Complex Natural Frequencies Research Articles

Surface effect on the biaxial buckling and free vibration of FGM nanoplate embedded in visco-Pasternak standard linear solid-type of foundation

In this study, nonlocal elasticity theory in conjunction with Gurtin–Murdoch elasticity theory is employed to investigate biaxial buckling and free vibration behavior of nanoplate made of functionally graded material (FGM) and resting on a visco-Pasternak standard linear solid-type of the foundation. The material characteristics of simply supported FGM nanoplates are assumed to be varied continuously as a power law function of the plate thickness. Hamilton’s principle is implemented to derive the non-classical governing equations of motion and related boundary conditions, which analytically solved to obtain the explicit closed-form expression for complex natural frequencies and buckling loads. Finally, attention is focused on considering the influences of various parameters on variation of damped natural frequency and buckling load ratio such as nonlocal parameter, surface effects, geometric parameters, power law index and properties of visco-Pasternak foundation and it is clearly demonstrated that these factors highly affect on vibration and buckling behavior.

Problem on natural vibrations of electroviscoelastic bodies with external electric circuits and finite element relations for its implementation

In this paper the mathematical statement for solving of the problem of natural vibrations of piecewise homogeneous electro-viscoelastic bodies with a passive external circuit of different configurations is proposed. The results of the problem solution are complex natural vibration frequencies, which characterize the natural vibration frequencies of bodies and their damping rates. These characteristics can be efficiently used to analyze dynamic processes in smart systems, i.e. materials exhibiting a piezoelectric effect. Recently these systems have found many applications in different branches of science and industry. Their capability to control and govern the dynamic behavior of structures can be greatly expanded by connecting external circuits of different combinations of resistive, capacitive and inductive elements to piezoelectric elements. The search for the optimal parameters of smart-systems is based on the appropriate choice of piezoelectric element arrangements, electric circuit schemes and circuit elements. When considering problems with a large number of parameters, the modeling is one of the efficient approaches for finding the optimal solution. Application software packages currently available for solving natural vibration problems do not provide direct solution to the problem formulated here. Therefore, this study involves finite-element relations, including those for the original finite element representing an external circuit, which, taking into account the well-known types of finite elements for continuous media description, form the basis for constructing numerical algorithms. As an example, calculations of the complex natural vibration frequencies of a half cylinder with an attached piezoelectric element and an external series RL-circuit are performed.

Analytical Solution for Thermomechanical Vibration of Double-Viscoelastic Nanoplate-Systems Made of Functionally Graded Materials

In this article, based on the nonlocal elasticity theory of Eringen, dynamic characteristics of a double-FGM viscoelastic nanoplates-system subjected to temperature change with considering surface effects (surface elasticity, tension and density) is studied. Two Kirchhoff nanoplates are coupled by an internal Kelvin–Voigt viscoelastic medium and also are limited to the external Pasternak elastic foundation. The material properties of the simply supported functionally graded nanoplates are assumed to follow power law distribution in the thickness direction. The governing equations of motion for three cases (out-of-phase vibration, in-phase vibration and one nanoplate fixed) are derived from Hamilton's principle. The analytical approach is employed to determine explicit closed-form expression for complex natural frequencies of the system. Numerical results are presented to show variations of the frequency of double-FGM viscoelastic nanoplates corresponding to various values of the nonlocal parameter, temperature change, power law index, aspect ratio and transverse and shear stiffness coefficients of the Pasternak elastic foundation. Moreover, influence of higher order modes, viscoelastic structural damping and damping coefficient of the viscoelastic medium on vibration characteristics are investigated. Numerical results show that natural frequency is greatly influenced by surface elastic modulus and residual surface stress.

