Assumption Of Harmonic Motion Research Articles

Free vibration of a functionally graded annular sector plate integrated with piezoelectric layers

Based on the first order shear deformation theory, free vibration behavior of functionally graded (FG) annular sector plates integrated with piezoelectric layers is investigated. The distribution of electric potential along the thickness direction of piezoelectric layers which is assumed to be a combination of linear and sinusoidal functions, satisfies both open and closed circuit electrical boundary conditions. Through a reformulation of governing equations and harmonic motion assumption, a novel decoupling method is suggested to transform the six second order coupled partial differential equations of motion into two eighth order and fourth order equations. A Fourier series method is then employed to present analytical solutions for free vibration of smart FG annular sector plates with simply supported radial edges and arbitrarily supported circular edges. The results, which can be used as a benchmark and suitable for design purposes, are verified with those reported in the literature. Finally, by presenting extensive ranges of frequencies, the effects of geometric parameters, power law index, FG and piezoelectric materials, electrical and mechanical boundary conditions as well as the piezoelectric layer thickness on vibration response of smart annular sector plates are discussed in detail.

Vibration characteristics of cylindrical shells filled with fluid based on first-order shell theory

In this article, an analytical method to study the natural frequencies of free vibrations for a thick cylindrical shell filled with fluid is proposed. Mindlin’s first-order shell theory is extended to derive the equations of motion and corresponding boundary conditions by Hamilton’s principle. Linearized potential flow theory is used to derive the hydrodynamic force. Moreover, the internal fluid pressure acting on the shell wall is obtained by the assumption of a non-penetration condition. The dispersion equations are obtained under the assumption of harmonic motion. The derived shell theory is used to calculate the natural frequencies of cylindrical shells with various thicknesses and lengths, and the results are compared with Flugge’s shell theory and finite-element method (FEM). As a result, the proposed shell theory shows improved accuracy and good agreement with published experimental results.

A dynamic homogenization model for long-wavelength wave propagation in corrugated sandwich plates

In the present work, a new dynamic homogenization model is developed to investigate the long-wavelength wave propagation in a corrugated sandwich plate. With the harmonic motion assumption and using a shifting operator, the governing equations of the plate are firstly represented in a state-space form. Then, a dynamic homogenization model is developed via the two-scale homogenization method. Based on this model and considering the propagation of sinusoidal waves, the dispersion relations and corresponding wave modes can be easily obtained. In order to validate the developed homogenization model, the obtained dispersion relations are compared with those predicted by the spectral element method. It is found that the present method gives accurate results in low frequency range. Furthermore, the effects of some geometric and material parameters on the dispersion relations for the corrugated sandwich plate are also discussed. The developed homogenization model is expected to be helpful in the prediction and control of dynamic responses of corrugated or even lattice sandwich structures.

An analytical approach for nonlinear vibration and thermal stability of shape memory alloy hybrid laminated composite beams

Abstract In this article, large amplitude vibration and thermal post-buckling of shape memory alloy (SMA) fiber reinforced hybrid composite beams with symmetric and asymmetric lay-up are analytically investigated. To predict the behavior of the smart laminated beam, the Euler–Bernoulli beam theory and the nonlinear von-Karman strain field are employed. Also, one-dimensional Brinson SMA model is utilized to calculate the recovery stress of SMA fibers in the case of restrained strain. Nonlinear governing equations of motion are derived via the Hamilton principle. Using an analytical approach based on the Galerkin procedure together with the simple harmonic motion assumption, a closed-form solution is obtained for the thermal post-buckling and nonlinear free vibration analysis of SMA fiber reinforced hybrid composite beams. Due to lack of any results on the free vibration and thermal stability of SMA fiber reinforced composite beams, the results obtained from the present solution for laminated composite beams without SMA fiber are compared with counterpart data in the open literature, which validate the present solution. Then, a set of parametric study is carried out to show the influence of SMA volume fraction, amount of prestrain in the SMA fiber, orientation of composite fiber, SMA-reinforced layer thickness to total thickness ratio, location of SMA layer, vibration amplitude, boundary conditions and temperature on the vibration characteristic of the laminated beam reinforced with SMA in the pre- and post-buckled domains.

