AN APPROXIMATE METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS WITH MOVING BOUNDARIES IN THE DEVELOPED SOFTWARE PACKAGE TB-ANALISYS Текст научной статьи по специальности «Математика»

НАУЧНОЕ ПРОГРАММНОЕ ОБЕСПЕЧЕНИЕ В ОБРАЗОВАНИИ И НАУКЕ / SCIENTIFIC SOFTWARE IN EDUCATION AND SCIENCE

udc 534.11 hhfifflaiaehaib

doi: 10.25559/sitito.17.202102.432-441

An Approximate Method for Solving Boundary Value Problems with Moving Boundaries in the Developed Software Package TB-Analisys

V. L. Litvinovab*, K. V. Litvinovab

a Samara State Technical University, Samara, Russian Federation 244 Molodogvardeyskaya St., Samara 443100, Russian Federation b Lomonosov Moscow State University, Moscow, Russian Federation 1 Leninskie gory, Moscow 119991, GSP-1, Russian Federation * [email protected]

Abstract

The problem of vibrations of objects with moving boundaries, formulated as a differential equation with boundary and initial conditions, is a nonclassical generalization of a hyperbolic problem. To facilitate the construction of the solution to this problem and to justify the choice of the form of the solution, equivalent integro-differential equations with symmetric and time-dependent kernels and time-varying integration limits are constructed. The advantages of the method of integro-differential equations are revealed in the transition to more complex dynamic systems carrying concentrated masses vibrating under the action of moving loads. The method is extended to a wider class of model boundary value problems that take into account the bending stiffness, the resistance of the external environment, and the stiffness of the base of an oscillating object. Particular attention is paid to the consideration of the most common case in practice, when external disturbances act at the borders. The solution is made in dimensionless variables accurate to second-order values of relatively small parameters characterizing the velocity of the boundary. An approximate solution is found to the problem of transverse vibrations of a rope of a lifting device, which has bending rigidity, one end of which is wound on a drum, and a load is fixed on the other. The results obtained for the amplitude of oscillations corresponding to the nth dynamic mode are presented. The phenomenon of steady-state resonance and passage through resonance is investigated using numerical methods. The solution is made in dimensionless variables using the TB-Analisys software package developed in the Matlab environment, which allows using the results obtained for calculating a wide range of technical objects.

Keywords: resonance properties, oscillations of systems with moving boundaries, laws of motion of boundaries, integro-differential equations, amplitude of oscillations

The authors declare no conflict of interest.

For citation: Litvinov V.L., Litvinova K.V. An Approximate Method for Solving Boundary Value Problems with Moving Boundaries in the Developed Software Package TB-Analisys. Sovremennye informa-cionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2021; 17(2):432-441. DOI: https://doi.org/10.25559/SITITO.17.202102.432-441

Контент доступен под лицензией Creative Commons Attribution 4.0 License. The content is available under Creative Commons Attribution 4.0 License.

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Том 17, № 2. 2021 ISSN 2411-1473 sitito.cs.msu.ru

НАУЧНОЕ ПРОГРАММНОЕ ОБЕСПЕЧЕНИЕ В ОБРАЗОВАНИИ И НАУКЕ

Приближенный метод решения краевых задач с подвижными границами в разработанном программном комплексе ТВ-ЛпаНБуБ

В. Л. Литвинов1'2*, К. В. Литвинова2

1 ФГБОУ ВО «Самарский государственный технический университет», г. Самара, Российская Федерация

443100, Российская Федерация, г. Самара, ул. Молодогвардейская, д. 244

2 ФГБОУ ВО «Московский государственный университет имени М. В. Ломоносова», г. Москва, Российская Федерация

119991, Российская Федерация, г. Москва, ГСП-1, Ленинские горы, д. 1 * [email protected]

