Electrodynamic long-stroke drive as an inertial compensator of low-frequency vibroactive forces

The article discusses various possibilities of using an electrodynamic drive with an increased displacement of a movable permanent magnet as a low-frequency inertial compensator of vibroactive forces.The main attention is paid to the analysis of an active dynamic vibration damper (ADVD) when installing an electrodynamic compensator (EDC) on the base near vibration supports and when connecting the EDC moving mass with a vibroactive mass using a spring. It is shown that the considered ADVD is effective as a vibration isolator in the low-frequency range.


Introduction
The active vibration isolation system includes a passive system (the vibroactive mechanism mass located on the elastic-dissipative supports) and a power drive (actuator) with a control system [1 -5].
To implement the dynamic inertial compensation principle [6] of vibration forces transmitted to the base through elastic-dissipative supports, it is advisable to use an electrodynamic drive. It can provide a sufficiently large displacement of the moving mass (7). As it is shown by the analysis of scientific and technical literature, for the low-frequency vibroactive forces range (0,5 -20 Hz), a promising direction for the vibration isolation purposes is the use of vibroactive forces inertial compensation by an electrodynamic compensator. It ensures the movement of the mass in antiphase with the vibroactive mechanism oscillations (7 -19).

Problem formulation
Application of the principle of vibroactive forces inertial compensation in the low-frequency range (1 -20 Hz) for active vibration isolation systems has shown both the effectiveness of this principle and the need to create a fairly simple and cheap actuator that ensures the operation of the active vibration isolation system. The principle of inertial compensation is that the actuator provides the displacement of the mass, which creates inertial forces in antiphase with vibroactive forces. An electrodynamic drive with an increased displacement of a movable permanent magnet can be used as such an actuator. A schematic diagram of such an actuator is shown in fig. 1 Figure 1. Actuator schematic diagram 1 is the housing made of soft magnetic material, 2 is the base, 3 is the control coil, 4 is the guide made of non-magnetic material, 5 is the sleeve made of non-magnetic material, 6 is the permanent neodymium magnet, 7 is the magnetic circuit, 8 are the springs, 9 is the alternating voltage source.
When an alternating voltage from 9 is applied to the coil 3, the movable link (5, 6, 7) makes a translational movement along the guide 4 with an amplitude proportional to the current of 3 and with the frequency of the applied voltage. The inertial force F is transmitted to the base and is equal in amplitude = • • 2 , where m is the mass of the moving part, x is the amplitude of the moving part, ω is the circular vibration frequency. The use of the proposed electrodynamic actuator to create inertial forces is possible in the following three options.
The first option -the actuator is installed in the immediate vicinity to the passive system elasticdissipative supports or is performed in one structure with the support. A schematic diagram of such an active vibration isolation system is shown in Fig. 2.

Figure 2.
Schematic diagram F(t) is vibroactive force, 1 is the vibroactive unit mass, 2 is mass of the actuator moving part, 3 is electromotive force developed by the actuator, 4 is regulator with a transfer ratio K, 5 is forcemeasuring device, m0 is the vibroactive unit mass, m1 is the actuator moving part mass, c0, c1are the elastic elements stiffness, b0, b1 are the viscous friction coefficients Force Fact is determined in accordance with the following equations:  is voltage on the winding of the control coil, is current strength, is electrodynamic force, , is inductance and resistance of the coil, is magnetic induction, is total length of the coil conductor. Study of the active system efficiency in accordance with fig. 2 [7] [8] showed that the reduction in the transmission of vibration force to the base is 20 -40 dB in the frequency range 1 -20 Hz. The next option for the actuator is shown in fig. 3, where the actuator is based at mass 0 .

Figure 3.
Design scheme 1 is the vibroactive unit mass, 2 is mass of the actuator moving part, 3 is electromotive force developed by the actuator, 4 is regulator with a transfer ratio K, 5 is force-measuring device, m0 is the vibroactive unit mass, m1 is the actuator moving part mass, c0, c1are the elastic elements stiffness, b0, b1are the viscous friction coefficients In this case, the electromechanical system in fig. 3 is a vibration isolation system with an active dynamic vibration damper. The efficiency of such a system has been studied in detail in [11] [14]. It is shown there that the reduction in force transmission to the base is 20-40 dB in the frequency range 1-20 Hz. In addition, it was shown that the system remains effective in the non-stationary mode of the vibroactive unit, for example, "start-stop" [14]. An active dynamic vibration damper with an electrodynamic actuator can be implemented by installing the EDC on the base and connecting the moving mass of the EDC by a spring with a vibroactive mass.

Theory
To compile a mathematical model of an active dynamic vibration damper, a schematic diagram is presented in Fig. 4 Figure 4. Schematic diagram Designations 1 -8 correspond to Fig. 1, 9 is the force-measuring device, is the controller, 0 is the vibroactive unit mass, 0 , 1are the elastic elements stiffness, 0is the viscous friction coefficients In accordance with fig. 4, the design scheme is shown in Fig. 5.

Figure 5. Schematic diagram 1 is regulator, 2 is electromotive force, 3 is force-measuring device
Assuming that the two-mass system in fig. 6 performs unidirectional motion about the equilibrium position, the equations system describing the dynamics of the system will have the form:   fig. 8 shows the values of the force on the base without the EDC and with the EDC turned on at a frequency of the vibroactive force of 1 Hz.

Figure 8.
( ) dependency graph Analysis of the equations system (3) showed that the system is stable at any positive values of K. With an increase in K, the efficiency of vibration isolation in the low-frequency range increases, the oscillation index and the period of natural oscillations increase.

Conclusion
The study of the effectiveness of an active vibration damper when installing an EHD on the basis showed that this direction is promising for vibration isolation in the low-frequency region of 1 -20 Hz, including at the natural frequency of a passive system. Analysis of possible options for including EHD in the active vibration isolation system showed that when controlled from a force-measuring device, their efficiency is the same and is 20 -40 dB in the range of 1 -20 Hz. The choice of one or another option for using the EDC depends on the design features of the construction of a passive vibration isolation system.