The influence of a supporting canopy on the dynamics of opening of a parachute

A mathematical model of the opening of a circular parachute under the action of a supporting canopy was executed on a computer. The proposed model is based on the simultaneous integration of the equations of nonlinear aerodynamics and the theory of soft membranes. It is shown that the supporting canopy may both improve the characteristics of the parachute system and make them worse. The numerical results are presented in the form of graphs. One figure.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 23–26.     

Numerical study of forebody wake effect on axisymmetric parachute opening shock and drag reduction

ABSTRACT

Parachute–forebody distance is a parameter which is amongst the most critical factors to be considered in forebody wake effect. In this study, a new axisymmetric parachute–forebody coupling model is developed. Axisymmetric wrinkling membrane element is built to assess the dynamic response of the parachute canopy membrane under fluid pressure. Besides, fluid model and its further implementation on the fluid structure analysis are discussed. With the proposed method, the wake effect on both the opening shock during inflation state and the drag reduction during steady state can be obtained efficiently. Finally, numerical model is validated with published experimental result and further employed to investigate the influence of distance parameters on fluid–parachute coupling behaviour. On the basis of numerical results, failure distance during the inflation process and critical forebody–parachute distance are determined. The results show that forebody–parachute distance has a strong influence on flow behaviour around the parachute in both inflation state and steady descent state.     

The use of the simplex method in the design of parachute systems with use of a prototype

We develop a formal mathematical apparatus that makes it possible using the simplex method (linear programming) to carry out the design of parachute systems with use of a prototype, the experimental development of parachute systems using a computer, the analysis of a technical problem, and the identification of mathematical models of an automated parachute design system.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 46–49.     

State of stress and strain of brake-class parachutes

The article describes the special features of the operation of brake-class parachutes. It substantiates the possibility of replacing the solution of the dynamic problem of opening of a brake parachute by the static calculation of its state of stress and strain (SSS). The derivation of the equation of motion of a soft carcassed shell is based on the finite element method. The steady-state solution is obtained by the method of adjustment. As an example the results of calculation of the characteristics of the SSS of a cross-shaped brake parachute are presented. It is shown that in the zone of its lower edge considerable concentrations of tension arise in the tissue, the tissue gradually joins the operation of the carcass; the transverse carcass is unsubstantially loaded. The results of the calculations agree with the experimental data.Translated from Dinamicheskie Sistemy, No. 5, pp. 32–36, 1986.     

Opening of a thin axisymmetric shell in a flow of ideal fluid

This article examines the opening, in an ideal fluid, of an axisymmetric impermeable shell modeling the canopy of a circular parachute. The form of the shell is determined in the class of shapes representing the combination of a truncated cone and an ellipsoid. The forces of interaction of the shell and the fluid are calculated on the basis of the method of discrete vortices. A Lagrange equation of the second kind is used to determine the time-varying coordinates of the generatrix. Numerical results are presented.Translated from Dinamicheskie Sistemy, No. 5, pp. 37–41, 1986.     

Principles of development of an interactive computing system for analyzing the stability and inflatibility of a parachute

This article describes the structure of a software package developed to study the stationary modes of motion of parachute systems and analyze their stability. A brief discussion of optimality criteria for such packages is given. A classification of the instabilities in the mechanics of a parachute is presented.Translated fromDinamicheskie Sistemy, No. 6, pp. 3–10, 1987.     

The large-particle method in arbitrary nets and its application to the solution of problems in the aeroelasticity of a parachute

Modifications are introduced into the large-particle method to make it possible to use the method in arbitrary nonrectangular computing nets in the study of the flow of an ideal compressible gas over mobile thin permeable membranes. A problem in the aeroelasticity of a parachute is stated. An example is given of the solution of a problem in the stationary formulation for an axially symmetric parachute in the wake of a cylindrical leading body for supersonic flow.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 16–22     

Generalization of results of the computational modeling of streamline flow of a parachute by an undisturbed high-speed flow of gas using the method of large particles

On the basis of analysis of the numerical solution of the problem of aerodynamics of a parachute using the method of large particles, a simple analytical method of determining the pressure gradient on the canopy and the resistance coefficient of the parachute in a high-speed flow of gas is given. The results of computations coincide well with the experimental ones.Translated from Dinamicheskie Sistemy, No. 9, pp. 81–86, 1990.     

Studies of the dynamics of space motion of a class of parachute systems

A mathematical model of space motion of a mechanical system that consists of a load and a parachute is presented. With the help of the apparatus of the Lyapunov functions, a particular case of the model is studied when the load can be considered as a mass point that does not have resistance. A sufficient condition of stability is obtained. The results can be applied to the model/ing and construction of parachute systems.Translated from Dinamicheskie Sistemy, No. 9, pp. 99–102, 1990.     

Analytic integration of a system of nonlinear ordinary differential equations with an irregular type singularity

Summary We shall discuss how to construct analytic expressions for bounded solutions of nonlinear ordinary differential equations of the form(A) which tend to0 with the order of certain positive powers of x as x approaches the origin through a sector. It will be shown that the opening angle of this sector is maximal. Entrata in Redazione il 1^o luglio 1971.     