Аннотация

Задача о колебаниях объектов с движущимися границами, сформулированная в виде дифференциального уравнения с граничными и начальными условиями, является неклассическим обобщением гиперболической задачи. Для облегчения построения решения этой задачи и обоснования выбора формы решения построены эквивалентные интегро-дифференциальные уравнения с симметричными и зависящими от времени ядрами и изменяющимися во времени пределами интегрирования. Преимущества метода интегро-дифференциальных уравнений раскрываются при переходе к более сложным динамическим системам, несущим сосредоточенные массы, колеблющиеся под действием подвижных нагрузок. Метод распространяется на более широкий класс модельных краевых задач, которые учитывают жесткость при изгибе, сопротивление внешней среды и жесткость основания колеблющегося объекта. Особое внимание уделяется рассмотрению наиболее распространенного на практике случая, когда на границах действуют внешние возмущения. Решение принимается в безразмерных переменных с точностью до значений второго порядка относительно малых параметров, характеризующих скорость границы. Найдено приближенное решение задачи о поперечных колебаниях каната подъемного устройства, обладающего жесткостью на изгиб, один конец которого намотан на барабан, а на другом закреплен груз. Представлены результаты, полученные для амплитуды колебаний, соответствующей п-му динамическому режиму. Явление установившегося резонанса и прохождения через резонанс исследуется с помощью численных методов. Решение выполнено в безразмерных переменных с использованием программного пакета TB-Analysis, разработанного в среде МаЙаЬ, что позволяет использовать полученные результаты для расчета широкого спектра технических объектов.

Ключевые слова: резонансные свойства, колебания систем с подвижными границами, законы движения границ, интегро-дифференциальные уравнения, амплитуда колебаний

Авторы заявляют об отсутствии конфликта интересов.

Для цитирования: Литвинов, В. Л. Приближенный метод решения краевых задач с подвижными границами в разработанном программном комплексе TB-Analisys / В. Л. Литвинов, К. В. Литвинова. - DOI 10.25559^ШТО.17.202102.432-441 // Современные информационные технологии и ИТ-образование. - 2021. - Т. 17, № 2. - С. 432-441.

Modern Information Technologies and IT-Education

I. Introduction

Among all the many problems of the dynamics of elastic systems from the point of view of technical applications, the problems of oscillations in systems with time-varying geometric dimensions are very relevant. Systems, the boundaries of which are moving, are widespread in technology (ropes of hoisting installations [1-3, 7, 8,

II, 13, 18, 23, 24], flexible transmission links [7, 11, 12], solid fuel rods [9, 10], drill strings [9], etc.). The studies of many authors on the dynamics of lifting ropes have led to the need to formulate new problems in mechanics concerning the dynamics of one-dimensional objects of variable lengths. In the mathematical formulation, this is reduced to new problems of mathematical physics - to the study of the corresponding equations of hyperbolic type in variable ranges of variation of both arguments. The presence of moving boundaries causes significant difficulties in the description of such systems; therefore, approximate solution methods are mainly used here [1-5, 7-19].

Of the analytical methods, the most effective is the method proposed in [6, 20], which consists in the selection of new variables that stop the boundaries and leave the wave equation invariant. In [21], the solution is sought in the form of a superposition of two waves traveling towards each other. The method used in [22], which consists in replacing the geometric variable with a purely imaginary variable, is also effective, which makes it possible to reduce the wave equation to the Laplace equation and apply the methodology of the theory of functions of a complex variable for the solution. However, the exact solution methods are limited by the wave equation and relatively simple boundary conditions.

Of the approximate methods, the most effective method is the Kan-torovich-Galerkin method [9, 14, 25], as well as the method for constructing solutions of integro-differential equations described in this paper. The problem of vibrations of objects with moving boundaries, formulated as a differential equation with boundary and initial conditions, is a nonclassical generalization of a hyperbolic problem. To facilitate the construction of the solution to this problem and to justify the choice of the form of the solution, equivalent integro-differential equations with symmetric and time-dependent kernels and time-varying integration limits are constructed. The construction of integro-differential equations of motion of objects of variable length is based on the direct integration of differential equations in combination with the standard replacement of the required function with a new variable.

In trivial cases, the methods of integral equations do not have an advantage over the method of differential equations when applied to the study of oscillations of a system with an infinite number of degrees of freedom [7, 11]. The advantages of the method of inte-gro-differential equations are revealed in the transition to more complex dynamic systems carrying concentrated masses, vibrating under the action of moving loads, etc. These methods can be very fruitful when applied to the dynamics of strands of variable lengths and other mechanical objects with varying boundaries. In this paper, the method of constructing solutions of integro-dif-ferential equations is extended to a wider class of model boundary value problems that take into account the flexural rigidity of an os-

cillating object [8, 10, 13], the resistance of the external environment [9] and the rigidity of the base (substrate) of the object [7, 11] ... Particular attention is paid to the consideration of the most common case in practice, when external disturbances act at the borders. For a fixed length of the object, the constructed integro-differential equations go over into the classical Fredholm equations of the second kind. The solution is made in dimensionless variables using the TB-Analisys1 software package developed in the Matlab environment, which allows using the results obtained for calculating a wide range of technical objects.

