## 1 Introduction  
In many theoretical and practical studies on the dynamics of rotor systems containing liquid, only oscillations of the rotor with liquid are considered, without taking into account the electromagnetic properties of the liquid and the mobility of the foundation [1]. This assumption leads to certain errors in assessing the dynamic and kinematic characteristics of the rotor system [2]. Studies of dynamic systems such as rotary systems show the importance of taking into account the electromagnetic properties of the fluid, the nonlinear properties of the shaft supports, foundation vibration and the need to develop measures to reduce them [3, 4]. The intensive development of magnetic and electrohydrodynamics (hereinafter referred to as MEHD) started in the 1960s by the [PERSON] group in the USA. In Europe, these studies were started by French and Spanish scientific centers [5] and others [6], where the issues of the possibility of using electrohydrodynamic effects in production and the importance of developing a general theory of MEHD were considered. In 1966, [PERSON] was the first to show how the application of a uniform electric field can deform the shape of a weakly conductive liquid depending on its electrohydrodynamic properties [7]. In the USSR, this direction was developed by authors such as [PERSON] and [PERSON] [8]. As a result, already in the 1970-1980s, the main provisions and systems of MEHD equations were developed, and the conditions under which various MEHD models are valid were considered. Intensive MEHD studies using applied physics methods were carried out in [9], where gas-dynamic flows with similarly charged particles were studied and new directions in MEHD were analyzed: MEHD turbulence, unsteady effects, and new methods for diagnosing MEHD flows. Some MEHD problems on the stability of an inhomogeneously heated low-conductivity liquid were studied in [10]. A large amount of work on the study of electrophysical processes in dielectric liquids initiated by high-intensity fields was carried out by the authors of [11]. The results of recent MEHD studies are presented in [12]. Oscillations of the free surface of a viscous fluid in a rotating cylindrical vessel and stability of rotational motions were first considered by [PERSON] [13] and [PERSON]; further, their studies were continued in the works of [PERSON] [14], [PERSON] [15], and [PERSON]. In these papers, linearized equations of a system acting on a vessel from the liquid side are studied, where it is assumed that the free surface of the liquid differs little from the unperturbed cylindrical shape. A similar formulation of the problem and a similar mathematical model was used in [14; 15; 16; 17], where the oscillations of a rotating cylinder partially filled with an ideal and viscous fluid were studied. This problem originates in the works of [PERSON] (1877, 1880), [PERSON] (1945), [PERSON] (1948, 1949) and [PERSON] (1957), and was first solved in a general form by [PERSON] in 1951 [18]. In 1952, [PERSON], independent of [18], obtained similar equations of motion, which can be found in more detail in [19]. Among the works of modern authors devoted to various aspects of this direction, we should mention the following papers: in [20], the analytical and numerical results obtained in the study of the motion of a system consisting of a rigid body with a cavity filled with a viscous fluid are presented; in [21; 22; 23], chaotic motions of a rigid body and with a cavity filled with a liquid are studied; in [24], the experimental work was carried out to study the oscillations and displacements of the free surface of a liquid for the case when the cavity is filled with two different liquids; in [25], a quasi-analytical model of oscillations of magnetic fluids under low gravity under the action of external inhomogeneous magnetic fields is presented [26], and the problem of free and forced oscillations is solved for models with axisymmetric geometries and loads in a linear formulation by the Ritz and finite element methods.