Exact Characteristic Equations Research Articles

Stability of the twisted states in a ring of oscillators interacting with distance-dependent delays

We consider a ring of phase oscillators interacting with distance-dependent time delays in general forms. It is found that the distance-dependent interaction delay is an essential driving mechanism for the twisted state to occur as a generic state. We carry out rigorous stability analysis for the existence and stability of the twisted states along the Ott–Antonsen invariant manifold and derive the exact characteristic equations for the stability conditions for arbitrarily chosen distance-dependent delay function. The complete stability diagrams are illustrated for two types of distance-dependent delay schemes, the propagational and step-wise interaction delays. Our theoretical results are verified using the direct numerical simulations of the model system.

Investigation on Optimal Switching Oscillation Suppression for SiC MOSFET by Inductively Coupled Damping

SiC MOSFET has been revolutionizing power electronics but producing more severe switching oscillation due to its inherent faster switching speed and lower damping characteristics. The inductively coupled damping approach is promising since no additional circuit element is inserted into the power processing loop. Thus, reliability of this inductively coupled fashion is higher compared to other existing snubbers. However, prior-art analytical design techniques are mainly based on scrutinizing the peak impedance and only discussed under a certain specific coupling, which limited its suppression effect. In this article, by viewing from the terminals of both upper and lower SiC MOSFETs in a half-leg circuit, two-port networks are linearized from switching oscillation and developed to deduce the exact characteristic equations. Then, the optimal snubber configuration along with two key parameters, i.e., the inductance factor and the secondary resistance, are determined via the root locus method. It is revealed that the damping effect of RL in the strong coupling region is even better than that of the secondary RLC resonator. What is more, applying the coupled RL damping circuit could not only remarkably dampen the switching oscillations, but also greatly lower switching losses. Finally, discussion on selecting the optimal snubber parameters is illustrated in detail.

Free vibration of circular and annular nanoplates with surface and flexoelectric effects

Free vibration of circular/annular piezoelectric nanoplates with flexoelectric and surface effects is studied. Using Hamilton’s principle, the governing equation and boundary conditions of circular/annular nanoplates are derived. Exact characteristic equations of clamped and simply-supported circular nanoplates are obtained. The static/dynamic flexoelectric and surface effects on the natural frequencies are analyzed. The results show that the flexoelectric and surface effects have obvious size effects. Positive surface stress increases the natural frequencies as if the nanoplate is tightened, and vice versa. There is a competitive mechanism between the influence of the flexoelectric and surface effects on the natural frequencies.

Free vibration of radially graded hollow cylinders subject to axial force via a higher-order shear deformation beam theory

The dynamic behavior of radially functionally graded tubes is studied when subjected to axial tensile and compressive forces. Differing from the Euler–Bernoulli/Timoshenko beam theories, a higher-order shear beam deformation model that does neither require a shear correction factor nor need a planar cross-section assumption after deformation. The cross-section’s warping is constructed to meet the shear-free condition on tube’s inner and outer surfaces. A governing partial differential equation is derived. Exact characteristic equations for determining the natural frequencies are given. For typical end supports including hinged-hinged, clamped–clamped, clamped-free, and clamped-hinged ends, the natural frequencies are calculated. By letting the frequency vanish, the critical buckling loads under compression can be exactly determined by solving the characteristic equation. A comparison of the present buckling loads and the natural frequencies with the classical ones and with finite element results is made and verifies the efficiency of the proposed model. The frequency-load interaction is plotted for typical boundary conditions. Axial tensile force causes the natural frequencies to increase and axial compressive force decreases the natural frequencies. The effects of the radial gradient, tube’s thickness and length, and the cross-sectional warping shape on the buckling loads and the natural frequencies are elucidated.

Resonant and antiresonant frequencies of multiple cracked bar

The natural frequencies or related resonant frequencies have been widely used for crack detection in structures by the vibration-based technique. However, antiresonant frequencies, the zeros of frequency response function, are less involved to use for the problem because they have not been thoroughly studied. The present paper addresses analysis of antiresonant frequencies of multiple cracked bar in comparison with the resonant ones. First, exact characteristic equations for the resonant and antiresonant frequencies of bar with arbitrary number of cracks are conducted in a new form that is explicitly expressed in term of crack severities. Then, the conducted equations are employed for analysis of variation of resonant and antiresonant frequencies versus crack position and depth. Numerical results show that antiresonant frequencies are indeed useful indicators for crack detection in bar mutually with the resonant ones.

