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1
Huang, Hui Rong, Ji Ping Hao, Hai Xia Zhang, and Yi Huang. "Displacement Governing Equations of Moderately Thick Cylindrical Shallow Shells by Transverse Shearing Deformation and the General Solution." Advanced Materials Research 291-294 (July 2011): 2066–70. http://dx.doi.org/10.4028/www.scientific.net/amr.291-294.2066.
Повний текст джерелаАнотація:
Displacement fundamental equations of the moderately thick cylindrical shallow shells concerning five independent variables, i.e. five middle surface displacements are established based on the displacement fundamental equations of the moderately thick shells by transverse shearing deformation and basic hypothesis on shallow shells. Three assistant displacement functions are introduced to solve the equations, which are tenth-order differential equations with variable coefficient; and then five second-order differential equations are converted into a second-order differential equation and two fourth-order transition differential equations using the Cauchy-Riemann condition, afterwards another assistant displacement function is introduced to build its decoupled governing differential equations, finally five displacement components through four assistant displacement functions are obtained.
2
Huang, Hui Rong, Ji Ping Hao, Hai Xia Zhang, and Yi Huang. "Displacement Fundamental Equations and Analysis of Governing Equations of the Circular Moderately Thick Shallow Spherical Shells." Advanced Materials Research 291-294 (July 2011): 2071–75. http://dx.doi.org/10.4028/www.scientific.net/amr.291-294.2071.
Повний текст джерелаАнотація:
Displacement fundamental equations of moderately thick shallow spherical shells in polar coordinates concerning five independent variables, i.e. five middle surface displacements are established, based on the displacement fundamental equations of the moderately thick shells by transverse shearing deformation and basic hypothesis on shallow shells. Four assistant displacement functions are introduced to solve displacement fundamental equations of circular moderately thick shallow spherical shells , which are tenth-order differential equations with variable coefficient, then the decoupled governing differential equations are built up, and five displacement components through four assistant displacement functions are obtained.
3
Bharath, S., B. C. Nakra, and K. N. Gupta. "Mathematical Model of a Railway Pneumatic Brake System With Varying Cylinder Capacity Effects." Journal of Dynamic Systems, Measurement, and Control 112, no. 3 (September 1, 1990): 456–62. http://dx.doi.org/10.1115/1.2896164.
Повний текст джерелаАнотація:
Governing equations for the analysis of pressure transient are derived from the principle of conservation of mass and momentum for a pneumatic brake system, which consists of a train pipe connected to a number of linear actuators (brake cylinders with piston displacement). The governing one-dimensional non-linear partial differential equations for the train pipe, non-linear ordinary differential equations for the brake cylinders, and second-order differential equation of motion for piston displacement are solved to determine the pressure transients in the brake system for a step change in pressure at the inlet. The governing equations are nondimensionalized and reduced to a set of ordinary nonlinear differential difference equations and integrated by standard numerical methods. The flow is considered isothermal, and the friction effects for turbulent and laminar flow are evaluated by quasi-steady state approximation. The auxiliary reservoir volume effect is also included. The results are compared with the experimental data obtained on a brake test rig.
4
Tuzcu, Ilhan, Joshua K. Moua, and Joe G. Olivares. "Control of a thermoelastic beam using heat actuation." Journal of Vibration and Control 23, no. 20 (January 29, 2016): 3309–26. http://dx.doi.org/10.1177/1077546316629251.
Повний текст джерелаАнотація:
This paper explores the idea of using heat as an actuator to simultaneously control vibration and temperature of a thermoelastic beam. We first model the beam as a slender, uniform cantilever beam of rectangular cross-section subject to heat through heat patches on the lower and upper surfaces at some discrete spanwise locations. The governing equations of the model are two coupled partial differential equations: one governing the elastic bending displacement and one governing the two-dimensional heat conduction of the beam. Through a discretization, the partial differential equations are replaced by a set of ordinary differential equations in a compact state-space form. We show that the coupling is actually between elastic displacement and those components of temperature contributing to the thickness-wise gradient at the midplane. The linear quadratic regulator in conjunction with the Kalman–Bucy filter is used for the control design to simultaneously damp out the displacement and the gradient. In a numerical example, we show the presence of thermoelastic damping due to the coupling. We also show that the displacement and gradient can simultaneously be controlled by using displacement measurements only, and that for less control effort it is also necessary to include some temperature measurements in the feedback.
5
Chang, Tai Ping. "Stochastic Nonlinear Vibration of Fluid-Loaded Double-Walled Carbon Nanotubes." Applied Mechanics and Materials 284-287 (January 2013): 362–66. http://dx.doi.org/10.4028/www.scientific.net/amm.284-287.362.
Повний текст джерелаАнотація:
This paper investigates the stochastic dynamic behaviors of nonlinear vibration of the fluid-loaded double-walled carbon nanotubes (DWCNTs) by considering the effects of the geometric nonlinearity and the nonlinearity of van der Waals (vdW) force. The nonlinear governing equations of the fluid-conveying DWCNTs are formulated based on the Hamilton’s principle. The Young’s modulus of elasticity of the DWCNTs is assumed as stochastic with respect to the position to actually describe the random material properties of the DWCNTs. By utilizing the perturbation technique, the nonlinear governing equations of the fluid-conveying can be decomposed into two sets of nonlinear differential equations involving the mean value of the displacement and the first variation of the displacement separately. Then we adopt the harmonic balance method in conjunction with Galerkin’s method to solve the nonlinear differential equations successively. Some statistical dynamic response of the DWCNTs such as the mean values and standard deviations of the amplitude of the displacement are computed. It is concluded that the mean value and standard deviation of the amplitude of the displacement increase nonlinearly with the increase of the frequencies.
6
Lee, Sen Yung, and Shueei Muh Lin. "Bending Vibrations of Rotating Nonuniform Timoshenko Beams With an Elastically Restrained Root." Journal of Applied Mechanics 61, no. 4 (December 1, 1994): 949–55. http://dx.doi.org/10.1115/1.2901584.
Повний текст джерелаАнотація:
Without considering the Coriolis force, the governing differential equations for the pure bending vibrations of a rotating nonuniform Timoshenko beam are derived. The two coupled differential equations are reduced into two complete fourth-order differential equations with variable coefficients in the flexural displacement and in the angle of rotation due to bending, respectively. The explicit relation between the flexural displacement and the angle of rotation due to bending is established. The frequency equations of the beam with a general elastically restrained root are derived and expressed in terms of the four normalized fundamental solutions of the associated governing differential equations. Consequently, if the geometric and material properties of the beam are in polynomial forms, then the exact solution for the problem can be obtained. Finally, the limiting cases are examined. The influence of the coupling effect of the rotating speed and the mass moment of inertia, the setting angle, the rotating speed and taper ratio on the natural frequencies, and the phenomenon of divergence instability (tension buckling) are investigated.
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Gupta, Bipin K., and Dipanjan Basu. "Nonlinear solutions for laterally loaded piles." Canadian Geotechnical Journal 57, no. 10 (October 2020): 1566–80. http://dx.doi.org/10.1139/cgj-2019-0341.
