On one solution of the problem of vibrations of mechanical systems with moving boundaries

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Introduction
One-dimensional systems, the boundaries of which move, are widely used in engineering: ropes of lifting
installations [1–9], flexible transmission links [1; 10–14], solid fuel rods [15], drill strings [3], etc. The presence
of moving boundaries causes significant difficulties in describing such systems; therefore, approximate methods
Вестник Самарского университета. Естественнонаучная серия 2024. Том 30, № 1. С. 40–49
Vestnik of Samara University. Natural Science Series 2024, vol. 30, no. 1, pp. 40–49 41
of solution are mainly used here [1–3; 10; 14–21]. Among the analytical methods, the most effective is the
method proposed in [11], which consists in the se-lection of new variables that stop the boundaries and
leave the wave equation invariant. In [22], the solution is sought in the form of a superposition of two
waves running towards each other. The method used in [23] is also effective, which consists in replacing
the geometric variable with a purely imaginary variable, which allows us to apply the wave equation to the
Laplace equation and apply the method of the theory of functions of a complex variable to the solution.
In this article an analytical method for solving the wave equation that describes the oscillations of
systems with moving boundaries is proposed. By replacing 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 which makes
it possible to approximate quite diverse laws of boundary motion by laws obtained from solving the inverse
problem. New particular solutions have been obtained for a fairly wide range of boundary motion laws. A
direct asymptotic method for the approximate solution of a functional equation is considered. The errors of
the approximate method are estimated, depending on the speed of the boundary movement. This approach
successfully combines the methodology used in [11; 22; 24–27].
1. Statement of the problem
Let us consider free oscillations in a system with moving boundaries.
utt(x, t) − a2uxx(x, t) = 0. (1.1)
The boundary conditions at the fixed ends have the form
u (l1(t), t) = 0; u (l2(t), t) = 0. (1.2)
(l1(0) 6 x 6 l2(0))
Here, u(x, t) is the displacement of the point of the object with the coordinate x at time t; a is the velocity
of wave propagation in the system; l1(x), l2(x) are the laws of boundary motion.
In works [11; 22] Vesnitsky A.I. proposed a fairly general method for selecting new variables for the wave
equation. Following this method, the replacement of variables is performed in the following form:
ξ = φ(t + x/a) − ψ(t − x/a);
τ = a−1 [φ(t + x/a) + ψ(t − x/a)] ,
(1.3)
where φ and ψ are some functions. As a result of such a replacement, the original equation remains invariant
(wave), and φ, ψ are determined from the condition of constancy ξ at the boundaries.
In new variables ξ, τ , defined by relation (1.3), the initial problem (1.1)–(1.2) is reduced to the following
Uττ (ξ, τ ) − Uξξ(ξ, τ ) = 0 (1.4)
under boundary conditions
U(ℓ1(τ ), τ ) = 0; Uξ(ℓ2(τ ), τ ) = 0; (1.5)
(ℓ1(τ ) 6 ξ 6 ℓ2(τ )).
Here τ, ξ are a dimensionless time (τ > 0) and a dimensionless spatial coordinate; U(ξ, τ ) = u(x, t) ; ℓi(τ ) —
the laws of movement of borders.
Boundary conditions (1.5) in variables ξ, τ are set on new, generally speaking, moving boundaries, the
position of which depends on two functions φ and ψ. Since they φ and ψ are arbitrary, one can require that
the boundary conditions should be written on fixed boundaries, i. e. ℓ1 = const and ℓ2 = const (ℓ2 > ℓ1).
For this, it is necessary that φ and ψ necessary to satisfy the system of functional equations:
{
φ(τ + ℓ1(τ )) − ψ(τ − ℓ1(τ )) = ℓ1;
φ(τ + ℓ2(τ )) − ψ(τ − ℓ2(τ )) = ℓ2,
(1.6)
which uniquely determine the functions of φ and ψ through the known laws of boundary motion. When
