Optimal Synthesis and Implementation of Resonant Vibratory Systems

The problems of synthesis and substantiation of elasticity parameters of the resonant vibratory device with electromagnetic drive and one flat spring are considered. At first, the harmonic systems with oscillation frequencies of 50 Hz and 100 Hz were investigated. Then, various asymmetric piecewise linear characteristics of elasticity were carried into effect on one flat spring using auxiliary intermediary fixed cylindrical supports. Due to this, the corresponding vibro-impact operation modes were obtained. The resonant systems characterized by improved functioning efficiency were carried into effect using the new technique of optimization synthesis of elasticity parameters. The resonant systems being investigated were implemented in practice. The basic experimental investigations of their kinematic, dynamic and energetic parameters were carried out. The fundamental result of the investigation consists in confirmation of the improved dynamic efficiency of vibro-impact systems with pulsed electromagnetic excitation designed according to the new technique. The proposed systems may be used in technological processes of materials compaction and screening, of surface treatment of machine parts and in processes associated with nanotechnology.


Introduction
Vibratory systems and machines are useful for various technological purposes [1] and are commonly used as multiform technical means. The field of application of vibratory machines covers new progressive methods of surface and bulk treatment of materials. The implementation of multi-frequency oscillations ensures the use of vibratory technological systems in the fields of nanotechnology and in problematic energy-intensive industries. However, the development of effective systems requires the search for new designs and calculation methods in order to use energy-efficient resonant modes [2]. For this purpose, it is expedient to use advanced methods of dynamic analysis, synthesis and optimization of parameters [3] with simultaneous ensuring of the specified technological factors.

Methodology of Synthesis and Optimization
The research is aimed at a generalized analysis and synthesis of resonant two-mass systems with harmonic and vibro-impact modes. The first stage of the research was the selection of criteria (indexes, parameters) that form a complex of requirements to the class of resonant systems. Taking into account the structure of the mechanical system, the vibratory machines may have a different number of oscillating masses. As a rule, only one of these masses is considered as technological one. Therefore, in terms of technological expediency, in order to evaluate the system, it is enough to use the maximum value of acceleration only of the working mass max 1 a (or g-force g a / max 1 = Γ ) and consumed (intake) power P . In this case, the target energy criterion has the following form: The index defines the energy efficiency of the operation in order to attain the key kinematic parameter.
The criterion of 2 ς is the efficiency factor, which defines the degree of losses of electromechanical oscillatory system 1 η or technological system 2 η (if one takes into account the technological process) ( The criterion of 3 ς defines the occurrence of the vibro-impact mode by the value of the coefficient of acceleration asymmetry a k of the working mass (for single-frequency systems ( The criterion of 4 ς is the width of the resonance zone, which evaluates the technological stability of the vibratory system It is expedient to implement the vibro-impact systems using the asymmetric piecewise linear elastic characteristics. The value of the natural frequency (self-frequency) of oscillations for the system with asymmetric piecewise linear elastic characteristic 0 ω without the gap is the function of inertia and stiffness parameters [4]: where the corresponding natural frequencies of oscillations corresponds to the directions of action of corresponding stiffness coefficients of the springs. Herewith, the coefficients 1 c and 2 c , which are in rather wide limits, remain unknown and may be presented as follows: In order to simplify the synthesis procedure, it is proposed to determine the actual values of the natural frequencies of oscillations by the relationships: here 0 / ω ω = z is setting of the resonance; frequency coefficient Θ and natural frequencies ratio Λ are the syntheses parameters.
Synthesis is carried out in accordance with the kinematic characteristics of the vibro-impact mode and ensuring the predefined value of the natural frequency of oscillations (the width of the resonance zone) within the appropriate limits. For example: The conditions in Eq. (8) are specified taking into account the technological purpose of the vibratory system.
It is expedient to implement the synthesis problem using the optimization condition according to the power criterion: The proposed method of calculation allows to reduce