Characteristic equation for antiresonant frequencies of multiple cracked bars and application for crack detection

ABSTRACTThe present paper addresses study and use of antiresonant frequencies for multi-crack detection in bar. First, there is an established new form of the exact characteristic equations for resonant and antiresonant frequencies in longitudinal vibration of bars with a finite number of cracks. Some asymptotic approximations of the equations are therefore derived and examined to evaluate their applicability in solving the forward problem of the multiple cracked bar. Then, the characteristic equations established in this study are used for analysis of change in antiresonant frequencies due to cracks. Finally, based on the characteristic equations and analysis, an efficient procedure is proposed to detect a number of cracks from given resonant and/or antiresonant frequencies. Numerical examples have been examined to illustrate and validate the proposed theory applied for detecting single crack in fixed-free end bar and double cracks in bar with free ends.

On vibrations of nonlocal rods: Boundary conditions, exact solutions and their asymptotics

The size effects on the dynamic behaviors of rods are investigated within the framework of the nonlocal strain gradient elastic theory. The variationally consistent boundary conditions are derived by using the weighted residual method with respect to the known equation of motion of rods. Similar to the fourth-order differential equation of motion for classical Euler–Bernoulli beams, the variationally consistent boundary conditions of nonlocal strain gradient rods are clarified for the first time. The exact characteristic equations for determining the longitudinal frequencies are presented for four types of boundary value problems. To gain insight into the asymptotic behaviors of rods, the reduced boundary values problems are also analytically given. The numerical results show that both the softening effect and the stiffening effect can be captured by adjusting the two material length parameters. However, when the two material length parameters are the same, the present results cannot recover to the corresponding classical ones for rods with general boundary conditions, which are different from the previous findings reported in the literature.

Buckling analysis of Euler–Bernoulli beams using Eringen’s two-phase nonlocal model

The inconsistency of Eringen’s nonlocal differential model, as applied to investigate nanostructures, has recently triggered the study of nonlocal integral models. In this paper we adopt Eringen’s two-phase nonlocal integral model to carry out an analytical study on the buckling problem of Euler-Bernoulli beams. By using a reduction method rigorously proved in the previous work, the resulting integro-differential equation for the problem is firstly reduced to a fourth order differential equation with mixed boundary conditions. Exact characteristic equations are then obtained for four types of boundary conditions. Further, after some detailed asymptotic analysis, asymptotic solutions for the critical buckling loads are obtained, which are shown to have a good agreement with the numerical solutions. The analytical solutions show clearly that the nonlocal effect reduces the buckling loads. It is also found that the effect could be first-order or second order depending on the boundary conditions.

Exact Characteristic Equations in Closed-Form for Vibration of Completely Free Timoshenko Beams

Huang 1 presented first in 1961 the characteristic equations and normal mode equations for all six common types of simple, finite Timoshenko beams in closed-form. Unfortunately, there exist several errors, not typographical, in the frequency in the characteristic equations for a Timoshenko beam free at both ends. In this paper, the exact characteristic equations in closed-form for completely free Timoshenko beams are derived based on Ref. 1. In order to justify the solutions obtained by amending herein Huang’s equations, both the closed-form exact solution and the one obtained by the Ritz method will be adopted as the references. Using the characteristic equations derived by Huang 1 , we can obtain only frequencies for the flexural modes, while using the closed-form exact method and Ritz method, we can obtain frequencies for the thickness-shear modes, as well as for the bending modes. The purpose of the present study is to identify the errors in Ref. 1, correct them, and provide some numerical results.

