1. bookVolume 25 (2021): Issue 1 (January 2021)
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Force analysis of the two-satellite planetary mechanism with elliptical gears

Published Online: 25 Jul 2021
Page range: 39 - 46
Received: 12 Dec 2020
Accepted: 29 Mar 2021
Journal Details
License
Format
Journal
eISSN
2354-0192
First Published
28 Dec 2018
Publication timeframe
1 time per year
Languages
English
Abstract

Non-circular gears can be used in modern machines and mechanisms for the implementation of various types of motion and have high strength and compactness compared to linkage mechanisms. This article presents the force analysis of non-circular gear on the example of the planetary mechanism with elliptical gears, providing the rotationally reciprocating motion of the impeller of the stirred tank. Based on the calculation schemes of the links, kinetostatic balance equations for each link of the mechanism are compiled and solved. Reaction forces in kinematic pairs and balancing moment on the input shaft of the mechanism are found. The results can be used in the synthesis and analysis of various machines with the proposed kinematic scheme of the mechanism.

Keywords

Introduction

Transmissions by non-circular gears have been known for a long time and are of great interest to scientists and engineers [1, 2]. The first schemes of mechanisms with non-circular wheels were found in the works of Leonardo da Vinci, and some of the first models of such devices were made in the 19th century by the German scientist Ferdinand Redtenbacher. Widespread research and dissemination of these gears has long been hindered by the lack of simple and accurate manufacturing methods for non-circular gears.

Currently, there are a large number of papers devoted to geometric and kinematic analysis, design and practical application of gears by non-circular gears [3, 4, 5, 6, 7, 8, 9]. For example, there are proposed: chain drives with variable ratio for bicycles [3]; a mechanical device containing a Maltese cross mechanism and a gearing to achieve intermittent motion [4]; the gear trains used for velocity variation and function generation [5]; constructions of non-circular gears as a part of rotor hydraulic machines [6] and high-performance pumps [7]; a knee motion assist device for biomechatronic exoskeleton having grooved cams and non-circular wheels [8]; single planetary non-circular gears with one internal and one external gear for bicycles with high efficiency [9]. It was shown in [10, 11, 12, 13, 14] that planetary gears with non-circular wheels allow to change the sign of the transfer function and, therefore, provide different types of motion, for example, intermittent motion [10, 11, 12] or rotationally reciprocating motion [13, 14]. The proposed mechanisms are more compact and durable compared to linkages, and the improvement of high-precision equipment for machining and additive production stimulates the development of this issue of the mechanism and machine theory.

Non-circular gears can have various shapes, but elliptical gears are the most researched and widespread at the moment [15, 16, 17]. A large number of mechanisms and devices based on elliptical gears have been developed, their geometry and kinematics have been investigated [18, 19, 20], and various problems of their manufacture have been solved [21, 22, 23].

However, most of the papers on the creation of non-circular gears are devoted only to the geometry and kinematics of these mechanisms, while the issues of their dynamics are much more complex and have not been studied. Nevertheless, some applied problems of the dynamics of non-circular gears have been solved. For example, Xing Liu et al. [24] carried out a theoretical and experimental study of the dynamic performance of elliptical gears with rotational axes at the focus and center of the pitch ellipse. Nan Gao et al. [17] investigated parametric vibrations and instabilities of elliptical gears caused by loading torque and eccentricity vibrations. Zhiqin Cai and Chao Lin [25] presented and investigated a generalized nonlinear dynamic model of a curved gear drive based on Lagrange-Bondon graphs. Many researchers consider the issues of dynamics in relation to practical applications of mechanical devices [24, 25, 26] containing non-circular gears, since dynamic models include the parameters of technological load on the working body of the machine.

The purpose of this article is the kinetostatic analysis of a planetary mechanism with two external gears operating as part of a drive of a rotationally reciprocating stirred tank. The idea of imparting various non-stationary types of motion, including rotationally reciprocating motion, to the impeller of the stirred tank has been supported by many researchers around the world [27, 28, 29]. Despite the increasing complexity of the drive constructions, the prospects of this research area are due to a significant increase in the mixing efficiency, which has been experimentally confirmed using various practical examples [30, 31].

Formulation of the problem

The object of this research is the actuator of the stirred tank (Figure 1a), in which the rotational motion of the motor shaft is converted into the reciprocating motion of the impeller due to a planetary mechanism with elliptical gears (Figure 1b).