Exact solution for free vibration of coupled double viscoelastic graphene sheets by viscoPasternak medium

Based on nonlocal theory, this article discusses vibration of CDVGS11Coupled double viscoelastic graphene sheets. systems. The properties of each single layer graphene sheet (SLGS) are assumed to be orthotropic and viscoelastic. The two SLGSs are simply supported and coupled by an enclosing viscoelastic medium which is simulated as a Visco-Pasternak layer. This model is aimed at representing dynamic interactions in nanocomposite materials with dissipation effect. By considering the Kirchhoff plate theory and Kelvin–Voigt model, the governing equation is derived using Hamilton's principle. The equation is solved analytically to obtain the complex natural frequency. The parametric study is thoroughly performed, concentrating on the series effects of viscoelastic damping structure, aspect ratio, visco-Pasternak medium, and mode number. In this system, in-phase (IPV) and out-of-phase (OPV) vibrations are investigated. The numerical results of this article show a perfect correspondence with those of the previous researches.

Vibration analysis of damped stiffened beams and plates by complex modes and frequencies

Stiffened beams and plates are used in design of structures forced by machinery excitation, which result in flexural motion having distorted normal modes due to the effect of both stiffeners and damping. Such plates and beams are often damped externally or internally, which leads in turn to complex modes and natural frequencies. The modes, which are normal to the direction or axis of the stiffeners can be calculated by different techniques, such as the “three moments” method for stiffened plates or beams. The effect of the stiffeners is given by finding the changes in frequencies and normal modes as related to the ratio of the stiffeners'- and plate's or beam's rigidities. The paper gives insight into complex modes and the eigenvalue for each such mode and also their relation to the common definition of modal shapes magnitude and frequencies and concludes with solved examples for stiffened plates and beams models.

Nonlocal mass-nanosensor model based on the damped vibration of single-layer graphene sheet influenced by in-plane magnetic field

Nano-materials such as graphene sheets have a great opportunity to be applied in development of a new generation of nanomechanical sensors and devices due to their unique physical properties. Based on the nonlocal continuum theory and vibration analysis, the single-layered graphene sheet with attached nanoparticles affected by in-plane magnetic field is proposed as a new type of the mass-nanosensor. The nonlocal Kirchhoff-Love plate theory is adopted to describe mechanical behavior of single-layered graphene sheet as an orthotropic nanoplate. The equation of motion of a simply supported orthotropic nanoplate is derived, where the influence of Lorentz magnetic force is introduced through classical Maxwell׳s equations. Complex natural frequencies, damped frequency shifts and relative shift of damping ratio for nanoplate with attached nanoparticles are obtained in the explicit form. The influences of the nonlocal and magnetic field parameter, different mass weights and positions of attached nanoparticles and damping coefficients on the relative damped frequency shift and relative shift of damping ratio are examined. The presented results can be useful in the analysis and design of nanosensors applied in the presence of strong magnetic field. Our results show that magnetic field could be successfully used to improve sensibility performances of nanomechanical sensors.

Dynamics of multiple viscoelastic carbon nanotube based nanocomposites with axial magnetic field

Nanocomposites and magnetic field effects on nanostructures have received great attention in recent years. A large amount of research work was focused on developing the proper theoretical framework for describing many physical effects appearing in structures on nanoscale level. Great step in this direction was successful application of nonlocal continuum field theory of Eringen. In the present paper, the free transverse vibration analysis is carried out for the system composed of multiple single walled carbon nanotubes (MSWCNT) embedded in a polymer matrix and under the influence of an axial magnetic field. Equivalent nonlocal model of MSWCNT is adopted as viscoelastically coupled multi-nanobeam system (MNBS) under the influence of longitudinal magnetic field. Governing equations of motion are derived using the Newton second low and nonlocal Rayleigh beam theory, which take into account small-scale effects, the effect of nanobeam angular acceleration, internal damping and Maxwell relation. Explicit expressions for complex natural frequency are derived based on the method of separation of variables and trigonometric method for the “Clamped-Chain” system. In addition, an analytical method is proposed in order to obtain asymptotic damped natural frequency and the critical damping ratio, which are independent of boundary conditions and a number of nanobeams in MNBS. The validity of obtained results is confirmed by comparing the results obtained for complex frequencies via trigonometric method with the results obtained by using numerical methods. The influence of the longitudinal magnetic field on the free vibration response of viscoelastically coupled MNBS is discussed in detail. In addition, numerical results are presented to point out the effects of the nonlocal parameter, internal damping, and parameters of viscoelastic medium on complex natural frequencies of the system. The results demonstrate the efficiency of the suggested methodology to find the closed form solutions for the free vibration response of multiple nanostructure systems under the influence of magnetic field.

