Set Of Explicit Formulas Research Articles

And optimal designs of tuned inerter dampers for base motion excited structures with inherent damping

Tuned inerter damper (TID) has recently gained increasing attention as a new structural control mechanism for seismic protection of structures. Currently, theoretical investigations are undertaken by researchers to reveal its fundamentals and to understand its underline principles in altering the structural performances of structures against dynamic loadings. However, the comprehensive study of optimization design of TID for undamped structures is lacking and the majority of the research focuses on the optimization of TID for structures without any damping. This research evaluates the and optimal designs of TIDs on the structures with damping. Using SDOF structure as an example, the frequency response function of the system equipped with TID underground motion excitation is obtained. The and designs of TID for structures without damping are derived considering various response parameters using analytical method. A numerical search method is utilized for the and designs of TID for structures with damping; meanwhile, a set of explicit formulae are obtained by curve-fitting for convenience in the application. Finally, the relative motion response of the inerter is explored, and an optimal design formula of TID which can reduce the displacements of primary mass and inerter simultaneously, is proposed.

Real Harmonic Analysis on the Special Orthogonal Group

This paper presents theoretical analysis and software implementation for real harmonics analysis on the special orthogonal group. Noncommutative harmonic analysis for complex-valued functions on the special orthogonal group has been studied extensively. However, it is customary to treat real harmonic analysis as a special case of complex harmonic analysis, and there have been limited results developed specifically for real-valued functions. Here, we develop a set of explicit formulas for real-valued irreducible unitary representations on the special orthogonal group, and provide several operational properties, such as derivatives, sampling, and Clebsch-Gordon coefficients. Furthermore, we implement both of complex and real harmonics analysis on the special orthogonal group into an open source software package that utilizes parallel processing through the OpenMP library. The efficacy of the presented results are illustrated by benchmark studies and an application to spherical shape matching.

Formulization of Three-Dimensional Stress and Strain Fields at Elliptical Holes in Finite Thickness Plates

Stress raisers such as holes are inevitable in structures at which stress concentration occurs and the static as well as fatigue strength of the structures can be significantly weakened. Therefore, to accurately evaluate the stress concentration factor and stress fields at holes is of essential importance for structure design and service life prediction. Although stress and strain concentration and fields at holes in finite thickness plates strongly change with and along the thickness, manuals of stress concentration for engineering design are mainly based on two-dimensional theory and no explicit formula is available even for circular holes in finite thickness plates. Here we obtain for the first time a complete set of explicit formulae for stress and strain concentration factors and the out-of-plane constraint factor at circular as well as elliptical holes in finite thickness plates by integrating comprehensive three-dimensional finite element analyses and available theoretical solutions. The three-dimensional stress distributions ahead of holes can also be predicted by the obtained formulae. With their accuracy and the corresponding applicable range being analyzed and outlined in detail, the formulae can serve as an important fundamental solution for three-dimensional engineering structure design and guideline for developing three-dimensional analytical methods.

Radiative corrections beyond the ultra relativistic limit in unpolarized ep elastic and Møller scatterings for the PRad Experiment at Jefferson Laboratory

The clear 7σ discrepancy between measurements of the proton charge radius from muonic hydrogen Lamb shifts and those from hydrogen Lamb shift and electron scattering lead to both intense theoretical and experimental efforts to understand and explain this difference. In this regard, a new experiment (PRad) based on unpolarized ep elastic scattering cross section measurements normalized to Møller scattering is underway at Jefferson Laboratory to extract the proton charge radius based on new proton electric form factor down to values of momentum transfer squared Q 2, as low as 10−4 GeV/c)2. To reach the precision of the experiment in such a small Q 2 region requires reliable knowledge of radiative corrections. In this paper, we present a complete calculation of radiative corrections for unpolarized elastic ep and Møller scatterings performed within a covariant formalism resulting in the set of explicit formulas beyond the ultra relativistic approximation (m 2 ≪ Q 2), and numerical results for the kinematics of the PRad experiment.

Wave Kinematics and Pressure Field of Third-Order Theory for Bichromatic Bi-Directional Waves in Water of Finite Depth

Based on the third-order theory for bichromatic bi-directional waves in water of finite depth, a set of explicit formulas for the state-of-the art quantities of wave kinematics for horizontal and vertical particle displacements, velocities and accelerations, and wave pressure field is developed, and would be much more accurate and realistic in the design of harbor, coastal and offshore structures and their structural members.

