scholarly journals Calculation of the deflection of an arched truss with suspended elements depending on the number of panels

2020 ◽  
Vol 16 (3) ◽  
pp. 179-184
Author(s):  
Mikhail N. Kirsanov
Keyword(s):  
Asymptotic Properties ◽  
Analytical Calculation ◽  
General System ◽  
Uniform Load ◽  
Equilibrium Equations ◽  
Rounding Errors ◽  
Symbolic Form ◽  
Total Load ◽  
Independent Parameters ◽  
System Of Equilibrium Equations

The aim of the work - to propose a scheme and analytical calculation of a statically definable planar truss with a suspended lower belt. Methods. The formula for the dependence of the deflection of the truss under the action of a uniform load on the lower belt on its size and the number of panels is derived in the computer mathematics system Maple. The forces in the rods are found from the solution of the general system of equilibrium equations of all nodes in symbolic form. The deflection is calculated using the Maxwell - Mohr's formula. Generalization of a number of formulas for deflection obtained by increasing the number of panels sequentially to an arbitrary number is performed by double induction using two independent parameters. In this case, special operators of the Maple system are used, allowing for a sequence of coefficients in the desired formula to create and solve recurrent equations that satisfy the elements of the sequences. Results. The obtained solutions have a polynomial form for the number of panels. Curves of deflection dependence on the number of panels are constructed and analyzed. Asymptotic properties of solutions are found in the case of a fixed span length of the structure and a given total load. The proposed scheme is a statically determinate structure with two independent parameters of regularity allows for the finding of a fairly simple analytical solution. The resulting formula is most effective in calculating systems with a large number of elements, where numerical methods tend to accumulate rounding errors.

scholarly journals ANALYTICAL CALCULATION OF DEFLECTION OF A MULTI-LATTICE TRUSS WITH AN ARBITRARY NUMBER OF PANELS

2020 ◽  
Vol 16 (3) ◽  
pp. 23-29
Author(s):  
Mikhail Kirsanov
Keyword(s):  
Asymptotic Solution ◽  
Arbitrary Number ◽  
Analytical Calculation ◽  
Analytical Form ◽  
Linear Recurrence ◽  
Equilibrium Equations ◽  
Recurrence Equations ◽  
Cutting Out ◽  
Homogeneous Linear ◽  
System Of Equilibrium Equations

The scheme of a planar externally statically indeterminate truss with four supports is proposed. In analytical form, for several types of loads, the problem of forces in the rods and deflectionof the structure is solved, depending on the number of panels, the size and intensity of the load. The solution uses the Maple computer mathematics system. The deflectionat Midspan is determined using Maxwell – Mohr's formula, the forces in the rods – the method of cutting out nodes from the system of equilibrium equations for all nodes, which includes four reactions of the supports. By induction, a series of solutions for trusses with a consistently increasing number of panels is generalized to an arbitrary number of panels. For the elements of the sequences of coefficientare developed and are solved by homogeneous linear recurrence equations. The resulting formulas for the deflectio of the structure under various loads have the form of polynomials in the number of panels. A linear asymptotic solution for the number of panels is found. The kinematic degeneration of the structure and the distribution of node speeds corresponding to this case were found. The dependences of the reaction of supports and forces in the most compressed and stretched rods on the number of panels are determined.

scholarly journals ANALYTICAL CALCULATION OF THE DEFLECTIONOF A SPATIAL HINGE-ROD FRAME WITH AN ARBITRARY NUMBER OF PANELS

2020 ◽  
pp. 65-75
Author(s):  
M. N. Kirsanov
Keyword(s):  
Arbitrary Number ◽  
Numerical Solutions ◽  
Optimization Problems ◽  
Analytical Calculation ◽  
Induction Method ◽  
Symbolic Form ◽  
Regular Type ◽  
Proposed Model ◽  
Different Types ◽  
Independent Parameters

Statement of the problem. The task is to obtain in symbolic form the dependence of the deflection of the proposed scheme of a statically definable spatial truss of a regular type on the number of panels under various loads, including the load from the truss plane. A truss has two independent parameters that define its proportions.Results. For several types of loading according to the Maxwell - Mohr formula, analytical dependences of the deflections of the structure on the number of panels, load, and dimensions are derived. When generalizing a series of partial solutions with a given number of panels to an arbitrary number of panels, together with operators of the Maple computer mathematics system, the induction method is used. Asymptotic approximations of solutions are obtained.Conclusions. The proposed model of a spatial frame with two independent numbers of panels that define the proportions of the structure allows an analytical solution of the problem of deflection under different types of loading. The derived formulas can be used as test formulas for evaluating approximate numerical solutions and for optimization problems.

