Showing papers on "Normal mode published in 2021"

Vibrations and Stability: Advanced Theory, Analysis, and Tools

Abstract: 1 Vibration Basics.- 2 Eigenvalue Problems of Vibrations And Stability.- 3 Nonlinear Vibrations: Classical Local Theory.- 4 Nonlinear Multiple-DOF Systems: Local Analysis.- 5 Bifurcations.- 6 Chaotic Vibrations.- 7 Special Effects of High-Frequency Excitation.- Appendix A - Performing Numerical Simulations.- A.1 Solving Differential Equations.- A.2 Computing Chaos-Related Quantities.- A.3 Interfacing with the ODE-Solver.- A.4 Locating Software on the Internet.- Appendix B - Major Exercises.- B.1 Tension Control of Rotating Shafts.- B.1.1 Mathematical Model.- B.1.2 Eigenvalue Problem, Natural Frequencies and Mode Shapes.- B.1.3 Discretisations, Choice of Control Law.- B.1.5 Quantitative Analysis of the Controlled System.- B.1.6 Using a Dither Signal for Open-Loop Control.- B.1.7 Numerical Analysis of the Controlled System.- B.1.8 Conclusions.- B.2 Vibrations of a Spring-Tensioned Beam.- B.2.1 Mathematical Model.- B.2.2 Eigenvalue Problem, Natural Frequencies and Mode Shapes.- B.2.3 Discrete Models.- B.2.4 Local Bifurcation Analysis for the Unloaded System.- B.2.5 Quantitative Analysis of the Loaded System.- B.2.6 Numerical Analysis.- B.2.7 Conclusions.- B.3 Dynamics of a Microbeam.- B.3.1 System Description.- B.3.2 Mathematical Model.- B.3.3 Eigenvalue Problem, Natural Frequencies and Mode Shapes.- B.3.4 Discrete Models, Mode Shape Expansion.- B.3.5 Local Bifurcation Analysis for the Statically Loaded System.- B.3.6 Quantitative Analysis of the Loaded System.- B.3.7 Numerical Analysis.- B.3.8 Conclusions.- Appendix C - Mathematical Formulas.- C.1 Formulas Typically Used in Perturbation analysis.- C.1.1 Complex Numbers.- C.1.2 Powers of Two-Term Sums.- C.1.3 Dirac's Delta Function (?).- C.1.4 Averaging Integrals.- C.1.5 Fourier Series of a Periodic Function.- C.2 Formulas for Stability Analysis.- C.2.1 The Routh-Hurwitz Criterion.- C.2.2 Mathieu's Equation:Stability of the Zero-Solution.- Appendix D - Vibration Modes and Frequencies for Structural Elements.- D.1 Rods.- D.1.1 Longitudinal Vibrations.- D.1.2 Torsional Vibrations.- D.2 Beams.- D.2.1 Bernoulli-Euler Theory.- D.2.2 Timoshenko Theory.- D.3 Rings.- D.3.1 In-Plane Bending.- D.3.2 Out-of-Plane Bending.- D.3.3 Extension.- D.4 Membranes.- D.4.1 Rectangular Membrane.- D.4.2 Circular Membrane.- D.5 Plates.- D.5.1 Rectangular Plate.- D.5.2 Circular Plate.- D.6 Other Structures.- Appendix E - Properties of Engineering Materials.- E.1 Friction and Thermal Expansion Coefficients.- E.2 Density and Elasticity Constants.- References.

Nonlinear secondary resonance of FG porous silicon nanobeams under periodic hard excitations based on surface elasticity theory

