Determination of inertia-stiffness parameters and motion modelling of three-mass vibratory system with crank excitation mechanism

Vibratory technological equipment is widely used in various industries. The vast majority of existing vibratory machines are equipped by singleor double-mass oscillatory systems and inertial or electromagnetic vibration exciters. The novelty of the present study consists in development and investigation of the three-mass oscillatory system with crank excitation mechanism. Such a system can be effectively implemented in various designs of vibratory equipment, e.g., conveyers, separators, feeders, shakers, batchers, sieves, etc. Based on the mathematical model derived in the form of differential equations of the system’s motion, there are deduced the analytical expressions for determining its inertia-stiffness parameters ensuring the energy-efficient resonance operation mode. Using the solid model of the vibratory conveyer-separator designed in SolidWorks software, there is determined the input data for calculating the parameters of the oscillatory system. Based on the results of calculations, the numerical modelling of the system’s motion is carried out in MathCad software. In order to verify the correctness of the theoretical investigations, the simulation of the system’s motion is carried out in SolidWorks Motion software. The comparative analysis of the results of numerical modelling and computer simulation is performed, and the prospects of their implementation are considered.


Introduction
The problems of reducing the energy consumption of various technological equipment are currently of significant interest among the scientists and designers. One of the ways of improving the energy efficiency of vibratory machines consists in implementation of three-mass oscillatory systems ensuring the possibilities of operation in so-called "multi-frequency" and "inter-resonance" modes [1][2][3]. Herewith, the vibrations can be excited by electromagnetic vibrators [1], air-operated vibrators [2], inertial drives [3][4][5][6], eccentric-type and crank mechanisms [7][8][9][10]. Each type of vibration exciter has its specific advantages and drawbacks, as well as the areas of implementation. The problems of parametric synthesis and dynamic behavior analysis of various oscillatory systems with electromagnetic and inertial vibrators are thoroughly investigated in numerous publications, e.g. [1,[3][4][5][6], whereas the air-operated vibrators are not widely used in industrial vibratory equipment because of their design complexity and low efficiency [2].
The crank excitation mechanisms are effectively used for actuating single-and double-mass oscillatory systems of vibratory machines [7][8][9][10]. However, the problems of exciting the oscillations of three-mass systems with the help of crank mechanisms are currently of significant interest. The novelty of the present paper consists in developing the technique of determining the inertia-stiffness parameters of the three-mass vibratory system excited by the crank mechanism, as well as modelling and simulation of its motion under near-resonance conditions.

Mathematical model and technique of determining the inertia-stiffness parameters of the three-mass vibratory system with crank excitation mechanism
The discrete three-mass vibratory system to be analyzed is presented in Fig. 1 considered as vibration isolators. The excitation mechanism consists of the crank 1 hinged to the intermediated mass , and the connecting rod 2 simultaneously hinged to the crank 1 and to the slider 3. The lower mass is connected to the slider 3 of the crank mechanism by the spring . Let us consider the case when the masses , , oscillate translationally relative to one another due to the uniform rotation of the crank 1 described by the constant angular speed (circular frequency) . The motion of the system can be completely described by the coordinates , and , which define the displacements of the masses , , from the corresponding equilibrium positions at any time . The differential equations of motion of the considered three-degree-of-freedom system can be written in the following form: where , are the lengths of the crank and of the connecting rod, respectively. Let us assume that ≫ ; this allows us to consider the periodic (harmonic) excitation of the vibratory system. In addition, let us neglect the stiffness of vibration isolators = 0 , because its value is considerably smaller than the values of and ≪ ; ≪ . In such a case, the system of differential Eq. (1) can be rewritten as follows: In the case when the system is subjected to the harmonic force = cos , we assume the steady-state solutions to be as follows: where , , are the amlitude values of displacements of the corresponding oscillating masses that depend on and on the system's parameters.
Substitution of Eq. (3) into Eq. (2) leads to: System Eq. (4) represents three algebraic equations in the unknowns , , . The determinant of the coefficients of , , is equal to: Equalizing the determinant ∆ in Eq. (5) to zero, let us derive the frequency (characteristic) equation, whose positive roots are the natural frequencies of the system's oscillations: Setting the values of the natural frequencies, and substituting them into Eq. (6), let us derive the analytical expressions for determining the stiffness coefficients and : where: Therefore, the input parameters for calculating the inertia-stiffness parameters of the three-mass vibratory system are following: the values of two natural frequencies , , and two masses , . The value of the mass can be estimated taking into account the following assumptions: the radicand in Eq. (8) and the numerators in Eq. (7) must take positive values: VIBROENGINEERING PROCEDIA. MARCH 2021, VOLUME 36 To plot the frequency-response curves (or, so-called, amplitude-frequency characteristics) of the three-mass vibratory system, let us solve Eq. (4) for the unknowns , , : 3. Results of numerical modelling and computer simulation

