Dynamic model of micropolar elastic thin plates with independent fields of displacements and rotations

A general model of dynamic bending of isotropic micropolar elastic thin plates with independent fields of displacements and rotations is presented. The model has been justified asymptotically based on the solutions for special cases subject to simplifying assumptions. The model incorporates transverse shear deformations. Neglecting transverse shear, a model of the dynamics of micropolar elastic thin plates is also constructed. Then, we study free and forced oscillations and derive the natural frequencies, the amplitudes of the forced oscillations and the resonance conditions for micropolar elastic hinge-supported rectangular and circular plates. Finally, the basic characteristic features of micropolar plates are numerically analysed for different values of various elastic and inertial constants of the micropolar material.     

Model of micropolar thin shell oscillations

The paper gives the equations of a general applied two-dimensional theory for the dynamics of micropolar elastic thin shells with independent fields of motion and rotation that completely take into account all rotation-shift and related deformations. Problems on free and forced oscillations of micropolar elastic shells are studied on the basis of this general theory. Special features for the dynamic behavior of shells made of a micropolar elastic material are revealed on the basis of numerical analysis.     

General theory of micropolar elastic thin shells

The paper formulates general hypotheses of micropolar elastic thin shells that are given asymptotic validation. Using these hypotheses and three-dimensional Cosserat (micropolar, asymmetric) theory of elasticity, general two-dimensional applied models of micropolar elastic thin shells with independent displacement and rotation fields, constrained rotation and low shear rigidity are constructed to suit dimensionless physical parameters of the shell material. The constructed micropolar shell models take into complete account transverse shear strain and related strain. Models of micropolar elastic thin plates and beams are particular cases of the constructed micropolar shell models. An axially symmetric stress-strain state problem of a hinged cylindrical micropolar shell is considered. Numerical analysis is used to demonstrate effective strength and rigidity characteristics of micropolar elastic shells.     

Nonlinear vibration of micromachined asymmetric resonators

In this paper, the nonlinear dynamics of a beam-type resonant structure due to stretching of the beam is addressed. The resonant beam is excited by attached electrostatic comb-drive actuators. This structure is modeled as a thin beam-lumped mass system, in which an initial axial force is exerted to the beam. This axial force may have different origins, e.g., residual stress due to micro-machining. The governing equations of motion are derived using the mode summation method, generalized orthogonality condition, and multiple scales method for both free and forced vibrations. The effects of the initial axial force, modal damping of the beam, the location, mass, and rotary inertia of the lumped mass on the free and forced vibration of the resonator are investigated. For the case of the forced vibration, the primary resonance of the first mode is investigated. It has been shown that there are certain combinations of the model parameters depicting a remarkable dynamic behavior, in which the second to first resonance frequencies ratio is close to three. These particular cases result in the internal resonance between the first and second modes. This phenomenon is investigated in detail.     

Non-linear vibrations of three-layer beams with viscoelastic cores,II: Experiment

Experimental results are presented for large amplitude, forced motion of damped, three-layer beams. The beams are constructed with a viscoelastic material constrained between stiff, elastic, outer layers. The sandwich beam is axially restrained; therefore large amplitude displacements cause non-linear response. When the beam is forced at one-half of the lateral vibration resonant frequency, superharmonic response occurs. The experiment is briefly described and frequency response characteristics, spatial shapes and a measure of superharmonic response are presented. The results are compared with predictions from a previously developed theory.     

Coupled bending and torsional vibration of a beam with in-span and tip attachments

In this paper, the coupled flexural-torsional free and forced vibrations of a beam with tip and/or in-span attachments are studied. First, a mathematical model is established, which consists of a beam with several tip attachments, i.e, a tip mass of non-negligible dimensions, a linear spring grounding the tip mass, and a torsional spring connected at the end of the beam. The modal functions of this model and the orthogonality condition among them are derived. For the purpose of verification the properties of the tip attachments are changed, and the numerical results obtained are compared with those given in the relevant literature. Effects of tip mass and distributed mass in-span on natural frequencies and modes are investigated for two cantilever beams with different cross sections. An application of the orthogonality condition in the case of a beam with tip mass is also presented for a forced vibration example.     