the borders move at a speed higher than the speed of wave propagation, the solution of the wave equation
becomes incorrect, therefore, a restriction is imposed on the speed of the boundaries |ℓ′
i(τ )| < 1. Constants
ℓi can be arbitrary, but not equal values (for example, ℓ1 = 0, ℓ2 = 1). Then system (1.6) will take the
form: {
φ(τ + ℓ1(τ )) = ψ(τ − ℓ1(τ ));
φ(τ + ℓ2(τ )) = ψ(τ − ℓ2(τ )) + 1,
(1.7)
The existence of a solution to this system was proved in [11].
42
Litvinov V.L., Litvinova K.V. On one solution of the vibration problem of mechanical systems...
Литвинов В.Л., Литвинова К.В. Об одном решении задачи о колебаниях механических систем...
Solution (1.4)–(1.5) is found by the Fourier method [27]:
U(ξ, τ ) =
∞Σ
n=1
sin (ω0nξ) (Dn cos (ω0nτ ) + En sin (ω0nτ )) =
=
∞Σ
n=1
rn {sin (ω0n(τ + ξ) + αn) − sin (ω0n(τ − ξ) + αn)} ,
(1.8)
where ω0n(ε0τ ) = πn
ℓ2−ℓ1
; rn = 1
2
√
D2
n + E2
n; αn = arctg (En/Dn) .
The solution obtained in [1–3; 10–13; 22–24] has a form similar to (1.8).
Returning to the variables x and t, we get
u(x, t) =
∞Σ
n=1
rn {sin (ω0nφ(t + x) + αn) − sin (ω0nψ(t − x) + αn)} . (1.9)
Here φ and ψ, they are found from the solutions of the system of functional equations (1.7) according to
the known laws of boundary motion, and the constants Dn, En are determined from the initial conditions.
Generally speaking, it is not easy to solve system (1.7). There are two different approaches to solving it:
— inverse problems [3, 11, 15, 22-26], i. e. according to the given "phases" of natural oscillations φ and ψ,
finding the laws of motion of the boundaries ℓi(τ );
— direct problems [15; 28], i. e. finding the "phases" of natural oscillations according to the given laws
of motion of the boundaries ℓi(τ ).
2. Solution of the inverse problem
To solve system (1.7) A.I. Vesnitsky [11] used the inverse method, i. e. to the given φ and ψ from the
resulting system of equations, the laws of boundary motion ℓ1(τ ) and ℓ2(τ ) are found. When solving the inverse
problem, the equations of system (1.7) are reduced to the study of algebraic or transcendental equations
with respect to ℓi(τ ), which in many cases admit exact solutions. Based on the inverse problem Vesnitsky
A.I. and Potapov A.I. [11; 22] solutions for a fairly wide range of laws of boundary motion are obtained.
System (1.7) has infinitely many solutions, since on the interval [0, 1] the function φ(z) and on the interval
[–1,0] the function ψ(z) can be set arbitrarily, and using the method of successive approximations [27], 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 = φ(z); y2 = ψ(z) Of all the solutions, we are only
interested in monotone ones, and monotone solutions in the case of boundary movement at a speed lower
than the wave propagation speed (|ℓ
′
1(τ )| < 1; |ℓ
′
2(τ )| < 1) can only be monotonously increasing.
Lemma. If the function φ(z) – is monotonously increasing (decreasing), then the function ψ(z) is also
monotonously increasing (decreasing).
Proof. Indeed, from the first equation of system (1.7) at τ = τ0, it follows that
φ(τ0 + ℓ1(τ0)) = ψ(τ0 − ℓ1(τ0)).
Now suppose that τ1 > τ0 and the function φ(z) also increases (decreases), then in the case of boundary
motion at a speed lower than the wave propagation speed (|ℓ
′
1(τ )| < 1; |ℓ
′
2(τ )| < 1), we will have:
τ1 + ℓ1(τ1) > τ0 + ℓ1(τ0);
τ1 − ℓ1(τ1) > τ0 − ℓ1(τ0)
Since the function φ(z) in this case increases (decreases), then in order to perform the first equality of
system (1.7) at τ = τ1, it is necessary that the function ψ(z) increases (decreases), i.e. the function ψ(z) is
also increasing (decreasing).
Let us also show that the monotonic solution of system (1.7) in the case of boundary motion at a speed
lower than the wave propagation velocity can only be increasing.
Indeed, given the inequality ℓ1(τ ) < ℓ2(τ ) we get:
τ + ℓ1(τ ) < τ + ℓ2(τ ); τ − ℓ1(τ ) > τ − ℓ2(τ );
Suppose that φ(z) and ψ(z) they decrease, then we can write:
φ(τ + ℓ2(τ )) < φ(τ + ℓ1(τ )) = ψ(τ − ℓ1(τ )) < ψ(τ − ℓ2(τ )).