Design peculiarities of the vibratory machine
The vibratory conveyer-separator was designed in SolidWorks software and implemented as an experimental prototype (see Fig. 2

Numerical modelling of the system's motion in MathCad software
Using Eq. (11), let us plot the frequency-response curves (or, so-called, amplitude-frequency characteristics) of the considered three-mass vibratory system with the help of MathCad software (see Fig. 3). Based on the obtained results, it can be concluded that the resonance effects take place at the frequencies = 95 rad/s, = 104 rad/s, which correspond to the ones prescribed by the input parameters. Numerical solution of the differential Eq. (2) has been obtained with the help of the Runge-Kutta method using MathCad software taking into account the following initial conditions: 0 = 0; 0 = 0; 0 = 0; 0 = 0; 0 = 0; 0 = 0. As an example, the plot of forced-vibration response of the upper oscillating mass (time dependence of the mass's displacement from its equilibrium position) is presented in Fig. 4. The vibratory system excited at the forced frequency = 99.5 rad/s runs into the steady-state operation mode in 2-3 s after the starting. The maximal displacement of the upper oscillating mass from its equilibrium position, i.e., the amplitude of vibration of the conveying-separating tray, is equal to 5 mm.

Computer simulation of the system's motion in SolidWorks software
In order to verify the correctness of the results obtained by theoretical investigations and numerical modelling, let us carry out computer simulation of the vibratory system's motion in SolidWorks Motion software. The corresponding solid model is presented in Fig. 5

Conclusions
Based on the carried out investigations, the following conclusions can be drawn: 1) The three-mass oscillatory systems with crank excitation mechanisms are of significant interest among the researchers and designers of vibratory equipment because of their design simplicity, reduced energy consumption, improved possibilities of frequency and amplitude regulation in accordance with the technological requirements etc.
2) The simplified diagram of the three-mass vibratory system has been considered; the differential equations describing the motion of the oscillating masses have been derived; the analytical expressions allowing determination of the inertia-stiffness parameters of the system have been deduced; the frequency-response dependencies have been proposed.
3) Prescribing the input parameters obtained on the basis of solid modelling of the vibratory conveyer-separator in SolidWorks software and implementing it as an experimental prototype, the numerical and computer simulation of the system's motion has been carried out; the results of theoretical investigations (numerical modelling) have been compared with the results of computer simulation, and the conclusion about their satisfactory agreement has been drawn.
Prescribing the masses = 83.7 kg, = 62.1 kg, and the natural frequencies = 95 rad/s, = 104 rad/s, there have been determined the values of the lower mass = 0.313 kg, and the stiffness coefficients = 3.794•10 5 N/m, = 2.865•10 3 N/m. The oscillations of the three-mass vibratory system of the conveyer-separator have been excited by the crank mechanism characterized by the lengths of the crank and the connecting rod = 0.019 m, = 0.078 m. The numerical modelling and computer simulation showed that at the forced frequency = 99.5 rad/s the amplitude of vibration of the upper tray reaches 5 mm.
The results of numerical modelling and computer simulation can be used in further investigations on the subject of the present paper while analyzing all the other kinematic parameters of the masses' oscillations, in particular, the velocity and acceleration of the tray, which characterize the conveying speed and the conveying regime (detached (lifted-off) or non-detached) for different types of products (bulky, loose, piecewise etc.).