Exact compact characteristic equations and new results for free vibrations of orthotropic rectangular Mindlin plates

This paper presents correct results for a paper by Liu and Xing (2011) [1] that had errors, greatly simplifies its formulations, overcomes its numerical stability problem, and provides some new results. In their early paper, Liu and Xing obtained exact closed-form solutions for free vibrations of orthotropic rectangular Mindlin plates by using the separation of variables method. The equations presented therein are correct, but the numerical values of the frequencies listed have some errors because the shear rigidities of the three-ply laminates were not correctly computed. In addition, the exact characteristic equations of the early paper can also be greatly simplified and the compact equations are presented herein. The characteristic equations in the early paper may have computational stability problem when plates become very thin. This paper overcome such computational difficulty. Some new exact frequency parameters are also included in this work.

A new model for composite beams with partial interaction

A theoretical formulation for the deflection of composite beams with consideration of interface slip is derived. The exact closed-form characteristic equations for composite beams with interlayer slip are derived for the boundary conditions. The equations also consider deflections and slips in the cross-section, leading to determinations of internal forces in the cross-section. Based on the research, practical applications are compared with the corresponding transformed section method. The results show that the theory is correct and convenient for solving problems of the analysis of steel–concrete composite box girders. Arbitrary cross-internal forces of the steel–concrete composite box girders are also obtained. The results provide a theoretical basis for the safe design of steel–concrete composite box girders.

The vibration modes of concentrically supported free circular plates

Abstract Vibration of a circular plate internally concentrically supported is basic in structural mechanics. The complete first six frequencies and mode shapes are determined for clamped, simply-supported, sliding, and ring supports using exact characteristic frequency equations. Complex mode switches occur as the support radius is varied. Asymptotic formulas for large support radii are derived. Different types of singularities as the support radius shrinks to zero are distinguished and classified.

An exact analytical approach for in-plane and out-of-plane free vibration analysis of thick laminated transversely isotropic plates

In this article, the governing equations of motion of thick laminated transversely isotropic plates are derived based on Reddy's third-order shear deformation theory. These equations are exactly converted to four uncoupled equations to study the in-plane and out-of-plane free vibrations of thick laminated plates without any usage of approximate methods. Based on the present analytical approach, exact Levy-type solutions are obtained for thick laminated transversely isotropic plates and, for some boundary conditions, the exact characteristic equations hitherto not reported in the literature are given. Also, the in-plane and out-of-plane deformed mode shapes are plotted for different boundary conditions. The present solutions can accurately predict both the in-plane and out-of-plane natural frequencies and mode shapes of thick laminated transversely isotropic plates.

Exact characteristic equations for free vibrations of thin orthotropic circular cylindrical shells

This paper presents an analytical procedure and closed-form vibration solutions with analytically determined coefficients for orthotropic circular cylindrical shells having classical boundary conditions. This analysis is based upon the Donnell–Mushtari shell theory. This is the simplest thin shell theory and its results for the lowest frequencies of a closed cylinder may not be as accurate. It is known that the exact procedure is complicated for orthotropic shells and this complexity has apparently prevented most researchers from getting results. Using the separation of variables method, the closed-form natural frequencies are successfully obtained in this work. They are found in a compact form. Moreover, the characteristics of the eigenvalues are examined. The exact solutions are validated through numerical comparisons with available solutions in literatures and the semi-analytical differential quadrature finite element method (S-DQFEM) solutions calculated by the authors.

An Analytical Method for Free Vibration Analysis of Composite Beams Subjected to Initial Thermal Stresses

The purpose of this paper is to present exact solutions for the free vibration of symmetrically laminated composite beams. The present analysis includes the first shear deformation theory and the rotary inertia. The analytical solutions take into account the thermal effect on the free vibration characteristics of the composite beams. In particular, the aim of this work is to derive the exact closed-form characteristic equations for common boundary conditions. The different parameters that could affect the natural frequencies are included as factors (aspect ratio, thermal load-to-shear coefficient, ply orientation) to better perform dynamic analysis to have a good understanding of dynamic behavior of composite beams. In order to derive the governing set of equations of motion, the Hamilton’s principle is used. The system of ordinary differential equations of the laminated beams is then solved and the natural frequencies’ equations are obtained analytically for different boundary conditions. Numerical results are presented to show the influence of temperature rise, aspect ratio, boundary conditions and ply orientation on the natural frequencies of composite beams.