Повний текст джерелаАнотація:
A nonlinear analysis framework for laterally loaded piles is presented that is as accurate as equivalent three-dimensional nonlinear finite element analysis, but computationally one order of magnitude faster. The nonlinear behavior of sands and clays are account for by using hyperbolic modulus–reduction relationships. These nonlinear–elastic constitutive models are used to calculate the reduced modulus at different points in the soil based on the soil strains induced by lateral pile displacement. The reduced modulus at different points in the soil domain are spatially integrated to calculate the reduced soil resistance parameters associated with the differential equation governing the lateral pile displacement. The differential equations governing the lateral displacements of pile and soil under equilibrium are obtained by applying the principle of virtual work to a continuum-based pile–soil system. These coupled differential equations are solved using the one-dimensional finite difference method following an iterative algorithm. The accuracy of the analysis is verified against equivalent three-dimensional nonlinear finite element analysis, and the validity of the analysis in predicting the field response is checked by comparisons with multiple pile load test results. Parametric studies are performed to gain insights into the lateral pile response.
8
Lee, S. Y., and J. C. Chao. "Exact Solutions for Out-of-Plane Vibration of Curved Nonuniform Beams." Journal of Applied Mechanics 68, no. 2 (May 16, 2000): 186–91. http://dx.doi.org/10.1115/1.1346679.
Повний текст джерелаАнотація:
The governing differential equations for the out-of-plane vibrations of curved nonuniform beams of constant radius are derived. Two physical parameters are introduced to simplify the analysis, and the explicit relations between the torsional displacement, its derivative and the flexural displacement are derived. With these explicit relations, the two coupled governing characteristic differential equations can be decoupled and reduced to one sixth-order ordinary differential equation with variable coefficients in the out-of-plane flexural displacement. It is shown that if the material and geometric properties of the beam are in arbitrary polynomial forms, then the exact solutions for the out-of-plane vibrations of the beam can be obtained. The derived explicit relations can also be used to reduce the difficulty in experimental measurement. Finally, two limiting cases are considered and the influence of taper ratio, center angle, and arc length on the first two natural frequencies of the beams are illustrated.
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Alibeigloo, A. "Static analysis of an anisotropic laminated cylindrical shell with piezoelectric layers using differential quadrature method." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 222, no. 6 (June 1, 2008): 865–80. http://dx.doi.org/10.1243/09544062jmes866.
Повний текст джерелаАнотація:
A three-dimensional solution is presented for the static analysis of an anisotropic laminated cylindrical shell embedded in piezoelectric layers with arbitrary conditions at the ends, using the differential quadrature method (DQM). With the Soong assumption, governing equations are reduced to differential equations with constant coefficients. By applying the DQM to the obtained governing differential equations and to the boundary conditions along the longitudinal direction, new state equations for state variables are derived at discrete points. Stress, displacement, and electric potential distributions are obtained by solving these state equations. Both direct and inverse piezoelectric effects are investigated, and the influence of piezoelectric layers on the mechanical behaviour of the shell is studied. The method is validated by comparing the numerical results for the shell with the simply supported edges, which can be solved analytically.
10
Wang, Fan. "Continuum Analysis for Plate-Cone Spherical Reticulated Shell." Advanced Materials Research 317-319 (August 2011): 124–27. http://dx.doi.org/10.4028/www.scientific.net/amr.317-319.124.
Повний текст джерелаАнотація:
Plate-cone reticulated shell is a new type of space structures with good mechanical behavior and technical economy. In this paper, a continuum analysis method for plate-cone spherical reticulated shell which is equated to three layers thin shell is put forward. Based on mechanical characteristics of plate-cone spherical reticulated shell, the equivalent stiffness is derived through the theory of elasticity. Then, plate-cone spherical reticulated shell is equated to a special type of three layers thin shell working interactively, the governing differential equations are derived on the basis of the theory of thin shells, these differential equations being from the displacement method and the mixed method of derivation. The solution of this system of differential equations gives the displacements and internal forces of plate-cone spherical reticulated shell.
11
Lin, S. M. "Dynamic Analysis of Rotating Nonuniform Timoshenko Beams With an Elastically Restrained Root." Journal of Applied Mechanics 66, no. 3 (September 1, 1999): 742–49. http://dx.doi.org/10.1115/1.2791698.
Повний текст джерелаАнотація:
A systematic solution procedure for studying the dynamic response of a rotating nonuniform Timoshenko beam with an elastically restrained root is presented. The partial differential equations are transformed into the ordinary differential equations by taking the Laplace transform. The two coupled governing differential equations are uncoupled into two complete fourth-order differential equations with variable coefficients in the flexural displacement and in the angle of rotation due to bending, respectively. The general solution and the generalized Green function of the uncoupled system are derived. They are expressed in terms of the four corresponding linearly independent homogenous solutions, respectively. The shifting relations of the four homogenous solutions of the uncoupled governing differential equation with constant coefficients are revealed. The generalized Green function of an nth order ordinary differential equation can be obtained by using the proposed method. Finally, the influence of the elastic root restraints, the setting angle, and the excitation frequency on the steady response of a beam is investigated.
12
Chang, Tai Ping. "Small Scale Effect on Nonlinear Vibration of Fluid-Loaded Double-Walled Carbon Nanotubes with Uncertainty." Applied Mechanics and Materials 479-480 (December 2013): 121–25. http://dx.doi.org/10.4028/www.scientific.net/amm.479-480.121.
Повний текст джерелаАнотація:
This paper investigates the statistical dynamic behaviors of nonlinear vibration of the fluid-loaded double-walled carbon nanotubes (DWCNTs) by considering the effects of the geometric nonlinearity and the nonlinearity of van der Waals (vdW) force. Besides, the small scale effects of the nonlinear vibration of the DWCNTs are studied by using the theory of nonlocal elasticity. The nonlinear governing equations of the fluid-conveying DWCNTs are formulated based on the Hamilton's principle. The Young's modulus of elasticity of the DWCNTs is assumed as stochastic to actually describe the random material properties of the DWCNTs. By utilizing the perturbation technique, the nonlinear governing equations of the fluid-conveying can be decomposed into two sets of nonlinear differential equations involving the mean value of the displacement and the first variation of the displacement separately. Then we adopt the harmonic balance method in conjunction with Galerkin's method to solve the nonlinear differential equations successively. Some statistical dynamic response of the DWCNTs such as the mean values and standard deviations of the amplitude of the displacement are computed; meanwhile the effects of small scale coefficients on the statistical dynamic response of the DWCNTs are investigated.
13
Daneshmehr, Alireza R., Samaun Nili, A. R. Nateghi, and Shirjan Hussaini. "Three-Dimensional Elasticity Study of Vibration of a Composite Shell Panel with Embedded Piezoelectric Sensors." Applied Mechanics and Materials 186 (June 2012): 87–97. http://dx.doi.org/10.4028/www.scientific.net/amm.186.87.