Boundary Integral Equation Methods for a Refined Model of Elastic Plates

2006 ◽  
Vol 11 (6) ◽  
pp. 642-654
Author(s):  
Radu Mitric ◽  
Christian Constanda
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Normal Strain ◽  
Elastic Plates ◽  
Equilibrium Equations ◽  
Neumann Problems ◽  
Integral Equation Methods ◽  
Transverse Normal Strain ◽  
System Of Equilibrium Equations

A theory of bending of elastic plates is considered, in which the effect of transverse shear deformation and transverse normal strain are taken into account through a specific form of the displacement field. It is shown that the system of equilibrium equations is elliptic and that Betti and Somigliana formulae can be established, which permit the solution of the interior and exterior Dirichlet and Neumann problems by means of boundary integral equation methods.

scholarly journals Analytical calculation of the deflection of the lattice truss

2018 ◽  
Vol 193 ◽  
pp. 03015 ◽  
Author(s):  
Mikhail Kirsanov ◽  
Dmitriy Tinkov
Keyword(s):  
Asymptotic Property ◽  
Analytical Calculation ◽  
Extreme Points ◽  
Horizontal Displacement ◽  
Vertical Force ◽  
Uniform Load

An algorithm is given for deriving the dependence of the deflection of a planar statically determinate beam truss on the number of panels, dimensions and load. Three load cases are considered: uniform load on the lower belt, upper belt and vertical force in the middle of the span. By induction, generalizing a series of solutions for trusses with a consecutively increasing number of panels, the desired formula is obtained for the deflection and horizontal displacement of the mobile support of the truss. All transformations are performed in the system of symbolic mathematics Maple. For a sequence of coefficients of the desired formula, using the special Maple operators, homogeneous recurrent equations are constructed and solved. The coefficients found are in the form of polynomials in the number of panels. The asymptotic property of the solution is found. On the graphs of the dependence of the deflection on the number of panels and on the height, extreme points are found. The solution can be used to test the calculations obtained numerically.

scholarly journals The Replication Hypothesis along the Take-Off Run and a System of Equilibrium Equations at the Lift-Off of a Protobird

Aerospace ◽  
2019 ◽  
Vol 6 (2) ◽  
pp. 21
Author(s):  
Phillip Burgers
Keyword(s):  
Experimental Data ◽  
Time Span ◽  
Flapping Wings ◽  
Equilibrium Equations ◽  
Scaling Down ◽  
Lift Off ◽  
To Come ◽  
System Of Equilibrium Equations

An extant bird resorts to flapping and running along its take-off run to generate lift and thrust in order to reach the minimum required wing velocity speed required for lift-off. This paper introduces the replication hypothesis that posits that the variation of lift relative to the thrust generated by the flapping wings of an extant bird, along its take-off run, replicates the variation of lift relative to the thrust by the flapping wings of a protobird as it evolves towards sustained flight. The replication hypothesis combines experimental data from extant birds with evidence from the paleontological record of protobirds to come up with a physics-based model of its evolution towards sustained flight while scaling down the time span from millions of years to a few seconds. A second hypothesis states that the vertical and horizontal forces acting on a protobird when it first encounters lift-off are in equilibrium as the protobird exerts its maximum available power for flapping, equaling its lift with its weight, and its thrust with its drag.

scholarly journals Form-Finding of Tensegrity Model with Triangular Cells

2018 ◽  
Vol 7 (3.36) ◽  
pp. 137
Author(s):  
Nur Farizah Filzah Naing ◽  
Oh Chai Lian ◽  
Ilyani Akmar Abu Bakar ◽  
Mohd Raizamzamani Md Zain
Keyword(s):  
Inverse Method ◽  
Equilibrium Equations ◽  
Form Finding ◽  
Tensegrity Structures ◽  
Weight Structure ◽  
Tensegrity Model ◽  
Generalized Inverse Method ◽  
Stage Class ◽  
System Of Equilibrium Equations ◽  
Finding Method

