Differential Quadrature Method: Application to Initial- Boundary-Value Problems

Tomasiello, Stefania

doi:10.1006/jsvi.1998.1833

Cited by 71 publications

(45 citation statements)

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“…141,142 In DQM the derivative of a function with respect to a variable at a given point is approximated as a weighted linear sum of the function values at all discrete points in the range of that variable. 138 In terms of dimensionless variables, the rth order derivative of a function w(z) at z ¼ z i , dened between 0 and 1 with N discrete grid points, is given by.…”

Section: Transient Phenomena and The Differential Quadrature Methods (mentioning

confidence: 99%

Single cycle and transient force measurements in dynamic atomic force microscopy

Gadelrab¹,

Santos²,

Font

et al. 2013

Nanoscale

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The monitoring of the deflection of a micro-cantilever, as the end of a sharp probe mounted at its end, i.e. the tip, interacts with a surface, forms the foundation of atomic force microscopy AFM. In a nutshell, developments in the field are driven by the requirement of obtaining ever increasing throughput and sensitivity, and enhancing the versatility of the instrument to simultaneously map the topography and quantify nanoscale processes and properties. In the most common dynamic mode of operation, the motion of the driven cantilever is monitored at a single point on its longitudinal axis. Here, we show that from this single point a waveform is obtained that contains all the details about conservative and dissipative interactions. Then a formalism that accounts for multiple arbitrary flexural modes is developed for an indirectly driven cantilever. The formalism is shown to allow recovery of the details of the interaction even in the presence of complex and relevant hysteretic forces when the cantilever oscillates in the steady state. In a different approach, we develop a formalism that monitors the wave profile of the cantilever, i.e. the waveform at five different points on its longitudinal axis. With this formalism the interaction can be reconstructed during a single oscillation cycle even in the transient state of oscillation. Finally, we discuss the potential and advantages of the proposed methods and future technical challenges. Other standard and state of the art techniques and methods are also discussed and compared with the ones presented here. This work should also provide insight into the current high throughput-high sensitivity developments dealing with multifrequency dynamic AFM where information is recovered from multiple eigenmodes.

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Section: Transient Phenomena and The Differential Quadrature Methods (mentioning

confidence: 99%

Single cycle and transient force measurements in dynamic atomic force microscopy

Gadelrab¹,

Santos²,

Font

et al. 2013

Nanoscale

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show abstract

“…However, after some investigations, it was found that the present nonlinear analysis is extremely sensitive to the sampling point distribution. Thus, Chebyshev-Gauss-Lobatto distribution is not a right choice for this type of problems, but the suggested distribution by Tomasiello (1998) can be used as an alternative choice. Then, the resulting ordinary differential equations are solved via the 4th order Runge-Kutta method.…”

Section: Resultsmentioning

confidence: 99%

“…Also, it was found that the nonlinear panel flutter analysis with GDQM is extremely sensitive to the grid point distribution. Therefore, the well-known Chebyshev-Gauss-Lobatto distribution is not suitable for this type of problems, but Tomasiello's distribution (Tomasiello, 1998) can be suggested as an efficient and suitable choice.…”

Section: Discussionmentioning

confidence: 99%

Aerothermoelastic Analysis of Functionally Graded Plates Using Generalized Differential Quadrature Method

Shahverdi

Khalafi

Noori

2016

Lat. Am. j. solids struct.

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In the present paper, the aerothermoelastic behavior of Functionally Graded (FG) plates under supersonic airflow is investigated using Generalized Differential Quadrature Method (GDQM). The structural model is considered based on the classical plate theory and the von Karman strain-displacement relations are utilized to involve the nonlinear behavior of the plate. To consider the supersonic aerodynamic loads on the plate, the first order piston theory is applied. The material properties of the FG panel are assumed to be temperature independent and alter in the thickness direction according to a power law distribution. The temperature distribution on the surface of the plate is assumed to be constant and in the thickness direction is obtained by one-dimensional steady conductive heat transfer equation. The discretized governing equations via GDQM are solved by the fourth order Runge-Kutta method. Comparison of the obtained results with those available in literature confirms the accuracy and ability of the GDQM to perform the aerothermoelastic analysis of FG plates. Also, the effect of some important parameters such as Mach number, inplane thermal load, plate aspect ratio and volume fraction index on the plate aerothermoelastic behavior is examined.

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“…Chen and Zhong [18] presented the study on the nonlinear computations of the differential quadrature method and differential cubature method. Tomasiello [19] applied the differential quadrature method to initialboundary-value problems. Wang et al [20] presented the free vibration analysis of circular annular plates with nonuniform thickness by the differential quadrature method.…”

Section: The Differential Quadrature Methodsmentioning

confidence: 99%

The Effect of Residual Stress on the Electromechanical Behavior of Electrostatic Microactuators

Hsu

2008

Active and Passive Electronic Components

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This work simulates the nonlinear electromechanical behavior of different electrostatic microactuators. It applies the differential quadrature method, Hamilton's principle, and Wilson-θ integration method to derive the equations of motion of electrostatic microactuators and find a solution to these equations. Nonlinear equation difficulties are overcome by using the differential quadrature method. The stresses of electrostatic actuators are determined, and the residual stress effects of electrostatic microactuators are simulated.

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Differential Quadrature Method: Application to Initial- Boundary-Value Problems

Cited by 71 publications

References 0 publications

Single cycle and transient force measurements in dynamic atomic force microscopy

Single cycle and transient force measurements in dynamic atomic force microscopy

Aerothermoelastic Analysis of Functionally Graded Plates Using Generalized Differential Quadrature Method

The Effect of Residual Stress on the Electromechanical Behavior of Electrostatic Microactuators

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