Methodology of Development, Calculation and Modeling
The structure of the vibratory device (see Fig. 1) used for carrying out the theoretical and experimental investigations is designed according to the two-mass scheme. The structure consists of the body 1 ( ) oscillating masses, which are attached by the flat spring 3. The intermediary fixed cylindrical supports 5 are attached to the body mass 1 with a help of the side plates 4. For power disturbance of oscillating masses, the electromagnetic drive of alternating current is used. The cores 6 of electromagnets are fastened on the body mass 1, and the armatures 7 are fixed on the reactive mass 2. The structure also includes the rubber vibration isolators and the profile frame. The circuit for switching on the electromagnets may be pulsed (for 50 Hz) or reactive (for 100 Hz).  The diagram of implementation of asymmetric piecewise linear elastic characteristic (see Fig. 3) allowed to obtain basic calculation formulas with a help of the finite element method [5]. In order to ensure the first natural frequency one may use the formula, which corresponds to the fastening diagram I of the flat spring: where ( ) The value of the second natural frequency depends on the position of the cylindrical supports and may be determined by the following formula for the fastening diagram II: If we take into account the vertical stiffness of the intermediary supports in the form of the coefficient c y , the corresponding value of the natural frequency of oscillations for the diagram II will be as follows: The mathematical model that describes the dynamic processes in the oscillatory system with power electromagnetic excitation is as follows [6,7]: U . The most widespread and the simplest exciting circuits of the vibratory devices are the pulse 50 Hz one and the reactive 100 Hz one (see Fig. 4). In the first circuit, the diode and the thyristor rectifiers are used. In the second circuit, the power supply of the electromagnets is carried out directly from the network. However, the new frequency-controlled circuits are currently more prospective [6]. The modelling of the system of ordinary nonlinear differential equations (see Eq. (12)) is carried out using the progressive numerical methods, in particular, Radau, BDF, AdamsBDF. The time dependencies of the all kinematics' and power parameters being defined: the g-force of oscillating mass m 1 is shown (see Fig. 5 and Fig. 6).
The main results of calculation using the methodology of optimization synthesis are presented in Table 1. The highest efficiency of operation is characteristic of vibro-impact system with the synthesis coefficients 0.75  At the next stages, the simulation of the system's operation and the selection of the appropriate value of the voltage 0 U should be carried out in order to ensure the necessary overload on the working mass. For introducing the system into practice, the bending stiffness of the flat spring is to be calculated using (see Eq.

Investigation of the Single-frequency Resonant Vibratory Systems
Experimental investigations were carried out using a two-channel USB-oscillograph with two accelerometers The implementation and investigation of single-frequency vibratory systems with traditionally calculated elastic parameters were the problems of the first priority. Obviously, for such systems, the coefficients of synthesis are equal . The single-frequency systems with 50 Hz (see Fig. 7) and 100 Hz (see Fig. 8) modes were implemented and investigated. : а) -oscillogram, b) -spectral analysis As a rule, the single-frequency systems with linear elastic characteristics have one harmonic. This fact is confirmed by the spectral analysis being carried out (see Fig. 8, b).

Investigation of the Vibro-impact Resonant Vibratory Systems
Due to the use of intermediary supports, the vibro-impact systems with the oscillations frequencies of 50 Hz and 100 Hz were implemented. For these systems, The next important stage of the investigation was the implementation of the vibro-impact system of the improved efficiency. In order to do this, the spring with the reduced stiffness for the value of the parameter 0.8 = Θ was made. The investigation was carried out for the value of the coefficient Fig. 11). The results presented in Table 2 indicate the significant power advantages of the vibro-impact systems calculated using the new technique.

Conclusions
Due to the system approach and the complex problems, we ascertained:  The expediency of use of multi-criteria evaluation of the quality of operation of vibratory machines of various types taking into account two partial technological parameters and two power indexes while developing and choosing the type of the machine for the certain technological process.  Asymmetric elastic characteristic with the highly efficient vibro-impact mode considering the maximal value of the acceleration of the working mass.  Qualitative conditions for ensuring the vibro-impact resonant modes on the basis of optimization problems with restraints on kinematic parameters (acceleration) and dynamic characteristics (natural frequency of oscillations). In this case, the proposed approach is of the generalized character and may be adapted for machines of various technological purposes.


The possibilities of implementation and experimental evaluation of single-frequency and vibro-impact modes with the excitation frequencies of 50 Hz and 100 Hz. In this case, the scheme of implementation of vibro-impact modes of improved efficiency (more than 2 times higher efficiency) using one flat spring with intermediary cylindrical supports has been confirmed.
Therefore, the developed complex means for evaluation, synthesis, optimization of parameters of multi-frequency resonant machines have a high practical value, which is confirmed by the corresponding theoretical results during the modelling (simulation) and partly in practice.