Повний текст джерелаАнотація:
In this paper, Free vibration analysis of a finite length composite shell panel with an embedded piezoelectric sensor, using three-dimensional elasticity solution, is presented. To this end, two different methods are applied to solve the governing equations of the problem. In the first method, the displacement field is derived using trigonometric function expansion in circumferential and longitudinal directions. Using the method of changing variables, the governing partial differential equations are reduced to ordinary differential equations. Then these equations are solved simultaneously with outer and inner boundary conditions to give the natural frequencies and shape modes of the shell panel. In the second method the highly coupled partial differential equations are reduced to ordinary differential equations by means of trigonometric function expansion in circumferential and axial directions and then the finite difference method is applied to evaluate the obtained differential equations in radial direction. Then, the natural frequencies of the multi-layered panel are calculated using the obtained ordinary differential equations. At last, some numerical examples are presented to compare the results obtained by these two different methods. Three layered laminated shell panel is assumed to be [0/90/P].
14
Asle-Zaeem, M., and S. D. Mesarovic. "Investigation of Phase Transformation in Thin Film Using Finite Element Method." Solid State Phenomena 150 (January 2009): 29–41. http://dx.doi.org/10.4028/www.scientific.net/ssp.150.29.
Повний текст джерелаАнотація:
Cahn-Hilliard type of phase field model coupled with elasticity is used to derive governing equations for the stress-mediated diffusion and phase transformation in thin films. To solve the resulting equations, a finite element (FE) model is presented. The partial differential equations governing diffusion and mechanical equilibrium are of different orders; Mixed-order finite elements, with C0 interpolation functions for displacement, and C1 interpolation functions for concentration are implemented. To validate this new numerical solver for such coupled problems, we test our implementation on thin film diffusion couples.
15
Ahmadi, Isa. "Edge stresses analysis in laminated thick sandwich cylinder subjected to distributed hygrothermal loading." Journal of Sandwich Structures & Materials 20, no. 4 (July 13, 2016): 425–61. http://dx.doi.org/10.1177/1099636216657681.
Повний текст джерелаАнотація:
The boundary layer hygrothermal stresses in the thick sandwich cylinder with laminated face are investigated. Uniform and through the thickness steady-state distribution for temperature and moisture content can be considered in the analysis. A displacement based layer-wise formulation is presented for analysis of thick sandwich composite cylinders subjected to hygrothermal loading conditions. Considering a general displacement field and employing a displacement based layer-wise theory, the governing equations of thick laminated sandwich cylinder are obtained. The displacement based formulation is derived for thick sandwich cylinder, which is subjected to non-uniform hygrothermal loading conditions. The faces of the sandwich cylinder are made of laminated composite with general layer stacking. The governing equations of the system include a set of coupled differential equations on the displacement components of the numerical surfaces. A semi-analytical solution is developed and the governing equations are solved for free edge boundary conditions. The accuracy of the numerical results is validated by the results of the finite element simulation and good agreements are seen between the predicted results. The free edge interlaminar stresses distributions are presented for thin and thick sandwich composite cylinders for uniform and non-uniform loading conditions. It is concluded that the presented layer-wise formulation is efficient and accurate method for analysis of thermal and hygroscopic stresses in thick and thin sandwich cylinders with general layer stacking.
16
Ibearugbulem, O. M., and Festus Chukwudi Onyeka. "Moment and Stress Analysis Solutions of Clamped Rectangular Thick Plate." European Journal of Engineering Research and Science 5, no. 4 (April 28, 2020): 531–34. http://dx.doi.org/10.24018/ejers.2020.5.4.1898.
Повний текст джерелаАнотація:
The bending solutions of rectangular thick plate with all four edges clamped (CCCC) were investigated in this study. The basic governing equations used for analysis are based on third-order shear deformation plate theory analysis under uniformly distributed load. Using a formulated total potential energy equation, the three coupled general governing differential equations for the determination of the out of plane displacement and shear deformations rotation along the direction of x and y coordinates were obtained. These equations as obtained are solved simultaneously after minimization to determine the coefficients of displacements of the plate and other the mentioned functions. By solving these equations, the analytic solutions of rectangular thick plate with all four edges clamped were derived. From the formulated expression, the formula for calculation of the maximum deflection, moment, stress and in-plane displacements were deduced. The proposed method obviates the need of shear correction factors, which is associated with Mindlin’s theory (FSDT) for the solution to the problem. Moreover, numerical comparison shows the correctness and accuracy of the results.
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Ibearugbulem, O. M., and Festus Chukwudi Onyeka. "Moment and Stress Analysis Solutions of Clamped Rectangular Thick Plate." European Journal of Engineering and Technology Research 5, no. 4 (April 28, 2020): 531–34. http://dx.doi.org/10.24018/ejeng.2020.5.4.1898.
Повний текст джерелаАнотація:
The bending solutions of rectangular thick plate with all four edges clamped (CCCC) were investigated in this study. The basic governing equations used for analysis are based on third-order shear deformation plate theory analysis under uniformly distributed load. Using a formulated total potential energy equation, the three coupled general governing differential equations for the determination of the out of plane displacement and shear deformations rotation along the direction of x and y coordinates were obtained. These equations as obtained are solved simultaneously after minimization to determine the coefficients of displacements of the plate and other the mentioned functions. By solving these equations, the analytic solutions of rectangular thick plate with all four edges clamped were derived. From the formulated expression, the formula for calculation of the maximum deflection, moment, stress and in-plane displacements were deduced. The proposed method obviates the need of shear correction factors, which is associated with Mindlin’s theory (FSDT) for the solution to the problem. Moreover, numerical comparison shows the correctness and accuracy of the results.
18
Rungamornrat, Jaroon, and Peerasak Tangnovarad. "Analysis of Linearly Elastic Inextensible Frames Undergoing Large Displacement and Rotation." Mathematical Problems in Engineering 2011 (2011): 1–37. http://dx.doi.org/10.1155/2011/592958.
Повний текст джерелаАнотація:
This paper presents an efficient semi-analytical technique for modeling two-dimensional, linearly elastic, inextensible frames undergoing large displacement and rotation. A system of ordinary differential equations governing an element is first converted into a system of nonlinear algebraic equations via appropriate enforcement of boundary conditions. Taylor's series expansion is then employed along with the co-rotational approach to derive the best linear approximation of such system and the corresponding exact element tangent stiffness matrix. A standard assembly procedure is applied, next, to obtain the best linear approximation of governing nonlinear equations for the structure. This final system is exploited in the solution search by Newton-Ralphson iteration. Key features of the proposed technique include that (i) exact load residuals are evaluated from governing nonlinear algebraic equations, (ii) an exact form of the tangent stiffness matrix is utilized, and (iii) all elements are treated in a systematic way via direct stiffness strategy. The first two features enhance the performance of the technique to yield results comparable to analytical solutions and independent of mesh refinement whereas the last feature allows structures of general geometries and loading conditions be modeled in a straightforward fashion. The implemented algorithm is tested for various structures not only to verify its underlying formulation but also to demonstrate its capability and robustness.
19
Kansal, Tarun. "Fundamental solution of the system of equations of pseudo oscillations in the theory of thermoelastic diffusion materials with double porosity." Multidiscipline Modeling in Materials and Structures 15, no. 2 (February 21, 2019): 317–36. http://dx.doi.org/10.1108/mmms-01-2018-0006.
