Theoretical modeling on the combination resonance of size-dependent microbeams

2021 ◽  
pp. 107754632110531
Author(s):  
Zhenkun Li ◽  
Qiyou Cheng ◽  
Zhizhuang Feng ◽  
Longtao Xing ◽  
Yuming He
Keyword(s):  
Boundary Conditions ◽  
Equations Of Motion ◽  
Deep Understanding ◽  
Perturbation Approach ◽  
Combination Resonance ◽  
Response Curves ◽  
Size Dependent ◽  
Time Histories ◽  
Different Sources ◽  
Special Case

The combination resonance of size-dependent microbeams is investigated. Two harmonic forces act on the microbeam, and combination resonance is observed while the excitation frequencies differ from the resonant frequency. Microbeams with two different sources of nonlinearities including three kinds of boundary conditions, clamped-free (nonlinearity comes from large curvature and nonlinear inertial), clamped-clamped, and hinged-hinged (nonlinearity originates from mid-plane stretching-bending coupling), are taken into consideration to have a deep understanding of this phenomenon. A traveling load acting on the microbeam is presented as a special case of combination resonance. The modal discretization technique is applied to discretize the equations of motion, and then the Lindstedt–Poincare method, a perturbation approach, is employed to solve the resultant equations. The conditions for combination resonance are presented, and frequency-response curves and time histories at the resonance point are obtained for microbeams of each boundary condition. Results reveal that different sources of nonlinearities result in different performances of combination resonance. The free vibration part constitutes a large percentage of the final response. Furthermore, the situation of coexistence of combination resonance and superharmonic (or subharmonic) resonance is determined. The special case demonstrates a higher amplitude than the common combination resonance for all the boundary conditions. Parametric studies are then carried out to discuss the effects of the length scale parameter, excitation force as well as its position, and damping on the performance of the microbeam.

scholarly journals Size-Dependent Free Vibration of Axially Moving Nanobeams Using Eringen’s Two-Phase Integral Model

2018 ◽  
Vol 8 (12) ◽  
pp. 2552 ◽  
Author(s):  
Yuanbin Wang ◽  
Zhimei Lou ◽  
Kai Huang ◽  
Xiaowu Zhu
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequency ◽  
Critical Velocity ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Integral Model ◽  
Two Phase ◽  
Size Dependent ◽  
Axially Moving

In this paper, vibration of axially moving nanobeams is studied using Eringen’s two-phase nonlocal integral model. Geometric nonlinearity is taken into account for the integral model for the first time. Equations of motion for the beam with simply supported and fixed–fixed boundary conditions are obtained by Hamilton’s Principle, which turns out to be nonlinear integro-differential equations. For the free vibration of the nanobeam, the critical velocity and the natural frequencies are obtained numerically. Furthermore, the effects of parameters on critical velocity and natural frequency are analyzed. We have found that, for the two-phase nonlocal integral model, regardless of the boundary conditions considered, both the critical velocity and the natural frequency increase with the nonlocal parameter and the geometric parameter.

scholarly journals On the Secondary Resonance of a Spinning Disk Under Space-Fixed Excitations

2004 ◽  
Vol 126 (3) ◽  
pp. 422-429 ◽  
Author(s):  
Jen-San Chen ◽  
Chin-Yi Hua ◽  
Chia-Min Sun
Keyword(s):  
Free Oscillation ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Multiple Scale ◽  
Combination Resonance ◽  
Secondary Resonance ◽  
Spinning Disk ◽  
Final Response ◽  
Steady State Solutions ◽  
Special Case