On the transient response of viscoelastic beams and plates on viscoelastic medium

In the present study, free vibration of isotropic viscoelastic beams and plates on viscoelastic medium is investigated. The Boltzmann superposition integral model with Dynamic Mechanical Analysis (DMA) results is considered for viscoelastic material of the beam and plate to describe more accurately the behavior of material rather than simple complex constants model. The viscoelastic beam and plate on viscoelastic medium modeling are used in railroad vibration with the damping layer on the lateral surface of the rail and the polymeric flooring on the floor as a viscoelastic medium for the beam and the plate, respectively. Weighted residual method and QZ algorithm are applied to obtain beam and plate natural frequency. In addition, medium properties׳ effects on the natural frequencies of the beam and plate are discussed. For the first time, critical damping of viscoelastic beam is presented due to the greater effect of medium viscosity at all boundary conditions. The unknown coefficients of dynamic response function are calculated by Fourier transform. Closed form expressions are obtained for lateral displacement of the beam and plate with various boundary conditions in terms of complex natural frequency, mode numbers, geometry parameters, initial conditions, material and medium properties. Results show excellent agreement with relative studies.

Transverse vibration of viscoelastic sandwich beam with time-dependent axial tension and axially varying moving velocity

Nonlinearly parametric resonances of axially accelerating moving viscoelastic sandwich beams with time-dependent tension are investigated in this paper. Based on the Kelvin differential constitutive equation, the controlling equation of the transverse vibration of a beam with large deflection is established. The system has been subjected to a time varying velocity and a harmonic axial tension. Here the governing equation of motion contains linear parametric terms and two frequencies, one is the frequency of axially moving velocity and the other one is the frequency of varying tension. The method of multiple scales is applied directly to the governing equation to obtain the complex eigenfunctions and natural frequencies of the system. The elimination of secular terms leads to the steady-state response and amplitude of vibrations. The influence of various parameters such as initial tension on natural frequencies and the amplitude of axial fluctuation, the phase angle between the two frequencies on response curves has been investigated for two different resonance conditions. With the help of numerical results, it has been shown that by using suitable initial tension, the amplitude of axial fluctuation, the phase angle, the vibration of the sandwich beam can be significantly controlled.

Short-Time Matrix Pencil Method for Chipless RFID Detection Applications

In this paper, the concept of short-time matrix pencil method (STMPM) is introduced and experimentally demonstrated as an efficient approach to extract a set of embedded poles in a chipless RFID. Here, by incorporating a few notches on the elliptical dipole antenna as a chipless RFID, the data is encoded as complex natural resonant frequencies of the structure. In this new detection approach, a sliding time window is introduced and the matrix pencil method (MPM) is applied to extract embedded poles and residues from the sliding time-window at each snapshot of time. The resulting complex poles and residues have a time index which allows us to process them as a set of time–frequency data. This time–frequency analysis enables us to discriminate late-time from early-time. Furthermore, averaging complex poles over the time increases the accuracy of the technique in a noisy environment. The measured data agrees well with our simulations and support the STMPM effectiveness in comparison to the MPM.

Extraction of electromagnetic target poles from multiple scattered fields with damped Min-norm method

In this study, the estimation of complex natural resonance frequencies (target poles) by using damped minimum-norm (DMin-norm) method is presented. The method mainly utilizes from the scattered fields belonging to multiple aspect angle/polarization cases of a target in resonance scattering region. By employing DMin-norm algorithm, the proposed method constitutes a function from which target poles can be efficiently extracted as its roots. However, the method in its conventional form should be executed as many times as the number of scattered fields to collect sufficient number of resonance frequencies. Therefore, the method is further improved in terms of computational time to acquire target poles with a single process for multiple scattered data. The described method is applied to a dielectric sphere having high number of target poles and good agreement between estimated and theoretical poles is observed.

