Dynamic damping of vibrations of technical object with two degrees of freedom

Approach to the solution of problems of dynamic damping for the technical object with two degrees of freedom on the elastic supports is developed. Such tasks are typical for the dynamics of technological vibrating machines, machining machine tools and vehicles. The purpose of the study is to justify the possibility of obtaining regimes of simultaneous dynamic damping of oscillations in two coordinates. The achievement of the goal is based on the use of special devices for the transformation of motion, introduced parallel to the elastic element. The dynamic effect is provided by the possibility of changing the relationships between the reduced masses of devices for transforming motion. The method of structural mathematical modeling is used, in which the mechanical oscillatory system is compared, taking into account the principle of dynamic analogies, the dynamically equivalent structural diagram of the automatic control system. The concept of transfer functions of systems interpartial relations and generalized ideas about the partial frequencies and frequencies dynamic damping is applied. The concept of a frequency diagram that determines the mutual distribution of graphs of frequency characteristics in the interaction of the elements of the system is introduced.


Introduction
Dynamic damping of oscillations as the effect of local balancing of the elements of a technical system or an object under the action of vibrational loads has been sufficiently well studied, which is reflected in the works [1 ÷ 3]. Design and technical solutions that implement solutions for adjustment of the active vibration absorbers are known [4,5]. These are applied in practice in solving the problems of improving the reliability and operation of various machines and equipment. [6].
At the same time, in most cases dynamic damping of oscillations is realized as a local effect created at one characteristic point of the object and a certain frequency of an external disturbance.
The idea of the proposed method is to use the fact that in a system with two degrees of freedom under the kinematic form of external action (for example, base vibration), the inputs of two partial systems are simultaneously excited. In this case, the system acquires, in contrast to the usual situation, additional dynamic capabilities.
I. Some general provisions; justification of the design scheme 1. The schematic diagram of the technical object is a solid body on elastic supports (k 1 , k 2 ) with parallel input devices for transformation of motion (UPD). Such devices can be realized with the help of non-self-locking screw mechanisms, in which there are nut-flywheels with reduced masses L 1 and L 2 [2,7,8]. The schematic diagram is shown in Figure 1. The motion of a rigid body with mass inertial  The initial mechanical oscillatory system ( Figure 1) is used to construct a system of linear differential equations based on Lagrange's equations of the second kind, which is transformed by Laplace under zero initial conditions. The technology of constructing mathematical models is described in detail in [3,7,8]. The block diagram of the partial system with main coupling and feedback and characteristics of external excitation is shown in Figure 2.  Figure 1 with kinematic perturbation; The notation p = jω is a complex variable ( 1 − = J ); the <-> icon above the variable means its Laplace image [9,10] 2. Using the block diagram in Figure 2, let us determine the transfer functions of the system; let us take z 1 = z 2 = z, then: It is important to note that α is introduced as a coupling coefficient between the values of L 1 and L 2 (L 2 = αL 1 ). Within the framework of the developed method, the frequency of the regime of dynamic damping of oscillations along the coordinates is determined from the conditions "zeroing" of the numerators of the transfer functions (1) and (2).
The equation for finding frequency dynamic damping in kinematic excitation for the coordinate y 1 is written as: In turn, for coordinate y 2 the following equation is obtained:  In Figure 4 point (1) determines the frequency of dynamic damping of oscillations simultaneously with respect to two coordinates 1 y and 2 y . The same point determines the value of the connectivity coefficient α.
The question of controlling the parameter α, that is, the coefficient of connectedness of the reduced masses of nuts-flywheels, is technically possible, but application of braking torque to nuts-flywheels is required. Such moment with respect to each element L 1 and L 2 , can be created by an appropriate control system. The frequency diagram in Figure 4, where p. (1) determines the desired regime.

1.
It is shown that under certain operating conditions of technical objects, the disturbance can be formed by simultaneous excitation of two partial systems. Under the conditions of compatibility of the action of disturbances, the introduction of relations between two force factors enables the linear formulation of the problem on the basis of the superposition principle to find the condition for simultaneous dynamic damping of oscillations with respect to two coordinates.
2. Taking into account the joint action of external disturbances, the regime of dynamic damping of oscillations can be considered in a generalized form, which correlates with the concepts of determining the corresponding frequencies from the condition of "zeroing" the numerators of the transfer functions.
3. It is shown that the frequencies of the regime of simultaneous dynamic damping of object oscillations are determined from the condition of simultaneous "zeroing" of the numerators of the transfer functions in two coordinates. 4. A method has been developed for determining the frequencies of dynamic damping of oscillations based on the use of frequency diagrams reflecting the complex character of the change in all frequencies of the system when the tuning parameters are changed.