In-Plane Free Vibrations of an Inclined Taut Cable

The investigation of the free vibrations of inclined taut cables has been a significant subject due to their wide applications in various engineering fields. For this subject, accurate analytical expression for the natural modes and the natural frequencies is of great importance. In this paper, the free vibration of an inclined taut cable is further investigated by accounting for the factor of the weight component parallel to the cable chord. Two coupled linear differential equations describing two-dimensional in-plane motion of the cable are derived based on Newton’s law. By variable substitution, the equation of the transverse motion becomes a Bessel equation of zero order when the equation of longitudinal motion is ignored. Solving the Bessel equation with the given boundary conditions, a set of explicit formulae is presented, which is more accurate for determining the natural frequencies and the modal shapes of an inclined taut cable. The accuracy of the proposed formulae is validated by numerical results obtained by the Galerkin method. The influences of two characteristic parameters λ and ε on the natural frequencies and modal shapes of an inclined taut cable are studied. The results are discussed and compared with those of other literatures. It appears that the present theory has an advantage over others in the aspect of accuracy, and may be used as a base for the correct analysis of linear and nonlinear dynamics of cable structures.

Explicit construction of high-pass filter sequence for orthogonal multiwavelets

For the construction of multiwavelets, there are no unified, explicit formulas as that in the scalar case available so far. In this paper, by studying the relationship between length 3 and length 4 filter sequences of orthogonal multiwavelets based on the result of Chui for the construction of length 3 orthogonal multiwavelets, a set of explicit formulas is given for the construction of length 4 high-pass filter sequence. Examples demonstrate that our proposed approach not only provides explicit formulas for the construction of length 4 high-pass filter sequence with multiplicity r, but also yields a set of new low-pass and high-pass filter sequences with length 3 and multiplicity 2 r .

The generalized triphasic correspondence principle for simultaneous determination of the mechanical properties and proteoglycan content of articular cartilage by indentation

The triphasic mixture theory has been used to describe the mechanical and physicochemical behaviors of articular cartilage under some specialized loading conditions. However, the mathematical complexities of this theory have limited its applications for theoretical analyses of experimental studies and models for predicting cartilage and other biological tissues’ deformational behaviors. A generalized correspondence principle has been established in the present study, and this principle shows that the equilibrium deformational behavior of a charged-hydrated material under loading is identical to that of an elastic medium without charge. A set of explicit formulas has been derived to correlate the mechanical properties of an equivalent material with the intrinsic elastic moduli, fixed charge density and free-ion concentration within the cartilage tissue. The validity of these formulas is independent of the deformation state of the elastic solid matrix under an infinitesimal strain. Therefore they can be employed for any loading conditions, such as confined or unconfined compression, tension, and indentation tests, etc. In the current study, the fixed charge density of bovine cartilage is determined from the indentation creep data using this generalized correspondence principle. The proteoglycan content results were then compared with those from biochemical assay, yielding a linear regression slope of 1.034. Additionally a correspondence principle within a framework of cubic symmetry and a bilinear response in tension-compression (the conewise linear elasticity model) has also been developed to demonstrate the potential application of current methodology for inhomogeneous, anisotropic and nonlinear situations.

Predicting nanovibration of multi-layered graphene sheets embedded in an elastic matrix

A continuum-based plate model is proposed to study the vibration behavior of multi-layered graphene sheets (MLGSs) that are embedded in an elastic matrix. A set of explicit formulas is derived to predict the natural frequencies and associated vibration modes of double-layered and triple-layered graphene sheets that are embedded in an elastic matrix. Numerical simulations are carried out to examine the effects of the van der Waals (vdW) interaction on the natural frequencies. The results show that for a given combination of wavenumbers ( m, n), the lowest natural frequency of a MLGS is independent of the vdW interaction, but all of the other higher natural frequencies depend significantly on the vdW interaction. The influence of the moduli of the surrounding matrix is also investigated, and the results indicate that the classical natural frequency depends significantly on these moduli. For lower-order resonant frequencies, the effect of the surrounding matrix is very small and can be neglected, but the higher-order resonant frequencies depend significantly on the moduli, and the influence of the surrounding matrix must thus be taken into consideration. In addition, the resonance modes of a MLGS system are dominated by the vdW interaction, whereas the moduli of the surrounding matrix only slightly affect the relative amplitude ratios of the sheets and do not change the vibration direction.

Explicit formulas of propagation characteristics for unilateral finline

In this paper, the unilateral fintine characteristics including finite metallization thickness and depth of mounting grooves are analysed by transverse resonance principle combining moment method. The numerical results are presented, the influence of the finite metallization thickness and depth of grooves on the propagation characteristics are investigated in detail. Finally, a set of explicit formulas for the propagation characteristics of unilateral fintine are developed. To be compared with the rigorous numerical results, the errors are within ±1% for phase constant and ±2% for characteristic impedance respectivelly.