Non-linear vibration analysis of specially orthotropic tapered micro-plates with arbitrary located crack: A non-classical analytical approach

The present work aims to study the non-linear vibrations in a cracked orthotropic tapered micro-plate. Linear and parabolic variation in the plate thickness is assumed in one as well as two directions. The partial crack is located in the centre, and it is continuous; this crack’s location is arbitrary and can be varied within the centre-line. Based on classical plate theory, the equilibrium principle is applied, and the governing equation of tapered orthotropic plate is derived. Additionally, the microstructure’s effect has been included in the governing equation using the non-classical modified couple stress theory. The simplified line spring model is used to consider the impact of partial crack on the plate dynamics and is incorporated using in-plane forces and bending moments. The introduction of Berger’s formulation brings the non-linearity in the model in terms of in-plane forces. Here, Galerkin’s method has been chosen for converting the derived governing equation into time-dependent modal coordinates, which uses an approximate solution technique to solve the non-linear Duffing equation. The crack is considered along the fibres and across the fibres to show the effect of orthotropy. Results are presented for an orthotropic cracked plate with non-uniform thickness. The effects of the variation of taper constants, crack location, crack length, internal material length scale parameter on the fundamental frequency are obtained for two different boundary conditions. The non-linear frequency response curves are plotted to show the effect of non-linearity on the system dynamics using the method of multiple scales, and the contribution of taper constants and crack parameters on non-linearity is shown with bending-hardening and bending-softening phenomenon. It has been found that vibration characteristics are affected by the taper parameters and fibre direction for a cracked orthotropic plate.

NEW SOLUTION TO THE PROBLEM OF A CRACK IN AN ORTHOTROPIC PLATE UNDER TENSION

Abstract— A classical plane problem of the theory of elasticity about a crack in a stretched orthotropic elastic unbounded plane is considered, which leads to a singular solution for stresses in the vicinity of the crack edge. The relations of the generalized theory of elasticity, including a small scale parameter, are given. The equations of the generalized theory are of a higher order than the equations of the classical theory and allow eliminating the singularity of the classical solution. The scale parameter is determined experimentally. The results obtained determine the effect of the crack length on the bearing capacity of the plate and are compared with the experimental results for plates made of fiberglass and carbon fiber reinforced plastic.

Sound Enhancement of Orthotropic Sound Radiation Plates Using Line Loads and Considering Resonance Characteristics

A new vibro-acoustic method is presented to analyze the sound radiation behavior of orthotropic panel-form sound radiators using strip-type exciters to exert line loads to the panels for sound radiation. The simple first-order shear deformation theory together with the Ritz method is used to formulate the proposed method that makes the vibro-acoustic analysis of elastically restrained stiffened orthotropic plates more computationally efficient than the methods formulated on the basis of the other shear deformation theories. An elastically restrained orthotropic plate consisting of two parallel strip-type exciters was tested to measure the experimental sound pressure level curve for validating the effectiveness and accuracy of the proposed method. The resonance characteristics (natural frequency and mode shape) detrimental to sound radiation are identified in the vibro-acoustic analysis of the orthotropic plate. For any orthotropic sound radiation plate, based on the detrimental mode shapes, a practical procedure is presented to design the line load locations on the plate to suppress the major sound pressure level dips for enhancing the smoothness of the plate sound pressure level curve. For illustration, the sound radiation enhancement of orthotropic plates with different fiber orientations for aspect ratios equal to 3, 2, and 1 subjected to one or two line loads is conducted using the proposed procedure. The results for the cases with two line loads perpendicular to the fiber direction and located at the nodal lines of the major detrimental mode shape may find applications in designing orthotropic panel-form speakers with relatively smooth sound pressure level curves.

Statement and solution the problem of designing of water-tight bulkheads web frames for large container vessels by using DNV-GL rules requirements and optimization techniques