However, this inequality contradicts the second equation of system (1.7). Therefore, functions
φ(z) and ψ(z)can only be monotonically increasing. The lemma is proved.
Note that from system (1.7) the functions φ(z) and ψ(z) are determined up to a constant in the sense
that if φ(z) and ψ(z) are the solution of system (1.7), then φ(z) + C and ψ(z) + C are also a solution
Вестник Самарского университета. Естественнонаучная серия 2024. Том 30, № 1. С. 40–49
Vestnik of Samara University. Natural Science Series 2024, vol. 30, no. 1, pp. 40–49 43
(here C — is an arbitrary constant). Therefore, for certainty, we can choose such a function ψ(z), that
ψ(−1) = −1. At the same time, from the second equation of system (1.7) for τ = 0, it follows that φ(1) = 0.
From the first equation of system (1.7) for τ = 0, we obtain
φ(0) = ψ(0).
When assigning functions φ(z) and ψ(z), several arbitrary constants are introduced into them. The
dependence of the found laws of motion ℓ1(τ ) and ℓ2(τ ) 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 reverse solutions is quite wide. The solutions below satisfy the relations:
ℓ1(0) = 0; ℓ2(0) = 1; ψ(−1) = −1.
The set of obtained laws of motion of boundaries is divided into classes:
1. The solutions shown in Table 2.1 belong to class A when the left boundary is fixed and φ(z) = ψ(z).
Solutions numbered 1, 2, 3, 6 were obtained by A.I. Vesnitsky and A.I. Potapov [11; 22], solutions 4, 5, 7
were obtained for the first time.
Table 2.1
Class A decisions
Таблица 2.1
Решения класса А
l2(τ ) φ(z) = ψ(z)
1 ντ + 1 Ln[(νz+1)/(1−ν)]
Ln[(1+ν)/(1−ν)]
− 1
2
√
Bτ + B2/ |B|
√
Bz + B + 0, 25−
−
√
B2 − B + 0, 25 − 1
3 1/(4Bτ + 1) Bz2 + 0, 5z − B − 0, 5
4 1
αarcsh
[
0,5
B1e −B2e−
] B1(eαz − e−α) + B2(e−αz − eα) − 1,
B1 = B2 + 1/(eα − e−α), α > 0
5
√
(τ + B)2(α2 − 1) + 1 + 2αB + B2 − α(τ + B)
Ln[(z+B)2+1+2αB+B2]
Ln[(1+α)/(1−α)]
−
Ln[(B−1)2+1+2αB+B2]
Ln[(1+α)/(1−α)]
− 1
6
1
α
[
−d +
√
1 + d2 + (ατ + B)2
]
,
d = 1+B2−α2
2α
arctg(αz+B)
arcctg[(1+B2−α2)/(2α)]
−
arctg(B−α)
arcctg[(1+B2−α2)/(2α)]
− 1
7 1
α
(
ln 1+
√
1+4A2e2
2A
)
− τ Aeαz + B, α = ln 1+
√
1+4A2
2A
2. The next class B is determined by the fact that the boundaries move according to the same law:
ℓ1(τ ) = ℓ(τ ); ℓ2(τ ) = 1 + ℓ(τ ); ℓ(0) = 0.
Since the movement of the boundaries is interconnected, there is also an interconnection between the
functions φ(z) and ψ(z). It is expressed by the functional equation:
φ( ¯φ(ψ(z)) + 1) − ψ(z − 1) = 1. (2.1)
System (1.7) in this case can only be satisfied by functions that are solutions of equation (2.1). Here are
two previously unknown solutions of class B:
1) ℓ = ντ ; φ(z) = (1 − ν)z/2 + (1 + ν)/2 − 1;
44
Litvinov V.L., Litvinova K.V. On one solution of the vibration problem of mechanical systems...
Литвинов В.Л., Литвинова К.В. Об одном решении задачи о колебаниях механических систем...
ψ(z) = (1 + ν)z/2 + (1 + ν)/2 − 1;
2) ℓ(τ ) = 1
α ln[(Be−ατ − Ceατ )/(B − C)];
φ(z) = B(e−αz − 1) − C(e−α − 1) − 1; B = C + 1/(e−α − 1);
ψ(z) = C(eαz − 1) − C(e−α − 1) − 1.
3. For class C solutions, the boundaries move symmetrically in different directions, i.e.
ℓ1(τ ) = −ℓ(τ ); ℓ2(τ ) = ℓ(τ ).
The equation of the relationship of functions φ(z)and ψ(z)here has the form:
φ(z) = ψ(z) + 0, 5
Class C solutions are obtained from class A solutions using the following formulas:
ℓ(τ ) = ℓA(τ ); ψ(z) =
1
2
ψA(z); φ(z) = ψ(z) + 0, 5 ,
where the corresponding functions of class A solutions are indicated with the index A.
4. A solution of class D is obtained for the case when both boundaries move uniformly:
ℓ1(τ ) = (B2 − B1)τ/(B2 + B1); ℓ2(τ ) = (B2e1/c − B1)τ/(B1 + B2e1/c) + 1;
φ(z) = C Ln(B1z + D) − C Ln(D − B2) − 1;
ψ(z) = C Ln(B1z + D) − C Ln(D − B2) − 1;
D = (B1 + B2e1/c)/(e1/c − 1).
The solution number one in Table 2.1 can be used to study the rope vibrations of load-lifting installations
at uniform ascent (descent) [1; 2; 4–9]. The above solutions of class B can be used in the study of oscillations
