The Mathematical Modelling of Nonlinear String Oscillations in an Isotropic Viscoelastic Medium Using the Example of a Long Power Line

Abstract

In this study, a nonlinear mathematical model of a thin string oscillating in an isotropic viscoelastic medium is developed. The model addresses external and internal mechanical energy dissipation in the string using components described by nonlinear exponential functions. The differential state equation of the studied item is based on a modified Hamilton–Ostrogradsky integral variation principle. The principle is modified by expanding the Lagrangian with two additional components: one addressing the external and internal mechanical energy dissipation in a system and the other addressing the energy of external non-potential forces acting on a system. To substantiate the existence and uniqueness of the solution to a mixed initial boundary problem, the general theory of nonlinear differential equations is applied. A long, single power line is used as an example; its elements oscillate between two support points of the wire. The computer simulation results for the nonlinear vibrations of the object are presented and analysed.

1. Introduction

The mathematical modelling of nonlinear oscillation processes in complex mechanical equipment is a topical problem that involves the effective creation of mathematical models that should be relatively simple, on the one hand, and address any implicit motions in the object under analysis as far as practicable, on the other hand. This is especially true of equipment and systems whose parts are analysed as media in one-, two-, or three-dimensional space. This analysis is often complicated by their operation in difficult conditions and complex environments where physical processes cannot be described with simple linear functions that fail to provide for satisfactory model adequacy. In such difficult conditions, the objects are, of course, subject to thermodynamic impacts, which change their internal energy and consequently affect their physical properties. To take a full account of such processes, nonlinear dependences must be applied, again involving nonlinear functions. The application of nonlinear functions obviously requires mathematical reasons for the existence and uniqueness of a solution to the mathematical model of an object.

To construct a mathematical model of a complex dynamic object, both external and internal nonlinear factors must be considered. In other words, external and internal energy dissipation must be taken into account to address the oscillatory process damping in an object; these functions are described with nonlinear equations.

A thin string is examined here, oscillating in complex conditions associated with both its heating and the external impacts of a nonlinear viscoelastic medium. We suggest that the oscillations of a part of a long overhead power line, where a wire between support points can be approximately treated as a thin string, are an instance of the practical application of the mathematical modelling of transient processes. To simplify the problem, the longitudinal vibrations of limits, which are the support points on the object’s insulation, are assumed to be lower, by some orders of magnitude, than the string’s length. The linear wire is also assumed to be homogeneous—in reality, this is a combination of aluminium and steel wires. If the shifting of the line connection points is taken into account, the so-called transversality conditions need to be applied [1]. Such models are quite complex and are not analysed here.

The problem is formulated as follows. The risk of wire icing arises in winter in long power lines, especially in northern regions. The ice is occasionally several times thicker than the diameter of a wire itself. This may snap the wires and cause grid breakdowns. The issue is approached with a variety of so-called ice-melting devices, such as zigzag transformers and an additional supply of direct current component to a line. The idea is to quickly heat a wire with high currents until the ice melts. The ice-melting devices are then disconnected. The icing process is associated with strong winds that increase the ice layer, on the one hand, and cause power lines to oscillate, on the other hand. Such adverse weather conditions affect the continuity of the electrical supply to potential users and thus the national energy security.

From the physical point of view, oscillations are damped mainly due to internal viscoelastic friction in the line continuum and external factors associated with external friction. Analysing the processes of line wire icing implies that two nonlinear components need to be added to the equation for wire vibrations or, in this case, string oscillations in order to fully address most physical factors. One considers the internal dissipation of mechanical energy, which relates to added wire heating, and the other addresses external factors: ice, snow, and rain residues on a wire, wind blasts, etc.

The modified integral Hamilton–Ostrogradsky variational principle for distributed-parameter systems serves to develop a mathematical model of the object examined [1]. The Euler–Poisson equation result can be seen as a mathematical interpretation of the model of nonlinear string vibrations. The general theory of nonlinear partial differential equations is also used to provide a mathematical substantiation of the theory, namely, confirmation of the existence and uniqueness of the solution to the mixed initial boundary problem.

It should be pointed out that the issue of mixed problem correctness, i.e., the existence and explicitness of its solution, is not new. Similar results have been reported since the 1980s. Our research is motivated by using results for the mixed problem correctness to substantiate the numerical convergence of the model being implemented. In effect, this helps to integrate the state equation of a studied object regardless of which specific numerical method is selected.

The modelling and analysis of string behaviour have long been studied, which has led to the formulation of classic wave equations. The resultant models are applicable to a variety of engineering areas, like the analysis of bridge wires and transmission line wires. The subject of mechanical material oscillations is also important in the context of container wall vibrations, like in LNG gas tanks or walls of oil-filled transformers (oscillations in two-dimensional space) [2]. The state of the art on modelling string oscillations and the applications of models to engineering practice are reviewed below.

The authors of [3] analyse the effectiveness of the SPH (Smoothed Particle Hydrodynamics) method in modelling coupled oscillating systems. Two ships sailing side by side are studied, considering the movements of their cargo, namely, LNG tanks. The oscillations of an LNG tank have a significant effect on the variations in pressure acting on internal tank walls. This may cause hazardous damage to the ship’s bulk. Refs. [4,5,6] examine membrane systems in place to determine system parameters, assuring safe operation. The methods of fracture mechanics and the S-N curve, as well as calculations aided by neural networks, are used and compared.

The authors of [7] analyse numerical solutions to one-dimensional wave equations. Both explicit and implicit finite difference schemes are employed. The stability of the proposed schemes is analysed as well.

Ref. [8] is dedicated to the analysis of the straight-line method serving to solve the wave equations of a vibrating string as a part of musical instruments. Graphical user interface software, developed using MATLAB R2022B, is utilised in the modelling. The results are presented as animations.

Refs. [9,10] examine the mathematical models of string oscillations. The former considers new, non-local boundary problems for wave equations and summarises the classic Goursat and Darboux problems. D’Alembert’s method is employed. The theory of loaded string oscillations is utilised as well. The latter article offers an asymptotic analysis of Carrier’s nonlinear model of string oscillations. The problem is reduced to two weakly nonlinear wave equations. An asymptotic approximation over a long-time interval is constructed. The results are presented as numerical experiments.

Refs. [11,12,13] analyse some models using viscoelastic media, including the responses of soft materials. The application of partial derivatives as tools for describing this effect is analysed. The boundary condition problem is solved for a wave equation containing a non-integral derivative relative to a space variable and then used to model oscillations in a viscoelastic medium, as the variations in polymer concrete deformation and durability.

Similar research is described in [14], whose authors also analyse a mathematical model of polymer concrete. A solution to the string vibration equation is introduced. The equation contains a partial derivative in relation to a space variable, while the derivative is defined by Caputo. The model serves to describe the mechanical process of oscillations in a viscoelastic medium. This theoretical solution is compared with experimental data.

The problem of determining forces in bridge cables, including flexible internal supports, is analysed in [15]. The paper focuses on the modelling of string vibrations with one or more elastic supports. PINN neural networks are suggested instead of the traditional method of finite elements. It is finally proven that, in the case of a string with restricted elastic supports, a weak solution improves the effectiveness of mathematical modelling and the clarity of solutions.

Ref. [16] concerns overhead lines and bent wires. Its authors analyse vibrations of low frequencies around 1 Hz. Two mathematical models are introduced that serve to calculate the vibrations of bent wires, assuming their constant tension. The first model describes the linear oscillations of an elastic conductor in the bending plane. Solutions to the Sturm–Liouville problem are produced as a result. The other model describes nonlinear vibrations in the sagging plane and oscillatory vibrations as the plane is excited.

The authors of [17] investigate the impact of string oscillations on structural stability. Bridge tie beams are an instance of this application. A method of damping multi-order resonance is introduced and analysed. The string damping is controlled by modifying its length. A model allowing for string length variations over time is used. The effects of variable string length on free and forced oscillations are analysed. String responses are solved using the finite difference method.

Ref. [18] describes nonlinear oscillation dampers produced using magnets. The nonlinear characteristics of the dampers for various magnet configurations are examined. The results suggest systems including more magnets have a greater capacity for rapid energy dispersion. A dynamic system model is presented, too, including motion equations for the main system displacement. The equations contain a nonlinear term from the dipole–dipole magnetic force. The equations are solved with the aid of MATLAB R2022B.

This brief review of literature implies that the modelling of oscillatory processes and the application of wave equations remain current areas of research in both engineering and mathematics.

The goals of this paper are (1) the mathematical modelling of nonlinear transients of a thin string vibrating in a viscoelastic medium based on the modified Hamilton–Ostrogradsky principle and (2) the mathematical substantiation of the existence and uniqueness of the solution to a mixed initial boundary problem using the instance of a part of a prototype long power line operating in adverse weather conditions (specifically, wire icing).