Tensegrity structures is a light-weight structure compared to concrete structures that are heavy and rigid in shape. The studies on form-finding for tensegrity configuration are still ongoing and have been extensively conducted. Additionally, many proposed tensegrity structures have not been built for real applications. This study aims to determine potential self-equilibrated configurations of three-stage Class I tensegrity model assemblage with triangular cells, which may be applied as deployable towers. The form-finding methodology involves phases in establishment of desired form and formulation for the self-equilibrated state. The system of equilibrium equations was solved by Moore-Penrose generalized inverse method.  A range of twist angles 10o – 50o for triangular cells was investigated in the form-finding process.  It was found that the form-finding method via changing of twist angles has successfully search self-equilibrated tensegrity models.  

On Some Classes of Correct Problems in the Membrane Theory of Convex Shells

2020 ◽  
pp. 25-29
Author(s):  
E.V. Tyurikov
Keyword(s):  
Smooth Boundary ◽  
Hilbert Problem ◽  
Boundary Value ◽  
Geometric Description ◽  
Membrane Theory ◽  
Equilibrium Equations ◽  
Wide Family ◽  
Boundary Solutions ◽  
System Of Equilibrium Equations ◽  
Riemann Hilbert Problem

On the basis of the theory of the modified Riemann-Hilbert problem for generalized analytic functions, a geometric description is given of a fairly wide family of correct by I. N. Vekua of boundary value problems of the membrane theory of convex hulls with a piecewise smooth boundary. Solutions to the corresponding Riemann-Hilbert problem for an elliptic system of equilibrium equations are found in the classes of N.I. Muskhelishvili and realize a state of tense equilibrium under the condition of stress concentration in corner points. An effective formula is given for calculating the index of the boundary condition, which allows us to formulate the results in a visible form. Families of shells are found for which the solvability picture of the main boundary-value problem coincides with the solvability picture of the Vekua problem for shells with a smooth border.

scholarly journals Lower estimate of the fundamental frequency of natural oscillations of a truss with an arbitrary number of panels

Vestnik MGSU ◽  
2019 ◽  
pp. 844-851
Author(s):  
Mikhail N. Kirsanov
Keyword(s):  
Fundamental Frequency ◽  
Arbitrary Number ◽  
Lower Estimate ◽  
Analytical Form ◽  
Equilibrium Equations ◽  
Rounding Errors ◽  
Natural Oscillations ◽  
Analytical Estimate ◽  
The Matrix ◽  
Analytical Result

Introduction: the paper deals with oscillations of a statically definable plane, truss with a double lattice of racks and descending braces with massive loads in the nodes of the lower chord. The weight of the truss rods is not taken into account. It is assumed that the freights are moved only vertically. The fundamental frequency of natural oscillations is estimated from the Dunkerley formula by the values of partial frequencies. Materials and methods: an analytical estimate is obtained by generalizing formulas obtained from a series of estimates for trusses with a consistently increasing number of panels. The stiffness of the truss was determined using the Mohr’s integral. The double lattice of the truss does not allow using the cross-section method; therefore, the forces in the rods were calculated (or estimated) in an analytical form using the method of cutting nodes with the compilation of a system of equilibrium equations simultaneously for all rods and three support reactions. The matrix of equilibrium equations was compiled in a software program written in the language of the Maple computer mathematics system based on the coordinates of the nodes and the values of the direction cosines of the forces. For a sequence of coefficients of the desired formula, linear homogeneous recurrent equations were found and solved by means of special operators of the Maple system. Results: the resulting formula estimating the relationship between the fundamental frequency and the panels number has the form of a sixth degree polynomial with coefficients depending on the parity of the number of panels. The analytical result is compared with the smallest frequency obtained numerically from the solution of the problem of oscillation of the cargo system. It is shown that the main frequency, depending on the truss height, has an extremum. Conclusions: the method of generalizing particular solutions using the Maple system operators allowed authors to obtain and analyze a formula for a lower estimate of the fundamental frequency of oscillation of a truss model with an arbitrary number of panels. The resulting estimate can be used as a test for numerically obtained solutions. The formula is especially efficient for systems with a large number of panels; as numerical methods for their calculation are time-consuming require considerable time and have a tendency for accumulating rounding errors.

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