Dynamics of a Micro Electrical Mechanical System Subject to Thermoelastic and Squeeze-Film Damping

The static and the dynamic behaviours of a MEMS subjected to thermoelastic and squeeze film effects have been investigated. Major attention is initially devoted to the modeling of this multi-physics system, including mechanical, electrical, thermoelastic and fluid dynamics behaviours, and their couplings. Then, static solutions have been obtained numerically by a finite-difference technique. Finally, an analysis of the vibrations is performed, and the complex natural frequencies are determined. The real part is related to the oscillatory behaviour, while the imaginary part provides the overall damping of the system.

ON BODY CONCEALED WEAPON DETECTION USING A PHASED ANTENNA ARRAY

The detection and identiflcation of metal items and, in particular weapons, of linear size ‚ 10cm, concealed upon the human body, is demonstrated as being entirely feasible by using a phased array of suitably ultra wide band transceivers. The complex natural resonances and especially the fundamental resonance, are excited by ultra wide band, stepped frequency continuous wave illumination of the target, using a phased array of antennae to focus the radiation. Broadband illumination of the target with microwave radiation of suitable frequency range (Typically 0.3{3GHz for handgun sized objects) excites low order complex natural resonances and the late time response of the concealed item can be spatially located using phased array imaging techniques. Further processing of the late time response enables classiflcation of the concealed object, based on the complex natural resonant frequencies of the object, so that threat items such as handguns and knives can be difierentiated from benign items such as mobile phone handsets and cameras.

An analytic formulation for blind modal identification

In this paper a complex-valued formulation of the modal superposition equation is provided and shown to be equivalent to the original, real-valued Blind Modal IDentification (BMID) problem. The complex-valued variant involves the analytic form of the physical and modal responses. The formulation is shown to be more concise and straightforward than the original. It is noted that complex-valued mode shapes can be obtained using a complex version of the two-step Joint Approximate Diagonalization (JAD) algorithm. Using this approach the modal response pairing step of the original BMID method is eliminated. Since the development of the original BMID method, several new, one-step JAD algorithms have been devised. Many of the algorithms can be extended to identify complex mixing matrices. A complex version of the one-step JAD method known as the Weighted Exhaustive Diagonalization with Gauss itErations algorithm is utilized to solve for the complex mode shapes and modal responses. By using this simplified formulation, the whitening step is eliminated, as well as the modal response pairing step, which is necessary in the original BMID algorithm. Performance of the new Complex BMID (CBMID) algorithm is evaluated by application to synthesized data from a three-degrees-of-freedom system with complex modes, application to measured laboratory data on a structural frame and application to measured output-only data from the Heritage Court Tower building. It is seen that the CBMID method results in essentially the same estimates of modal responses, complex mode shapes, natural frequencies and modal damping compared to results from BMID. Furthermore, it is shown that modal parameters from BMID and CBMID are very consistent with those obtained from state-of-the-art methods, such as the Eigensystem Realization Algorithm and the covariance-driven Stochastic Subspace Identification method.

Divergence and Flutter Instability of Damped Laminated Beams Subjected to a Triangular Distribution of Nonconservative Forces

The divergence and flutter instability behavior of the damped laminated beams subjected to a triangular distribution of subtangential forces are investigated based on the finite element model. The formal engineering approach of the mechanics for the laminated beam theory is presented and the extended Hamilton's principle is employed to obtain the mass, damping, elastic stiffness, geometric stiffness matrices, and the load correction stiffness matrix due to the nonconservative forces, respectively. The methods for the evaluation of the divergence and flutter loads of the nonconservative systems are briefly introduced in case of considering and neglecting external and internal damping effects. Throughout numerical examples, the influence of various parameters on the dynamic instability behavior of the laminated beams is newly investigated: 1) the variation of the divergence and flutter loads due to the nonconservativeness with respect to the fiber angle change, 2) the influence the boundary condition of beams on the instability region of the divergence-flutter system, 3) the influence of external and internal damping on the flutter load by analysis of complex natural frequencies.