2. Methods

2.1. Non-Conservative Lagrangian

The conservative force Lagrange function is the key part of the Hamilton–Ostrogradsky principle [19], which will be expressed as follows:

$$S={\displaystyle \underset{t_1}{\overset{t_2}{\int }}{L}_1^{*}dt }, L_1^{*}=L_1^{*}(q_k,{\dot{q}}_k,t)=T^{*}-P^{*}, \delta S=\delta {\displaystyle \underset{t_1}{\overset{t_2}{\int }}{L}_1^{*}dt =0}$$

(1)

where $S$—action according to Hamilton; $L_1^{*}$—the conservative force Lagrange function, $T^{*}$—kinetic energy (coenergy), $P^{*}$—potential energy, $q_k,{\dot{q}}_k$—generalised coordinates and velocities, k—the number of degrees of freedom, t—time, $\delta$—variation symbol.

Such an interpretation of the Hamilton–Ostrogradsky principle (1) is not normally used for the purposes of mathematical modelling of dynamic processes in real objects, since in this case, the conservative Lagrange function (Lagrangian) fails to address losses on energy dissipation or the effect of external non-potential forces acting on the object.

It is clear that the limitations of the conservative Lagrangian make the variational approach to mathematical modelling non-competitive compared with the well-known classic methods based on the law of energy conservation. In order to equalise the competitive capacities of the classic and variation approaches to modelling, therefore, the latter needs to be improved as appropriate. The literature review suggests a successful attempt is made in [19], whose authors rely on Helmholtz’s idea that the Lagrange function need not necessarily be presented in its classic format, that is, as the difference between kinetic and potential energies, and can be treated as any dependence between the generalised coordinates and velocities. Consequently, the authors add to the kinetic energy a component addressing energy dissipation across the system, naming it non-conservative kinetic energy (coenergy), and add to the potential energy a component addressing the impact of external non-potential forces acting on the system, naming it non-conservative potential. The resultant expanded Lagrange function is named the non-conservative force Lagrange function, cf. (2), as distinct from the conservative function, see (1). Note that the two components added to [19] are solely empirical, and no mathematical reasons are given for this procedure.

A somewhat different approach is suggested in [1], where a mathematical substantiation of the expanded non-conservative Lagrangian is given:

$$L^{*}(t)=T^{*}-P^{*}+{\Phi }^{*}-D^{*} , {\Phi }^{*}={\displaystyle \underset{0}{\overset{t}{\int }}{\Phi }_R^{*}{(\dot{q},t)}_{\left|t=\tau \right.}d\tau }$$

(2)

where $L^{*}$—modified (non-conservative) force Lagrange function; $D^{*}$—the energy of non-potential active and passive forces (in [19], the component is referred to as non-conservative potential); ${\Phi }^{*}$—the function of external and internal energy dissipation (in [19], the component is referred to as non-conservative kinetic energy (coenergy)); ${\Phi }_R^{*}$—the dissipative function of the dynamic system (most commonly, the Rayleigh function); $\tau$—additional integrating variable.

The idea of arriving at (2) in the monograph [1] is based on the well-known Lagrange equations of the second type, whose authenticity is undoubted. This is the idea of producing the non-conservative Lagrange function.

Let us formulate the Lagrange equations of the second type for holonomic systems [1,19]:

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial T^{*}}{\partial {\dot{q}}_k}}-{\displaystyle \frac{\partial T^{*}}{\partial q_k}}=Q_k-{\displaystyle \frac{\partial P^{*}}{\partial q_k}}-{\displaystyle \frac{\partial {\Phi }_R^{*}}{\partial {\dot{q}}_k}}$$

(3)

where $Q_k$—generalised force, k—the number of degrees of freedom.

Let us assume the system is in a cyclical motion. In line with Helmholtz’s theory, this means energy dissipation will only depend on generalised velocity and time, which is acceptable to a prevailing majority of applied dynamics problems:

$$T^{*}=T^{*}(q_k,{\dot{q}}_k,t) , P^{*}=P^{*}(q_k,t) , {\Phi }^{*}={\Phi }^{*}({\dot{q}}_k,t) , D^{*}=D^{*}(q_k,{\dot{q}}_k,t)$$

(4)

Based on the rule of superposition, this will be expressed in the vector format:

$$Q=Q_1+Q_2$$

(5)

where $Q_1$—generalised active forces, $Q_2$—generalised passive forces.

According to Newton’s second law, the generalised active forces will be calculated as follows:

$$Q_1={\displaystyle \frac{dp}{dt}} , p={\displaystyle \frac{\partial D^{*}}{\partial {\dot{q}}_k}}$$

(6)

where $p$—generalised external impulse received by the dynamic system.

Non-potential generalised passive forces can be found as follows:

$$Q_2={\displaystyle \frac{\partial D^{*}}{\partial q_k}}$$

(7)

This results from the definition of force: $F=\nabla W$, where $W$—the energy of non-potential passive forces, $\nabla$—Hamilton operator.

Since the generalised active and passive forces are opposite, e.g., the friction force and the driving force applied, (5) in the vectoral projections, considering (6) and (7), will be written as follows:

$$Q=Q_1-Q_2={\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial D^{*}}{\partial {\dot{q}}_k}}-{\displaystyle \frac{\partial D^{*}}{\partial q_k}}$$

(8)

Substituting (8) into the Lagrange equations of the second type (3) and considering (2), the following results:

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial T^{*}}{\partial {\dot{q}}_k}}-{\displaystyle \frac{\partial T^{*}}{\partial q_k}}={\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial D^{*}}{\partial {\dot{q}}_k}}-{\displaystyle \frac{\partial D^{*}}{\partial q_k}}-{\displaystyle \frac{\partial P^{*}}{\partial q_k}}-{\displaystyle \frac{\partial }{\partial {\dot{q}}_k}}{\displaystyle \frac{d{\Phi }^{*}}{dt}}$$

(9)

As per the definition, let us express the full function derivative from the arguments of: generalised coordinates, generalised velocity, and time: $f=f(q,\dot{q},t)$:

$$\begin{gathered} {\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial f(q,\dot{q},t)}{\partial \dot{q}}}={\displaystyle \frac{{\partial }^{2}f}{\partial q\partial \dot{q}}}{\displaystyle \frac{dq}{dt}}+{\displaystyle \frac{{\partial }^{2}f}{\partial {\dot{q}}^2}}{\displaystyle \frac{d\dot{q}}{dt}}+{\displaystyle \frac{{\partial }^{2}f}{\partial t\partial \dot{q}}}= \\ =-{\displaystyle \frac{\partial f}{\partial q}}+{\displaystyle \frac{\partial }{\partial \dot{q}}}\left({\displaystyle \frac{\partial f}{\partial q}}{\displaystyle \frac{dq}{dt}}+{\displaystyle \frac{\partial f}{\partial \dot{q}}}{\displaystyle \frac{d\dot{q}}{dt}}+{\displaystyle \frac{\partial f}{\partial t}}\right)={\displaystyle \frac{\partial }{\partial \dot{q}}}{\displaystyle \frac{df}{dt}}-{\displaystyle \frac{\partial f}{\partial q}} \end{gathered}$$

(10)

Considering the system’s cyclical motion, we shall write the following:

$${\displaystyle \frac{\partial }{\partial \dot{q}}}{\displaystyle \frac{df(\dot{q},t)}{dt}}={\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial f(\dot{q},t)}{\partial \dot{q}}}$$

(11)

Another dependence will be expressed on the basis of (11) as follows:

$${\displaystyle \frac{d}{dt}}{\displaystyle \underset{0}{\overset{\dot{q}}{\int }}f(\dot{q},t)d\dot{q}}={\displaystyle \underset{0}{\overset{\dot{q}}{\int }}\frac{df(\dot{q},t)}{dt}d\dot{q}}$$

(12)

Once the components are grouped as appropriate, (9) considering (11) will be expressed as follows:

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial T^{*}}{\partial {\dot{q}}_k}}+{\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial {\Phi }^{*}}{\partial {\dot{q}}_k}}-{\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial D^{*}}{\partial {\dot{q}}_k}} ={\displaystyle \frac{\partial T^{*}}{\partial q_k}}-{\displaystyle \frac{\partial P^{*}}{\partial q_k}}-{\displaystyle \frac{\partial D^{*}}{\partial q_k}}$$

(13)

Based on (4), the following will result:

$${\displaystyle \frac{\partial P^{*}(q_k,t)}{\partial {\dot{q}}_k}}\equiv 0 , {\displaystyle \frac{\partial {\Phi }^{*}({\dot{q}}_k,t)}{\partial q_k}}\equiv 0$$

(14)

Considering (14), (13) will be transformed as follows:

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial T^{*}}{\partial {\dot{q}}_k}}-{\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial P^{*}}{\partial {\dot{q}}_k}}+{\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial {\Phi }^{*}}{\partial {\dot{q}}_k}}-{\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial D^{*}}{\partial {\dot{q}}_k}} ={\displaystyle \frac{\partial T^{*}}{\partial q_k}}-{\displaystyle \frac{\partial P^{*}}{\partial q_k}}+{\displaystyle \frac{\partial {\Phi }^{*}}{\partial q_k}}-{\displaystyle \frac{\partial D^{*}}{\partial q_k}}$$

(15)

or, in its final form, as follows:

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial L^{*}}{\partial {\dot{q}}_k}}-{\displaystyle \frac{\partial L^{*}}{\partial q_k}}=0, L^{*}=T^{*}-P^{*}+{\Phi }^{*}-D^{*}$$

(16)

(16) represents the Euler–Lagrange equations for non-conservative dispersing systems considering the effect of external non-potential forces [1,19].

