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Scientific article DOI: 10.18287/2541-7525-2023-29-3-72-81
Submited: 12.07.2023
Revised: 16.08.2023
Accepted: 30.10.2023
V.L. Litvinov
Samara State Technical University, Samara, Russian Federation Moscow State University, Moscow, Russian Federation E-mail: vladlitvinov@rambler.ru. ORCID: http://orcid.org/0000-0002-6108-803X
K.V. Litvinova
Moscow State University, Moscow, Russian Federation E-mail: kristinalitvinova900@rambler.ru. ORCID: http://orcid.org/0000-0002-1711-9273
ON ONE SOLUTION OF THE PROBLEM OF VIBRATIONS OF MECHANICAL SYSTEMS WITH MOVING BOUNDARIES
ABSTRACT
An analytical method for solving the wave equation describing the oscillations of systems with moving boundaries is considered. By changing the variables that stop the boundaries and leave the equation invariant, the original boundary value problem is reduced to a system of functional-difference equations, which can be solved using direct and inverse methods. An inverse method is described that makes it possible to approximate quite diverse laws of boundary motion by laws obtained from solving the inverse problem. New particular so-lutions are obtained for a fairly wide range of laws of boundary motion. A direct asymptotic method for the approximate solution of a functional equation is considered. An estimate of the errors of the approximate method was made depending on the speed of the boundary movement.
Key words: wave equation; boundary value problems; oscillations of systems with moving boundaries; change of variables; laws of motion of boundaries; functional equations.
Citation. Litvinov V.L., Litvinova K.V. On one solution of the problem of vibrations of mechanical systems with moving boundaries. Vestnik Samarskogo universiteta. Estestvennonauchnaya seriya / Vestnik of Samara University. Natural Science Series, 2023, vol. 29, no. 3, pp. 72-81. DOI: http://doi.org/10.18287/2541-7525-2023-29-3-72-81. (In Russ.)
Information about the conflict of interests: authors and reviewers declare no conflict of interests.
© Litvinov V.L., Litvinova K.V., 2023 Vladislav L. Litvinov — Candidate of Technical Sciences, head of the Department of General-Theoretical Disciplines, assistant professor, Samara State Technical University, 244, Molodogvardeyskaya street, Samara, 443100, Russian Federation; doctoral student, Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russian Federation.
Kristina V. Litvinova — student, Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russian Federation
Introduction
One-dimensional systems, the boundaries of which move, are widely used in engineering: ropes of lifting installations [1; 4; 8; 11-15; 21], flexible gear links [1; 2; 5; 16; 19; 20], solid fuel rods [22], drill strings [8], etc. The presence of moving boundaries causes significant difficulties in describing such systems; therefore, here, approximate methods of solution are mainly used [1-4; 8; 10; 16; 17; 20-22; 25-29]. Of the analytical
methods, the most effective is the method proposed in [5], which consists in the selection of new variables that stop the boundaries and leave the wave equation invariant. In [6], the solution is sought in the form of a superposition of two waves running towards each other. The method used in [7] is also effective, which consists in replacing the geometric variable with a purely imaginary variable, which makes it possible to reduce the wave equation to the Laplace equation and apply the method of the theory of functions of a complex variable to the solution.
This article considers an analytical method for solving the wave equation that describes the oscillations of systems with moving boundaries. By changing the variables that stop the boundaries and leave the equation invariant, the original boundary value problem is reduced to a system of functional-difference equations that can be solved using direct and inverse methods. An inverse method is described that makes it possible to approximate quite diverse laws of boundary motion by laws obtained from solving the inverse problem. New particular solutions are obtained for a fairly wide range of laws of motion of boundaries. A direct asymptotic method for the approximate solution of a functional equation is considered. An estimate of the errors of the approximate method is made depending on the speed of the boundary movement. This approach successfully combines the technique used in [5; 6; 9; 18; 23; 24].
1. Statement of the problem
Let us consider free oscillations in a system with moving boundaries.
utt(x,t) - a2uxx(x,t) =0. (1)
The boundary conditions at the fixed ends have the form
u (li (t),t) = 0; u (l2(t),t) = 0. (2)
(li(0) < x < l2(0))
Here, u(x,t) — is the displacement of the point of the object with the coordinate x at time t; a — velocity of wave propagation in the system; l1(x), l2(x) — the laws of movement of borders.