Повний текст джерелаАнотація:
PurposeThe purpose of this paper to construct the fundamental solution of partial differential equations in the generalized theory of thermoelastic diffusion materials with double porosity.Design/methodology/approachThe paper deals with the study of pseudo oscillations in the generalized theory of thermoelastic diffusion materials with double porosity.FindingsThe paper finds the fundamental solution of partial differential equations in terms of elementary functions.Originality/valueAssuming the displacement vector, volume fraction fields, temperature change and chemical potential functions in terms of oscillation frequency in the governing equations, pseudo oscillations have been studied and finally the fundamental solution of partial differential equations in case of pseudo oscillations in terms of elementary functions has been constructed.
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Babaei, Hadi, and M. Reza Eslami. "Nonlinear Snap-Through Instability of FGM Shallow Micro-Arches with Integrated Surface Piezoelectric Layers Based on Modified Couple Stress Theory." International Journal of Structural Stability and Dynamics 19, no. 08 (August 2019): 1950088. http://dx.doi.org/10.1142/s0219455419500883.
Повний текст джерелаАнотація:
Based on the modified couple stress theory, an attempt is made in this study to analyze the nonlinear snap-through instability of shallow sandwich arches. The microstructure-dependent functionally graded material (FGM) arch with surface bonded piezoelectric actuator layers is analyzed. The piezo-FGM sandwich arch is subjected to uniform transverse pressure load in thermo-electrical environment. All material properties of the FGM micro arch are assumed to be temperature- and position-dependent. The governing equilibrium equations of the piezo-FGM sandwich arch are established with the aid of virtual displacement principle and the uncoupled thermoelacticity theory. The obtained governing differential equations are based on the first-order shear deformation shallow arch theory of the Timoshenko and von Kármán nonlinear assumptions. These equilibrium equations contain three coupled ordinary differential equations in terms of displacements. The nondimensional governing equations are solved for the cases of piezo-FGM sandwich arches with simply supported and clamped boundary conditions by using the two-step perturbation technique. Analytical closed-form solutions are derived to give the deflected shape of the piezo-FGM sandwich arch with immovable ends. Comparison is made with the existing results for the cases of FGM arch without couple stress and piezoelectric layers, where good agreement is obtained. The nonlinear behavior of the sandwich arches is highly affected by the couple stress, piezoelectric layers, temperature change, volume fraction index, and geometrical properties of the arch.
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Bosiakov, Sergei, and Gennadi Mikhasev. "MATHEMATICAL MODEL FOR ANALYSIS OF TRANSLATIONAL DISPLACEMENTS OF TOOTH ROOT." Mathematical Modelling and Analysis 20, no. 4 (July 20, 2015): 490–501. http://dx.doi.org/10.3846/13926292.2015.1068877.
Повний текст джерелаАнотація:
Analytical modeling of stress-strain state of a periodontal ligament in the case of the translational displacement of a tooth root was carried out. The tooth root was assumed as a rigid body. The boundary conditions corresponding to the translational displacement of the root and fixed external surface of the periodontal ligament in the dental alveolus were considered. The system of differential equations describing the periodontal ligament’s plane-strain state induced by the translational motion of the tooth were used as the governing equations. An analytical solution was found for the governing equations in the explicit form. Comparative analysis of the concentrated force generated by the prescribed translational motion of the tooth root was performed using the obtained analytical solution and the model of an incompressible periodontal ligament in the form of a circular paraboloid and hyperboloid. The mathematical model developed in this paper can be used to analyze stresses and strains in the periodontal tissue during orthodontic movement.
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Zhao, Bao Sheng, Jia Lian Shi, Ying Tao Zhao, and Yang Gao. "The General Solution for Torsional Circular Shaft of Cubic Quasicrystal." Advanced Materials Research 213 (February 2011): 206–10. http://dx.doi.org/10.4028/www.scientific.net/amr.213.206.
Повний текст джерелаАнотація:
Gregory’s decomposed theorem of isotropic plate is extended to investigate torsional circular shaft of cubic quasicrystal with homogeneous boundary conditions, and the theory of equivalence that Cheng’s refined theory and Gregory’s decomposed theorem is extended to the cylindrical coordinate. The general solution of torsional circular shaft on cubic quasicrystal with homogeneous boundary conditions is proposed on the basis of the classical elasticity theory and stress-displacement relations of cubic quasicrystal without ad hoc assumptions. At first expressions are obtained for all the displacements and stress components in term of some 1D functions. Using Lur’e method, the exact equations were given. And the exact equations for the torsional circular shaft on cubic quasicrystal without surface loadings consist of four governing differential equations: two harmonic equations and two transcendental equations. Using basic mathematic method and the general solutions, an example is examined.
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Zhao, Bao Sheng, Ying Tao Zhao, and Yang Gao. "The Decomposed Theorem of Torsional Circular Shaft with Two-Dimensional Dodecagonal Quasicrystal." Advanced Materials Research 341-342 (September 2011): 1–5. http://dx.doi.org/10.4028/www.scientific.net/amr.341-342.1.
Повний текст джерелаАнотація:
Gregory’s decomposed theorem of isotropic plate is extended to investigate torsional circular shaft for two-dimensional dodecagonal quasicrystal (2D dodecagonal QCs)with homogeneous boundary conditions, and the theory of equivalence that Cheng’s refined theory and Gregory’s decomposed theorem is extended to the cylindrical coordinate. The decomposed theorem of torsional circular shaft of 2D dodecagonal QCs with homogeneous boundary conditions is proposed on the basis of the classical elasticity theory and stress-displacement relations of 2D dodecagonal QCs without ad hoc assumptions. At first expressions are obtained for all the displacements and stress components in term of some 1D functions. Using Lur’e method, the exact equations were given. And the exact equations for the torsional circular shaft on 2D dodecagonal QCs without surface loadings consist of four governing differential equations: two harmonic equations and two transcendental equations.
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GAO, YANG, and LAN-GE SHANG. "GOVERNING EQUATIONS AND GENERAL SOLUTIONS OF PLANE ELASTICITY OF TWO-DIMENSIONAL DECAGONAL QUASICRYSTALS." International Journal of Modern Physics B 25, no. 20 (August 10, 2011): 2769–78. http://dx.doi.org/10.1142/s0217979211101065.
Повний текст джерелаАнотація:
Two-dimensional problem is systematically investigated for the coupled equations in two-dimensional decagonal quasicrystals, and three general solutions are presented by an operator method. To simplify governing equations, each general solution shall take three different forms expressed in terms of three displacement functions, which satisfy partial three differential equations of second order. To illustrate its utility, the closed-form solutions are obtained for problems of point phonon forces and phason force acting at the apex of a quasicrystal wedge, all in terms of the general solutions. Furthermore, these solutions can be degenerated to those of problems of point forces applied on the boundary of a quasicrystal half-plane.
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Peng, Jian She, Gang Xie, Liu Yang, and Yu Quan Yuan. "A Time-Domain DQ Approach for Vibration Analysis of Beams." Advanced Materials Research 631-632 (January 2013): 957–61. http://dx.doi.org/10.4028/www.scientific.net/amr.631-632.957.