This method of determining the non-conservative Lagrange function is also applicable to distributed-parameter systems. In this case, the Lagrange function is presented as its density. There are three types of Lagrangian density in real problems: linear, superficial, and volumetric. Depending on the type of problem, the relevant density of the function is applied. In the case of a string, the expanded Lagrange function (16) will be represented as a linear density.

Differentiating (16) for the spatial variable and considering the initial conditions will result in the following:

$${\displaystyle \frac{\partial }{\partial x}}\left(L^{*}=T^{*}-P^{*}+{\Phi }^{*}-D^{*}\right)\Rightarrow L=T-P+\Phi -D$$

(17)

where L—the linear density of the non-conservative Lagrange function, T, P, Φ, D—the relevant linear densities of the non-conservative Lagrange function.

(1) becomes more complicated in the case of distributed-parameter systems:

$$S={\displaystyle \underset{t_1}{\overset{t_2}{\int }}{\displaystyle \underset{l}{\int }L}dldt}, I={\displaystyle \underset{l}{\int }L}dl, \delta S=\delta {\displaystyle \underset{t_1}{\overset{t_2}{\int }}{\displaystyle \underset{l}{\int }L}dldt}={\displaystyle \underset{t_1}{\overset{t_2}{\int }}\delta {\displaystyle \underset{l}{\int }L}dldt}$$

(18)

where I—internal energy functional.

If the internal energy functional variation is zero, the variation in the action functional as per Hamilton will obviously be zero, too. This means the internal energy functional must be analysed according to its extreme in order to produce the Euler–Poisson equation for distributed-parameter systems.

2.2. The Mathematical Model of Nonlinear String Vibrations

To develop a mathematical model of string vibrations, all four linear density elements of the non-conservative Lagrangian will be written. It must be remembered that the number of degrees of freedom is infinite in the case of distributed-parameter systems. Therefore, the concept of generalised coordinates and velocity cannot be introduced and will be replaced with the notion of the generalised coordinate and velocity functions. For the string, these functions will be as follows: the function describing the transverse motion of string points, the function describing the velocity of string point motion, and the current coordinate along the string.

The string motion equation can be expressed in two ways. The first version applies to the linear dissipation of mechanical energy across the string, while the other applies to the nonlinear dissipation. The linear density elements of the non-conservative Lagrangian will be written [1].

2.2.1. The Linear Dissipation of Mechanical Energy Across the String

$${\displaystyle \frac{\partial T^{*}}{\partial x}}\equiv T={\displaystyle \frac{\mathsf{\rho }S}{2}}{\left({\displaystyle \frac{\partial u}{\partial t}}\right)}^2, {\displaystyle \frac{\partial P^{*}}{\partial x}}\equiv P={\displaystyle \frac{N}{2}}{\left({\displaystyle \frac{\partial u}{\partial x}}\right)}^2 , {\displaystyle \frac{\partial D^{*}}{\partial x}}\equiv D={\displaystyle \underset{0}{\overset{u}{\int }}{f}_O(u)du} , {\displaystyle \frac{\partial u}{\partial t}}\equiv v$$

(19)

$${\displaystyle \frac{\partial {\Phi }^{*}}{\partial x}}\equiv \Phi ={\displaystyle \underset{0}{\overset{t}{\int }}{({\Phi }_{R1}+{\Phi }_{R2})}_{\left|t=\tau \right.}d\tau }={\displaystyle \underset{0}{\overset{t}{\int }}{\left(\frac{{\nu }_e}{2}{\left(\frac{\partial u}{\partial t}\right)}^2+\frac{\xi }{2}{\left(\frac{{\partial }^{2}u}{\partial x\partial t}\right)}^2\right)}_{\left|t=\tau \right.}d\tau }={\displaystyle \underset{0}{\overset{t}{\int }}{\left(\frac{{\nu }_e}{2}{v}^2+\frac{\xi }{2}{\left(\frac{\partial v}{\partial x}\right)}^2\right)}_{\left|t=\tau \right.}d\tau }$$

(20)

where $\mathsf{\rho }$—the density of string material; $N$—the constant force of string tension; ${\nu }_e$—the coefficient of external string dissipation; $\xi$—the coefficient of internal material dissipation; ${\Phi }_{R1} , {\Phi }_{R2}$—the linear densities of external and internal dissipation function, respectively; $f_O$—the distributed external force acting on the string along the transverse motion direction.

The following is the energy functional for the string, (16) and (18):

$$I={\displaystyle \underset{l}{\int } \left[\frac{\mathsf{\rho }S}{2}{v}^2-\frac{N}{2}{\left(\frac{\partial u}{\partial x}\right)}^2\right.}+{\displaystyle \underset{0}{\overset{t}{\int }}{\left(\frac{{\nu }_e}{2}{v}^2+\frac{\xi }{2}{\left(\frac{\partial v}{\partial x}\right)}^2\right)}_{\left|t=\tau \right.}d\tau }-\left.{\displaystyle \underset{0}{\overset{u}{\int }}{f}_O(u)du}\right]dl$$

(21)

Let us write the functional (21) variation and compare it to zero, cf. (18), remembering that the procedures of variation and differentiation (integration) are independent:

$$\begin{gathered} \delta I=0={\displaystyle \underset{l}{\int } \left[\frac{\mathsf{\rho }S}{2}\delta v^2-\frac{N}{2}\delta u_x^2\right.}+\delta {\displaystyle \underset{0}{\overset{t}{\int }}{\left(\frac{{\nu }_e}{2}{v}^2+\frac{\xi }{2}{v}_x^2\right)}_{\left|t=\tau \right.}d\tau }-\left.\delta {\displaystyle \underset{0}{\overset{u}{\int }}{f}_O(u)du}\right]dl= \\ ={\displaystyle \underset{l}{\int } \left[\mathsf{\rho }Sv\delta v-N{u}_x\delta u_x\right.}+{\displaystyle \frac{{\nu }_e}{2}}{\displaystyle \frac{\partial }{\partial v}}{\displaystyle \underset{0}{\overset{t}{\int }}{\left(v^2\right)}_{\left|t=\tau \right.}d\tau }\delta v+{\displaystyle \frac{\xi }{2}}{\displaystyle \frac{\partial }{\partial v_x}}{\displaystyle \underset{0}{\overset{t}{\int }}{\left(v_x^2\right)}_{\left|t=\tau \right.}d\tau }\delta v_x-{\displaystyle \frac{\partial }{\partial u}}{\displaystyle \underset{0}{\overset{u}{\int }}{f}_O(u)du}\delta u]dl=0 \end{gathered}$$

(22)

Let us express all the components in (22) relying on the Ostrogradsky–Gauss theorem, the principle of integration by parts, and Equations (11) and (12):

$${\displaystyle \underset{l}{\int } \mathsf{\rho }Sv\delta v}dl=\mathsf{\rho }S{\displaystyle \underset{l}{\int } u_t\frac{\partial }{\partial t}\delta u}dl=\mathsf{\rho }S\left[{\displaystyle \underset{l}{\int } \frac{\partial }{\partial t}\left(u_t\delta u\right)}dl-{\displaystyle \underset{l}{\int } \frac{\partial u_t}{\partial t}\delta u}dl\right]=\mathsf{\rho }S\left[-{\displaystyle \underset{l}{\int } u_{tt}\delta u}dl+{\Omega }_1\right]$$

(23)

$$-{\displaystyle \underset{l}{\int } N{u}_x\delta u_x}dl=-N{\displaystyle \underset{l}{\int } u_x\frac{\partial }{\partial x}\delta u}dl=-N\left[{\displaystyle \underset{l}{\int } \frac{\partial }{\partial x}\left(u_x\delta u\right)}dl-{\displaystyle \underset{l}{\int } u_{xx}\delta u}dl\right]=-N\left[-{\displaystyle \underset{l}{\int } u_{xx}\delta u}dl+{\Omega }_2\right]$$