We investigate the possibility of secondary resonance of a spinning disk under space-fixed excitations. Von Karman’s plate model is employed in formulating the equations of motion of the spinning disk. Galerkin’s procedure is used to discretize the equations of motion, and the multiple scale method is used to predict the steady state solutions. Attention is focused on the nonlinear coupling between a pair of forward (with frequency ωmn¯) and backward (with frequency ωmn) traveling waves. It is found that combination resonance may occur when the excitation frequency is close to 2ωmn+ωmn¯,ωmn+2ωmn¯, or 1/2ωmn¯+ωmn. When the combination resonance does occur, the frequencies of the free oscillation components are shifted slightly from the respective natural frequencies ωmn¯ and ωmn. The final response is therefore quasiperiodic. However, in the case when the excitation frequency is close to 1/2ωmn¯−ωmn, no combination resonance is possible. In the case when the excitation frequency is close to 1/3ωmn and 1/2ωmn¯−ωmn simultaneously, internal resonance between the forward and backward modes can occur. The frequencies of the free oscillation components are exactly three times and five times that of the excitation frequency. In this special case both saddle-node and Hopf bifurcations are observed.

Powered-Flight Trajectories of Rockets Under Constant Transverse Thrust

1963 ◽  
Vol 30 (4) ◽  
pp. 559-561 ◽  
Author(s):  
Chong-Hung Zee
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Series Expansion ◽  
Practical Problem ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Variable Time ◽  
Truncation Errors ◽  
Independent Variable ◽  
Special Case

The second-order nonlinear differential equations of motion in the case of a rocket in drag-free powered-flight under constant transverse (normal to the focal radius) thrust are solved by series expansion developed to the seventh power of the independent variable “time.” The coefficients of the powers of time are in terms of given boundary conditions. The truncation errors of the series are estimated, hence the accuracy for any practical problem based on the analysis presented in this paper could be well established. The case of constant transverse thrust acceleration, which may be conceived as a special case of the present analysis, is also solved.

Experimental and analytical study on the superharmonic resonance of size-dependent cantilever microbeams

2019 ◽  
Vol 25 (21-22) ◽  
pp. 2733-2748
Author(s):  
Zhenkun Li ◽  
Yuming He ◽  
Jian Lei ◽  
Song Guo ◽  
Dabiao Liu
Keyword(s):  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Variation Principle ◽  
Response Curves ◽  
Stress Theory ◽  
Superharmonic Resonance ◽  
Size Dependent ◽  
Time Histories ◽  
Modified Couple Stress ◽  
Good Agreement

The superharmonic resonance (SR) of size-dependent cantilever microbeams is investigated experimentally and analytically. Nickel cantilever microbeams are employed with concentrated harmonic force on the tip. The SR of order two, three, four, five, and six are observed. The frequency–response curves (FRCs) near the SR frequencies as well as the time histories are obtained. The FRCs indicate that the superharmonic resonant frequencies are different from the classical situation. Furthermore, a nonlinear model within the framework of modified couple stress theory is derived to interpret the observations by aid of Hamilton's variation principle. The resulting partial differential equation of motion is discretized into a series of ordinary differential equations (ODEs) by a two-mode Galerkin scheme. The ODEs are then solved analytically with the multi-dimensional Lindstedt–Poincaré method. Analytical results are in good agreement with experimental results in SR of order three. The effects of different nonlinear terms, length scale parameter, and damping coefficients on the nonlinear system are then discussed.

Model of vortex flow of charge in a vessel with turbine impeller

1987 ◽  
Vol 52 (8) ◽  
pp. 1888-1904
Author(s):  
Miloslav Hošťálek ◽  
Ivan Fořt
Keyword(s):  
Boundary Conditions ◽  
Equations Of Motion ◽  
Theoretical Data ◽  
Eddy Diffusion ◽  
Quantitative Agreement ◽  
Creeping Flow ◽  
Model Parameters ◽  
Mean Flow ◽  
Two Dimensional Flow ◽  
Vorticity Transport