Abstract: To impart desirable material properties, functionally graded (FG) porous silicon has been produced in which the porosity changes gradually across the material volume. The prime objective of this work is to predict the influence of the surface free energy on the nonlinear secondary resonance of FG porous silicon nanobeams under external hard excitations. On the basis of the closed-cell Gaussian-random field scheme, the mechanical properties of the FG porous material are achieved corresponding to the uniform and three different FG patterns of porosity dispersion. The Gurtin–Murdoch theory of elasticity is implemented into the classical beam theory to construct a surface elastic beam model. Thereafter, with the aid of the method of multiple time-scales together with the Galerkin technique, the size-dependent nonlinear differential equations of motion are solved corresponding to various immovable boundary conditions and porosity dispersion patterns. The frequency response and amplitude response associated with the both subharmonic and superharmonic hard excitations are obtained including multiple vibration modes and interactions between them. It is found that for the subharmonic excitation, the nanobeam is excited within a specific range of the excitation amplitude, and this range shifts to higher excitation amplitude by incorporating the surface free energy effects. For the superharmonic excitation, by taking surface stress effect into account, the excitation amplitude associated with the peak of the vibration amplitude enhances. Moreover, in the subharmonic case, it is demonstrated that by increasing the porosity coefficient, the value of the excitation frequency at the joint point of the two branches of the frequency-response curve reduces. In the superharmonic case, it is revealed that an increment in the value of porosity coefficient leads to decrease the peak of the oscillation amplitude and the associated excitation frequency.

Quantum Effects in Chemical Reactions under Polaritonic Vibrational Strong Coupling.

Abstract: The electromagnetic field in an optical cavity can dramatically modify and even control chemical reactivity via vibrational strong coupling (VSC). Since the typical vibration and cavity frequencies are considerably larger than thermal energy, it is essential to adopt a quantum description of cavity-catalyzed adiabatic chemical reactions. Using quantum transition state theory (TST), we examine the coherent nature of adiabatic reactions in cavities and derive the cavity-induced changes in eigenfrequencies, zero-point energy, and quantum tunneling. The resulting quantum TST calculation allows us to explain and predict the resonance effect (i.e., maximal kinetic modification via tuning the cavity frequency), collective effect (i.e., linear scaling with the molecular density), and selectivity (i.e., cavity-induced control of the branching ratio). The TST calculation is further supported by perturbative analysis of polariton normal modes, which not only provides physical insights to cavity-catalyzed chemical reactions but also presents a general approach to treat other VSC phenomena.

A Review of Antenna Analysis Using Characteristic Modes

Abstract: Characteristic modes can be used to solve many radiation and scattering problems involving fully conducting structures. The basic concept is to use eigenvalues and eigenvectors of the characteristic mode equations of a certain topology, and to use these eigenmode currents to control the radiating structure’s behavior. This is accomplished by understanding the overall reflection and radiation characteristics across the operating frequency band. More specifically, characteristic modes are effective for example in determining the best location for the excitation of a radiating structure. In this paper, a comprehensive overview is provided to the different studies and their results obtained for different antennas that are based on the characteristic mode analysis method. A future perspective is given regarding the potential of this method in antenna analysis and design.

Benchmarks for higher-order modes evaluation in the free vibration response of open thin-walled beams due to the cross-sectional deformations

Abstract: Highly flexible thin-walled beams with complex open cross-sections are sensitive to torsional and warping effects. The analysis of higher-order vibration modes in these structures needs more accurate and precise methods in order to achieve reliable results and detect the cross-sectional deformations in the structures’ free vibration response. This paper analyzes higher vibration modes in a series of thin-walled beams, which were proposed by Chen as benchmark problems. These are all open-section thin-walled beams with complex geometries. Global vibration modes, such as bending and torsion, related to the rigid cross-sectional deformations can be detected via classical and shear refined theories. However, cross-sectional deformations appear at higher frequencies, and these modes are mixed with the global ones. To highlight this fact, this paper compares classical beam theories with refined ones based on the Carrera Unified Formulation (CUF) and the shell results using the commercial finite element (FE) software and the data available from the literature. The CUF FEs based on the power of cross-sectional deformation coordinates (x, z) and those based on the Lagrangian polynomials are implemented and compared using Modal Assurance Criterion. A number of interesting conclusions are drawn about the effectiveness of classical and CUF-based results. The need for models capable of detecting cross-sectional deformations is outlined. In fact, many modes are lost by classical beam theories; on the other hand, they show rigid cross-section modes that do not really exist. This fact is also confirmed by the shell models, which are more expensive in terms of computational costs regarding the efficient CUF ones proposed here.