Free vibration analysis of elastically connected multiple-beams by using the Adomian modified decomposition method

The Adomian modified decomposition method (AMDM) is employed in this paper to investigate the free vibrations of N elastically connected parallel Euler–Bernoulli beams, which are continuously joined by a Winkler-type elastic layer. The proposed AMDM method can be used to analyze the vibration of beam system consisting of an arbitrary number of beams. By using boundary conditions the natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The numerical results for different boundary conditions, beam numbers and the stiffness of the Winkler-type elastic layer are presented. It is shown that the AMDM offers an accurate and effective method of free vibration analysis of multiple-connected beams with arbitrary boundary conditions.     

Forced transverse vibrations of an elastically connected complex simply supported double-beam system

The present paper is devoted to analyzing undamped forced transverse vibrations of an elastically connected complex double-beam system. The problem is formulated and solved in the case of simply supported beams. The classical modal expansion method is applied to ascertain dynamic responses of beams due to arbitrarily distributed continuous loads. Several cases of particularly interesting excitation loadings are investigated. The action of stationary harmonic loads and moving forces is considered. In discussing vibrations caused by exciting harmonic forces, conditions of resonance and dynamic vibration absorption are determined. The beam-type dynamic absorber is a new concept of a continuous dynamic vibration absorber (CDVA), which can be applied to suppress excessive vibrations of corresponding beam systems. A numerical example is presented to illustrate the theoretical analysis.     

IMPROVEMENT OF THE SEMI-ANALYTICAL METHOD, FOR DETERMINING THE GEOMETRICALLY NON-LINEAR RESPONSE OF THIN STRAIGHT STRUCTURES: PART II—FIRST AND SECOND NON-LINEAR MODE SHAPES OF FULLY CLAMPED RECTANGULAR PLATES

In a previous series of papers, a semi-analytical model based on Hamilton's principle and spectral analysis has been developed for geometrically non-linear free vibrations occurring at large displacement amplitudes of clamped-clamped beams and fully clamped rectangular homogeneous and composite plates. In Part I of this series of papers, concerned with geometrically non-linear free and forced vibrations of various beams, a practical simple “multi-mode theory”, based on the linearization of the non-linear algebraic equations, written in the modal basis, in the neighbourhood of each resonance has been developed. Simple explicit formulae, ready and easy to use for analytical or engineering purposes have been derived, which allows direct calculation of the basic function contributions to the first three non-linear mode shapes of the beams considered. Also, various possible truncations of the series expansion defining the first non-linear mode shape have been considered and compared with the complete solution, which showed that an increasing number of basic functions has to be used, corresponding to increasingly sized intervals of vibration amplitudes; starting from use of only one function, i.e., the first linear mode shape, corresponding to very small amplitudes, for which the linear theory is still valid, and ending by the complete series, involving six functions, corresponding to maximum vibration amplitudes at the beam middle point up to once the beam thickness. For higher amplitudes, a complementary second formulation has been developed, leading to reproduction of the known results via the solution of reduced linear systems of five equations and five unknowns. The purpose of this paper is to extend and adapt the approach described above to the geometrically non-linear free vibration of fully clamped rectangular plates in order to allow direct and easy calculation of the first, second and higher non-linear fully clamped rectangular plate mode shapes, with their associated non-linear frequencies and non-linear bending stress patterns. Also, numerical results corresponding to the first and second non-linear modes shapes of fully clamped rectangular plates with an aspect ratio α=0·6 are presented. Data concerning the higher non-linear modes, the aspect ratio effect, and the forced vibration case will be presented later.     

General dynamic theory of micropolar elastic thin plates with free rotation and special features of their natural oscillations

A general applied two-dimensional theory for the dynamics of micropolar elastic thin plates that takes into account rotationally shear and related deformations is developed on the basis of the method of hypotheses with asymptotic confirmation. A problem of natural oscillations of micropolar elastic thin rectangular plates is solved using it. Special features of the dynamic characteristics of elastic thin plates made of a micropolar elastic material are revealed within the framework of numerical analysis.     