In the works [5; 6] Vesnitsky A.I. a fairly general method for selecting new variables for the wave equation was proposed. Following this method, the change of variables is made in the following form:
£ = y(t + x/a) — ф(t — x/a); t = a-1 [y(t + x/a) + ф(t — x/a)]
-U.„n , „лл , „ш ^ (3)
where y and ^ — are some functions. As a result of such a replacement, the original equation remains invariant (wave), and y, ^ are determined from the condition of constancy £ at the boundaries.
In new variables £, t, defined by relation (3), the original problem (1)-(2) is reduced to the following
utt(£,t) - %(£,t)=0 (4)
under boundary conditions
U(iI(t),t) = 0; Ut(12(t),t)=0; (5)
(ii(t) < £ < ¿2(t)).
Here t, £ dimensionless time (t > 0) and dimensionless spatial coordinate; U(£,t) = u(x,t) ; Ii(t) — the laws of movement of borders.
Boundary conditions (5) in variables £, t are set on new, generally speaking, moving boundaries, the position of which depends on two functions y and Since they y and ^ are arbitrary, one can require that the boundary conditions are written on fixed boundaries, i.e. l1 = const and = const (l2 >11). For this, it is y and ^ necessary to satisfy the system of functional equations:
{ y(T + li(T)) - $(t - li(T)) = li;
1 y(T + l2(T)) - t(T - l2(T)) = l2, (6)
which uniquely determine the functions y and ^ through the known laws of boundary motion. When the borders move at a speed greater than the speed of wave propagation, the solution of the wave equation becomes incorrect, therefore, a restriction is imposed on the speed of the borders ¡¿¡(t)| < 1. Constants £i can be arbitrary, but not equal values (for example, l1 =0, l2 = 1). Then system (6) will take the form:
{ y(T + li(T)) = ^(t - li(T)); (7)
\ y(T + ¿2 (t ))= ^(t - ¿2 (t )) + 1, (7)
The existence of a solution to this system was proved in [5].
Solution (4)-(5) is found by the Fourier method [24]:
œ
U(i,T) = E sin (w0nO (Dn cos (wg„r) + En sin (wq„t)) =
œ n=1 (8)
= E r'n {sin (^Qn(r + £) + an) - sin (^Qn(r - £) + an)} ,
n=1
where u0n (e0 t) = ; rn = Dn + En ; an = arctg (En/Dn). The solution obtained in [1-6; 8-10] has a form similar to (8). Returning to the variables x and t, we get
œ
u(x, t) = rn {sin (w0n^(t + x) + an) - sin (w0n^(t - x) + an)} . (9)
n=1
Here p and they are found from the solutions of the system of functional equations (7) according to the known laws of motion of the boundaries, and the constants Dn, En are determined from the initial conditions.
Generally speaking, it is not easy to solve system (7). There are two different approaches to solving it:
— is the inverse problem [5; 6; 8; 9; 18; 22; 23], i.e. according to the given "phases" of natural oscillations p and finding the laws of motion of the boundaries ¿¿(t);
— is the direct problem [17; 22], i.e., finding the "phases" of natural oscillations according to the given laws of motion of the boundaries ¿¿(t).
2. Solution of the inverse problem
To solve system (7) A.I. Vesnitsky [5] used the inverse method, i.e. given p and ^ from the resulting system of equations, the laws of motion of the boundaries t\(r) and 12(t) are found. When solving the inverse problem, the equations of system (7) are reduced to the study of algebraic or transcendental equations with respect to li(t), which in many cases admit exact solutions. Based on the inverse problem Vesnitsky A.I. and Potapov A.I. [5; 6] obtained solutions for a fairly wide range of laws of boundary motion.
System (7) has infinitely many solutions, since on the interval [0,1] the function y>(z) and on the interval [-1,0] the function ^(z) can be set arbitrarily, and using the method of successive approximations [24], the values of functions in other areas are found. It is enough for us to find one particular solution that determines the one-to-one correspondence of points z and points y1 = <^(z); y2 = ^(z) Of all the solutions, we are only interested in monotonic ones, and monotonic solutions in the case of boundary movement at a speed lower than the wave propagation speed (|11(t)| < 1; |12(t)| < 1) can only be monotonically increasing.