(24)

$$\begin{gathered} {\displaystyle \underset{l}{\int } }{\displaystyle \frac{{\nu }_e}{2}}{\displaystyle \frac{\partial }{\partial v}}{\displaystyle \underset{0}{\overset{t}{\int }}{\left(v^2\right)}_{\left|t=\tau \right.}d\tau }\delta vdl={\displaystyle \frac{{\nu }_e}{2}}{\displaystyle \underset{l}{\int } }{\displaystyle \frac{\partial }{\partial u_t}}{\displaystyle \underset{0}{\overset{t}{\int }}{\left(u_t^2\right)}_{\left|t=\tau \right.}d\tau }\delta u_{t}dl={\nu }_e{\displaystyle \underset{l}{\int } }{\displaystyle \underset{0}{\overset{t}{\int }}{u}_{t\left|t=\tau \right.}d\tau }{\displaystyle \frac{\partial }{\partial t}}\delta udl= \\ ={\nu }_e\left[{\displaystyle \underset{l}{\int } }{\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \underset{0}{\overset{t}{\int }}{u}_{t\left|t=\tau \right.}d\tau }\delta u\right)dl-{\displaystyle \underset{l}{\int } }{\displaystyle \frac{\partial }{\partial t}}{\displaystyle \underset{0}{\overset{t}{\int }}{u}_{t\left|t=\tau \right.}d\tau }\delta udl\right]= \\ ={\nu }_e\left[-{\displaystyle \underset{l}{\int } }{\displaystyle \frac{\partial }{\partial t}}{\displaystyle \underset{0}{\overset{t}{\int }}{u}_{t\left|t=\tau \right.}d\tau }\delta udl+{\Omega }_3\right]={\nu }_e\left[-{\displaystyle \underset{l}{\int } }{u}_t\delta udl+{\Omega }_3\right] \end{gathered}$$

(25)

$$\begin{gathered} {\displaystyle \underset{l}{\int } }{\displaystyle \frac{\xi }{2}}{\displaystyle \frac{\partial }{\partial v_x}}{\displaystyle \underset{0}{\overset{t}{\int }}{\left(v_x^2\right)}_{\left|t=\tau \right.}d\tau }\delta v_{x}dl=\xi {\displaystyle \underset{l}{\int } }{\displaystyle \underset{0}{\overset{t}{\int }}{u}_{xt\left|t=\tau \right.}d\tau }\delta u_{xt}dl= \\ =\xi \left[{\displaystyle \underset{l}{\int } }{\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \underset{0}{\overset{t}{\int }}{u}_{xt\left|t=\tau \right.}d\tau }\delta u_t\right)dl\right.-\left.{\displaystyle \underset{l}{\int }\frac{\partial }{\partial x} }{\displaystyle \underset{0}{\overset{t}{\int }}{u}_{xt\left|t=\tau \right.}d\tau }\delta u_{t}dl\right]=\xi \left[-{\displaystyle \underset{l}{\int } }{\displaystyle \underset{0}{\overset{t}{\int }}{u}_{xxt\left|t=\tau \right.}d\tau }\delta u_{t}dl+{\Omega }_4\right] \end{gathered}$$

(26)

Let us apply the Ostrogradsky–Gauss theorem again to (26):

$$\begin{gathered} \xi \left[-{\displaystyle \underset{l}{\int } }{\displaystyle \underset{0}{\overset{t}{\int }}{u}_{xxt\left|t=\tau \right.}d\tau }\delta u_{t}dl+{\Omega }_4\right]=\xi \left[-{\displaystyle \underset{l}{\int } \frac{\partial }{\partial t}}\left({\displaystyle \underset{0}{\overset{t}{\int }}{u}_{xxt\left|t=\tau \right.}d\tau }\delta u\right)dl+{\displaystyle \frac{\partial }{\partial t}}{\displaystyle \underset{0}{\overset{t}{\int }}{u}_{xxt\left|t=\tau \right.}d\tau }\delta udl+{\Omega }_4\right]= \\ =\xi \left[u_{xxt}\delta udl+{\Omega }_4-{\Omega }_5\right] \end{gathered}$$

(27)

where ${\Omega }_i$—functions dependent on the boundary conditions of functional elements.

$$-{\displaystyle \frac{\partial }{\partial u}}{\displaystyle \underset{0}{\overset{u}{\int }}{f}_O(u)du}\delta u=-f_O(u)\delta u$$

(28)

(22) considering (23)–(28) can be expressed as follows:

$$\delta I=\left[{\displaystyle \underset{l}{\int }\left(-\mathsf{\rho }S{u}_{tt}\right.}+N{u}_{xx}-{\nu }_e{u}_t+\xi u_{xxt}-\left.f_O(u)\right)\delta udl\right]+\left[-\mathsf{\rho }S{\Omega }_1+N{\Omega }_2-{\nu }_e{\Omega }_3+\xi ({\Omega }_4-{\Omega }_5)\right]=0$$

(29)

While analysing (29), it can be easily seen that the expression in the first square brackets is a direct variation in energy functional (21) and that the expression in the second square brackets defines its boundary conditions. The theory of variations calculus says that, in order to arrive at the extreme of (21), both the expressions in (29) must reach zero (not the general expression in (29)) [1]. Let us analyse these expressions. The first becomes zero under the following conditions:

(1)

When the internal expression in the ordinary brackets is equal to zero.

(2)

The variation in the generalised coordinate function $\delta u$ is equal to zero.

(3)

Both the numerators are equal to zero.

Of course, the variation cannot equal zero a priori. Thus, the expression in the first square brackets can only become zero in the first case, namely

$$-\mathsf{\rho }S{u}_{tt}+N{u}_{xx}-{\nu }_e{u}_t+\xi u_{xxt}-f_O(u)=0$$

(30)

This is known as the Euler–Poisson equation. As far as the second expression in the square brackets in (29) is concerned, it is assumed to equal zero, because the boundary conditions in the general case are unknown and will be determined using not variational, but classic approaches.

(30) in its general format:

$${\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}=a^2{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{\xi }{\mathsf{\rho }S}}{\displaystyle \frac{{\partial }^{3}u}{\partial t\partial x^2}}-{\displaystyle \frac{v_e}{\mathsf{\rho }S}}{\displaystyle \frac{\partial u}{\partial t}}-{\displaystyle \frac{1}{\mathsf{\rho }S}}{f}_O(u), a^2={\displaystyle \frac{N}{\mathsf{\rho }S}}$$

(31)

where a—the velocity of elastic wave propagation in the continuous spring medium.

The string oscillation equations will include nonlinear expressions describing the external and internal mechanical energy dissipation in a string, for which the modified expressions of dissipation function linear densities will be presented.

2.2.2. Nonlinear Dissipation

Accordingly, (20) will change into [1]:

$$\begin{gathered} {\displaystyle \frac{\partial {\Phi }^{*}}{\partial x}}\equiv \Phi ={\displaystyle \underset{0}{\overset{t}{\int }}{\left({\nu }_e{\displaystyle \underset{0}{\overset{v}{\int }}{\left|v\right|}^{n-2}vdv}+\xi {\displaystyle \underset{0}{\overset{v_x}{\int }}{\left|v_x\right|}^{m-2}{v}_{x}d{v}_x}\right)}_{\left|t=\tau \right.}d\tau }= \\ ={\displaystyle \underset{0}{\overset{t}{\int }}{\left({\nu }_e{\displaystyle \underset{0}{\overset{u_t}{\int }}{\left|u_t\right|}^{n-2}{u}_{t}d{u}_t}+\xi {\displaystyle \underset{0}{\overset{u_{xt}}{\int }}{\left|u_{xt}\right|}^{m-2}{u}_{xt}d{u}_{xt}}\right)}_{\left|t=\tau \right.}d\tau } \end{gathered}$$

(32)

The string vibration equations, cf. (25)–(27), will be expressed in their nonlinear versions:

$$\begin{gathered} {\displaystyle \underset{l}{\int } }{\nu }_e{\displaystyle \frac{\partial }{\partial u_t}}{\displaystyle \underset{0}{\overset{t}{\int }}{\displaystyle \underset{0}{\overset{u_t}{\int }}{\left|u_t\right|}^{n-2}{u_t}_{\left|t=\tau \right.}}d{u}_{t}d\tau }\delta u_{t}dl={\nu }_e{\displaystyle \underset{l}{\int } }{\displaystyle \underset{0}{\overset{t}{\int }}{\left|u_t\right|}^{n-2}{u}_{t\left|t=\tau \right.}d\tau }\delta u_{t}dl= \\ ={\nu }_e\left[{\displaystyle \underset{l}{\int } }{\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \underset{0}{\overset{t}{\int }}{\left|u_t\right|}^{n-2}{u}_{t\left|t=\tau \right.}d\tau }\delta u\right)dl-{\displaystyle \underset{l}{\int } }{\displaystyle \frac{\partial }{\partial t}}{\displaystyle \underset{0}{\overset{t}{\int }}{\left|u_t\right|}^{n-2}{u}_{t\left|t=\tau \right.}d\tau }\delta udl\right]={\nu }_e\left[-{\displaystyle \underset{l}{\int } }{\left|u_t\right|}^{n-2}{u}_t\delta udl+{\Omega }_3^{\prime }\right] \end{gathered}$$

