerical examples are presented and corresponding parameter studies have been performed to illustrate the applicability and reliability of the proposed method. It is demonstrated that the proposed dynamic stiffness matrix-based method is effective even though the beam is considered as a single element without adding additional nodes.

This paper presents a topology optimization method for dynamic problems with an improved bi-directional evolutionary structural optimization (BESO) technique. The sensitivity derivation for the frequency optimization problem in the case of multiple eigenvalues and for the stiffness–frequency optimization problem is proposed. Algorithms for a filter scheme, sensitivity history-averaging, and sensitivity global-ranking are used in the present method. Techniques for adaptively removing alternative elements and eliminating singular and single-hinged elements are proposed. Solution-convergence and localized modes are discussed through numerical examples. Results show that the improved BESO method is capable of solving the frequency optimization problem and the multi-objective optimization problem for stiffness and frequency effectively.

A nonlocal Timoshenko curved beam model is developed using a modified couple stress theory and Hamilton's principle. The model contains a material length scale parameter that can capture the size effect, unlike the classical Timoshenko beam theory. Both bending and axial deformations are considered, and the Poisson effect is incorporated in the model. The newly developed nonlocal model recovers the classical model when the material length scale parameter and Poisson's ratio are both taken to be zero and the straight beam model when the radius of curvature is set to infinity. In addition, the nonlocal Bernoulli–Euler curved beam model can be realized when the normal cross-section assumption is restated. To illustrate the new model, the static bending and free vibration problems of a simply supported curved beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko curved beam model. Also, the differences in both the deflection and rotation predicted by the current and classical Timoshenko model are very large when the beam thickness is small, but they diminish with the increase of the beam height. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the nonlocal model is higher than that by the classical model, and the difference between them is significantly large only for very thin beams. These predicted trends of the size effect at the micron scale agree with those observed experimentally.