2. Formulation of the Problem

The differential equation of motion of mechanical objects of variable length has the form

U„(£,T)+m/(£,Ty} = 9(£,T). (1)

We write the boundary conditions in the following form

Yji [u((j(£t),t)] = F(t); (2)

i = 1, m; j = 1,2.

Here U,t) - offset function; L - linear homogeneous differential operator with respect to ^ order 2m (m " 2 - positive integer); Yji — linear homogeneous differential operators with respect to ^;

Fji(t) - specified class functions C and C2 respectively; ij (ST) - boundary motion law; s - small parameter (s = V / a,

V - border speed, a - vibration propagation speed).

The movement of the boundaries according to the law £ j corresponds to the slow movement mode.

When analyzing the resonance properties, the initial conditions are taken in the form U(|,0) = UT (|,0) = 0.

To eliminate in homogeneities in the boundary conditions, a new function is introduced into equation (1)

= + (3)

where 2

EE Artf.^w. (4)

k=1 r=l

the function satisfies the equation

LD i$,£r)] = 0 (5)

and conditions

Substituting (3) into equation (1) taking into account (4), (5), the function V(<H,t) is found as a solution to the following problem:

V„^,T) + L[V(4,T)] = ^,T)-ff„^,T); (6)

7..[F(^r),r)] = 0. (7)

In [11], an integro-differential equation corresponding to problem (6), (7) was obtained in the form

V (t,r) = -2 j) k (Z,C,ST) [ (i,d-9(Z,r) + HTT (Z,T)]dZ,

tl(ST)

(8)

fl,k = j A r = i;

[О, к Ф jv гФ i.

1 Litvinov V.L., Yashagin N.S., Anisimov V.N. Certificate of registration of the electronic resource "Automated Research Complex" TB - ANALISYS" in Joint Fund of the

Electronic Resources "Science and Education" No. 19517 dated September 26, 2013 and FGANU CITiS No. 130912114653 dated September 30, 2013. (In Russ.)

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where K(p,Z,ET) - symmetrical in p and Ç kernel time-dependent via parameter st .

Theorem 1. In a time interval At , commensurate with unity, the equation of oscillations of an object with a fixed parameter l = l (т0) = const differs from the corresponding equation of oscillations of an object with a variable parameter l = l(r) by terms proportional to the factor s , provided that the derivative of the kernel K(x, s, l) with respect to the parameter l(т) is bounded. Proof. Let us expand the right-hand side of Eq. (8) in terms of the parameter l(т) in the vicinity of some fixed value of the dimensionless length 1(t0) in a Taylor series. Assuming

l (t0 + At) = l (t0) + Al (t) +...

get

l To)

V(P,t) = - J K(P,Ç, l (To)) [ (Z,t)-ç(Ç,r)]dÇ-

0

-Al (T) {K (g, l (To), l (To)) [ [ (l (To), T) -ç(l (To), t)] +

(9)

l (To)

J)dK%fo»[[-ç(Z,T)]\-

(Al(T) )2

2!

dk (g, l (To), l (To))

dl (t)

(10)

l (To)

V(4,t) = - J K(4,Z,l(To))[ (Z,T)-ç(Z,T)]dÇ-

0

-e1'(t)6(t) {K (4, l (To), l (To)) [[ (l (To),T)-Ç(l (To),T)]

l (To)

0 [ZT)-v(z,T)]dA-

2(T)

e(T)

2!

dK (4, l (To), l (To)) dl(T)

3. The Solution of the Problem

The solutiog to problem (9) will be sought in the form of a series:

V (Z,T) = £ fn (T)X„ (4,T (11)

n=1

where Xn(4,et) - eigenfunctions, which are formally constructed

(12)

solutions of the integral equation

1

Xn(4,st) = w20n(st) | K(4,z,st)X„(£,er)dZ,

(¡(st)

where st considered as a parameter.

Solution (11) is exact if the boundaries are motionless.

Own functions Xn(J;,ST) satisfy the boundary conditions (8) and in

this case play the role of dynamic modes.

Using the results of [11], we expand the symmetric in 4 and Z kernel in a row by its own functions X (4,et) :

* ({.{.„)=l x ^ (13)

where a0n (st) - natural frequencies of the problem determined by the formula

(14)

:_- i i k(p r Cr) x (p x (r ST\dPdr

We will assume that the function l(t) is a function of slow time

l = l(Tj), Tj = st, i.e. is a function of time, the time derivative of which is proportional to some small parameter s . The differential of the length of the object AI(t1) in accordance with the rule of differentiation of the slow time function [7, 11] is calculated by the

formula Al T) = e ^^ At.

dT1

Let us choose the time interval At in the form

At = 6(t),

where 6(t) - is some function of order unity.