(33)

$$\begin{gathered} {\displaystyle \underset{l}{\int } }{\displaystyle \frac{\partial }{\partial u_{xt}}}{\displaystyle \underset{0}{\overset{t}{\int }}{\left(\xi {\displaystyle \underset{0}{\overset{u_{xt}}{\int }}{\left|u_{xt}\right|}^{m-2}{u}_{xt}d{u}_{xt}}\right)}_{\left|t=\tau \right.}d{u}_{xt}d\tau }\delta u_{xt}dl=\xi {\displaystyle \underset{l}{\int } }{\displaystyle \underset{0}{\overset{t}{\int }}{\left|u_{xt}\right|}^{m-2}{u}_{xt\left|t=\tau \right.}d\tau }\delta u_{xt}dl= \\ =\xi \left[{\displaystyle \underset{l}{\int } }{\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \underset{0}{\overset{t}{\int }}{\left|u_{xt}\right|}^{m-2}{u}_{xt\left|t=\tau \right.}d\tau }\delta u_x\right)dl-{\displaystyle \underset{l}{\int } }{\displaystyle \frac{\partial }{\partial t}}{\displaystyle \underset{0}{\overset{t}{\int }}{\left|u_{xt}\right|}^{m-2}{u}_{xt\left|t=\tau \right.}d\tau }\delta u_{x}dl\right]= \\ =\xi \left[-{\displaystyle \underset{l}{\int } }{\left|u_{xt}\right|}^{m-2}{u}_{xt\left|t=\tau \right.}\delta u_{x}dl+{\Omega }_4^{\prime }\right] \end{gathered}$$

(34)

The Ostrogradsky–Gauss theorem will once again be applied to (34):

$$\begin{gathered} \xi \left[-{\displaystyle \underset{l}{\int } }{\left|u_{xt}\right|}^{m-2}{u}_{xt\left|t=\tau \right.}\delta u_{x}dl+{\Omega }_4^{\prime }\right]=\xi \left[-{\displaystyle \underset{l}{\int } }{\displaystyle \frac{\partial }{\partial x}}\left({\left|u_{xt}\right|}^{m-2}{u}_{xt\left|t=\tau \right.}\delta u\right)dl+{\displaystyle \underset{l}{\int } }{\displaystyle \frac{\partial }{\partial x}}\left({\left|u_{xt}\right|}^{m-2}{u}_{xt\left|t=\tau \right.}\right)\delta udl\right]= \\ =\xi \left[{\displaystyle \underset{l}{\int } }{\displaystyle \frac{\partial }{\partial x}}\left({\left|u_{xt}\right|}^{m-2}{u}_{xt}\right)\delta udl +{\Omega }_4^{\prime }-{\Omega }_5^{\prime } \right] \end{gathered}$$

(35)

Executing the mathematical operations, like for the linear version, the string vibration equations will finally address the external and internal mechanical energy dissipation.

$${\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}=a^2{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{\xi }{\mathsf{\rho }S}}{\displaystyle \frac{\partial }{\partial x}}\left({\left|u_{xt}\right|}^{m-2}{u}_{xt}\right)-{\displaystyle \frac{{\nu }_e}{\mathsf{\rho }S}}{\left|u_t\right|}^{n-2}{u}_t-{\displaystyle \frac{1}{\mathsf{\rho }S}}{f}_O(u), a^2={\displaystyle \frac{N}{\mathsf{\rho }S}}$$

(36)

If the linear version of the mechanical energy dissipation, i.e., n = 2 and m = 2, is taken into account, (36) will become identical with (31).

We consider the practical case of (36), in which self-oscillating processes will occur in the oscillating system, namely the case of $f_o(u)=f(x,t).$ An example of such a function can be a periodic external forcing force. Thus, we will finally consider a nonlinear wave equation:

$${\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}=a^2{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{\xi }{\mathsf{\rho }S}}{\displaystyle \frac{\partial }{\partial x}}\left({\left|u_{xt}\right|}^{m-2}{u}_{xt}\right)-{\displaystyle \frac{v_e}{\mathsf{\rho }S}}{\left|u_t\right|}^{n-2}{u}_t-{\displaystyle \frac{1}{\mathsf{\rho }S}}f(x,t).$$

(37)

We study the nonlinear oscillations of the string for the initial deflection amplitude and zero initial velocity of the string points. We consider the case of a string rigidly fixed at the ends. Thus, we will investigate, for (13) in the domain $Q_T=[0,l]\times [0,T]$ (l is the string’s length), a mixed problem with the following initial conditions:

$$u(x,0)=u_0(x),\hspace{1em}{\displaystyle \frac{\partial u(x,0)}{\partial t}}=0$$

(38)

and the following boundary conditions:

$$u(0,t)=u(l,t)=0.$$

(39)

The mixed problem (37)–(39) is a problem for a wave equation that contains terms describing both nonlinear internal and nonlinear external dissipation in the oscillating system. These equations and the various related problems (initial, boundary value, inverse, etc.) have been extensively studied in the last two decades. Both the theoretical interest in the properties of the solutions to such problems [20] and their practical application and use in the mathematical modelling of technological oscillatory systems [21] should be noted.

Problems for wave equations with the presence of nonlinear dissipative terms are studied on the basis of general methods and approaches of the nonlinear theory of boundary value problems. An abstract approach to the existence, uniqueness, and regularity of solutions to some semilinear wave equation classes in Banach spaces is developed in [22]. The results are applied to the analysis of mathematical models of the so-called damping semilinear wave equations with self-adjoint operators in the main part. In [23], the global solution, uniqueness, and asymptotic behaviour of a one-dimensional wave equation with terms that describe the nonlinear nature of internal and external dissipation are investigated. To prove the existence and uniqueness, the Faedo–Galerkin approximation is used, and the asymptotic behaviour is obtained by means of the Nakao method.

Similar problems are considered by [24], in which the main emphasis is placed on the study of the blowup property for the solutions to the p-Laplacian-type wave equation with a weak dissipation of the m-Laplacian type. In [25], some bounded solutions to the wave equation with supercritical nonlinear sources and nonlinear damping are considered, and the results of the uniqueness of the weak global solutions are obtained. The authors of [26] study the existence and uniqueness of weak solutions and the existence of weakly compact attractors in the nonlinear models of Kirchhoff–Boussinesq plates. Ref. [27] presents a study of mixed problem solutions for the nonlinear wave equation with p-Laplacian damping and with Dirichlet boundary conditions. The existence of a global solution is proved, and the rate of energy decay is estimated. The authors of [28] prove that, under some natural conditions for nonlinear terms and initial data, the initial boundary value problem for a class of nonlinear wave equations with a dissipative term admits a global weak solution, which exponentially decays to zero at t → +∞ on condition of a sufficiently small initial energy. In particular, the uniqueness of the generalised solution is proved in the one-dimensional case. An estimate of the energy decay of the global solution to the wave equation with the p-Laplacian and dissipation is obtained in [29]. The method of multipliers in combination with nonlinear integral inequalities is used.

Ref. [30] considers a mixed problem for an equation of quasi-hyperbolic type with logarithmic nonlinearity. The Galerkin method is applied. The conditions for the nonexistence of a global solution are obtained, and an estimate of the existence time of local solutions is given. The authors of [31] consider the problem for a nonlinear wave equation with power-type dissipative terms and initial and Dirichlet boundary conditions. Under appropriate conditions, a global theorem on the nonexistence of a solution is proved depending on the exponents of the nonlinear term, powers of the equation, and the condition of negative initial energy. Ref. [32] considers the equation of a damped wave with impulse action and determines sufficient conditions for the existence of piecewise continuous almost periodic solutions.

In recent years, the problems of synthesis and optimisation of parameters for nonlinear oscillatory systems, based on the study of inverse problems for equations with dissipation, have been actively examined (see [33] and the references there). In particular, Ref. [33] considers the mixed and inverse coefficient problem for a semilinear hyperbolic equation with strong damping. Conditions for the existence and uniqueness of solutions to these problems in Sobolev spaces have been established. Starting with one of the first works [34] on nonlinear dissipative oscillatory systems with a fractional derivative, such research has also been actively conducted in recent years.