Evaluation of elastic modulus of cantilever beam by TA-ESPI

The paper proposes an evaluation technique for the elastic modulus of a cantilever beam by vibration analysis based on time average electronic speckle pattern interferometry (TA-ESPI) and Euler-Bernoulli equation. General approaches for the measurement of elastic modulus of a thin film are the Nano indentation test, Buldge test, Micro-tensile test, and so on. They each have strength and weakness in the preparation of the test specimen and the analysis of experimental results. ESPI is a type of laser speckle interferometry technique offering non-contact, high-resolution and whole-field measurement. The technique is a common measurement method for vibration mode visualization and surface displacement. Whole-field vibration mode shape (surface displacement distribution) at resonance frequency can be visualized by ESPI. And the maximum surface displacement distribution from ESPI can be used to find the resonance frequency for each vibration mode shape. And the elastic modules of a test material can be easily estimated from the measured resonance frequency and Euler-Bernoulli equation. The TA-ESPI vibration analysis technique can be used to find the elastic modulus of a material requiring simple preparation process and analysis.     

Determination of elastic modulus of a beam by using electronic speckle pattern interferometry

This paper proposes a sonic resonance test for an elastic modulus measurement which is based on the electronic speckle pattern interferometry. Previous measurement technique of elastic constant has the limitation of application for thin film or polymer material because contact to specimen affects the result. It has been developed as a non-contact optical measurement technique which can visualize resonance vibration mode shapes with whole field. The maximum vibration amplitude at each vibration mode shape is a clue to find the resonance frequencies. The dynamic elastic constant of test material can be easily determined from vibration of a beam equation using the measured resonance frequencies. The proposed technique is able to give high accurate elastic modulus of materials through a simple experimental set-up and analysis. The experimental result also compared to the theoretical result.     

Nonlinear finite amplitude vibrations of sharp-edged beams in viscous fluids

In this paper, we study flexural vibrations of a cantilever beam with thin rectangular cross section submerged in a quiescent viscous fluid and undergoing oscillations whose amplitude is comparable with its width. The structure is modeled using Euler–Bernoulli beam theory and the distributed hydrodynamic loading is described by a single complex-valued hydrodynamic function which accounts for added mass and fluid damping experienced by the structure. We perform a parametric 2D computational fluid dynamics analysis of an oscillating rigid lamina, representative of a generic beam cross section, to understand the dependence of the hydrodynamic function on the governing flow parameters. We find that increasing the frequency and amplitude of the vibration elicits vortex shedding and convection phenomena which are, in turn, responsible for nonlinear hydrodynamic damping. We establish a manageable nonlinear correction to the classical hydrodynamic function developed for small amplitude vibration and we derive a computationally efficient reduced order modal model for the beam nonlinear oscillations. Numerical and theoretical results are validated by comparison with ad hoc designed experiments on tapered beams and multimodal vibrations and with data available in the literature. Findings from this work are expected to find applications in the design of slender structures of interest in marine applications, such as biomimetic propulsion systems and energy harvesting devices.     

Vibration characteristic analysis of buried pipes using the wave propagation approach

A theoretical analysis for the free vibration of simply supported buried pipes has been investigated using the wave propagation approach. The pipe modeled as a thin cylindrical shell of linear homogeneous isotropic elastic material buried in a linear isotropic homogeneous elastic medium of infinite extent. The vibrations of the pipe are examined by using Flüggle shell equation. The natural frequencies are obtained for the pipes surrounded by vacuo or elastic medium. The results are compared with those available in the literature and agreement is found with them. It is found that the free vibration frequency of the pipe does not appear for some of the axial or circular vibration modes and the real natural frequencies of the pipe are significantly dependent on the rigidity of the surrounding medium.     

