scholarly journals Asymptotic analysis of an elastic material reinforced with thin fractal strips

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mustapha El Jarroudi ◽  
Youness Filali ◽  
Aadil Lahrouz ◽  
Mustapha Er-Riani ◽  
Adel Settati
Keyword(s):  
Asymptotic Behavior ◽  
Elastic Material ◽  
Degrees Of Freedom ◽  
Hausdorff Metric ◽  
Sierpinski Gasket ◽  
Three Dimensional ◽  
Effective Energy ◽  
Energy Term ◽  
Sierpiński Gasket ◽  
Thin Boundary Layer

<p style='text-indent:20px;'>We study the asymptotic behavior of a three-dimensional elastic material reinforced with highly contrasted thin vertical strips constructed on horizontal iterated Sierpinski gasket curves. We use <inline-formula><tex-math id="M1">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-convergence methods in order to study the asymptotic behavior of the composite as the thickness of the strips vanishes, their Lamé constants tend to infinity, and the sequence of the iterated curves converges to the Sierpinski gasket in the Hausdorff metric. We derive the effective energy of the composite. This energy contains new degrees of freedom implying a nonlocal effect associated with thin boundary layer phenomena taking place near the fractal strips and a singular energy term supported on the Sierpinski gasket.</p>

scholarly journals SPECTRAL SELF-AFFINE MEASURES IN $\mathbb{R}^N$

2007 ◽  
Vol 50 (1) ◽  
pp. 197-215 ◽  
Author(s):  
Jian-Lin Li
Keyword(s):  
Large Class ◽  
Sierpinski Gasket ◽  
Three Dimensional ◽  
Iterated Function Systems ◽  
Sierpiński Gasket ◽  
Spectral Pair ◽  
Iterated Function ◽  
Spectral Pairs ◽  
The Given ◽  
Check Condition

AbstractThe aim of this paper is to investigate and study the possible spectral pair $(\mu_{M,D},\varLambda(M,S))$ associated with the iterated function systems $\{\phi_{d}(x)= M^{-1}(x+d)\}_{d\in D}$ and $\{\psi_{s}(x)=M^{\ast}x+s\}_{s\in S}$ in $\mathbb{R}^n$. For a large class of self-affine measures $\mu_{M,D}$, we obtain an easy check condition for $\varLambda(M,S)$ not to be a spectrum, and answer a question of whether we have such a spectral pair $(\mu_{M,D},\varLambda(M,S))$ in the Eiffel Tower or three-dimensional Sierpinski gasket. Further generalization of the given condition as well as some elementary properties of compatible pairs and spectral pairs are discussed. Finally, we give several interesting examples to illustrate the spectral pair conditions considered here.

Spectrality of self-affine measures on the three-dimensional Sierpinski gasket

2012 ◽  
Vol 55 (2) ◽  
pp. 477-496 ◽  
Author(s):  
Jian-Lin Li
Keyword(s):  
Spectral Measure ◽  
Sierpinski Gasket ◽  
Three Dimensional ◽  
Standard Basis ◽  
The Self ◽  
Exponential Functions ◽  
Sierpiński Gasket

AbstractThe self-affine measure μM, D corresponding to M = diag[p1, p2, p3] (pj ∈ ℤ \ {0, ± 1}, j = 1, 2, 3) and D = {0, e1, e2, e3} in the space ℝ3 is supported on the three-dimensional Sierpinski gasket T(M, D), where e1, e2, e3 are the standard basis of unit column vectors in ℝ3. We shall determine the spectrality and non-spectrality of μM, D, and show that if pj ∈ 2ℤ \ {0, 2} for j = 1, 2, 3, then μM, D is a spectral measure, and if pj ∈ (2ℤ + 1) \ {±1} for j = 1, 2, 3, then μM, D is a non-spectral measure and there exist at most 4 mutually orthogonal exponential functions in L2(μM, D), where the number 4 is the best possible. This generalizes the known results on the spectrality of self-affine measures.

Non-spectrality of self-affine measures on the three-dimensional Sierpinski gasket

2019 ◽  
Vol 31 (6) ◽  
pp. 1447-1455 ◽  
Author(s):  
Zheng-Yi Lu ◽  
Xin-Han Dong ◽  
Peng-Fei Zhang
Keyword(s):  
Sierpinski Gasket ◽  
Three Dimensional ◽  
Diagonal Matrix ◽  
Exponential Functions ◽  
Sierpiński Gasket ◽  
Digit Set ◽  
Orthogonal Set