2.3. The Mathematical Aspects of the Existence and Uniqueness of the Solution: The Galerkin and Monotonicity Methods

In this section, we will briefly describe a substantiation scheme for the existence and uniqueness of the solution to the mixed problem in the above mathematical model of nonlinear oscillations under the impact of external and internal dissipative forces. Note that the scheme is based on the application of general approaches from the nonlinear theory of boundary value problems—the Galerkin method and the monotonicity method [35,36]. In addition, note that the main result of this section is a theoretical justification of the possibility of applying an appropriate numerical method to finding an approximate solution, and the choice of a numerical method itself is fundamental only from the point of view of the efficiency of a computational procedure.

Assume that the initial deviations, $u_0(x)\in H_0^1(0,l)$, $m>2$, $n>2$, $f(x,t)\in L^{m^{\prime }}(Q_T)$, $n^{\prime }={\displaystyle \frac{n}{n-1}}$, $m^{\prime }={\displaystyle \frac{m}{m-1}}$ are numbers conjugated to $n$ and $m$ accordingly.

We establish a generalised solution to (37)–(39) in $Q_T$ the function $u \in C([0,T]; H_0^1(0,l))$ such that

$${\displaystyle \frac{\partial u}{\partial t}} \in C([0,T]; L^2(0,l))\cap L^m((0,T); W_0^{1,m}(0,l))\cap L^n((0,T); L^n(0,l))$$

(40)

which satisfies conditions (38) and the integral identity

$$\begin{gathered} {\displaystyle \underset{0}{\overset{l}{\int }}{\displaystyle \underset{0}{\overset{\tau }{\int }}\left[-u_t{v}_t+\frac{\xi }{\mathsf{\rho }S}{\left|u_{xt}\right|}^{m-2}{u}_{xt}{v}_x+a^2{u}_x{v}_x+\frac{1}{\mathsf{\rho }S}f(x,t)v+\frac{v_e}{\mathsf{\rho }S}{\left|u_t\right|}^{n-2}{u}_{t}v\right]dxdt}}+ \\ +{\displaystyle \underset{0}{\overset{l}{\int }}\left[u_t(x,\tau )-u_1(x)v(x,0)\right]}dx=0 \end{gathered}$$

(41)

for an arbitrary function $v \in L^m((0,T); W_0^{1,m}(0,l))\cap L^n((0,T); L^n(0,l))$ such that $v_t \in L^2((0,T); L^2(0,l))$ and for arbitrary $\tau \in \left(0,T\right].$

The formulation of the main result: In the conditions of the mathematical model of nonlinear oscillations given in the previous section, there is a unique generalised solution to (37)–(39) in $Q_T$, so the property of global well-posedness of the solution is proved.

We will describe the general scheme of using the Galerkin method to justify the main result.

Existence. We will use the justification scheme for applying the Galerkin method to the problem presented in [35]. Consider a sequence of Galerkin approximations $u^N(x,t)={\displaystyle \sum _{k=1}^N}{C}_k^N(t){\omega }_k(x),$ N = 1, 2, … in the domain $Q_T,$ where $\left\{{\omega }_k\right\}$ is orthonormalised in $L^2(0,l)$ system of linearly independent elements of space $W_0^{1,m}(0,l)\cap L^n(0,l)$ such that linear combinations $\left\{{\omega }_k\right\}$ are dense in $W_0^{1,m}(0,l)\cap L^n(0,l)$. At the same time, we define functions $C_k^N(t)$ as a Cauchy problem solution to a system of ordinary differential equations:

$${\displaystyle \underset{0}{\overset{l}{\int }}\left[u_{tt}^N{\omega }_k+\frac{\xi }{\mathsf{\rho }S}{\left|u_{xt}^N\right|}^{m-2}{u}_{xt}^N{\omega }_{kx}+a^2{u}_x^N{\omega }_{kx}-\frac{1}{\mathsf{\rho }S}f(x,t){\omega }_k-\frac{v_e}{\mathsf{\rho }S}{\left|u_t^N\right|}^{n-2}{u}_t^N{\omega }_k\right]dx}=0$$

(42)

with the initial conditions $C_k^N(0)=u_{0,k}^N$, $u_0^N(x) = {\displaystyle \sum _{k=1}^N}{u}_{0,k}^N{\omega }_k(x)$, $C_{k,t}^N(0)=0$, ${‖u_0^N-u_0‖}_{H_0^1(0,l)} \to 0,$ $N\to +\infty ,$ $k=1,\dots ,N$. On the basis of Caratheodori’s theorem [37], there exists a continuous solution to a Cauchy problem with a continuous time derivative in $\left(0,T\right]$. We multiply each equation of (42) by $C_{k,t}^N$, sum all equations from 1 to $N$ by $k$ and integrate the result by time on the interval $\left(0,\tau \right]$, $\tau \le T$. As a result, we obtain the following equality:

$${\displaystyle \underset{0}{\overset{\tau }{\int }}}〈u_{tt}^N,u_t^N〉dt+{\displaystyle \underset{0}{\overset{\tau }{\int }}{\displaystyle \underset{0}{\overset{l}{\int }}\left[\frac{\xi }{\mathsf{\rho }S}{\left|u_{xt}^N\right|}^m+\frac{{\nu }_e}{\mathsf{\rho }S}{\left|u_t^N\right|}^n+a^2{u}_x^N{u}_{xt}^N-f(x,t)u_t^N\right]}}dxdt=0.$$

(43)

Note that (37) implies that $u_{tt}^N\in L^2\left((0,T); W_0^{-1,m}\left(0,l\right) + L^{n^{\prime }}\left(0,l\right)\right)$, where $W_0^{-1,m}\left(0,l\right) + L^{n^{\prime }}\left(0,l\right)$ is the space conjugate to $W_0^{1,m}(0,l)\cap L^n(0,l)$. That is, the value of the scalar product $〈u_{tt}^N,u_t^N〉$ between a dual pair of spaces $W_0^{-1,m}\left(0,l\right) + L^{n^{\prime }}\left(0,l\right)$ and $W_0^{1,m}(0,l)\cap L^n(0,l)$ is correctly defined. Therefore, it is possible, in particular, to apply the formula of integration by parts when calculating the definite integral from the indicated scalar product [35]. We will carry out the transformations and evaluations of the integrals of equality (43). The possibility of integration in parts will result from the fact stated above.

$${\displaystyle \underset{0}{\overset{\tau }{\int }}}〈u_{tt}^N,u_t^N〉dt={\displaystyle \frac{1}{2}}{\displaystyle \underset{0}{\overset{l}{\int }}{\left(u_t^N(x,\tau )\right)}^2}dx.$$

Let us now evaluate the other integrals of equality (43):

$$\begin{gathered} {\displaystyle \underset{0}{\overset{\tau }{\int }}{\displaystyle \underset{0}{\overset{l}{\int }}{a}^2{u}_x^N{u}_{xt}^N}}dxdt={\displaystyle \frac{a^2}{2}}{\displaystyle \underset{0}{\overset{\tau }{\int }}{\displaystyle \underset{0}{\overset{l}{\int }}{\left({\left(u_x^N\right)}^2\right)}_t}}dxdt={\displaystyle \frac{a^2}{2}}{\displaystyle \underset{0}{\overset{l}{\int }}{\left(u_x^N\right)}^2}(x,\tau )dx; \\ {\displaystyle \underset{0}{\overset{\tau }{\int }}{\displaystyle \underset{0}{\overset{l}{\int }}f(x,t)u_t^N}}dxdt\le \delta {\displaystyle \underset{0}{\overset{\tau }{\int }}{\displaystyle \underset{0}{\overset{l}{\int }}{\left|u_t^N\right|}^n}}dxdt+C{\displaystyle \underset{0}{\overset{l}{\int }}{\left|f(x,t)\right|}^{n^{\prime }}}dxdt, \end{gathered}$$

where $\delta$, an arbitrary sufficiently small positive, is constant; $C$ is some positive constant that depends on $p$, $\delta$. The Cauchy–Bunyakovsky inequality serves to obtain the last estimate.