Random vibrations of stress-driven nonlocal beams with external damping

Abstract: Stochastic flexural vibrations of small-scale Bernoulli–Euler beams with external damping are investigated by stress-driven nonlocal mechanics. Damping effects are simulated considering viscous interactions between beam and surrounding environment. Loadings are modeled by accounting for their random nature. Such a dynamic problem is characterized by a stochastic partial differential equation in space and time governing time-evolution of the relevant displacement field. Differential eigenanalyses are performed to evaluate modal time coordinates and mode shapes, providing a complete stochastic description of response solutions. Closed-form expressions of power spectral density, correlation function, stationary and non-stationary variances of displacement fields are analytically detected. Size-dependent dynamic behaviour is assessed in terms of stiffness, variance and power spectral density of displacements. The outcomes can be useful for design and optimization of structural components of modern small-scale devices, such as micro- and nano-electro-mechanical-systems.

Solar inertial modes: Observations, identification, and diagnostic promise

Abstract: The oscillations of a slowly rotating star have long been classified into spheroidal and toroidal modes. The spheroidal modes include the well-known 5-min acoustic modes used in helioseismology. Here we report observations of the Sun’s toroidal modes, for which the restoring force is the Coriolis force and whose periods are on the order of the solar rotation period. By comparing the observations with the normal modes of a differentially rotating spherical shell, we are able to identify many of the observed modes. These are the high-latitude inertial modes, the critical-latitude inertial modes, and the equatorial Rossby modes. In the model, the high-latitude and critical-latitude modes have maximum kinetic energy density at the base of the convection zone, and the high-latitude modes are baroclinically unstable due to the latitudinal entropy gradient. As a first application of inertial-mode helioseismology, we constrain the superadiabaticity and the turbulent viscosity in the deep convection zone.

Solar inertial modes: Observations, identification, and diagnostic promise.

Abstract: The oscillations of a slowly rotating star have long been classified into spheroidal and toroidal modes. The spheroidal modes include the well-known 5-min acoustic modes used in helioseismology. Here we report observations of the Sun's toroidal modes, for which the restoring force is the Coriolis force and whose periods are on the order of the solar rotation period. By comparing the observations with the normal modes of a differentially rotating spherical shell, we are able to identify many of the observed modes. These are the high-latitude inertial modes, the critical-latitude inertial modes, and the equatorial Rossby modes. In the model, the high-latitude and critical-latitude modes have maximum kinetic energy density at the base of the convection zone, and the high-latitude modes are baroclinically unstable due to the latitudinal entropy gradient. As a first application of inertial-mode helioseismology, we constrain the superadiabaticity and the turbulent viscosity in the deep convection zone.

Fundamental frequency optimization of variable stiffness composite skew plates

Abstract: In this study, natural frequencies and vibrational mode shapes of variable stiffness composite skewed plates are optimized applying a genetic algorithm. The variable stiffness behavior is obtained by altering the fiber angles continuously according to two selected curvilinear fiber path functions in the composite laminates. Fundamental frequency and related mode shapes of the plates are optimized for two different fiber path functions using the structural model obtained based on the virtual work principle. A three-layer composite skewed plate with four types of boundary conditions and different plate geometries is considered as case study in this research. Diverse sweptback angles as well as different aspect ratios are considered as various plate geometries. The present study aims to calculate the best fiber path with maximized fundamental frequency or in-plane strengths for a composite skewed plate. The generalized differential quadrature method of solution is employed to solve the governing equations of motion. Moreover, the linear kinematic strain assumptions are used, and the first-order shear deformation theory is employed to generalize the formulation for the case of moderately thick plates including transverse shear effects. Numerical results demonstrate the effect of the fiber angles, boundary conditions, and diverse geometries on the natural frequencies of the composite plate. The optimal fiber angles of each layer are presented for the above cases in free vibration analysis. It is verified that the application of optimized curvilinear fibers instead of the traditional straight fibers introduces a higher degree of flexibility, which can be used to adjust frequencies and mode shapes.