Free vibrations of a laminated beam by a microstructure theory

The free vibrations of a laminated beam are considered within the framework of a theory that models the composite beam as a macrohomogeneous beam with microstructure. The beams are assumed to consist of several parallel alternating layers of two homogeneous, isotropic elastic materials. The system of three coupled partial differential equations is solved exactly, and attention is devoted to the determination of natural frequencies of vibration of laminated beams with (i) hinged-hinged ends and (ii) clamped-clamped ends. For the sake of comparison, the same boundary value problems are also solved within the framework of the so-called effective modulus theory, which treats the composite as a transversely isotropic and “fictitiously” homogeneous Timoshenko beam, with effective moduli and density. For relatively long beams, i.e., in the low frequency range, the natural frequencies obtained from the two theories are in excellent agreement, but as the depth-to-length ratio, ζ, increases the microstructure frequencies are observed to be much lower than the effective modulus frequencies, the magnitude of the effect becoming more pronounced with increasing mode number n.     

Mechanical detection of carbon nanotube resonator vibrations

Bending-mode vibrations of carbon nanotube resonators were mechanically detected in air at atmospheric pressure by means of a novel scanning force microscopy method. The fundamental and higher order bending eigenmodes were imaged at up to 3.1 GHz with subnanometer resolution in vibration amplitude. The resonance frequency and the eigenmode shape of multiwall nanotubes are consistent with the elastic beam theory for a doubly clamped beam. For single-wall nanotubes, however, resonance frequencies are significantly shifted, which is attributed to fabrication generating, for example, slack. The effect of slack is studied by pulling down the tube with the tip, which drastically reduces the resonance frequency.     

An analogy between free vibration of a plate and of a particle of mass

In this paper, the free flexural vibration of an elastic circular thin plate with an initial imperfection is investigated. Approximate solution of the problem for the fundamental frequency of vibration, of large amplitude and with the plate imperfection, leads to a non-linear ordinary differential equation with time as the independent variable. It is shown that this equation also represents the free vibration of a particle of mass on a shallow curve of fourth degree.. With this similarity in view, it is possible to draw an analogy between these two vibrations. A numerical analysis is made with particular reference to this analogy and the results are given in various figures which represent the vibratory motion and the period of vibration versus the initial amplitude of the plate or of the particle of mass.     

Free vibration analysis of a rotating hub–functionally graded material beam system with the dynamic stiffening effect

A comprehensive dynamic model of a rotating hub–functionally graded material (FGM) beam system is developed based on a rigid–flexible coupled dynamics theory to study its free vibration characteristics. The rigid–flexible coupled dynamic equations of the system are derived using the method of assumed modes and Lagrange's equations of the second kind. The dynamic stiffening effect of the rotating hub–FGM beam system is captured by a second-order coupling term that represents longitudinal shrinking of the beam caused by the transverse displacement. The natural frequencies and mode shapes of the system with the chordwise bending and stretching (B–S) coupling effect are calculated and compared with those with the coupling effect neglected. When the B–S coupling effect is included, interesting frequency veering and mode shift phenomena are observed. A two-mode model is introduced to accurately predict the most obvious frequency veering behavior between two adjacent modes associated with a chordwise bending and a stretching mode. The critical veering angular velocities of the FGM beam that are analytically determined from the two-mode model are in excellent agreement with those from the comprehensive dynamic model. The effects of material inhomogeneity and graded properties of FGM beams on their dynamic characteristics are investigated. The comprehensive dynamic model developed here can be used in graded material design of FGM beams for achieving specified dynamic characteristics.     

ASYMPTOTIC APPROACH FOR NON-LINEAR PERIODICAL VIBRATIONS OF CONTINUOUS STRUCTURES

An asymptotic approach for determining periodic solutions of non-linear vibration problems of continuous structures (such as rods, beams, plates, etc.) is proposed. Starting with the well-known perturbation technique, the independent displacement and frequency is expanded in a power series of a natural small parameter. It leads to infinite systems of interconnected non-linear algebraic equations governing the relationships between modes, amplitudes and frequencies. A non-trivial asymptotic technique, based on the introduction of an artificial small parameter is used to solve the equations. An advantage of the procedure is the possibility to take into account a number of vibration modes. As examples, free longitudinal vibrations of a rod and lateral vibrations of a beam under cubically non-linear restoring force are considered. Resonance interactions between different modes are investigated and asymptotic formulae for corresponding backbone curves are derived.