Embedded Singularity Chipless RFID Tags

Every structure scatters an impulse plane wave in a unique fashion. The structural information of an object can be extracted by analyzing the late-time scattered field as the impulse-response of the structure. The late-time scattered field, which represents the source-free response of the structure, contains a summation of damped sinusoids. The frequency and damping factor of these damped sinusoids are uniquely associated with the structural information, and can be used to identify an unknown object. We propose to create uniquely identifiable scattered fields from an object by incorporating “notches” in the structure giving rise to specific damped sinusoids in the source-free scattered field of the structure. In this manner, data can be embedded into the structure of an object which is detectable using electromagnetic waves, allowing a metallic object to serve as a chipless radio-frequency identification tag (RFID). Data is encoded as complex natural resonant frequencies (referred to as poles) in the structure and is retrieved from the scattered field. Data retrieval is based on Singularity Expansion Method (SEM) analysis using target identification techniques. Each complex-frequency pole provides two-dimensional data (real and imaginary) which can be extracted from the late-time impulse response of the structure using a numerical technique such as the Matrix Pencil Method. We have designed and prototyped a 6-bit (3-pole) tag. The tag is analyzed using simulations and measurements. The tag is successfully read remotely via its scattered fields. The measured data are compared with simulation, and are in close agreement.

Dynamical analysis of axially moving plate by finite difference method

The complex natural frequencies for linear free vibrations and bifurcation and chaos for forced nonlinear vibration of axially moving viscoelastic plate are investigated in this paper. The governing partial differential equation of out-of-plane motion of the plate is derived by Newton’s second law. The finite difference method in spatial field is applied to the differential equation to study the instability due to flutter and divergence. The finite difference method in both spatial and temporal field is used in the analysis of a nonlinear partial differential equation to detect bifurcations and chaos of a nonlinear forced vibration of the system. Numerical results show that, with the increasing axially moving speed, the increasing excitation amplitude, and the decreasing viscosity coefficient, the equilibrium loses its stability and bifurcates into periodic motion, and then the periodic motion becomes chaotic motion by period-doubling bifurcation.

Vibrations and Stability of an Axially Moving Rectangular Composite Plate

The vibrations and stability are investigated for an axially moving rectangular antisymmetric cross-ply composite plate supported on simple supports. The partial differential equations governing the in-plane and out-of-plane displacements are derived by the balance of linear momentum. The natural frequencies for the in-plane and out-of-plane vibrations are calculated by both the Galerkin method and differential quadrature method. It can be found that natural frequencies of the in-plane vibrations are much higher than those in the out-of-plane case, which makes considering out-of-plane vibrations only is reasonable. The instability caused by divergence and flutter is discussed by studying the complex natural frequencies for constant axial moving velocity. For the axially accelerating composite plate, the principal parametric and combination resonances are investigated by the method of multiple scales. The instability regions are discussed in the excitation frequency and excitation amplitude plane. Finally, the axial velocity at which the instability region reaches minimum is detected.

Detection of handguns by their complex natural resonant frequencies

The decomposition of the late time, transient response of metallic objects illuminated with broadband microwave energy into damped sinusoidal signals which are characterised by aspect-independent poles, is used extensively as a method of radar identification of aircraft and missiles. This method is utilised here for the detection and discrimination of handguns and non-threatening items, such as mobile phone handsets, cameras, keys and pens. It is of critical importance to explore the limits of this method, if it is to be utilised for the detection of such objects concealed on the human body or in bags carried on or close to the body that the limits of this technique is explored. Here, investigation is made by computer modelling of the complex natural resonances of two different handguns and these are compared with those for two non-threat items: a mobile phone handset and a fountain pen. Dependence of pole positions on small details present on a handgun is also investigated as is the effect of combining threat and non-threat items in close proximity. A way of enhancing the natural resonances by the use of a cross polarised receiver is also investigated.

Vibrations and stability of axially traveling laminated beams

In this paper, vibrations and stability of an axially traveling laminated composite beam are investigated analytically via the method of multiple scales. Based on classical laminated beam theory, the governing equations of motion for a time-variant axial speed are obtained using Newton’s second law of motion and constitutive relations. The method of multiple scales, an approximate analytical method, is applied directly to the gyroscopic governing equations of motion and complex eigenfunctions and natural frequencies of the system are obtained. The stability boundaries of the system near resonance are determined via the Routh–Hurwitz criterion. Finally, a parametric study is conducted which considers the effects of laminate type and configuration as well as the mean speed and amplitude of speed fluctuations on the vibration response, natural frequencies and stability boundaries of the system.