Повний текст джерелаАнотація:
This paper presents a new time-domain DQ (differential quadrature) method for structural vibration analysis. It adopts differential quadrature method both in space domain and in time domain on the basis of governing partial differential equation and initial-boundary value condition of vibration problems of structures, and gets new differential quadrature linear equations with complete initial-boundary value conditions for solving all parameters of the displacement-field. The examples in this paper show the time-domain differential quadrature method is a useful and efficient tool for structural vibration analysis.
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Yang, L., and S. G. Hutton. "Nonlinear Vibrations of Elastically-Constrained Rotating Disks." Journal of Vibration and Acoustics 120, no. 2 (April 1, 1998): 475–83. http://dx.doi.org/10.1115/1.2893854.
Повний текст джерелаАнотація:
An analysis of nonlinear vibrations of an elastically-constrained rotating disk is developed. The equations of motion, which are two coupled nonlinear partial differential equations corresponding to the transverse force equilibrium and to the deformation compatibility, are first developed by using von Karman thin plate theory. Then the stress function is analytically solved from the compatibility equation by assuming a multi-mode transverse displacement field. Galerkin’s method is applied to transform the force equilibrium equation into a set of coupled nonlinear ordinary differential equations in terms of time functions. Finally, numerical integration is used to solve the time governing equations, and the effects of nonlinearity on the vibrations of a rotating disk are discussed.
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Rhim, Y., and J. Tichy. "The Flow of Layered Liquid Crystals in a Thin Wedge With Dislocations." Journal of Tribology 113, no. 3 (July 1, 1991): 492–97. http://dx.doi.org/10.1115/1.2920650.
Повний текст джерелаАнотація:
The velocity, displacement, and stress field of layered liquid crystals with dislocations in a lubricating gap are investigated numerically. Galerkin’s weighted residual finite element method is employed to solve a set of five highly nonlinear coupled differential equations in terms of two spatial coordinates. In addition to the usual continuity and momentum equations, equations are required for a body force term governing permeation through the layers and elastic displacement of the layers. The last equation contains a fourth derivative term which gives rise to dislocations and numerical complications. The results show a strong increase in load relative to an equivalent viscous lubricant. The load increase depends strongly on the magnitude of a permeation parameter.
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Li, Yuan, CuiYing Fan, Qing-Hua Qin, and MingHao Zhao. "Closed-form solutions of an elliptical crack subjected to coupled phonon–phason loadings in two-dimensional hexagonal quasicrystal media." Mathematics and Mechanics of Solids 24, no. 6 (October 22, 2018): 1821–48. http://dx.doi.org/10.1177/1081286518807513.
Повний текст джерелаАнотація:
An elliptical crack subjected to coupled phonon–phason loadings in a three-dimensional body of two-dimensional hexagonal quasicrystals is analytically investigated. Owing to the existence of the crack, the phonon and phason displacements are discontinuous along the crack face. The phonon and phason displacement discontinuities serve as the unknown variables in the generalized potential function method which are used to derive the boundary integral equations. These boundary integral equations governing Mode I, II, and III crack problems in two-dimensional hexagonal quasicrystals are expressed in integral differential form and hypersingular integral form, respectively. Closed-form exact solutions to the elliptical crack problems are first derived for two-dimensional hexagonal quasicrystals. The corresponding fracture parameters, including displacement discontinuities along the crack face and stress intensity factors, are presented considering all three crack cases of Modes I, II, and III. Analytical solutions for a penny-shaped crack, as a special case of the elliptical problem, are given. The obtained analytical solutions are graphically presented and numerically verified by the extended displacement discontinuities boundary element method.
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Zhou, Yong, Timo Nyberg, Gang Xiong, Shi Li, Hongbo Zhou, and Sen Bao. "Analytical Solution of Thick Piezoelectric Curved Beams with Variable Curvature Considering Shearing Deformation." International Journal of Applied Mechanics 09, no. 01 (January 2017): 1750006. http://dx.doi.org/10.1142/s1758825117500065.
Повний текст джерелаАнотація:
In this paper, an analytical method based on Timoshenko theory is derived for obtaining the in-plane static closed-form general solutions of deep curved laminated piezoelectric beams with variable curvatures. The equivalent modulus of elasticity is utilized to take into account the material couplings in the laminated beam. The linear piezoelectric effect is considered to develop the static governing equations. The governing differential equations are formulated as functions of the angle of tangent slope by introducing the coordinate system defined by the arc length of the centroidal axis and the angle of tangent slope. To solve the governing equations, defined are the fundamental geometric properties, such as the moments of the arc length with respect to horizontal and vertical axes. As the radius is known, the fundamental geometric quantities can be calculated to obtain the static closed-form solutions of the axial force, shear force, bending moment, rotation angle, and displacement fields at any cross-section of curved beams. The closed-form solutions of the circle beams covered with piezoelectric layers under various loading cases are presented. The results show the consistency in comparison with finite results. Solutions of the non-dimensional displacements for the laminated circular and spiral curved beams with different lay-ups are available. The non-dimensional displacements with geometry and material parameters are also investigated.
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Kamaloo, Abbas, Mohsen Jabbari, Mehdi Yarmohammad Tooski, and Mehrdad Javadi. "Nonlinear free vibration analysis of delaminated composite circular cylindrical shells." Journal of Vibration and Control 26, no. 19-20 (January 30, 2020): 1697–707. http://dx.doi.org/10.1177/1077546320902556.
Повний текст джерелаАнотація:
This study aims to present an analysis of nonlinear free vibrations of simply supported laminated composite circular cylindrical shells with throughout circumference delamination. Governing equations of motion are derived by applying energy methods; using Galerkin’s method reduced the nonlinear partial differential equations to a system of coupled nonlinear ordinary differential equations, which are subsequently solved using a numerical method. This research examines the effects of delamination on the oscillatory motion of delaminated composite circular cylindrical shells and then the effects of increase in delamination length, shell middle surface radius, number of layers, and orthotropy as changes in material properties on the nonlinearity of these types of shells. The results show that delamination leads to a decrease in frequency of oscillations and displacement. An increase in delamination length, shell middle surface radius, and orthotropy of layers decreases nonlinearity and displacement, whereas an increase in the number of layers increases nonlinearity and displacement. It is also observed that an increase in the circumferential wave number can decrease the effect of delamination.
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Sladek, Jan, and Vladimir Sladek. "Crack Analysis in Piezoelectric Semiconductors." Key Engineering Materials 627 (September 2014): 269–72. http://dx.doi.org/10.4028/www.scientific.net/kem.627.269.
Повний текст джерелаАнотація:
Mechanical and electric loads are considered for 2-d crack problems in conducting piezoelectric materials. The electric displacement in conducting piezoelectric materials is influenced by the electron density and it is coupled with the electric current. The coupled governing partial differential equations (PDE) for stresses, electric displacement field and current are satisfied in a local weak-form on small fictitious subdomains. Nodal points are spread on the analyzed domain and each node is surrounded by a small circle for simplicity. Local integral equations are derived for a unit function as the test function on circular subdomains. All field quantities are approximated by the moving least-squares (MLS) scheme.
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WALSH, TIMOTHY, and MONICA TORRES. "FINITE ELEMENT METHODS FOR NONLINEAR ACOUSTICS IN FLUIDS." Journal of Computational Acoustics 15, no. 03 (September 2007): 353–75. http://dx.doi.org/10.1142/s0218396x0700338x.