1 t2(ST)t2(ST)

— = J J K(P,Z,ST)X„(P,ST)X„(Z,ST)dpdZ.

"0n (£'':) (¡(st) (¡(st)

Let us differentiate series (11) by time:

№

V(P,T) = Y[f:(T)Xn(4,ET) + eX„T (4,ET)fn(t)].

n=1

After repeated differentiation, we get

v„ = ¿1 fn (T) Xn (Z,ET) + IcX^^T)/; (t) X^ (l;,£r)f (t)}.

(15)

Substitute series (11), (13), (15) into equation (9) taking into account the orthogonality of the functions xn(4,et) on the interval [ 1(et); i 1(et)\ with weight g(p) and replacements

2 m

Substituting (10) into (9), we find that in a time interval At , of the order of unity, expansion (9) has the form

where

<2(£T)

0* («> - --"

| x2n(i,£T)g(4)di

Note that if we expand the function H(4,t) in a Fourier series:

№

H (P,T) = (T) X (P,ET), (16)

where

12E)

12(")

Ç„ (т) = J H (P,T)Xn (P,ET)(P)dp/ J X2JP,ET)g (p)dp,

Taking into account the condition of the theorem on the bounded-ness of the derivative of the kernel K(x,s,l) with respect to the parameter 1(t) and comparing the results obtained, we find that an equation with a fixed parameter l = l(t0) = const differs from an equation with a variable parameter in the range of At ^ 1 terms proportional to the factor. This proves the theorem.

ll(ET)

(et)

here g(p) - is a weight function, then the replacement can be made in a simpler form

fn (?) = (r) -9„ (T).

In resonance phenomena, the amplitudes of all dynamic modes,

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+

+

n=1

-fc

436 НАУЧНОЕ ПРОГРАММНОЕ ОБЕСПЕЧЕНИЕ В ОБРАЗОВАНИИ И НАУКЕ В ПЛ' Литвинов,

К. В. Литвинова

except for the resonant one, are small. Therefore, the nonresonant terms of series (11), (15) are neglected due to their smallness. In this case, we obtain a split system of ordinary differential equations with variable coefficients

A„ (st)^ (T) + 2sA2n (ST)£ (T) + s2 ^ (ST)^ (T) + A„ (sT)pn (T) =0n (T), (17) where

2 m 2 m (18)

i=1 r=1 Jt=l r=1

i-1 r-1

Here:

<2cjr) (19)

4„(«) = j

7 X^,ST)Xn^,ST)g(4)d^, (20)

12(ZT)

s2A3n(st) = | ^ (§,£t)X„(§>£t)g(21)

ll(ST) t,(et)

tf (4,т) = (t)X„ (ç,et).

Proof. The quantities qn (st) determined by the expression

t2 (т

l2(ST)

For simplicity, we introduce into equation (23) a new function

Vn (t) = An y„ (т),

sAn (ST)

where

An (ж) = exP

-J-

Ain (£T)

dr .

Then equation (23) will not contain a term with y 'to

yn (t) + w02„ (st) y„ (t) = d„ (t) /[an (st) an (st)]

(25)

(26)

(27)

Let be

^,r) = 50(£)cosJF0(r); _ _ (28)

¿^^cos^r); i = \,m, (29)

where BJt constant values; W0(r), Wfl(j) monotonically increasing functions; — function characterizing the intensity of the distributed load.

Taking into account equalities (28), (29), expression (18) takes the form

. A?, , = MM(ex)cosW»(T) + £¿M,(er)cosW„Jl(t), (30)

A0n (eT)Ain (eT) 1-1 i-1

where t2(sT) M„0(st) = j b0(ç)xn(ç,er)g(ç)dç/[a»„(st)an(st)];

t,(sr)

*2C„t(*r) = j \Qnt, (t, ST) + /)„(£ £T)]„ -X. (4, ez)g(4)d4;

id")

<2 <«)

System (17) up to values of the order of smallness s2 will look like

an(£T)ß"n (t) + 2sAln (Si)ß'„(t) + 4„(sT)rn20n(si)ß„ (t) = dn(t), (23) where

(t) = öo2„ (t an (zt)ç„ (t) + en (t). Taking into account (11), (16) solution (3) will have the form:

U(4,T) = (T)X (4,ET) + j j^Fk, (TlDkr (4,ET) + jj QnjET)Xn (4,ET)1 (24)

n=1 k=1 r=1 n=1

Theorem 2. The solution to problem (1) - (2) can be represented in the form

mnß (st) =

-ban (st)q„ß (st) ain (st)

(31)

We restrict ourselves to considering the case when expression (30) has the form

= J*.(CT)cos W.to. (32)

where Wn(f) - monotonically increasing function. Equality (30) can be replaced by equality (32) in the cases described in [8, 9, 14].