Taking into account the above estimates, we arrive at:

$$\begin{gathered} {\displaystyle \frac{1}{2}}{\displaystyle \underset{0}{\overset{l}{\int }}\left[{\left(u_t^N(x,\tau )\right)}^2+{\left(u_x^N(x,\tau )\right)}^2\right]}+\left({\displaystyle \frac{v_e}{\mathsf{\rho }S}}-\delta \right){\displaystyle \underset{0}{\overset{\tau }{\int }}{\displaystyle \underset{0}{\overset{l}{\int }}{\left|u_t^N\right|}^n}}dxdt+{\displaystyle \frac{\xi }{\mathsf{\rho }S}}{\displaystyle \underset{0}{\overset{\tau }{\int }}{\displaystyle \underset{0}{\overset{l}{\int }}{\left|u_{xt}^N\right|}^m}}dxdt\le \\ \le C{\displaystyle \underset{0}{\overset{\tau }{\int }}{\displaystyle \underset{0}{\overset{l}{\int }}{\left|f\right|}^{n^{\prime }}}}dxdt. \end{gathered}$$

After relabeling and choosing a sufficiently small constant $\delta$ and taking into account the conditions of the initial deviation and the function $f(x,t)$, the inequality results in the following:

$${\displaystyle \underset{0}{\overset{l}{\int }}\left[{\left(u_t^N(x,\tau )\right)}^2+{\left(u_x^N(x,\tau )\right)}^2\right]}+{\displaystyle \underset{0}{\overset{\tau }{\int }}{\displaystyle \underset{0}{\overset{l}{\int }}\left[{\left|u_t^N\right|}^n+{\left|u_{xt}^N\right|}^m\right]}}dxdt\le C_1,$$

(44)

where $C_1$ is some positive constant that depends on the right-hand side and does not depend on $u^N$. The a priori estimate (44) is correct for arbitrary $\tau \in \left(0,T\right]$; (44) implies the existence of some subsequence $\left\{u^{N_k}\right\}$ of the sequence $\left\{u^N\right\}$ such that

$$u^{N_k}\to u*—\mathrm{weakly} \mathrm{in} L^{\infty }(\left(0,T\right); H_0^1(0,l)),$$

(45)

$$u_t^{N_k}\to u_t*—\mathrm{weakly} \mathrm{in} L^{\infty }(\left(0,T\right); L^2(0,l)),$$

(46)

$$u_t^{N_k}\to u_t—\mathrm{weakly} \mathrm{in} L^m(\left(0,T\right); W_0^{1,m}(0,l)),$$

(47)

$$u_t^{N_k}\to u_t—\mathrm{weakly} \mathrm{in} L^n(\left(0,T\right); L^n(0,l))$$

(48)

$N_k\to \infty$. Let us denote now

$$A(u_t)={\displaystyle \frac{\xi }{\mathsf{\rho }S}}{\displaystyle \frac{\partial }{\partial x}}\left({\left|u_{xt}\right|}^{m-2}{u}_{xt}\right)-{\displaystyle \frac{v_e}{\mathsf{\rho }S}}{\left|u_t\right|}^{n-2}{u}_t.$$

(49)

Using the above conditions (in particular, $n, m>2$), we conclude the nonlinear operator $A$, defined by (49), is uniformly continuous, monotonic, and coercive [36,38]. Thus, taking into account (44), it is possible to express this as follows:

$$‖A(u_t^{N_k})‖\le C_2, C_2=const>0.$$

(50)

We conclude from inequality (50) (passing to subsequences if necessary) that $A(u_t^{N_k})\to \zeta ,$$\zeta \in L^m((0,T); W_0^{1,m}(0,l))\cap L^n((0,T); L^n(0,l))$ [35].

It follows from (45)–(48) in the corresponding Sobolev functional spaces that the defined function $u$ satisfies conditions (38)–(39) and the integral identity:

$$\begin{gathered} {\displaystyle \underset{0}{\overset{l}{\int }}{\displaystyle \underset{0}{\overset{\tau }{\int }}\left[-u_t{v}_t+\frac{\xi }{\mathsf{\rho }S}{\left|\zeta \right|}^{m-2}\zeta v_x+a^2{u}_x{v}_x+\frac{1}{\mathsf{\rho }S}f(x,t)v+\frac{v_e}{\mathsf{\rho }S}{\left|\zeta \right|}^{n-2}\zeta v\right]dxdt}}+ \\ +{\displaystyle \underset{0}{\overset{l}{\int }}\left[u_t(x,\tau )-u_1(x)v(x,0)\right]}dx=0 \end{gathered}$$

(51)

for an arbitrary function $v \in L^m((0,T); W_0^{1,m}(0,l))\cap L^n((0,T); L^n(0,l))$ such that $v_t \in L^2((0,T); L^2(0,l))$ and for an arbitrary $\tau \in \left(0,T\right].$ Using [31], we derive $u \in C([0,T]; H_0^1(0,l))$ and ${\displaystyle \frac{\partial u}{\partial t}} \in C([0,T];L^2(0,l))\cap L^m((0,T); W_0^{1,m}(0,l))\cap L^n((0,T); L^n(0,l))$ from (51). To complete the justification of the existence, it remains to show that $\zeta =A(u_t).$ If $A$ is a monotone operator, then ${\displaystyle \underset{0}{\overset{T}{\int }}}〈A(u_t^N)-A(v),u_t^N-v〉dt\ge 0$ for an arbitrary function $v$ with the necessary properties. In addition, using (45)–(49), it is possible to conclude that for such a function, $v$ the inequality is correct:

$${\displaystyle \underset{0}{\overset{T}{\int }}}〈\zeta -A(v),u_t-v〉dt\ge 0.$$

(52)

Let us substitute $v=u_t-\lambda w,$ $\lambda >0$ to the inequality (52), $w \in L^m((0,T); W_0^{1,m}(0,l))\cap L^n((0,T); L^n(0,l))$ is an arbitrary function. Then, we will produce $\lambda {\displaystyle \underset{0}{\overset{T}{\int }}}〈\zeta -A(u_t-\lambda w),w〉dt\ge 0$. Using Lebesgue’s theorem when $\lambda \to +0,$ we have

$${\displaystyle \underset{0}{\overset{T}{\int }}}〈\zeta -A(u_t-\lambda w),w〉dt\ge 0,$$

(53)

$w$ is an arbitrary function. Due to the arbitrariness of $w$, it is clear from (53) that $\zeta =A(u_t),$ i.e., $u$ is a generalised solution to problem (37)–(39) for which condition (41) is fulfilled.

Uniqueness. Let us take $w=u^1-u^2$, where $u^1,$ $u^2$ are two generalised solutions to (37)–(39). As noted above, ${\displaystyle \underset{0}{\overset{\tau }{\int }}}〈w_{tt},w_t〉dt$ exists for an arbitrary $\tau \in \left(0,T\right].$ In addition, for the integral ${\displaystyle \underset{0}{\overset{\tau }{\int }}}〈w_{tt},w_t〉dt$, it is possible to apply the formula of integration by parts. Using the above considerations and zero initial conditions, an estimate similar to (44) can be obtained:

$${\displaystyle \underset{0}{\overset{l}{\int }}\left[{\left(u_t^N(x,\tau )\right)}^2+{\left(u_x^N(x,\tau )\right)}^2\right]}+{\displaystyle \underset{0}{\overset{\tau }{\int }}{\displaystyle \underset{0}{\overset{l}{\int }}\left[{\left|u_t^N\right|}^n+{\left|u_{xt}^N\right|}^m\right]}}dxdt\le 0$$

(54)

for an arbitrary $\tau \in \left(0,T\right].$ Thus, (54) immediately implies that $u^1=u^2$ almost everywhere in $Q_T$. Uniqueness is proved.

According to the definition, the spatial derivative will be written out to express internal dissipation from (36):

$${\displaystyle \frac{\xi }{\mathsf{\rho }S}}{\displaystyle \frac{\partial }{\partial x}}\left({\left|u_{xt}\right|}^{m-2}{u}_{xt}\right)={\displaystyle \frac{\xi }{\mathsf{\rho }S}}\left[(m-2){\left|u_{xt}\right|}^{m-3}\left|u_{xxt}\right|u_{xt}+{\left|u_{xt}\right|}^{m-2}{u}_{xxt}\right]$$

(55)

(55) is expressed as follows:

$${\displaystyle \frac{\xi }{\mathsf{\rho }S}}\left[(m-2){\left|{\displaystyle \frac{\partial v}{\partial x}}\right|}^{m-3}\left|{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}\right|{\displaystyle \frac{\partial v}{\partial x}}+{\left|{\displaystyle \frac{\partial v}{\partial x}}\right|}^{m-2}{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}\right], {\displaystyle \frac{\partial u}{\partial t}}\equiv u_t=v, {\displaystyle \frac{\partial u}{\partial x}}\equiv u_x$$

(56)

(56) is discretised using the straight-line method [1]:

$${\displaystyle \frac{1}{{(\Delta x)}^2}}{\displaystyle \frac{\xi }{\mathsf{\rho }S}}\left[(m-2)\left|v_{i+1}-2v_i+v_{i-1}\right|{\left|{\displaystyle \frac{v_{i+1}-v_{i-1}}{2\Delta x}}\right|}^{m-3}\left({\displaystyle \frac{v_{i+1}-v_{i-1}}{2\Delta x}}\right)+{\left|{\displaystyle \frac{v_{i+1}-v_{i-1}}{2\Delta x}}\right|}^{m-2}\left(v_{i+1}-2v_i+v_{i-1}\right)\right]$$