Parametric study on supersonic flutter of angle-ply laminated plates using shear deformable finite element method

The influence of fiber orientation, flow yaw angle and length-to-thickness ratio on flutter characteristics of angle-ply laminated plates in supersonic flow is studied by finite element approach. The structural model is established using the Reissner-Mindlin theory in which the transverse shear deformation is considered. The aerodynamic pressure is evaluated by the quasi-steady first-order piston theory. The equations of motion are formulated based on the principle of virtual work. With the harmonic motion assumption, the flutter boundary is determined by solving a series of complex eigenvalue problems. Numerical study shows that (1) The flutter dynamic pressure and the coalescence of flutter modes depend on fiber orientation, flow yaw angle and length-to-thickness ratio; (2) The laminated plate with all fibers aligned with the flow direction gives the highest flutter dynamic pressure, but a slight yawing of the flow from the fiber orientation results in a sharp decrease of the flutter dynamic pressure; (3) The angle-ply laminated plate with fiber orientation angle equal to flow yaw angle gives high flutter dynamic pressure, but not the maximum flutter dynamic pressure; (4) With the decrease of length-to-thickness ratio, an adverse effect due to mode transition on the flutter dynamic pressure is found.

A simple approximate method for free vibration analysis of framed tube structures

SUMMARYA simple approximate method is developed for determining natural frequency of tall buildings. Timoshenko's beam model, which considers the influence of shear and flexural deformation, was used in modelling framed tube structures. In this paper, natural frequency and mode shape of framed tube structures were calculated based on the flexural and shear rigidities along with the effects of rotational inertia. Dynamic model of Timoshenko's beam can be obtained by writing equilibrium equations of forces acting on an infinite element. The solution of the dynamic model was obtained by first applying the separation of variables to time and space, followed by the assumption of harmonic motion in time, the steady state eigen system was obtained. Natural frequencies of framed tube structures were calculated by solving the eigenproblem. A numerical example has been presented to demonstrate the ease of application and accuracy of the proposed method. The structure's fundamental frequency was computed using ETABS V9.0.0 (Computer and Structures, Berkeley, California, USA) and compared with the result obtained from proposed method, which shows that the percentage of error was low and acceptable. The proposed method can be adopted as an alternative procedure to evaluate the natural frequency of framed tube in the preliminary stages of structural design. Copyright © 2010 John Wiley & Sons, Ltd.

Large amplitude vibration analysis of composite beams: Simple closed-form solutions

Large amplitude vibration analysis of laminated composite beam with axially immovable ends is investigated with symmetric and asymmetric layup orientations by using the Rayleigh–Ritz (R–R) method. The displacement fields used in the analytical formulation are coupled by using the homogeneous governing static axial equilibrium equation of the beam. Geometric nonlinearity of von-Karman type is considered which accounts for the membrane stretching action of the beam. The simple closed-form solutions are presented for the nonlinear harmonic radian frequency as function of central amplitude of the beam using the R–R method. The nonlinear harmonic radian frequency results obtained from the closed-form solutions of the R–R method in general show good agreement with the results obtained from simple iterative finite element formulation. Furthermore, the closed-form expressions are corrected for the harmonic motion assumption from the available literature results on the existence of quadratic and cubic nonlinearity. It is interesting to note that the composite beams can result in asymmetric frequency vs. amplitude curves depending upon the nature of direction of displacement in contrast to isotropic beams which exhibit cubic nonlinearity only and leads to symmetric frequency vs. amplitude curves with respect to sign of the amplitude.

Estimating Damping Parameters in Multi-Degree-of-Freedom Vibration Systems by Balancing Energy

A method of estimating damping parameters for multi-degree-of-freedom vibration systems is outlined, involving a balance of dissipated and supplied energies over a cycle of periodic vibration. The power is formulated as the inner product between the velocity and force terms and integrated over a cycle. Conservative terms (mass and stiffness) drop out of the formulation. The displacement response and the input are measured, and the damping coefficients are estimated without knowledge of the mass and stiffness, which can be nonlinear, as illustrated in one example. The identification equations are also obtained with a modal reduction based on proper orthogonal decomposition. The method can be applied with a harmonic motion assumption or by simple numerical integration. The method is illustrated with linear and quadratic damping, in simulations of a four degree-of-freedom system, and in a string with and without noise.