Taking into account the above, equation (27) takes the form

y'n (t) + щ>„ (st)yn (t) = Mn (st) cos Wn (t).

(33)

Solution of equation (33) with zero initial conditions j>(0) = 0; /(0) = 0 has the form [12]:

' (34)

Qnkr (T = - J Dkr(M,ST)X„(Ç,sr)g/ J XK^ET)g

are for the function -Dkr(^,ST) the coefficients of the Fourier series expansion in the system of orthogonal weights g(Ç) eigen-functions Xn(Ç,st) on the interval [11(et), t2(et)], i.e.

XQnr (£T)Xn(4,£T) = -D»(4,ET).

Therefore, the expression in square brackets of equality (24) is equal to zero. The theorem is proved.

where

,T = -yu (j)yin (O - yu (Ç)yir, W " y^Qy\M)-y[n(Qy,n(O '

y^yin — linearly independent solutions of the homogeneous equation corresponding to (33), i.e.

y'n (T) + wt (sT)yn (T) = 0. (35)

Linearly independent solutions (35) have the form [9, 12]: ^„(r) = «,,(«')sinw„(T-); (36)

(37)

Here the functions an(er) and w„(r) are determined up to values of E2 the order of from the following system of equations obtained using the asymptotic method considered in [12]:

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SCIENTIFIC SOFTWARE IN EDUCATION AND SCIENCE

437

dwn (t)

dz dan (et)

= a0n (et);

an (eT) d®0n (et)

dz 2a0n (ez) dz The solution of this system, up to a constant, has the form

T

wn (t) = jfflo„ (sT)dT;

0

an = I ^

V®0„ (ZT)

(38)

(39)

a2(t) = 14 4l (zT)a2(£T) J ¡F„ (does $ „(ZVZ + Jf„ (Z)oos ®aZ)dZ

IL o °

+ T\Fn (<)sin $ „i(Z)dZ + )f„ (<)sin ®„2(Z)dZ

where

r ^ 1 Mn EZ)

an(et) = , = ; Fn(sZ) = - n

an (sfK (Z)

; wn (t) = |®on(et)^t;

Mnii (ez) =

4>„ ^

; On (O = w„(Z) - (Z).

^„(OcosO^CO^

J^cosiiKtjo^

(41)

to calculate the resonance properties of the bearing links of a wide range of lifting machines.

The equation that takes into account the bending stiffness and tension of the vibrating link has the form [9]

U„ (x, t) + — Um (x, t) - aU (x, t) = 0. P

Border conditions

U(0, t) = 0; Uxx (0, t) = 0; U (lo (t), t ) = B cos W0 (w0t ); ^ (/0 (t), t ) = 0.

(42)

(43)

(44)

As ffl0n(£t) > 0, then it follows from equalities (38), (39) that w„(r) - monotonically increasing function.

Returning to solution (34), taking into account (36), (37), we get:

, i . , MJsC) cos WJC) cos wJC) ,, y» to = a. (to sin w. (r) J " , " ~

, x , , \Mn «) cos W„ (Z) sin w (O J „

-0. («") COS W. (z) J ", "

Using the well-known trigonometric formulas taking into account (25), we obtain the following expression for the total amplitude of oscillations corresponding to the nth dynamic mode:

Integrals containing sin$ n1(Z), cos$ n1(Z), increase during the entire period while the resonance phenomenon is observed, and make the main contribution to the amplitude. Ignoring members containing ®„2(i), we obtain the following expression for the vibration amplitude:

4(T) = \4„(£T)a2„(£T)<

In problem (42) - (44), the following designations are used: U (x, t)

- lateral displacement of the link point with coordinate x at time t; I

- axial moment of inertia of the rope section; p - linear mass density; a ~\jT / p - the minimum speed of wave propagation; T - is the tensile force; l0(t) = L0 - v0t - the law of movement of the border of the rope, L0 - the initial length of the rope, v0 - the speed of movement of the border; W0(z) - class function C2; B,ma - constant values; E - is the modulus of elasticity of the rope material. Introduce dimensionless variables into problem (42) - (44):

x/a; z=rn0t + -; U(x,t) = BV(%,r).

vo

Then the problem will take the form

V„ (4,z) + p2V^(i;,z) - Vg (4,r) = 0; (45)

V(0,z) = 0; V^ (0,t) = 0; (46)

V(l(ez),z) = cos W(z); (ez),z) = 0, (47)

where

Ei 2

P2 = — % 1(ez) = 1 + ez; W(t) = W0 (z-y0); p a v 7

7o ; e=-vja.