The interactive system “PARACHUTE”

The structure and uses of an interactive system are described. The results of computations of the resistance and inflatibility of corrugated parachute canopies are presented.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 30–35.     

Nonstationary breakaway flow over a permeable parachute

The properties of a uniformly permeable surface that models the canopy of a real parachute are presented. A modernization of the discrete vortex method is proposed that makes it possible to apply it to calculate the flow over uniformly permeable surfaces. The results of computations of the aerodynamic characteristics of permeable parachutes are presented and compared with experimental data.Translated fromDinamichekie Sistemy, No. 6, pp.3–10, 1987.     

Shape of the parachute canopy

The article presents new equations of equilibrium of an axisymmetric parachute canopy generalizing Kh. A. Rakhmatulin's well-known equations. The equations take the biaxial nature of the state of stress near the pole of the canopy into account, and this leads to a finite value of the tension of the tissue and to a finite value of the radius of curvature at the pole.Translated from Dinamicheskie Sistemy, No. 5, pp. 21–25, 1986.     

Stress-strain state of a soft braking system

Conclusions It was shown that step-by-step use of models 1 and 2 to determine the stress-strain state of an opening circular parachute on the basis of equations of the theory of soft shells makes it possible to conduct parametric studies of problems of the statics and dynamics of such systems on a computer of moderate capacity and to significantly (by one or two orders of magnitude) reduce computing time.Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 839–845, September–October, 1987.     

The influence of the properties of elasticity of a soft axisymmetrical shell on the characteristics of its filling in the flow of a compressible gas

For a soft axisymmetrical shell, which represents a conical canopy of a parachute with a cylindrical insert, some regularities of the process of its filling in a subsonic and supersonic flow under different levels of load and stiffness of the shell are considered. The computations are carried out with the help of a specially developed aeroelastic model.Translated from Dinamicheskie Sistemy, No. 9, pp. 94–98, 1990.     

Pulsations of the parachute canopy

The problem of pulsations of the canopy of a parachute is considered. Equations which are linearized in a neighborhood of the stationary state of motion are given, the algorithm of solution that is based on the composition of the methods of discrete curls and finite differences is described. The results of computations of the characteristics of frequency of the canopy are given in the form of graphs.Translated from Dinamicheskie Sistemy, No. 9, pp. 87–94, 1990.     

Mathematical model of the dynamics of the canopy of an axisymmetric parachute

The article presents a mathematical model of the dynamics of the canopy of an axisymmetric parachute taking into account the biaxiality of the state of complex stress of the tissue, the effect of the rigging lines of the annular carcass, viscous deformation, riffling. For the solution a threelayered symmetric implicit difference schema is suggested. The method of accelerated attainment of a stable solution without distortion of the dynamic pattern is explained.Translated from Dinamicheskie Sistemy, No. 5, pp. 25–32, 1986.     

On maximal commutative subrings of non-commutative rings

We observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exactly three (resp., four) maximal commutative subrings are characterized. If R has acc or dcc on its commutative subrings containing the center, whose intersection with the nontrivial summands is trivial, then R is Dedekind-finite. It is observed that every Artinian commutative ring R, is a finite intersection of some Artinian commutative subrings of a non-commutative ring, in each of which, R is a maximal subring. The intersection of maximal ideals of all the maximal commutative subrings in a non-commutative local ring R, is a maximal ideal in the center of R. A ring R with no nontrivial idempotents, is either a division ring or a right ue-ring (i.e., a ring with a unique proper essential right ideal) if and only if every maximal commutative subring of R is either a field or a ue-ring whose socle is the contraction of that of R. It is proved that a maximal commutative subring of a duo ue-ring with finite uniform dimension is a finite direct product of rings, all of which are fields, except possibly one, which is a local ring whose unique maximal ideal is of square zero. Analogues of Jordan-Hölder Theorem (resp., of the existence of the Loewy chain for Artinian modules) is proved for rings with acc and dcc (resp., with dcc) on commutative subrings containing the center. A semiprime ring R has only finitely many maximal commutative subrings if and only if R has a maximal commutative subring of finite index. Infinite prime rings have infinitely many maximal commutative subrings.     

On intrinsic ergodicity of piecewise monotonic transformations with positive entropy

We consider a class of piecewise monotonically increasing functionsf on the unit intervalI. We want to determine the measures with maximal entropy for these transformations. In part I we construct a shift-space Σ _f ⁺ isomorphic to (I, f) generalizing the \-shift and another shift Σ _M over an infinite alphabet, which is of finite type given by an infinite transition matrixM. Σ _M has the same set of maximal measures as (I, f) and we are able to compute the maximal measures of maximal measures of. In part II we try to bring these results back to (I, f). There are only finitely many ergodic maximal measures for (I, f). The supports of two of them have at most finitely many points in common. If (I, f) is topologically transitive it has unique maximal measure.