An extended energy balance method for resonance prediction in forced response of systems with non-conservative nonlinearities using damped nonlinear normal mode

Abstract: The dynamic analysis of systems with nonlinearities has become an important topic in many engineering fields. Apart from the forced response analyses, nonlinear modal analysis has been successfully extended to such non-conservative systems thanks to the definition of damped nonlinear normal modes. The energy balance method is a tool that permits to directly predict resonances for a conservative system with nonlinearities from its nonlinear modes. In this work, the energy balance method is extended to systems with non-conservative nonlinearities using the concept of the damped nonlinear normal mode and its application in a full-scale engineering structure. This extended method consists of a balance between the energy loss from the internal damping, the energy transferred from the external excitation and the energy exchanged with the non-conservative nonlinear force. The method assumes that the solution of the forced response at resonance bears resemblance to that of the damped nonlinear normal mode. A simplistic model and full-scale structure with dissipative nonlinearities and a simplistic model showing self-excited vibration are tested using the method. In each test case, resonances are predicted efficiently and the computed force–amplitude curves show a great agreement with the forced responses. In addition, the self-excited solutions and isolas in forced responses can be effectively detected and identified. The accuracy and limitations of the method have been critically discussed in this work.

On the first frequency of reinforced partially hinged plates

Abstract: We consider a partially hinged rectangular plate and its normal modes. The dynamical properties of the plate are influenced by the spectrum of the associated eigenvalue problem. In order to improve...

Locally Resonant Periodic Wave Barriers for Vibration Isolation in Subway Engineering

Abstract: Subway transportation is being promoted worldwide to effectively solve urban congestion However, the vibration induced by subway traffic has caused a major adverse impact on building safety, precision instrument operation and human health Wave barriers have been proven effective in mitigating ground vibration, whereas they have some limitations in achieving ideal attenuation zone and high efficiency to cover the low-frequency vibration in underground railway system Based on locally resonant phononic crystals theory, this paper designs three-component locally resonant periodic wave barriers (LRPWBs), and investigates the effects of geometrical and material parameters on the bandgap features in detail The band structures are calculated using improved plane wave expansion (IPWE), the transmission spectra and vibration modes are obtained by finite element method (FEM) The results indicate LRPWBs are able to give lower and wider bandgap to cover the main frequency of subway environment, which is proved by time and frequency domain analysis For the bandgap mechanism, the local resonance features of LRPWBs result in the energy conversion between kinetic energy and elastic strain energy, thus the elastic wave energy is localized in resonance unit and then the locally resonant bandgap is created In addition, the bandgap can be adjusted by carefully selecting proper geometrical and material parameters to actualize low-frequency broadband attenuation Further studies about multi-oscillator system indicate that the appropriate combination of multiple LRPWBs are conductive to diverse and broad bandgaps The investigations can provide inspiration for periodic wave barriers design in multi-frequency vibration attenuation field

Normal ordering normal modes

Abstract: In a soliton sector of a quantum field theory, it is often convenient to expand the quantum fields in terms of normal modes. Normal mode creation and annihilation operators can be normal ordered, and their normal ordered products have vanishing expectation values in the one-loop soliton ground state. The Hamiltonian of the theory, however, is usually normal ordered in the basis of operators which create plane waves. In this paper we find the Wick map between the two normal orderings. For concreteness, we restrict our attention to Schrodinger picture scalar fields in 1+1 dimensions, although we expect that our results readily generalize beyond this case. We find that plane wave ordered n-point functions of fields are sums of terms which factorize into j-point functions of zero modes, breather and continuum normal modes. We find a recursion formula in j and, for products of fields at the same point, we solve the recursion formula at all j.

Dynamic monitoring and data analysis of a long-span arch bridge based on high-rate GNSS-RTK measurement combining CF-CEEMD method

Abstract: The purpose of this article is to develop a combined data analysis method of Chebyshev filter (CF) and complementary ensemble empirical mode decomposition (CEEMD) for weakening the influence of the background noise of global navigation satellite system (GNSS) sensors. To test the effect of noise reduction using the proposed CF-CEEMD method, a nonlinear signal with additive noise is first introduced. Then, the GNSS measured signal of a long-span arch bridge is analyzed using CF-CEEMD. Moreover, the dynamic characteristic parameters (i.e. natural frequencies, mode shapes and damping ratios) of the bridge are extracted from the de-noised signal employing the data-driven stochastic subspace identification (DD-SSI) algorithm. Meanwhile, the finite element (FE) model of the bridge is established to predict the natural frequencies and the mode shapes via modal analysis. Finally, the results depict that GNSS-RTK technique is applicable to monitor the dynamic response of long-span bridges with reasonable accuracy via CF-CEEMD analysis. Furthermore, the natural frequencies and mode shapes derived experimentally via DD-SSI analysis have good agreement with the predicted values based on FE model.