(57)

Considering (57), the discrete form of string vibration equations is expressed for:

2.3.1. Linear Dissipation

$${\displaystyle \frac{d{v}_i}{dt}}=a^2{\displaystyle \frac{u_{i+1}-2u_i+u_{i-1}}{{(\Delta x)}^2}}+{\displaystyle \frac{\xi }{\mathsf{\rho }S}}{\displaystyle \frac{v_{i+1}-2v_i+v_{i-1}}{{(\Delta x)}^2}}-{\displaystyle \frac{{\nu }_e}{\mathsf{\rho }S}}{v}_i-{\displaystyle \frac{1}{\mathsf{\rho }S}}{f}_O(u), {\displaystyle \frac{d{u}_i}{dt}}=v_i$$

(58)

2.3.2. Nonlinear Dissipation

$$\begin{gathered} {\displaystyle \frac{d{v}_i}{dt}}=a^2{\displaystyle \frac{u_{i+1}-2u_i+u_{i-1}}{{(\Delta x)}^2}}+{\displaystyle \frac{1}{{(\Delta x)}^2}}{\displaystyle \frac{\xi }{\mathsf{\rho }S}}\left[(m-2)\left|v_{i+1}-2v_i+v_{i-1}\right|{\left|{\displaystyle \frac{v_{i+1}-v_{i-1}}{2\Delta x}}\right|}^{m-3}\left({\displaystyle \frac{v_{i+1}-v_{i-1}}{2\Delta x}}\right)\right]+ \\ +{\displaystyle \frac{1}{{(\Delta x)}^2}}{\displaystyle \frac{\xi }{\mathsf{\rho }S}}\left[{\left|{\displaystyle \frac{v_{i+1}-v_{i-1}}{2\Delta x}}\right|}^{m-2}\left(v_{i+1}-2v_i+v_{i-1}\right)\right]-{\displaystyle \frac{{\nu }_e}{\mathsf{\rho }S}}{\left|v_i\right|}^{n-2}{v}_i-{\displaystyle \frac{1}{\mathsf{\rho }S}}{f}_O(u), {\displaystyle \frac{d{u}_i}{dt}}=v_i \end{gathered}$$

(59)

3. Computer Simulation Results

The following assumptions underlie the analysis of nonlinear oscillatory processes in a string vibrating in a viscoelastic medium. A prototype overhead power line, where a wire between support points can, at the first approximation, be treated as a thin string, serves as an instance of practical processes. These are the parameters of the mathematical string model. Note that a simplified long-line prototype is used in this paper to demonstrate the effect of the nonlinearity of the external and internal dissipation function on the line wires. These are the parameters of the mathematical string model 85-A1/49-ST1A high-voltage cable with the following parameters: distributed weight: M₀ = 0.638 kg/m, cable cross-section: S_Al = 84.82 mm², S_Fe = 49.48 mm², total cable diameter without considering the cable fill factor: $Ø$ = 15.0 mm, maximum tensile strength: Fg = 70.33 kN, the distance between supports: L = 100 m, as well as N = 4 kN, $\xi$ = 200 kg·m/s, $v$ = 3 kg/s·m, m = 2.1, n = 2.9, $f_O(u)$ = 0, $i=1, 2,\dots , 11$, $\Delta x$ = 10 m.

A calculation scheme that visualises the discrete string nodes is shown in Figure 1. It can be seen that 22 nonlinear ordinary differential equations are integrated in the temporal space: 11 of node displacements and 11 of node velocities. As for the initial conditions, the node velocities are zero. The initial conditions for the equations of the displacement function are calculated on the basis of the geometrical structure—Figure 2 and

Table 1. The amplitude of discrete displacement nodes.

u1 = u11, m	u2 = u10, m	u3 = u9, m	u4 = u8, m	u5 = u7, m	u6, m
0.0	0.8	1.6	2.4	3.2	4.0

.

Ordinary equations discretised with the straight-line method were integrated using the explicit Runge–Kutta method and implicit Euler method, considering the Seidel method, producing nearly identical results. The integration step in the time domain was h = 10⁻⁵ s.

Figure 3, Figure 4, Figure 5 and Figure 6 present the oscillatory processes across the central string discretisation node for four experiments. The first, shown in Figure 3, concerns the analysis of string oscillatory processes in the case of linear dependences for the components of external and internal mechanical energy dissipation.

The second experiment in Figure 4 consists of the process analysis in the case of a nonlinear dependence of external dissipation, that is, where the string oscillates in a viscoelastic medium.

The third experiment in Figure 5 involves the analysis of transient processes in the case of a linear dependence of the component, including external dissipation, whereas the internal dissipation components are nonlinear, namely, the string is subject to a thermodynamic impact (the line wire is heated).

Figure 6 depicts the oscillatory process of the central string node, where both types of energy dissipation are nonlinear.

Figure 7 presents the results for all four experiments. The curve numbers correspond to the number of an experiment.

In the simplest case, oscillatory processes persist in a power line for linear dissipation dependences. If a string in a viscoelastic medium is affected by weather factors acting on a line, the mechanical wave vanishes more intensely. If the line has been heated, the mechanical dissipation reduces, and the vanishing of the mechanical wave is exacerbated. Both energy dissipations are present in experiment four.

The velocity waveform of the central string discretisation node is illustrated in Figure 8 (with both the dissipation types being nonlinear).

As the mechanical wave arises, the oscillatory processes include higher harmonics (in the function expansion into a Fourier series) owing to the nonlinearity of the displacement function; see Figure 6. In time, the velocity oscillation becomes damped and smoother.

In Figure 9 and Figure 10, some families of the discrete node waveforms for the function of string displacement and velocity in the time interval $t\in \left(0 ; 5\right)$ s (with both the dissipation types being nonlinear). Figure 9 shows a displacement function of all the discrete string nodes.

Figure 9 shows how a mechanical wave and some significantly nonlinear oscillatory processes across the string boundary discretisation nodes are generated.

A transient process for the velocity function for the 6th, 4th, and 2nd nodes is presented in Figure 10.

The spatial waveforms of string point displacement and velocity for different times are depicted in Figure 11 and Figure 12 (with both the dissipation types being nonlinear).

The initial conditions of the string motion described in Table 1 and Figure 2, and shown in Figure 11 and Figure 12, are asymmetrical from the string centre by x = 50 m. It might seem the string oscillations should be symmetrical for distant points to the left and right of its centre, too. However, this is not the case. In transient processes, the mechanical wave executes complicated motions of a constant velocity; see (31) and (36). That motion introduces some asymmetry to the string’s oscillations. Nonlinear processes complicate the functions analysed even further. String oscillations become symmetrical in a steady process. This is even clearer in the next figure.

The temporal–spatial waveforms of string displacement for two time intervals—$t\in \left(0; 3\right)$ s and $t\in \left(9; 12\right)$ s—are illustrated in Figure 13 and Figure 14 (with both the dissipation types being nonlinear).

These figures suggest a complete picture of the complicated physical processes across a string, including the nonlinear external and internal mechanical energy dissipations at the time of string oscillations. Figure 13 and Figure 14 should be analysed in conjunction with Figure 9 and Figure 11.

4. Conclusions

The mathematical reasons, based on the Lagrange equations of the second type, for expanding the classic Lagrange function (conservative Lagrangian) by introducing two components—one including the dissipation of external and internal energy and the other including the energy of external non-potential forces—fully corroborate Helmholtz’s concept of introducing the modified integral variational Hamilton–Ostrogradsky principle to the theory of mathematical modelling.

Introducing the modified integral variational Hamilton–Ostrogradsky principle to the theory of mathematical modelling allows for placing variational and classic approaches on the same footing. In the event, the variational approaches are fully competitive with the classic approaches.

The proposed theory of modifying the Hamilton–Ostrogradsky principle helps to construct a mathematical model of real physical objects based solely on an energetic approach, regardless of the scientific field. Thus, the variational methods can legitimately be treated as interdisciplinary approaches to modelling.

The equation of nonlinear string oscillations in a viscoelastic medium, including two nonlinear components (external and internal dissipations of mechanical energy), needs mathematical substantiation of the existence and uniqueness of its solution, which is demonstrated in this paper.

Following a comparative analysis of computer simulation results, we believe it is reasonable to address the nonlinear components of external and internal dissipation in the string equation. This improves the adequacy of the object under discussion. Oscillatory processes during ice melting on an electric wire of a long power line operated in complex weather conditions are analysed as an example.

Since the model of line cable oscillations is analysed in its simplified version, it obviously ignores specific damping processes across the line.