Note that the value of the quantity p in technical problems usually does not exceed 0,25.

Integrating equation (45) over p and getting rid of inhomogene-ities in the boundary conditions by analogy with (6) - (7), we obtain an integro-differential equation for transverse vibrations of a rope of variable length in the form:

i (t

V(4,z) = - | K(4,z,ez)[[(Z,T) + H„(Z,r)]dZ. (48)

The kernel of equation (48) in the case under consideration will be determined by the function

Some Results of Numerical Studies of Lateral Vibrations of a Rope of a Handling Unit

As an example, consider the transverse vibrations of a rope of a lifting installation, one end of which is wound on a drum, and a load is hinged on the other. With the help of the given model, it is possible

K (Ç,Ç,er) =

¡(EZ)-4) f I (EZ)-4 , S-Z

ß

¡(ez)-Z ) f l(ez)-Z

ß

2

Z-4

(49)

Function (49) is also symmetric with respect to arguments p and Z and depends on time through the parameter contained in it ez. When fixed l(ez) = const function (49) coincides with the func-

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2

0

tion of the influence of deflections of a rope of constant length. Thus, problem (45) - (47) is reduced to an integro-differential equation (48) with a symmetric time-varying kernel (49) and time-varying integration limits.

The solution of problem (48) will be carried out in dimensionless variables in accordance with the method described above. As a result, for the amplitude of oscillations corresponding to the nth dynamic mode, we obtain the following expression:

T a b l e 1. Dependence of the vibration amplitude An on s and 0 when passing through resonance in the first and second dynamic modes

A2(t) = En (er)

where

72/

ЕП (er) = -

Jfh (eZ) cos Ф, (QdÇ + \Fn (eZ )sin Ф, (Z)dZ

; Ф„ (С) = и-, (Z) - Wn (Z);

1

=-£

2ß2

1-1+ V1 + 4ß2

■nn -1

A1ii(t1,t1) = £1(et1) \ ¡Fn «)cos$„ (Ç)dÇ + ¡Fn (Osin $n (Ç)dÇ

ß \ £ 0,02 0,04 0,06 0,08

и T3 0,01 17,3 10,7 8,8 6,7

s rH 0,2 14,1 9,2 7,3 5,4

и тз 0,01 12,5 7,7 5,1 4,2

s CN 0,2 9,3 5,4 4,3 3,7

4 A„ 0n

Fn(<) = Qni «W^ZK^Z

The phenomenon of steady-state resonance in the system under consideration is observed if

Wn to = Wn to + Y>

where y - is a constant.

Under the action of a harmonic perturbation on the system with a frequency aa, when W(t) = T, on any of the dynamic modes, the phenomenon of passage through a resonance may occur. The point of the resonance regiont0, in which 0'n(t0) = 0, is approximately determined by the following formula:

To study the phenomenon of passage through resonance, it is necessary to find the values of t and t2 at which the square of the amplitude

(50)

has a maximum.

Using the TB-Analisys2 software complex, developed in the Matlab environment, the dependence of the maximum amplitude of the transverse vibrations of the rope when passing through resonance in the first and second dynamic modes on the relative velocity of the boundary movement at various values of the dimensionless coefficient characterizing the bending stiffness of the object was established (Table 1).

Analysis of the results obtained allows us to draw the following conclusions:

- with a decrease s the amplitude of the oscillations increases;

- when £ —^ 0 the amplitude of oscillations tends to infinity;

- with an increase in the mode number and bending stiffness of the object, the maximum vibration amplitude

decreases.