Lemma. If a function y>(z) - is monotonically increasing (decreasing), then the function ^(z) - is also monotonically increasing (decreasing).
Proof. Indeed, from the first equation of system (7) at t = to, it follows that
^(tO + iI(tO)) = ^(tO - iI(tO)).
Now suppose that t1 > t0 the function y>(z) also increases (decreases), then in the case of boundaries moving at a speed lower than the wave propagation speed (|11(t)| < 1; |12(t)| < 1), we will have:
t 1 + li(tI) > tO + iI(tO);
ti - ii(tI) > to - iI(tO)
Since the function y>(z) in this case increases (decreases), then in order to fulfill the first equality of system (7) at t = t1, it is necessary that the function ^(z) increases (decreases), i.e. the function ^(z) is also increasing (decreasing).
Let us also show that the monotonic solution of system (7) in the case of boundary motion at a speed lower than the wave propagation velocity can only be increasing.
Indeed, given the inequality 11(t) < 12(t) we get:
t + ii(t) <t + I2(t); t - li(t) >t - I2(t);
Suppose that y>(z) and ^(z) they decrease, then we can write:
v(t + l2(t)) < ¥>(t + li(T)) = ^(t - li(T)) < ^(t - l2(t)).
However, this inequality contradicts the second equation of system (7). Therefore, functions y>(z) and ^(z)can only be monotonically increasing. The lemma is proved. □
Note that from system (7) the functions y(z) and ф(г) are determined up to a constant in the sense that if y(z) and ip(z) the solution to system (7) is y(z) + C and ф(z) + C also a solution (here C — an arbitrary constant). Therefore, for definiteness, we can choose such a function ф(z), that ф( —1) = — 1. At the same time, from the second equation of system (7) for t = 0, it follows that ф(1) = 0. From the first equation of system (7) for t = 0, we obtain
ф(0) = ф(0).
When assigning functions y(z) and ip(z), several arbitrary constants are introduced into them. The dependence of the laws of motion £1(t) and £2(t) found on the values of these constants makes it possible to approximate quite diverse laws of motion of the boundaries by laws obtained from solving the inverse problem.
The set of inverse solutions is quite wide. The solutions below satisfy the relations:
li(0)=0; £2(0) = !; ф( — 1) = — 1.
The set of obtained laws of motion of boundaries is divided into classes:
1. The solutions shown in Table 1 are class A when the left boundary is fixed and y(z) = ip(z). Solutions numbered 1, 2, 3, 6 were obtained by A.I. Vesnitsky and A.I. Potapov [5; 6], solutions 4, 5, 7 were obtained for the first time.
Table 1
Class A decisions
Таблица 1
Решения класса А
l2(T ) Ф(£) = ф(z)
1 VT +1 Ln[(vz+1)/(1-v)] . Ln[(1+v)/(1-v)\ 1
2 Vbt + B2 /\B\ л/Bz + B + 0, 25— — л/B2 — B + 0, 25 — 1
3 1/(4bt + 1) Bz2 +0, 5z — B — 0, 5
4 a arcsh [в1еат-Б2 е-ат\ B1(eaz — e-a) + B2(e-az — ea) — 1, B1 = B2 + 1/(ea — e-a), a > 0
5 ■s/(t + B)2(a2 — 1) + 1 + 2aB + B2 — а(т + B) Ln[(z+B)2 + 1+2aB+B2] Ln[(1+a)/(1-a)\ Ln[(B-1)2 + 1+2aB+B2] 1 Ln[(1+a)/(1-a)\ 1
6 arctg(az+B)
a [—d + y/1 + d2 + (ат + B)2] , d = 1+B2-a2 d = 2a arcctg[(1+B2-a2)/(2a)\ arctg(B-a) -| arcctg[(1+B2-a2)/(2a)\ 1
7 a (in ) — т Aeaz + в, a = ln 1+^+4Л2
2. The following class B is determined by the fact that the boundaries move according to the same law:
¿i(T)= ¿(t); ¿2(t) = 1+ ¿(t); ¿(0)=0.