Exact free vibration study of rectangular Mindlin plates with all-over part-through open cracks

Based on Mindlin plate theory (MPT), a set of exact closed-form characteristic equations incorporating shear deformation and rotary inertia are proposed for the first time to analyze free vibration problem of moderately thick rectangular plates with an arbitrary number of all-over part-through cracks. The proposed rectangular plates have two opposite edges simply supported while six possible combinations of free, simply supported and clamped boundary conditions are taken into account for two other edges. The crack is assumed to be open, non-propagating and perpendicular to two opposite simply supported edges. A continuously distributed line-spring model is used to describe the elastic behavior of an all-over part-through crack. The accuracy of the current approach is investigated through comparing the present exact natural frequencies with those of 3D finite element method obtained by ABAQUS software package. A parametric study is undertaken to show the effect of crack depth, crack location, number of cracks and thickness-to-length ratio on natural frequencies of rectangular moderately thick plates with different boundary conditions in tabular and graphical forms. Finally, the effect of shearing and tearing modes on the modeling of cracks located at the nodal line is shown.

Vibration Equations of Thick Rectangular Plates Using Mindlin Plate Theory

Problem statement: Rectangular steel plates are widely used in variou s steel structures and steel industries. For a proper design of steel plate structures and efficient use of material, the behavior, strength, buckling and post-buckling char acteristics of plates should be accurately determined. Approach: Considering the significance of this matter, later al vibration of thick rectangular plates was studied on the basis of mind lin plate theory. The exact characteristic equations for a plate which is single supported in two opposite edges are available in the literature. S-C-S-F boundary condition which covers all possible situations is selected in this study. Results: The plate frequencies were calculated for this boundary condition for a wide range of plate sizes and thicknesses. The plate mode shapes were obtained fo r different cases and the effect of changes in boundary conditions; size ratio and thickness on th e vibration behavior of rectangular steel plates ar e studied. Conclusion/Recommendations: Since the results of this study is exact and witho ut any approximation, the presented values can be used as a proper criteria to evaluate the error value of approximate methods which are used by engineers for design of steel plates. These results can provide a good gridline for efficient design and prevention of using high safety factors. Considering the wide range of steel rectangular plates, more sizes and t hicknesses of plates can be studied. The behavior o f plates with other boundary conditions can also be s tudied for future research.

Mode structure and attenuation characteristics of hollow parabolic waveguides

The mode structure and attenuation constants in parabolic hollow waveguides with arbitrary parabolic domains are investigated based on the exact vector field expressions and characteristic equations. Normalized attenuation charts are provided for a variety of mode numbers, parities, and polarizations. The analysis is not restricted to parabolic waveguides with a symmetric cross section.

Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates

The dimensionless equations of motion are derived based on the Mindlin plate theory to study the transverse vibration of thick rectangular plates without further usage of any approximate method. The exact closed form characteristic equations are given within the validity of the Mindlin plate theory for plates having two opposite sides simply supported. The six distinct cases considered involve all possible combinations of classical boundary conditions at the other two sides of rectangular plates. Accurate eigenfrequency parameters are presented for a wide range of aspect ratio η and thickness ratio δ for each case. The three dimensional deformed mode shapes together with their associated contour plots obtained from the exact closed form eigenfunctions are also presented. Finally, the effect of boundary conditions, aspect ratios and thickness ratios on the eigenfrequency parameters and vibratory behavior of each distinct cases are studied in detail. It is believed that in the present work, the exact closed form characteristic equations and their associated eigenfunctions, except for the plates with four edges simply supported, for the rest of considered six cases are obtained for the first time.

Analysis of transmission characteristics of doubly clad fibers with an inner cladding made of uniaxial crystal materials

A doubly clad optical fiber with an inner cladding made of a uniaxial crystal material whose optical axis is parallel to the fiber axis was proposed, and exact characteristic equations of vector modes were derived. The influence of the ratio ( k cl) of the extraordinary to the ordinary ray indexes upon the waveguide dispersion was examined in detail. In view of the impossibility to deduce the expression of waveguide dispersion directly due to the complexity of the characteristic equations, a feasible approach to calculate waveguide dispersion was established. The calculated results indicate that the values of waveguide dispersion can be effectively changed through variation of k cl without changing the geometrical and optical parameters ( S and R). The influences of k cl, S and R on the propagation and cutoff characteristics of the low order modes are also analyzed.