Natural frequencies and critical velocities of fixed-fixed laminated circular cylindrical shells conveying fluids

The natural frequencies and critical velocities of laminated circular cylindrical shells with fixed-fixed ends conveying fluids are studied. Equations of motion are derived by the Hamilton principle under the scope of the Mindlin-type first-order transverse shear deformable cylindrical shell theory. Fluid pressure acting on the wall is obtained through the nonpenetration condition and the assumption of ideal flow. Dynamic characteristic equations are then obtained under the assumption of harmonic motion. Using linear superposition, the natural frequencies corresponding to each flow velocity are found by satisfying dynamic characteristic equation and boundary conditions. Critical velocities are those where the natural frequencies vanish, wherein the static divergence, i.e. buckling, occurs. Numerical examples are presented, in which the parameter studies include stacking angle, length-to-thickness and radius-to-thickness ratios.

Travelling Wave Ultrasonic Motors, Part II: A Numerical Method For The Flexural Vibrations Of The Stator

A finite difference scheme is used to calculate the forced transverse vibrations of thin annular plates with radially varying thickness. Under the assumption of harmonic motion, the governing partial differential equation reduces to a system of ordinary differential equations with variable coefficients. The numerical scheme that solves this system for various boundary conditions is successfully tested vs. known analytical solutions, and it is subsequently used to compute the forced vibrations of plates with more complicated thickness variations. The numerical results for these plates are compared to experimental results and found to be in good agreement. The effects of the rotatory inertias are studied and found to be important at high frequencies.

On the derivation of dynamic information from diffraction data

Publisher Summary This chapter discusses the significance of the anisotropic thermal parameters. The thermal motion of atoms in crystals is described in terms of the normal modes of vibration of the entire crystal, but this formally correct and comprehensive approach cannot usually be followed in crystal structure analysis because it is too cumbersome and too difficult to apply to any but the simplest systems. Instead, the enormously simplified statistical model is used, in which the atoms in the structure are assumed to move independently, that is, with no coupling between them. It is noted that this approximation does not impair the model for the intensities of the Bragg reflections although it prevents any sensible analysis of the non-Bragg, thermal diffuse scattering). An alternative approach to the derivation of a theoretical model for the motions of a macromolecule is to attempt to calculate the normal modes of vibration for a complete molecule by consideration of the restoring force on each atom. This represents a fearsome computational problem and returns to the simplifying assumption of harmonic motion, but there is some hope that the major displacements of the polypeptide backbone may be accounted for by a small number of the lowest frequency modes.

Class of shockfree airfoils producing the same surface pressure

Writing the loads in this fashion emphasizes that it is not the airfoil section pitch and heave motions (a and h) that must be identified, but rather the mean and linear components of the normal velocity distribution over the airfoil chord. Observe that the a terms actually arise from two sources: from the time derivative of (h + Ua) as well as from the pitch rate. It is worthwhile noting that in this form the result is applicable as well to the case of time-varying freestream velocity £7 (see Ref. 2), except that the Theodorsen lift deficiency function must be generalized to account for the expansion and compression of the shed wake vorticity. The result is also applicable for arbitrary motion, again except for the lift deficiency function; indeed, the derivation of Ref. 1 does not find it necessary to introduce the assumption of harmonic motion until the final step of evaluating C( k) [Eq. (5-308) of Ref. 1]. The most common approach in helicopter analyses has been to identify h as the normal velocity up at the rotor blades, and a as the pitch rate 0 + Q/3 (or sometimes even just 0). The numerical results for the lift will not be greatly influenced by this error, since the unsteady lift is small compared to the steady component. The steady moment component is normally small however, so the calculation of the moment will be seriously affected if the unsteady terms are incorrect. Another consequence of the incorrect identification of h and a, important for the lift as well as for the moment, is that the equivalence of flapping and feathering motion for an articulated rotor blade will be violated. As an example, consider the rigid flap and rigid pitch motion of an articulated rotor blade in forward flight. Unsteady airfoil theory requires (h + Ua), which is the air velocity normal to the airfoil section, at the pitch axis; and d, which is the equivalent pitch rate or camber of the airfoil. Hence for the present case h + Ua = u T6 - u p = ( Qr + QRfjLsin\l/ ) d