Since the movement of the boundaries is interconnected, there is also an interconnection between the functions y(z) and ^(z). It is expressed by the functional equation:
y(y(^(z)) + 1) - ^(z - 1) = 1. (10)
System (7) in this case can only be satisfied by functions that are solutions of equation (10). Here are two previously unknown solutions of class B:
1) I = VT; p(z) = (1 - v)z/2+(1 + v)/2 - 1; V>(z) = (1 + v)z/2 + (1 + v)/2 - 1;
2) 1(t) = a ln[(Be-aT - CeaT)/(B - C)];
p(z) = B(e-az - 1) - C(e-a - 1) - 1; B = C + 1/(e-a - 1); ^(z) = C(eaz - 1) - C(e-a - 1) - 1.
3. For class C solutions, the boundaries move symmetrically in different directions, i.e.
ii(t ) = -1(T ); 12(t ) = 1(T ). The equation for the relationship of functions y>(z) and ^(z) here has the form:
p(z) = ^(z) +0, 5
Class C solutions are obtained from class A solutions using the following formulas:
1(t) = Ia(t); ^(z) = 1 Va(z); ^(z) = V(z)+0, 5 ,
where the index A denotes the corresponding functions of solutions of the class A.
4. A solution of class D is obtained for the case when both boundaries move uniformly:
Ii(t) = (B2 - Bi)t/(B2 + Bi); 12(t) = (B2e1/c - Bx)t/(Bx + B2e1/c) + 1; p(z) = C Ln(B1z + D) - C Ln(D - B2) - 1; ^(z) = C Ln(B1z + D) - C Ln(D - B2) - 1;
D = (Bi + B2e1/c)/(e1/c - 1).
The solution number one in Table 1 can be used to study the vibrations of the ropes of load-lifting installations with a uniform ascent (descent) [1; 4; 11-15; 21]. The given solutions of class B can be used in the study of oscillations of flexible gear links [16; 19; 20]. The rest of the solutions are model.
The class of inverse solutions is limited, for example, no solution was obtained for the uniformly accelerated motion of the boundary 1(t) = 1 + vt2. Obtaining the indicated solution is relevant when describing the longitudinal and transverse vibrations of the ropes of load-lifting installations at the acceleration stage [1].
3. Solution of the direct problem
The solution of the direct problem, as a rule, faces great difficulties, because well-known methods for solving functional equations, although sometimes they can be found p and ^ from known ones li(T), but in a limited range of argument values and in a form that is not very suitable for analytical research.
In this regard, we consider an approximate solution of the functional equation
p(T + l(T)) - p(T - l(T)) = 1. (11)
For an approximate solution of equation (11), it is proposed to use the asymptotic method [17].
For fixed boundaries 1(t) = l, the solution to (11) is the linear function
1
Ps(z) = + const.
In the case of a slow motion of the boundary 1(t), the "phase" of the wave p(z) during its run through the system changes insignificantly with respect to ps(z). It is assumed that p(z) it has derivatives of any order, and writing p(t + 1(t)) in the form of power series in 1(t), after substituting them into (1), we obtain a differential equation for slowly changing current "phase" p(t)
~ lfc+1 dk+1p = ( )
^(k + 1)! ' d,Tk+1 =L (12)
k=0 v '
Since p(t) it deviates little from the linear law ps(z = t) during the wave travel time, each next term on the left side of equation (12) is much smaller than the previous one, and its solution must be sought in the form of a series
P(T) = E Pn(T). (13)
n=0
Substituting (13) into (12) and equating the terms of the same order of smallness individually to zero, we obtain for the zeroth approximation
T
po(T ) = 1/ W).
0
In the case of a linear law of motion of the boundary l(t) = 1 + vt, the phase of dynamic natural oscillations is equal to
ф) = H(VZ +1W + V)]. (14)
2v
Values (14) were compared with the values obtained using the exact solution (Table 1):
( Ln [(vz + 1)/(1 - v)] 1
The values of the maximum absolute errors A of the asymptotic method, depending on the speed of the boundary movement v, are given in Table 2.
Table 2
Error of the asymptotic method depending on the velocity of the boundary
Таблица 2
Погрешность асимптотического метода в зависимости от скорости границы
v 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9
A 0,002 0,006 0,013 0,023 0,036 0,053 0,073 0,100 0,139
In the interval v £ [0,1; 0,6] of error of the approximate method are small. The increase in the error when approaching v unity is explained by the fact that the function (15) becomes infinitely large when v ^ 1.