A theoretical model is described of the mean two-dimensional flow of homogeneous charge in a flat-bottomed cylindrical tank with radial baffles and six-blade turbine disc impeller. The model starts from the concept of vorticity transport in the bulk of vortex liquid flow through the mechanism of eddy diffusion characterized by a constant value of turbulent (eddy) viscosity. The result of solution of the equation which is analogous to the Stokes simplification of equations of motion for creeping flow is the description of field of the stream function and of the axial and radial velocity components of mean flow in the whole charge. The results of modelling are compared with the experimental and theoretical data published by different authors, a good qualitative and quantitative agreement being stated. Advantage of the model proposed is a very simple schematization of the system volume necessary to introduce the boundary conditions (only the parts above the impeller plane of symmetry and below it are distinguished), the explicit character of the model with respect to the model parameters (model lucidity, low demands on the capacity of computer), and, in the end, the possibility to modify the given model by changing boundary conditions even for another agitating set-up with radially-axial character of flow.

scholarly journals ON CANONICAL QUANTIZATION OF THE GAUGED WZW MODEL WITH PERMUTATION BRANES

2011 ◽  
Vol 26 (26) ◽  
pp. 4647-4660
Author(s):  
GOR SARKISSIAN
Keyword(s):  
Riemann Surface ◽  
Phase Space ◽  
Boundary Conditions ◽  
Canonical Quantization ◽  
Simons Theory ◽  
Gauge Fields ◽  
Time Line ◽  
Chern Simons ◽  
Special Case ◽  
Chern Simons Theory

In this paper we perform canonical quantization of the product of the gauged WZW models on a strip with boundary conditions specified by permutation branes. We show that the phase space of the N-fold product of the gauged WZW model G/H on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of the double Chern–Simons theory on a sphere with N holes times the time-line with G and H gauge fields both coupled to two Wilson lines. For the special case of the topological coset G/G we arrive at the conclusion that the phase space of the N-fold product of the topological coset G/G on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of Chern–Simons theory on a Riemann surface of the genus N-1 times the time-line with four Wilson lines.

scholarly journals Actions, topological terms and boundaries in first-order gravity: A review

2016 ◽  
Vol 25 (04) ◽  
pp. 1630011 ◽  
Author(s):  
Alejandro Corichi ◽  
Irais Rubalcava-García ◽  
Tatjana Vukašinac
Keyword(s):  
Boundary Conditions ◽  
Symplectic Structure ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Action Principle ◽  
Internal Boundary ◽  
Four Dimensions ◽  
First Order ◽  
Conserved Charges ◽  
Boundary Terms

In this review, we consider first-order gravity in four dimensions. In particular, we focus our attention in formulations where the fundamental variables are a tetrad [Formula: see text] and a [Formula: see text] connection [Formula: see text]. We study the most general action principle compatible with diffeomorphism invariance. This implies, in particular, considering besides the standard Einstein–Hilbert–Palatini term, other terms that either do not change the equations of motion, or are topological in nature. Having a well defined action principle sometimes involves the need for additional boundary terms, whose detailed form may depend on the particular boundary conditions at hand. In this work, we consider spacetimes that include a boundary at infinity, satisfying asymptotically flat boundary conditions and/or an internal boundary satisfying isolated horizons boundary conditions. We focus on the covariant Hamiltonian formalism where the phase space [Formula: see text] is given by solutions to the equations of motion. For each of the possible terms contributing to the action, we consider the well-posedness of the action, its finiteness, the contribution to the symplectic structure, and the Hamiltonian and Noether charges. For the chosen boundary conditions, standard boundary terms warrant a well posed theory. Furthermore, the boundary and topological terms do not contribute to the symplectic structure, nor the Hamiltonian conserved charges. The Noether conserved charges, on the other hand, do depend on such additional terms. The aim of this manuscript is to present a comprehensive and self-contained treatment of the subject, so the style is somewhat pedagogical. Furthermore, along the way, we point out and clarify some issues that have not been clearly understood in the literature.

scholarly journals Surface Wave Speed of Functionally Graded Magneto-Electro-Elastic Materials with Initial Stresses

