Solving dynamic equations in dye transport

Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

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Sufficient conditions for hyperbolicity and consistency of the dynamic equations for fluid-saturated solids

Archive of Applied Mechanics ◽

10.1007/s00419-021-01904-6 ◽

2021 ◽

Author(s):

Vladimir A. Osinov

Keyword(s):

Boundary Value Problem ◽

Wave Speed ◽

Boundary Value ◽

Sufficient Conditions ◽

Positive Definiteness ◽

Dynamic Equations ◽

Necessary Condition ◽

Systems Of Equations ◽

Acoustic Tensor ◽

Ill Posed

AbstractPrevious studies showed that the dynamic equations for a porous fluid-saturated solid may lose hyperbolicity and thus render the boundary-value problem ill-posed while the equations for the same but dry solid remain hyperbolic. This paper presents sufficient conditions for hyperbolicity in both dry and saturated states. Fluid-saturated solids are described by two different systems of equations depending on whether the permeability is zero or nonzero (locally undrained and drained conditions, respectively). The paper also introduces a notion of wave speed consistency between the two systems as a necessary condition which must be satisfied in order for the solution in the locally drained case to tend to the undrained solution as the permeability tends to zero. It is shown that the symmetry and positive definiteness of the acoustic tensor of the skeleton guarantee both hyperbolicity and the wave speed consistency of the equations.

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Free vibration and sound insulation of functionally graded honeycomb sandwich plates

Journal of Sandwich Structures & Materials ◽

10.1177/10996362211020440 ◽

2021 ◽

pp. 109963622110204

Author(s):

Fenglian Li ◽

Wenhao Yuan ◽

Chuanzeng Zhang

Keyword(s):

Free Vibration ◽

Poisson’S Ratio ◽

Functionally Graded ◽

Poisson's Ratio ◽

Dynamic Equations ◽

Sound Insulation ◽

Sandwich Plates ◽

Negative Poisson’S Ratio ◽

Honeycomb Sandwich ◽

Negative Poisson's Ratio

Based on the hyperbolic tangent shear deformation theory, free vibration and sound insulation of two different types of functionally graded (FG) honeycomb sandwich plates with negative Poisson’s ratio are studied in this paper. Using Hamilton’s principle, the vibration and vibro-acoustic coupling dynamic equations for FG honeycomb sandwich plates with simply supported edges are established. By applying the Navier’s method and fluid–solid interface conditions, the derived governing dynamic equations are solved. The natural frequencies and the sound insulation of FG honeycomb sandwich plates obtained in this work are compared with the numerical results by the finite element simulation. It is proven that the theoretical models for the free vibration and the sound insulation are accurate and efficient. Moreover, FG sandwich plates with different honeycomb cores are investigated and compared. The corresponding results show that the FG honeycomb core with negative Poisson’s ratio can yield much lower frequencies. Then, the influences of various geometrical and material parameters on the vibration and sound insulation performance are systematically analyzed.

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Dynamic Modeling and Characteristic Analysis of Cantilever Piezoelectric Bimorph

Mathematical Problems in Engineering ◽

10.1155/2019/3926906 ◽

2019 ◽

Vol 2019 ◽

pp. 1-9

Author(s):

Heng Chen ◽

Jun-shan Wang ◽

Chao Chen ◽

Shi-xiang Liu ◽

Hai-peng Chen

Keyword(s):

Axial Force ◽

Dynamic Characteristics ◽

Dynamic Equations ◽

Steady State Response ◽

Static Characteristics ◽

Piezoelectric Bimorph ◽

Characteristic Analysis ◽

Generalized Coordinates ◽

State Response ◽

Millisecond Range

The analytical model of an axially precompressed cantilever bimorph is established using the Hamilton’s principle in this study, and the static characteristics are obtained. The dynamic equations of the cantilever bimorph in generalized coordinates are established using a numerical method, and the dynamic characteristics are analyzed. Finally, simulations are performed and experiments are conducted to verify the validity of the theory. The results show that increase of axial force has significant amplification effects on the steady-state response amplitude of the displacement, and it reduces the resonance frequency. The response time is still in the millisecond range under a large axial force, which indicates that the bimorph has excellent dynamic characteristics as an actuator.

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