of flexible transmission links [12–14]. 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 l(τ ) = 1 + ντ 2. 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. Well-known methods for solving
functional equations, sometimes can find φ and ψ from known ones ℓi(τ ), 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
φ(τ + l(τ )) − φ(τ − l(τ )) = 1. (3.1)
For an approximate solution of equation (3.1), it is proposed to use the asymptotic method [28].
For fixed boundaries ℓ(τ ) = ℓ, the solution to (3.1) is the linear function
φs(z) =
1
2ℓ
z + const.
In the case of a slow motion of the boundary ℓ(τ ), the “phase” of the wave φ(z) during its run through
the system changes slightly with respect to φs(z). It is assumed that φ(z) has derivatives of any order,
and writing φ(τ +ℓ(τ )) in the form of power series in ℓ(τ ), after substituting them into (1.1), we obtain a
differential equation for slowly changing current "phase" φ(τ )
∞Σ
k=0
lk+1
(k + 1)!
· dk+1φ
dτk+1 = 1. (3.2)
Since φ(τ ) deviates slightly from the linear law φs(z = τ ) during the wave travel time, each next term
on the left side of equation (3.2) is much smaller than the previous one, and its solution must be sought
in the form of a series
φ(τ ) =
∞Σ
n=0
φn(τ ). (3.3)
Substituting (3.3) into (3.2) and equating the terms of the same order of smallness individually to zero,
we obtain for the zero approximation
φ0(τ ) =
1
2
∫τ
0
dt
ℓ(t)
.
Вестник Самарского университета. Естественнонаучная серия 2024. Том 30, № 1. С. 40–49
Vestnik of Samara University. Natural Science Series 2024, vol. 30, no. 1, pp. 40–49 45
In the case of a linear law of motion of the boundary ℓ(t) = 1 + ντ , the phase of dynamic natural
oscillations is equal to
φ(z) =
ln[(νz + 1)/(1 + ν)]
2ν
. (3.4)
Values (3.4) were compared with the values obtained using the exact solution (Table 2.1):
φ(z) =
Ln [(νz + 1)/(1 − ν)]
Ln [(1 + ν)/(1 − ν)]
− 1. (3.5)
The values of the maximum absolute errors Δ of the asymptotic method, depending on the speed of the
boundary movement ν, are given in Table 3.1.
Table 3.1
Error of the asymptotic method depending on the velocity of the boundary
Таблица 3.1
Погрешность асимптотического метода в зависимости от скорости границы
ν 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9
Δ 0,002 0,006 0,013 0,023 0,036 0,053 0,073 0,100 0,139
In the interval ν ∈ [0, 1; 0, 6] the errors of the approximate method are small. The increase of the error
when ν approaches unity is explained by the fact that the function (3.5) becomes infinitely large when ν → 1.
Insignificant errors make it possible to apply the described method to solve functional equation (3.1) in
cases where its exact solution is not known.
Conclusion
Using the analytical method of variable substitution, 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 (1.7). Vesnitsky A.I. proposed to solve it by the reverse method, i.e. to set functions φ and ψ 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 (1.7) 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 rope vibrations of lifting installations with a uniform ascent (descent), flexible links of transmission (for example, a belt drive), etc.

About the authors

V. L. Litvinov

Samara State Technical University;
Lomonosov Moscow State University

Author for correspondence.
Email: vladlitvinov@rambler.ru
ORCID iD: 0000-0002-6108-803X

Candidate of Technical Sciences, head of the Department of General-Theoretical Disciplines, assistant professor; doctoral student

244, Molodogvardeyskaya Street, Samara, 443100, Russian Federation; GSP-1, Leninskie Gory, 1, Moscow, 119991, Russian Federation

K. V. Litvinova

Lomonosov Moscow State University

Email: kristinalitvinova900@rambler.ru
ORCID iD: 0000-0002-1711-9273

student

GSP-1, Leninskie Gory, 1, Moscow, 119991, Russian Federation

References

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