2014 ◽  
Vol 44 (3) ◽  
pp. 49-64 ◽  
Author(s):  
Li Li ◽  
P. J. Wei
Keyword(s):  
Surface Wave ◽  
Boundary Conditions ◽  
Elastic Material ◽  
Short Circuit ◽  
Equations Of Motion ◽  
Open Circuit ◽  
Functionally Graded ◽  
Initial Stresses ◽  
Shear Surface ◽  
Traction Boundary Conditions

Abstract The shear surface wave at the free traction surface of half- infinite functionally graded magneto-electro-elastic material with initial stress is investigated. The material parameters are assumed to vary ex- ponentially along the thickness direction, only. The velocity equations of shear surface wave are derived on the electrically or magnetically open circuit and short circuit boundary conditions, based on the equations of motion of the graded magneto-electro-elastic material with the initial stresses and the free traction boundary conditions. The dispersive curves are obtained numerically and the influences of the initial stresses and the material gradient index on the dispersive curves are discussed. The investigation provides a basis for the development of new functionally graded magneto-electro-elastic surface wave devices.

Free nonlinear vibration analysis of nano-truncated conical shells based on modified strain gradient theory

2021 ◽  
pp. 146442072110419
Author(s):  
Alireza Sheykhi ◽  
Shahrokh Hosseini-Hashemi ◽  
Adel Maghsoudpour ◽  
Shahram E Haghighi
Keyword(s):  
Boundary Conditions ◽  
Strain Gradient ◽  
Couple Stress ◽  
Equations Of Motion ◽  
Differential Quadrature ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Modified Strain Gradient Theory ◽  
Conical Shells ◽  
Modified Couple Stress

In this study, the nonlinear free vibrations behaviour of nano-truncated conical shells was analysed, using the first-order shear deformable shell model. The analysis took into account the structure size through modified strain gradient theory, and differential quadrature and Fréchet derivative methods in von Kármán-Donnell-type approach to kinematic nonlinearity. The governing equations were obtained, utilizing Hamilton's principle. Partial differential equations plus the non-classical and classical boundary conditions were used to obtain the shells’ equations of motion. Discretizing the boundary conditions and equations of motion were performed based on a generalized differential quadrature analogy. The eigenvalue system was considered based on the harmonic balance technique. The Galerkin and Fréchet derivative approaches were used to determine the nonlinear free vibration behaviour of the carbon nano-cone, which was modelled in the simply- and clamped-supported boundary conditions. Comparisons were made between the findings from the new model versus the couple and classical stress theories, indicating that the classical and modified couple stress theories are distinct representations of modified strain gradient theory. The results also revealed that the degree of hardening of nano-truncated conical shells in the modified strain gradient theory is less than that of modified couple stress and classical theories. This led to a rise in the non-dimensional amplitude and frequency ratios. This study investigated the effect of size on free nonlinear vibrations of nano-truncated conical shells for various apex angles and lengths. Finally, we evaluated and compared our findings versus those reported by previous studies, which confirmed the precision and accuracy of our results.

Approximate Formulas for Cylindrical Shell Free Vibration Based on Vlasov’s and Enhanced Vlasov’s Semi-Momentless Theory

2018 ◽  
Author(s):  
Igor Orynyak ◽  
Yaroslav Dubyk
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Middle Surface ◽  
Axial Direction ◽  
Mode Shapes ◽  
Natural Mode ◽  
Transverse Deflection ◽  
Approximate Formulas

Simple approximate formulas for the natural frequencies of circular cylindrical shells are presented for modes in which transverse deflection dominates. Based on the Donnell-Mushtari thin shell theory the equations of motion of the circular cylindrical shell are introduced, using Vlasov assumptions and Fourier series for the circumferential direction, an exact solution in the axial direction is obtained. To improve the results assumptions of Vlasov’s semimomentless theory are enhanced, i.e. we have used only the hypothesis of middle surface inextensibility to obtain a solution in axial direction. Nonlinear characteristic equations and natural mode shapes, are derived for all type of boundary conditions. Good agreement with experimental data and FEM is shown and advantage over the existing formulas for a variety of boundary conditions is presented.

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