Повний текст джерелаАнотація:
In this paper, weak formulations and finite element discretizations of the governing partial differential equations of three-dimensional nonlinear acoustics in absorbing fluids are presented. The fluid equations are considered in an Eulerian framework, rather than a displacement framework, since in the latter case the corresponding finite element formulations suffer from spurious modes and numerical instabilities. When taken with the governing partial differential equations of a solid body and the continuity conditions, a coupled formulation is derived. The change in solid/fluid interface conditions when going from a linear acoustic fluid to a nonlinear acoustic fluid is demonstrated. Finite element discretizations of the coupled problem are then derived, and verification examples are presented that demonstrate the correctness of the implementations. We demonstrate that the time step size necessary to resolve the wave decreases as steepening occurs. Finally, simulation results are presented on a resonating acoustic cavity, and a coupled elastic/acoustic system consisting of a fluid-filled spherical tank.
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Hasrati, Emad, Reza Ansari, and Jalal Torabi. "Nonlinear Forced Vibration Analysis of FG-CNTRC Cylindrical Shells Under Thermal Loading Using a Numerical Strategy." International Journal of Applied Mechanics 09, no. 08 (December 2017): 1750108. http://dx.doi.org/10.1142/s1758825117501083.
Повний текст джерелаАнотація:
Employing an efficient numerical strategy, the nonlinear forced vibration analysis of composite cylindrical shells reinforced with single-walled carbon nanotubes (CNTs) is carried out. It is assumed that the distribution of CNTs along the thickness direction of the shell is uniform or functionally graded and the temperature dependency of the material properties is accounted. The governing equations are presented based on the first-order shear deformation theory along with von-Karman nonlinear strain-displacement relations. The vectorized form of energy functional is derived and directly discretized using numerical differential and integral operators. By the use of variational differential quadrature (VDQ) method, discretized nonlinear governing equations are obtained. Then, the time periodic differential operators are applied to perform the discretization procedure in time domain. Finally, the pseudo-arc length continuation method is employed to solve the nonlinear governing equations and trace the frequency response curve of the nanocomposite cylindrical shell. A comparison study is first presented to verify the efficiency and validity of the proposed numerical method. Comprehensive numerical results are then given to investigate the effects of the involved factors on the nonlinear forced vibration characteristics of the structure. The results show that the changes of fundamental vibrational mode shape have considerable effects on the frequency response curves of composite cylindrical shells reinforced with CNTs.
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Saviz, M. R., M. Shakeri, and M. H. Yas. "Three-dimensional elasticity analysis of a laminated cylindrical shell with piezoelectric layer under dynamic loads." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 221, no. 12 (December 1, 2007): 1507–19. http://dx.doi.org/10.1243/09544062jmes661.
Повний текст джерелаАнотація:
The elasticity solution for simply-supported, laminated cylindrical shell with piezoelectric layer is presented. The shell is subjected to various dynamic loads. The direct piezoelectric effect is considered. The governing differential equations are reduced to ordinary differential equations by means of trigonometric function expansion for displacement and electric potential. The loading function is expanded as a double Fourier series in axial and circumferential coordinates. The resulting equations are solved by Galerkin's finite element in radial direction. The static results and natural frequencies are compared with similar ones in the literature. The effect of radius to thickness ratio and band load width on dynamic behaviour is studied. Time responses are presented for [0/90/Piezo] lamination.
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Mirzade, F. Kh. "Small-Scale Effect on Longitudinal Wave Propagation in Laser-Excited Plates." Journal of Nanoscience 2014 (October 21, 2014): 1–8. http://dx.doi.org/10.1155/2014/513010.
Повний текст джерелаАнотація:
Longitudinal wave propagation in an elastic isotopic laser-excited solid plate with atomic defect (vacancies, interstitials) generation is studied by the nonlocal continuum model. The nonlocal differential constitutive equations of Eringen are used in the formulations. The coupled governing equations for the dynamic of elastic displacement and atomic defect concentration fields are obtained. The frequency equations for the symmetrical and antisymmetrical motions of the plate are found and discussed. Explicit expressions for different characteristics of waves like phase velocity and attenuation (amplification) coefficients are derived. It is shown that coupling between the displacement and defect concentration fields affects the wave dispersion characteristics in the nonlocal elasticity. The dispersion curves of the elastic-diffusion instability are investigated for different pump parameters and larger wave numbers.
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Wu, Xiu Gen, Bai Lin Zheng, Peng Fei He, and Shu Guang Liu. "The Behavior of Column during Constrained Buckling Process." Applied Mechanics and Materials 166-169 (May 2012): 385–91. http://dx.doi.org/10.4028/www.scientific.net/amm.166-169.385.
Повний текст джерелаАнотація:
The elastic Euler buckling of an inextensible column is confined in a plane, and subject to fixed end displacement, in the presence of rigid, frictionless side-walls which constrain overall lateral displacements. The whole deflection is divided into some typical columns because of the symmetry. The governing equations of constrained buckling mode and deflection about axial load are deduced, based on linearized differential equation of beam. Point and line contact models are introduced to describe the behavior of the column in constrained buckling process, including load capacity, buckling wave and reaction force. The analysis on the deflection of column is helpful to the research about possible post-buckling paths.
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Lee, Jongsuh, and Semyung Wang. "Vibration Analysis of a Partially Connected Double-Beam System with the Transfer Matrix Method and Identification of the Slap Phenomenon in the System." International Journal of Applied Mechanics 09, no. 07 (October 2017): 1750093. http://dx.doi.org/10.1142/s1758825117500934.
Повний текст джерелаАнотація:
This study concerns dynamic behavior of a partially connected double-beam system. This beam system is composed of two single beams and these beams are connected to each other by the distributed stiffness of which are located at certain regions. The governing equation of this double-beam system is composed of two fourth-order differential equations for each single beam. These two governing equations are coupled to each other by the partially distributed stiffness. And the coupling in the governing equations makes it difficult to analyze the dynamic behavior of this system. In this study, therefore, the difficulty has been overcome by using one assumption and one concept. The assumption is that the double beam is composed of two equivalent single beams. And the concept is concerned with the relative displacement between the two layers. In addition, the transfer matrix method is used to take problems concerned with discontinuity into account, which are generated by the distributed stiffness. And it is predicted that there will be the slap phenomenon in the double-beam system when the amplitude of the relative displacement is larger than the natural gap distance between the two layers. This phenomenon has been identified in experimental measurements.
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Al-Khaleefi, Abdulateef M., and Humayun R. H. Kabir. "On the Thermal Buckling Response of Shear-Flexible All-Edge Clamped Rectangular Plates." Journal of Vibration and Control 9, no. 5 (May 2003): 495–506. http://dx.doi.org/10.1177/1077546303009005001.
Повний текст джерелаАнотація:
Using an analytical approach, we investigate a thermal stability response for a rectangular plate with all-edge clamped boundary conditions. We consider the first-order shear deformation theory that utilizes shear flexible response, in order to obtain three highly coupled governing partial differential equations in three unknowns: one transverse displacement, and two independent rotations of the normal. The solution functions are assumed in the form of double Fourier series that satisfy the boundary conditions, as well as the partial differential equations. The results obtained from the analytical solution are compared with available finite element solutions. These analytically obtained results can be capitalized to check the accuracy of various approximate methods.