Closed-form solution for free vibration of variable-thickness cylindrical shells rotating with a constant angular velocity

Abstract: Based on the classical Donnell’s and Love’s shell theories, free vibration behavior of variable-thickness thin cylindrical shells rotating with a constant angular velocity is analyzed. The equations of motion and corresponding boundary conditions of rotating homogenous cylindrical shells with axisymmetric variation of thickness are derived using Hamilton’s principle. This formulation includes effects of initial hoop tension due to the centrifugal force as well as Coriolis and centrifugal accelerations. Considering the variation of stiffness coefficients in axial direction, the classical Love’s theory results in a coupled system of two second-order and one fourth-order partial differential equations with variable coefficients in terms of kinematic variables. The Galerkin’s method is used to obtain a closed-form expression of sixth-order frequency equation with coefficients in terms of rotation speed ( Ω ) for with boundary conditions. The natural frequencies of stationary and rotating cylindrical shells with constant thickness, as special cases, are validated with existing ones in the literature. Case studies are presented for three different patterns of thickness variation in forms of arbitrary power, axisymmetric modal and stepwise functions. It is seen that the influence of thickness variation on the natural frequencies and mode shapes is a function of different parameters including rotation speed, boundary conditions, and geometric ratios, especially this effect is much more pronounced for cantilever long shells with a high rotation speed. Also, differences in results predicted by the Donnell’s and Love’s theories become larger for long and cantilever shells in vibration mode shape corresponding to ( m = 1 , n = 2 ) , i.e., the ovalization of cross section.

Oscillatory flow around a vertical wall-mounted cylinder: Dynamic mode decomposition

Abstract: This study applies the Dynamic Mode Decomposition (DMD) to better understand and model the oscillatory flow across a vertical wall-mounted cylinder. At different Keulegan–Carpenter numbers, three-dimensional direct numerical simulations are performed to provide the flow details such as the snapshots of the coherent structures around the cylinder, vorticity fields, and bed shear stress. The selected fields are decomposed into dynamic modes. The characteristic flow features with relevant information including spatial mode shape, frequency, and mode amplitude are systematically investigated. The time series of flow fields is also reconstructed using the DMD analysis and compared against the original data to assess the efficacy of information in the low-dimensional system. The results show that the DMD analysis can capture the dynamic and nonlinear features of the oscillatory flow past the vertical wall-mounted cylinder and also efficiently reconstruct the relevant fields with reasonable accuracy. It provides a basis for the data-driven model of scour near the cylinder–wall junction relevant to coastal engineering applications.

Flexoelectric effects on the natural frequencies for free vibration of piezoelectric nanoplates

Abstract: Flexoelectricity is an electromechanical coupling phenomenon between polarization and strain gradient. Based on the Kirchhoff thin plate theory, the electromechanical coupling responses of nanoplates with the piezoelectric and flexoelectric effects are studied in this paper. Free vibration of a piezoelectric nanoplate with consideration of flexoelectricity is analyzed with emphasis on the influence of the dynamic flexoelectric effect on the natural frequencies. By means of Hamilton’s variational principle, the governing equation of rectangular plates together with associated boundary conditions is derived. The natural frequencies are evaluated for a nanoplate simply supported at two opposite edges, and exact frequency equations are obtained for the other two opposite edges being simply supported, clamped–clamped, clamped–free, simply supported–free, or clamped–simply supported. The influence of dynamic flexoelectricity on the natural frequencies is elucidated. The results show that the dynamic flexoelectric effect is also size-dependent; the smaller the plate thickness is, the more obvious the dynamic flexoelectric effect is. The results also show that the dynamic flexoelectric effect is more pronounced when the order of vibration modes is higher and nanoplate’s side ratio is larger. The positive and negative choice of static and dynamic flexoelectric coefficients have completely different effects on the natural frequencies. The influence of the dynamic flexoelectric effect on the natural frequencies is closely related to the side constraint and geometry of the plate. The piezoelectric effect does not alter the natural frequencies for free vibration of a homogeneous nanoplate.