AbstractLet {\mu_{M,D}} be a self-affine measure generated by an expanding diagonal matrix {M\in M_{3}(\mathbb{R})} with entries {\rho_{1},\rho_{2},\rho_{3}} and the digit set {D=\{(0,0,0)^{t},(1,0,0)^{t},(0,1,0)^{t},(0,0,1)^{t}\}}. In this paper, we prove that for any {\rho_{1},\rho_{2},\rho_{3}\in(1,\infty)}, if {\rho_{1},\rho_{2},\rho_{3}\in\{\pm x^{\frac{1}{r}}:x\in\mathbb{Q}^{+},r\in% \mathbb{Z}^{+}\}}, then {L^{2}(\mu_{M,D})} contains an infinite orthogonal set of exponential functions if and only if there exist two numbers of {\rho_{1},\rho_{2},\rho_{3}} that are in the set {\{\pm(\frac{p}{q})^{\frac{1}{r}}:p\in 2\mathbb{Z}^{+},q\in 2\mathbb{Z}^{+}-1% \text{ and }r\in\mathbb{Z}^{+}\}}. In particular, if {\rho_{1},\rho_{2},\rho_{3}\in\{\frac{p}{q}:p,q\in 2\mathbb{Z}+1\}}, then there exist at most 4 mutually orthogonal exponential functions in {L^{2}(\mu_{M,D})}, and the number 4 is the best possible.

scholarly journals Dimer Coverings on the Sierpinski Gasket

2008 ◽  
Vol 131 (4) ◽  
pp. 631-650 ◽  
Author(s):  
Shu-Chiuan Chang ◽  
Lung-Chi Chen
Keyword(s):  
Sierpinski Gasket ◽  
Sierpiński Gasket

scholarly journals Design, Fabrication, and Testing of a Novel 3D 3-Fingered Electrothermal Microgripper with Multiple Degrees of Freedom

Micromachines ◽  
2021 ◽  
Vol 12 (4) ◽  
pp. 444
Author(s):  
Guoning Si ◽  
Liangying Sun ◽  
Zhuo Zhang ◽  
Xuping Zhang
Keyword(s):  
Degrees Of Freedom ◽  
Dynamic Properties ◽  
Three Dimensional ◽  
Polyimide Film ◽  
Zebrafish Embryos ◽  
Multiple Degrees Of Freedom ◽  
Electrothermal Actuator ◽  
3D Space ◽  
Pick And Place

This paper presents the design, fabrication, and testing of a novel three-dimensional (3D) three-fingered electrothermal microgripper with multiple degrees of freedom (multi DOFs). Each finger of the microgripper is composed of a V-shaped electrothermal actuator providing one DOF, and a 3D U-shaped electrothermal actuator offering two DOFs in the plane perpendicular to the movement of the V-shaped actuator. As a result, each finger possesses 3D mobilities with three DOFs. Each beam of the actuators is heated externally with the polyimide film. The durability of the polyimide film is tested under different voltages. The static and dynamic properties of the finger are also tested. Experiments show that not only can the microgripper pick and place microobjects, such as micro balls and even highly deformable zebrafish embryos, but can also rotate them in 3D space.

scholarly journals Metric approximations of spectral triples on the Sierpiński gasket and other fractal curves

2021 ◽  
Vol 385 ◽  
pp. 107771
Author(s):  
Therese-Marie Landry ◽  
Michel L. Lapidus ◽  
Frédéric Latrémolière
Keyword(s):  
Sierpinski Gasket ◽  
Sierpiński Gasket ◽  
Spectral Triples

scholarly journals Asymmetry in Three-Dimensional Sprinting with and without Running-Specific Prostheses

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 580
Author(s):  
Anna Lena Emonds ◽  
Katja Mombaur
Keyword(s):  
Degrees Of Freedom ◽  
Body Position ◽  
Three Dimensional ◽  
Problem Formulation ◽  
Contact Phase ◽  
Individual Level ◽  
Lower Trunk ◽  
Floating Base ◽  
The Asymmetry ◽  
Limb Asymmetry

As a whole, human sprinting seems to be a completely periodic and symmetrical motion. This view is changed when a person runs with a running-specific prosthesis after a unilateral amputation. The aim of our study is to investigate differences and similarities between unilateral below-knee amputee and non-amputee sprinters—especially with regard to whether asymmetry is a distracting factor for sprint performance. We established three-dimensional rigid multibody models of one unilateral transtibial amputee athlete and for reference purposes of three non-amputee athletes. They consist of 16 bodies (head, ipper, middle and lower trunk, upper and lower arms, hands, thighs, shanks and feet/running specific prosthesis) with 30 or 31 degrees of freedom (DOFs) for the amputee and the non-amputee athletes, respectively. Six DOFs are associated with the floating base, the remaining ones are rotational DOFs. The internal joints are equipped with torque actuators except for the prosthetic ankle joint. To model the spring-like properties of the prosthesis, the actuator is replaced by a linear spring-damper system. We consider a pair of steps which is modeled as a multiphase problem with each step consisting of a flight, touchdown and single-leg contact phase. Each phase is described by its own set of differential equations. By combining motion capture recordings with a least squares optimal control problem formulation including constraints, we reconstructed the dynamics of one sprinting trial for each athlete. The results show that even the non-amputee athletes showed less symmetrical sprinting than expected when examined on an individual level. Nevertheless, the asymmetry is much more pronounced in the amputee athlete. The amputee athlete applies larger torques in the arm and trunk joints to compensate the asymmetry and experiences a destabilizing influence of the trunk movement. Hence, the inter-limb asymmetry of the amputee has a significant effect on the control of the sprint movement and the maintenance of an upright body position.

Sign in / Sign up

Export Citation Format

Share Document