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Lesieutre, G. A., and E. Bianchini. "Time Domain Modeling of Linear Viscoelasticity Using Anelastic Displacement Fields." Journal of Vibration and Acoustics 117, no. 4 (October 1, 1995): 424–30. http://dx.doi.org/10.1115/1.2874474.
Повний текст джерелаАнотація:
A time domain model of linear viscoelasticity is developed based on a decomposition of the total displacement field into two parts: one elastic, the other anelastic. The anelastic displacement field is used to describe that part of the strain that is not instantaneously proportional to stress. General coupled constitutive equations for (1) the total and (2) the anelastic stresses are developed in terms of the total and anelastic strains, and specialized to the case of isotropic materials. A key feature of the model is the absence of explicit time dependence in the constitutive equations. Apparent time-dependent behavior is described instead by differential equations that govern (1) the motion of mass particles and (2) the relaxation of the anelastic displacement field. These coupled governing equations are developed in a parallel fashion, involving the divergence of appropriate stress tensors. Boundary conditions are also treated: the anelastic displacement field is effectively an internal field, as it is driven exclusively through coupling to the total displacement, and cannot be directly affected by applied loads. In order to illustrate the use of the method, model parameters for a commonly-used high damping polymer are developed from available complex modulus data.
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Kaiser, Tobias, Samuel Forest, and Andreas Menzel. "A finite element implementation of the stress gradient theory." Meccanica 56, no. 5 (March 2, 2021): 1109–28. http://dx.doi.org/10.1007/s11012-020-01266-3.
Повний текст джерелаАнотація:
AbstractIn this contribution, a finite element implementation of the stress gradient theory is proposed. The implementation relies on a reformulation of the governing set of partial differential equations in terms of one primary tensor-valued field variable of third order, the so-called generalised displacement field. Whereas the volumetric part of the generalised displacement field is closely related to the classic displacement field, the deviatoric part can be interpreted in terms of micro-displacements. The associated weak formulation moreover stipulates boundary conditions in terms of the normal projection of the generalised displacement field or of the (complete) stress tensor. A detailed study of representative boundary value problems of stress gradient elasticity shows the applicability of the proposed formulation. In particular, the finite element implementation is validated based on the analytical solutions for a cylindrical bar under tension and torsion derived by means of Bessel functions. In both tension and torsion cases, a smaller is softer size effect is evidenced in striking contrast to the corresponding strain gradient elasticity solutions.
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HOSSEINI, M., S. A. FAZELZADEH, and P. MARZOCCA. "CHAOTIC AND BIFURCATION DYNAMIC BEHAVIOR OF FUNCTIONALLY GRADED CURVED PANELS UNDER AERO-THERMAL LOADS." International Journal of Bifurcation and Chaos 21, no. 03 (March 2011): 931–54. http://dx.doi.org/10.1142/s0218127411028738.
Повний текст джерелаАнотація:
This paper presents the nonlinear analysis of functionally graded curved panels under high temperature supersonic gas flows. The aerothermoelastic governing equations are determined via Hamilton's variational principle. The von Karman nonlinear strain–displacement relations are used to account for large deflections. The material properties are assumed to be temperature-dependent and varying through the thickness direction according to a power law distribution in terms of the volume fractions of the constituent components. The panel is assumed to be infinitely long and simply supported. The Galerkin method is applied to convert the partial differential governing equation into a set of ordinary differential equations and the resulting system of nonlinear equations is solved through a numerical integration scheme. The effects of volume fraction index, curved panel height-rise, and aerodynamic pressure, in conjunction with the applied thermal loading, on the dynamical behavior of the panel are investigated. Regular and chaotic motions regime are determined through bifurcation analysis using Poincaré maps of maximum panel deflection, panel time history, phase-space and frequency spectra as qualitative tools, while Lyapunov's exponents and dimension are used as quantitative tools.
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Kandasamy, Selvakumar, and Anand V. Singh. "Transient Vibration Analysis of Open Circular Cylindrical Shells." Journal of Vibration and Acoustics 128, no. 3 (November 28, 2005): 366–74. http://dx.doi.org/10.1115/1.2172264.
Повний текст джерелаАнотація:
A numerical method based on the Rayleigh-Ritz method has been presented for the forced vibration of open cylindrical shells. The equations are derived from the three-dimensional strain-displacement relations in the cylindrical coordinate system. The middle surface of the shell represents the geometry, which is defined by an angle that subtends the curved edges, the length, and the thickness. The displacement fields are generated with a predefined set of grid points on the middle surface using considerably high-order polynomials. Each grid point has five degrees of freedom, viz., three translational components along the cylindrical coordinates and two rotational components of the normal to the middle surface. Then the strain and kinetic energy expressions are obtained in terms of these displacement fields. The differential equation governing the vibration characteristics of the shell is expressed in terms of the mass, stiffness, and the load consistent with the prescribed displacement fields. The transient response of the shell with and without damping is sought by transforming the equation of motion to the state-space model and then the state-space differential equations are solved using the Runge-Kutta algorithm.
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Lin, Fan, Jian She Peng, and Liu Yang. "The Forced Vibration Analysis of Beams Based on a Time-Domain Differential Quadrature Method." Applied Mechanics and Materials 741 (March 2015): 108–12. http://dx.doi.org/10.4028/www.scientific.net/amm.741.108.
Повний текст джерелаАнотація:
Under driving forceF(x,t)=Q*sinwt, a time-domain DQ (differential quadrature) method for dynamic problems of beams with initial-boundary value conditions is presented in this paper. On the basis of governing partial differential equation, the discrete DQ method adopted both in space domain and time domain in this method gives rise to new differential quadrature linear equations with complete initial-boundary value conditions for solving all parameters of the displacement-field. The numerical examples show that the computational accuracy and efficiency of time-domain DQ method is better than finite element method based on time-domain difference.
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Vosoughi, AR, and MR Nikoo. "A new mixed method for nonlinear fuzzy free vibration analysis of nanobeams on nonlinear elastic foundation." Journal of Vibration and Control 24, no. 24 (May 16, 2016): 5765–73. http://dx.doi.org/10.1177/1077546316648491.
Повний текст джерелаАнотація:
A new mixed method for nonlinear fuzzy free vibration analysis of nanobeams on nonlinear elastic foundation is introduced. The governing equations are derived based on the first-order shear deformation theory (FSDT) in conjunction with the von-Kármán’s assumptions and the Eringen’s nonlocal elasticity theory. The differential quadrature method (DQM) is employed to discretize the governing equations and the related boundary conditions. The direct displacement control iterative method is used to solve the discretized system of equations. The fuzzy transformation method (FTM) is coupled with the solution to incorporate effects of different uncertainties such as the small scale effect parameter, nonlinear elastic foundation parameters and vibration amplitude of the nanobeam. Applicability, rapid rate of convergence and high accuracy of the presented method are shown and significant effects of the nonlinearity on the response of nanobeams are investigated via solving some examples.