Поперечные переборки крупнотоннажных контейнерных судов представляют собой сложные конструкции, при проектировании которых обычно используется методология проверочного расчета на основе метода конечных элементов (МКЭ). Для создания конечно-элементной модели необходимо знать размеры всех элементов конструкций, входящих в состав переборки. Поэтому такой подход к проектированию является итерационным, что обуславливает высокую трудоемкость процесса проектирования. На ранних стадиях проектирования размеры конструкций поперечных переборок контейнеровоза могут быть быстро и достаточно точно оценены на основе аппарата нелинейного программирования, относительно простой модели составной (конструктивно-ортотропной) пластины и нормативных требований Правил классификационных обществ. Такой подход применяется в Российской практике при проектировании двойных конструкций типа двойное дно или понтон плавучего дока. В статье предложено решение задачи проектирования рамного набора поперечной переборки крупнотоннажного контейнеровоза на нагрузки от контейнеров, действующие на переборку при качке судна. Конструкция переборки приводится к условной модели «коффердамного» типа. Затем используется методика приведения составной «конструктивно-ортотропной» пластины к изотропной пластине с несколько иным соотношением сторон, но с такими же параметрами изгиба. Это позволяет применить существующие табличные данные для определения расчетных изгибающих моментов и перерезывающих сил, которые после аппроксимации представляются в виде полиномов – аналитических зависимостей. Показана постановка оптимизационно-поисковой задачи математического программирования. Целевая функция – характеристика массы рамного набора. Ограничения задачи формируются на основе нормативных требований DNV-GL и математических зависимостей модели составной пластины. Для решения задачи используется инструмент MS Excel «Поиск решения» Представлены результаты тестового проектного расчета применительно к конструкции крупнотоннажного контейнеровоз с контейнерной вместимостью 18 тыс. TEU. Сопоставление результатов проектирования с оригинальными расчетами фирмы – проектанта показали удовлетворительное соответствие. The transverse bulkheads of the large container vessels are complex structures that are commonly designed using the finite element method (FEM) verification methodology. To create a finite element model, it is necessary to know the dimensions of all structural elements of the bulkhead. Therefore, this approach to design is iterative, which leads to a high complexity of the design process. At the early stages of design, the dimensions of the structures of the transverse bulkheads of a container vessel can be quickly and accurately estimated based on the nonlinear programming technique, a relatively simple model of a composite (structural-orthotropic) plate, and the regulatory requirements of the Rules of Classification Societies. This approach is used in practice in Russia when designing double structures such as a double bottom or pontoon of a floating dock. The article proposes a solution to the problem of the transverse bulkhead web frames designing in application to a large-tonnage container vessel for loads of containers acting on the bulkhead when the vessel is moving on the waves. The bulkhead structure is reduced to the conditional "cofferdam" type model. The technique is used to reduce a composite "structurally-orthotropic" plate to an isotropic one with a slightly different aspect ratio, but with the same bending parameters. This allows applying the existing tabular data to determine the design bending moments and shear forces, which, after approximation, are represented as polynomial analytical dependencies. The statement the optimization-search problem of mathematical programming is shown. The goal function is the characteristic of the bulkhead's webs mass. The constraints of the problem are formed on the DNV-GL regulatory requirements and mathematical relationships of the composite plate model. MS Excel tool "Solver" is used to solve the problem. The results of a test calculation are presented as applied to a large-capacity container ship with container capacity of 18000 TEU. Comparison of the design results with the original calculations of the designer’s company showed satisfactory agreement.

On the Green function of an orthotropic clamped plate in a half-plane

First, we calculate, in a heuristic manner, the Green function of an orthotropic plate in a half-plane which is clamped along the boundary. We then justify the solution and generalize our approach to operators of the form $$(Q(\partial ')-a^2\partial _n^2)(Q(\partial ')-b^2\partial _n^2)$$ ( Q ( ∂ ′ ) - a 2 ∂ n 2 ) ( Q ( ∂ ′ ) - b 2 ∂ n 2 ) (where $$\partial '=(\partial _1,\dots ,\partial _{n-1})$$ ∂ ′ = ( ∂ 1 , ⋯ , ∂ n - 1 ) and $$a>0,b>0,a\ne b)$$ a > 0 , b > 0 , a ≠ b ) with respect to Dirichlet boundary conditions at $$x_n=0.$$ x n = 0 . The Green function $$G_\xi $$ G ξ is represented by a linear combination of fundamental solutions $$E^c$$ E c of $$Q(\partial ')(Q(\partial ')-c^2\partial _n^2),$$ Q ( ∂ ′ ) ( Q ( ∂ ′ ) - c 2 ∂ n 2 ) , $$c\in \{a,b\},$$ c ∈ { a , b } , that are shifted to the source point $$\xi ,$$ ξ , to the mirror point $$-\xi ,$$ - ξ , and to the two additional points $$-\frac{a}{b}\xi $$ - a b ξ and $$-\frac{b}{a}\xi ,$$ - b a ξ , respectively.