Conclusion

An approximate method for constructing solutions of integro-dif-ferential equations is extended to a wider class of model boundary value problems on the vibrations of objects with moving boundaries in a linear formulation described by equations of hyperbolic type. This method allows taking into account the effect on the system of resistance forces of the external environment, bending stiffness and rigidity of the object substrate. The solution of the problem is brought to obtain the quadrature formulas of the amplitude of oscillations corresponding to the n-th dynamic mode. The phenomenon of steady-state resonance and passage through resonance is investigated by numerical methods for transverse vibrations of a rope of a hoisting machine. The above results allow at the design stage to prevent the possibility of high-amplitude oscillations in mechanical objects with moving boundaries. Using the TB-Analisys software complex, developed in the Matlab environment, the dependence of the maximum amplitude of the transverse vibrations of the rope when passing through resonance in the first and second dynamic modes on the relative velocity of the boundary movement at various values of the dimensionless coefficient characterizing the bending stiffness of the object was established.

References

[1] Kolosov L.B., Zhigula T.I. Longitudinal-transverse vibrations of the rope of the lifting installation. Izvestiya vysshikh uchebnykh zavedenii. Gornyi zhurnal = Minerals and Mining Engineering. 1981; (3):83-86. (In Russ.)

[2] Zhu W.D., Chen Y. Theoretical and Experimental Investigation of Elevator Cable Dynamics and Control. Journal of Vibration and Acoustics. 2006; 128(1):66-78. (In Eng.) DOI: https://doi.org/10.1115/1.2128640

[3] Shi Y.-J., Wu L.-L., Wang Y.-Q. Analysis on nonlinear natural frequencies of cable net. Journal of Vibration Engineering.

2 Ibid.

1

1

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2006; 19(2):173-178. (In Chine, abstract in Eng.)

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[5] Zhao Y., Wang L. On the symmetric modal interaction of the suspended cable: Three-to-one internal resonance. Journal of Sound and Vibration. 2006; 294(4-5):1073-1093. (In [18] Eng.) DOI: https://doi.org/10.1016/j.jsv.2006.01.004

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[7] Goroshko O.A., Savin G.N. Vvedenie v mehaniku deformirue-myh odnomernyh tel peremennoj dliny [Introduction to

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[9] Litvinov V.L., Anisimov V.N. Mathematical modeling and study of oscillations of one-dimensional mechanical systems with moving boundaries. SamSTU Publ., Samara; 2017. 149 p. Available at: https://www.elibrary.ru/item.as- [21] p?id=36581641 (accessed 20.05.2021). (In Russ., abstract

in Eng.)

[10] Lezhneva A.A. Svobodnye izgibnye kolebanija balki peremennoj dliny [Free bending vibrations of a beam of variable [22] length]. Uchenye zapiski Permskogo gosudarstvennogo uni-versiteta. 1966; (156):143-150. (In Russ.)

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[12] Litvinov V.L. Study free vibrations of mechanical objects with moving boundaries using asymptotical method. Zhurnal Srednevolzhskogo Matematicheskogo Obshchest-va = Middle Volga Mathematical Society Journal. 2014; 16(1):83-88. Available at: https://www.elibrary.ru/item. [24] asp?id=21797219 (accessed 20.05.2021). (In Russ., abstract in Eng.)

[13] Liu Z.-j., Chen G.-p. Planar non-linear free vibration analysis of stay cable considering the effects of flexural rigidity. Journal of Vibration Engineering. 2007; 20(1):57-60. (In Chine, [25] abstract in Eng.)

[14] Anisimov V.N., Korpen I.V., Litvinov V.L. Application of the Kantorovich-Galerkin Method for Solving Boundary Value Problems with Conditions on Moving Borders. Mechanics of Solids. 2018; 53(2):177-183. (In Eng.) DOI: https://doi. org/10.3103/S0025654418020085

[15] Berlioz A., Lamarque C.-H. A non-linear model for the dynamics of an inclined cable. Journal of Sound and Vibration. 2005; 279(3-5):619-639. (In Eng.) DOI: https://doi. org/10.1016/j.jsv.2003.11.069

[16] Sandilo S.H., van Horssen W.T. On variable length induced

vibrations of a vertical string. Journal of Sound and Vibration. 2014; 333:2432-2449. (In Eng.) DOI: https://doi. org/10.1016/j.jsv.2014.01.011

Zhang W., Tang Y. Global dynamics of the cable under combined parametrical and external excitations. International Journal of Non-Linear Mechanics. 2002; 37(3):505-526. (In Eng.) DOI: https://doi.org/10.1016/S0020-7462(01)00026-9

Palm J. et al. Simulation of mooring cable dynamics using a discontinuous Galerkin method. In: Ed. by B. Brinkmann, P. Wriggers. MARINE V: Proceedings of the V International Conference on Computational Methods in Marine Engineering. CIMNE, Hamburg, Germany; 2013. p. 455-466. Available at: http://hdl.handle.net/2117/333008 (accessed 20.05.2021). (In Eng.)