Insignificant errors make it possible to apply the described method to solve functional equation (11) in cases where its exact solution is not known.
Conclusion
Using the analytical method of change of variables, the original boundary value problem is reduced to a system of functional-difference equations. The solution of the original problem depends on whether it is possible to solve the given system (7). Vesnitsky A.I. proposed to solve it by the reverse method, i.e. to set functions y and ^ from the resulting system of equations to find the laws of motion of the boundaries. The paper presents five new inverse solutions of the system.
An approximate asymptotic method for solving the functional equations of system (7) is considered. Under conditions of slow motion of the boundaries, minor errors make it possible to apply this method in cases where the exact solution of the system of functional equations is not known.
The above solutions can be used in the study of vibrations of ropes of lifting installations with a uniform rise (descent), flexible links of transmission (for example, a belt drive), etc.
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[28] Litvinov V.L. Variational formulation of the problem on vibrations of a beam with a moving spring-loaded support // Theoretical and Mathematical Physics, 2023, vol. 215, issue 2, pp. 709-715. DOI: https://doi.org/10.1134/S0040577923050094. (In English; original in Russian)
Научная статья DOI: 10.18287/2541-7525-2023-29-3-72-81
УДК 517.958:531.12; 534.11 Дата: поступления статьи: 12.07.2023
после рецензирования: 16.08.2023 принятия статьи: 30.10.2023
В.Л. Литвинов
Самарский государственный технический университет, г. Самара, Российская Федерация
Московский государственный университет имени М.В. Ломоносова, г. Москва, Российская Федерация
E-mail: vladlitvinov@rambler.ru. ORCID: http://orcid.org/0000-0002-6108-803X
К.В. Литвинова
Московский государственный университет имени М.В. Ломоносова, г. Москва, Российская Федерация E-mail: kristinalitvinova900@rambler.ru. ORCID: http://orcid.org/0000-0002-1711-9273
ОБ ОДНОМ РЕШЕНИИ ЗАДАЧИ О КОЛЕБАНИЯХ МЕХАНИЧЕСКИХ СИСТЕМ С ДВИЖУЩИМИСЯ ГРАНИЦАМИ
АННОТАЦИЯ
В статье рассмотрен аналитический метод решения волнового уравнения, описывающего колебания систем с движущимся границами. Заменяя переменные, останавливающие границы и оставляющие уравнение инвариантным, исходная краевая задача сводится к системе функционально-разностных уравнений, которая решена прямым и обратным методами. Описан также обратный метод, позволяющий аппроксимировать весьма разнообразные законы движения границы законами, полученными в результате решения обратной задачи. Новые частные решения получены для достаточно широкого круга законов движения границы. С помощью прямого асимптотического метода рассматривается приближенное решение функционального уравнения. Оценка погрешностей приближенного метода производится в зависимости от скорости движения границы.
Ключевые слова: волновое уравнение; краевые задачи; колебания систем с подвижными границами; замена переменных; законы движения границ; функциональные уравнения.
Цитирование. Litvinov V.L., Litvinova K.V. On one solution of the problem of vibrations of mechanical systems with moving boundaries // Вестник Самарского университета. Естественнонаучная серия / Vestnik of Samara University. Natural Science Series. 2023. Т. 29, № 3. С. 72-81. DOI: http://doi.org/10.18287/2541-7525-2023-29-3-72-81.
Информация о конфликте интересов: авторы и рецензенты заявляют об отсутствии конфликта интересов.
© Литвинов В.Л., Литвинова К.В., 2023 Владислав Львович Литвинов — кандидат технических наук, заведующий кафедрой Общетеоретических дисциплин, доцент, Самарский государственный технический университет, 443100, Российская Федерация, г. Самара, ул. Молодогвардейская, 244; докторант МГУ имени М.В. Ломоносова, механико-математический факультет, ГСП-1, 119991, Российская Федерация, г. Москва, Ленинские горы.
Кристина Владимировна Литвинова — студент 4 курса, Московский государственный университет имени М.В. Ломоносова, ГСП-1, 119991, Российская Федерация, г. Москва, Ленинские горы.
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