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CARRERA, ERASMO, and GAETANO GIUNTA. "REFINED BEAM THEORIES BASED ON A UNIFIED FORMULATION." International Journal of Applied Mechanics 02, no. 01 (March 2010): 117–43. http://dx.doi.org/10.1142/s1758825110000500.
Повний текст джерелаАнотація:
This paper proposes several axiomatic refined theories for the linear static analysis of beams made of isotropic materials. A hierarchical scheme is obtained by extending plates and shells Carrera's Unified Formulation (CUF) to beam structures. An N-order approximation via Mac Laurin's polynomials is assumed on the cross-section for the displacement unknown variables. N is a free parameter of the formulation. Classical beam theories, such as Euler-Bernoulli's and Timoshenko's, are obtained as particular cases. According to CUF, the governing differential equations and the boundary conditions are derived in terms of a fundamental nucleo that does not depend upon the approximation order. The governing differential equations are solved via the Navier type, closed form solution. Rectangular and I-shaped cross-sections are accounted for. Beams undergo bending and torsional loadings. Several values of the span-to-height ratio are considered. Slender as well as deep beams are analysed. Comparisons with reference solutions and three-dimensional FEM models are given. The numerical investigation has shown that the proposed unified formulation yields the complete three-dimensional displacement and stress fields for each cross-section as long as the appropriate approximation order is considered. The accuracy of the solution depends upon the geometrical parameters of the beam and loading conditions.
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Li, Xiao Chuan, and Jin Shuang Zhang. "Hamiltonian Duality Equation on Three-Dimensional Problems of Magnetoelectroelastic Solids." Applied Mechanics and Materials 268-270 (December 2012): 1099–104. http://dx.doi.org/10.4028/www.scientific.net/amm.268-270.1099.
Повний текст джерелаАнотація:
Hamiltonian system used in dynamics is introduced to formulate the three-dimensional problems of the transversely isotropic magnetoelectroelastic solids. The Hamiltonian dual equations in magnetoelectroelastic solids are developed directly from the modified Hellinger-Reissner variational principle derived from generalized Hellinger-Ressner variational principle with two classes of variables. These variables not only include such origin variables as displaces, electric potential and magnetic potential, but also include such their dual variables as lengthways stress, electric displacement and magnetic induction in the symplectic space. Similar to the Hamiltonian formulation in classic dynamics, the z coordinate is treated analogous to the time coordinate so that the method of separation of variables can be used. The governing equations are a set of first order differential equations in z, and the coefficient matrix of the differential equations is Hamiltonian in (x, y).
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Yang, F., R. Sedaghati, and E. Esmailzadeh. "Free in-plane vibration of curved beam structures: A tutorial and the state of the art." Journal of Vibration and Control 24, no. 12 (August 28, 2017): 2400–2417. http://dx.doi.org/10.1177/1077546317728148.
Повний текст джерелаАнотація:
The study of free in-plane vibration of curved beams, using different beam theories, is more challenging than that of straight beams, since the structural deformations in curved beams depend not only on the rotation and radial displacements, but also on the coupled tangential displacement caused by the curvature of structures. A critical review of the publications on the free in-plane vibration of curved beams to demonstrate the state of the art has been presented. The governing differential equations of motion for the curved beams, based on different hypotheses (including and excluding the axial extensity, rotary inertia and the shear deformation), were discussed and different approaches to solve the developed equations of motion have been identified. Finally, a systematic comparison of the dynamic properties of curved beams evaluated with various forms of curvatures based on different hypotheses were presented.
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Naciri, I., A. Rguiti, L. Elmaimouni, J. E. Lefebvre, F. E. Ratolojanahary, J. G. Yu, Y. Belkassmi, and A. El Moussati. "Numerical Modelling of Vibration Characteristics of a Partially Metallized Micro Electromechanical System Resonator Disc." Acta Acustica united with Acustica 105, no. 6 (November 1, 2019): 1164–72. http://dx.doi.org/10.3813/aaa.919393.
Повний текст джерелаАнотація:
In this paper, we report an extension of a polynomial and numerical vibrational characterization of an annular piezoelectric disc resonator partially covered with electrodes. The three governing partial differential equations of motion are solved to provide the frequency response of the piezoelectric disc using a polynomial approach. This method makes use of Legendre polynomials series to express the mechanical displacement components and the electrical potential which are introduced into the equations of motion of the piezoelectric structure. The principal advantage of this method consists of incorporating the electrical source, the boundary and continuity conditions directly into the governing equations by the use of position-dependent physical constants and by a wise choice of the polynomial expansions for the independent variables, the mechanical displacement components and the electrical potential. Both harmonic and modal analyses were studied and are presented. Numerical calculations based on the foregoing method were performed to present resonance and anti-resonance frequencies, electromechanical coupling coefficient, field profiles and electrical input admittance for PIC151 and PZT5A disc resonators with various metallization rates. The high accuracy and reliability of our approach is confirmed via a comparison of our results with their counterparts reported in literature.
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Yaghoubshahi, M., E. Asadi, and S. J. Fariborz. "A higher-order shell model applied to shells with mixed boundary conditions." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 225, no. 2 (June 20, 2010): 292–303. http://dx.doi.org/10.1243/09544062jmes2050.
Повний текст джерелаАнотація:
By means of the principle of virtual work, the governing equations together with the required boundary conditions of a higher-order shear deformation theory are formulated for the analysis of laminated shells under static loads. A system of 31 first-order partial differential equations is performed for the determination of stress resultants and displacement components. These equations are then solved numerically, utilizing the generalized differential quadrature method for two isotropic cylindrical panels with equal arc length but different radii having S2-type simply supported boundary conditions. The results matched those of other theories. Another analysis is carried out for composite cylindrical panels with two lamination schemes, five different mixed boundary conditions, and two length-to-thickness ratios. The results are compared against solutions obtained from ANSYS finite-element software.
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Baferani, A. Hasani, A. R. Saidi, and H. Ehteshami. "On Free Vibration of Functionally Graded Mindlin Plate and Effect of In-Plane Displacements." Journal of Mechanics 29, no. 2 (January 29, 2013): 373–84. http://dx.doi.org/10.1017/jmech.2013.11.
Повний текст джерелаАнотація:
AbstractIn this paper, free vibration analysis of functionally graded rectangular plate is investigated based on the first order shear deformation theory and the effect of in-plane displacements on the natural frequencies of such plate is studied. The governing equations of motion are obtained, which are five coupled partial differential equations, without any simplification. Some mathematical manipulation leads us to decouple the equations. The decoupled equations are solved by the Levy's method for various boundary conditions. As the results show, in some boundary conditions the in-plane displacements cause a drastic change of frequencies. In other words, neglecting the in-plane displacement, which is assumed in some papers, is not proper for these boundary conditions. However, in the other boundary conditions, the natural frequencies are not significantly affected by the in-plane displacements. The results for various boundary conditions are discussed in detail and some interpretations for these differences are provided. Besides to the comparisons, the accurate natural frequencies of the plate for six different boundary conditions with several aspect ratios, thickness-length ratios and power law indices are presented. The natural frequencies of Mindlin functionally graded rectangular plates with considering the in-plane displacements are reported for the first time and can be used as benchmark.