Free vibrational characteristics of GNP-reinforced joined conical–conical shells with different boundary conditions

Abstract: The current study is devoted to analyzing the free vibration characteristics of joined conical–conical shells made of epoxy enriched with graphene nanoplatelets (GNPs). The mathematical modeling of the shell is performed utilizing the first-order shear deformation theory (FSDT) to incorporate the effects of shear deformations and rotational inertia. The effective modulus of elasticity is estimated using the Halpin–Tsai model and other effective mechanical properties are evaluated utilizing the rule of mixture. The set of the governing equations, associated compatibility conditions at the intersection of two shell segments, and boundary conditions are obtained using Hamilton’s principle. These equations are solved in the circumferential direction using trigonometric functions and a numerical solution is provided in the meridional direction utilizing the differential quadrature method (DQM). Through this semi-analytical solution, the natural frequencies of the shell and corresponding mode shapes are determined and the validity of the presented solution is confirmed via the benchmark results reported by the other authors. The effects of various parameters on the natural frequencies of the GNP-reinforced joined conical–conical​ shells are examined including the circumferential wave number, the boundary conditions, the length-to-small radius ratio and semi-vertex angle in two shell segments, thickness-to-small radius ratio of the shell, and also mass fraction, aspect ratio, and dispersion pattern of the GNPs.

Shear and Breathing Modes of Layered Materials.

Abstract: Layered materials (LMs), such as graphite, hexagonal boron nitride, and transition-metal dichalcogenides, are at the center of an ever-increasing research effort, due to their scientific and technological relevance. Raman and infrared spectroscopies are accurate, non-destructive approaches to determine a wide range of properties, including the number of layers, N, and the strength of the interlayer interactions. We present a general approach to predict the complete spectroscopic fan diagrams, i.e., the relations between frequencies and N for the optically active shear and layer-breathing modes of any multilayer comprising N ≥ 2 identical layers. In order to achieve this, we combine a description of the normal modes in terms of a one-dimensional mechanical model, with symmetry arguments that describe the evolution of the point group as a function of N. Group theory is then used to identify which modes are Raman- and/or infrared-active, and to provide diagrams of the optically active modes for any stack composed of identical layers. We implement the method and algorithms in an open-source tool to assist researchers in the prediction and interpretation of such diagrams. Our work will underpin future efforts on Raman and infrared characterization of known, and yet not investigated, LMs.

Normal mode decomposition and dispersive and nonlinear mixing in stratified fluids

Abstract: Motivated by the analysis of the propagation of internal waves in a stratified ocean, we consider in this article the incompressible Euler equations with variable density in a flat strip, and we study the evolution of perturbations of the hydrostatic equilibrium corresponding to a stable vertical stratification of the density. We show the local well-posedness of the equations in this configuration and provide a detailed study of their linear approximation. Performing a modal decomposition according to a Sturm–Liouville problem associated with the background stratification, we show that the linear approximation can be described by a series of dispersive perturbations of linear wave equations. When the so-called Brunt–Vaisala frequency is not constant, we show that these equations are coupled, hereby exhibiting a phenomenon of dispersive mixing. We then consider more specifically shallow water configurations (when the horizontal scale is much larger than the depth); under the Boussinesq approximation (i.e., neglecting the density variations in the momentum equation), we provide a well-posedness theorem for which we are able to control the existence time in terms of the relevant physical scales. We can then extend the modal decomposition to the nonlinear case and exhibit a nonlinear mixing of different nature than the dispersive mixing mentioned above. Finally, we discuss some perspectives such as the sharp stratification limit that is expected to converge towards two-fluid systems.