Faravelli L., Fuggini C., Ubertini F. Toward a hybrid control solution for cable dynamics: Theoretical prediction and experimental validation. Structural Control and Health Monitoring. 2010; 17(4):386-403. (In Eng.) DOI: https://doi. org/10.1002/stc.313

Anisimov V.N., Litvinov V.L., Korpen I.V. On a method of analytical solution of wave equation describing the oscillations sistem with moving boundaries. Journal of Samara State Technical University. Ser. Physical and Mathematical Sciences. 2012; (3):145-151. Available at: https://www.eli-brary.ru/item.asp?id=19092391 (accessed 20.05.2021). (In Russ., abstract in Eng.)

Vesnitskii A.I. The inverse problem for a one-dimensional resonator the dimensions of which vary with time. Radio-physics and Quantum Electronics. 1971; 14(10):1209-1215. (In Eng.) DOI: https://doi.org/10.1007/BF01035071 Barsukov K.A., Grigoryan G.A. Theory of a waveguide with moving boundaries. Radiophysics and Quantum Electronics. 1976; 19(2):194-200. (In Eng.) DOI: https://doi. org/10.1007/BF01038526

Anisimov V.N., Litvinov V.L. Calculation of eigen frequencies of a rope moving in longitudinal direction. Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva = Middle Volga Mathematical Society Journal. 2017; 19(1):130-139. Available at: https://www.elibrary.ru/item.asp?id=29783056 (accessed 20.05.2021). (In Russ., abstract in Eng.) Litvinov V.L. Longitudinal oscillations of the rope of variable length with a load at the end. Bulletin of Science and Technical Development. 2016; (1):19-24. Available at: https://www.elibrary.ru/item.asp?id=28765822 (accessed 20.05.2021). (In Russ., abstract in Eng.) Litvinov V.L. Solution of boundary value problems with moving boundaries using the method of change of variables in the functional equation. Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva = Middle Volga Mathematical Society Journal. 2013; 15(3):112-119. Available at: https://www.elibrary.ru/item.asp?id=21064140 (accessed 20.05.2021). (In Russ., abstract in Eng.)

Submitted 20.05.2021; approved after reviewing 14.06.2021; accepted for publication 19.06.2021.

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About-thejjjthorjШШШШШШШ^ШШ

Vladislav L. Litvinov, Head of the Department of General-Theoretical Disciplines, Deputy Dean of the Faculty of Mechanics, Syzran' Branch of the Samara State Technical University (45 Sovetskaya St., Syzran' 446001, Samara region, Russian Federation); Doctoral Student of the Faculty of Mechanics and Mathematics, Lomonosov Moscow State University (1 Leninskie gory, Moscow 119991, GSP-1, Russian Federation), Ph.D. in Technical Sciences, Associate Professor, ORCID: https://orcid.org/0000-0002-6108-803X, [email protected]

Kristina V. Litvinova, Student, Lomonosov Moscow State University (1 Leninskie gory, Moscow 119991, GSP-1, Russian Federation), ORCID: https://orcid.org/0000-0002-1711-9273

All authors have read and approved the final manuscript.

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V. L. Litoinw, SCIENTIFIC SOFTWARE IN EDUCATION AND SCIENCE

K. V. Litvinova

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Поступила 20.05.2021; одобрена после рецензирования 14.06.2021; принята к публикации 19.06.2021.

|об авторах:|

Литвинов Владислав Львович, заведующий кафедрой общетеоретических дисциплин, заместитель декана механического факультета филиала ФГБОУ ВО «Самарский государственный технический университет» в г. Сызрани (446001, Российская Федерация, Самарская обл., г. Сызрань, ул. Советская, д. 45); докторант механико-математического факультета, ФГБОУ ВО «Московский государственный университет имени М. В. Ломоносова» (119991, Российская Федерация, г. Москва, ГСП-1, Ленинские горы, д. 1), кандидат технических наук, доцент, ORCID: https://orcid.org/0000-0002-6108-803X, [email protected]

Литвинова Кристина Владиславовна, студент, ФГБОУ ВО «Московский государственный университет имени М. В. Ломоносова» (119991, Российская Федерация, г. Москва, ГСП-1, Ленинские горы, д. 1), ORCID: https://orcid.org/0000-0002-1711-9273

Все авторы прочитали и одобрили окончательный вариант рукописи.

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