Figure 1

Rotationally reciprocating stirred tank: a – scheme of the actuator; b – structural scheme of the planetary mechanism

The proposed mechanism (Figure 1b) consists of rack 0, three-vertex links 1, 2, 3, two-vertex link 4, 1-DOF kinematic pairs A, C, D, G and 2-DOF kinematic pairs B, E, F. The structural analysis carried out in [13] showed that the mechanism exists in a three-moving space (translational displacements along the x and y axes and rotation around the z axis), which is traditionally called “planar.” In the study of kinetostatics, we assume that all the kinematic pairs lie in the same plane. Figure 2 presents the calculation scheme of the mechanism. The angles of rotation of the input shaft together with the carrier and the output shaft are designated as φ1 and φ3, respectively.

Figure 2

Calculation scheme of the mechanism

The study of kinetostatics was carried out for the actuator of the stirred tank, operating in the following mode: the frequency of the rotationally reciprocating motion f = 10 Hz, the swing angle of the impeller α= 150°. For this mode, the following calculation data are considered known from [13, 14, 32]: active forces, moments of forces, masses and moments of inertia of links, as well as kinematic parameters (positions, velocities and accelerations). It is also assumed that the friction forces in the kinematic pairs are equal to zero.

Force analysis of the mechanism

Conducting a force analysis and determining reactions in kinematic pairs consists of writing and solving kinetostatic equations. In the proposed mechanism, for each link, it is necessary to compose 3 equations, in total 12 equations. The number of unknown reactions is also 12: 1 reaction in the kinematic pairs B, E, F and 2 components of the reaction in the kinematic pairs A, C, D, G, and also the balancing moment Mb.

Force analysis is carried out sequentially for each link, beginnign with conducting a force analysis from link 4, since when it is released from bonds, the smallest number of unknown reactions is formed. In accordance with the d’Alembert principle, the acting forces and moments, as well as the forces and moments of force inertiawere applied to link 4 (Figure 3).

Figure 3

Calculation scheme of link 4

Link 4 makes plane-parallel motion, the instantaneous center of velocity, of which is the point on the sun wheel, opposite point B. Dynamic analysis showed [14, 32] that the angular velocity of the input link 1 is not a constant value due to the changes in the load on the impeller. However, when using the flywheel, the speed fluctuations are insignificant and do not have much effect on the forces acting on the links of the machine. Therefore, for the convenience of calculations, we assume that ω1 = const, ω2 = const and ω4 = const.

Link 4 is affected by: a reaction R41 {\vec R_{41}} in kinematic pair D, the direction of which is not known in advance; reaction R43 {\vec R_{43}} in the kinematic pair E, which is directed at an angle α to the tangent O1O2; as well as inertial forces, which are applied to the center of mass of the link. Gravity forces are directed perpendicular to the plane of the figure, therefore, they are not projected on the x and y axes.

The lines O1O2, O3O4 (Figure 3(5)) make an angle γ with the axis Ox2. Since O3O4 is a tangent to the ellipse at the point F, to determine the angle γ, the equation of the tangent in the plane Ox2y2 is written as [33, 34]: x0xa2+y0yb2=1 {{{x_0} \cdot x} \over {{a^2}}} + {{{y_0} \cdot y} \over {{b^2}}} = 1 where x0 = FG · sin (φ1φ3), y0 = FG · cos (φ1φ3) − OG are the coordinates of the point F in the plane Ox2y2, a and b are the semiaxes of the ellipse, OG = c is the focal distance of the ellipse.

Transforming equation (1), we obtain an expression for finding γ: γ=arctg(x0y0b2a2). \gamma = {\rm{arctg}}\left({- {{{x_0}} \over {{y_0}}} \cdot {{{b^2}} \over {{a^2}}}} \right).

To determine the inertia forces, link 4 is considered as the sum of two bodies: body 4.1 (counterweight and satellite shaft), whose inertia force Fin41 {\vec F_{in41}} is applied to point D, and body 4.2 (elliptical wheel), whose inertia force Fin42 {\vec F_{in42}} is applied to point D1.

The accelerations of points D and D1 are defined as follows. Since ε1 = 0, then aD=aDn=ω12 {a_D} = a_D^n = \omega _1^2 . Taking point D as a pole, we obtain aD1=aD+aD1D {\vec a_{{D_1}}} = {\vec a_D} + {\vec a_{{D_1}D}} , where aD1Dn=ω22 a_{{D_1}D}^n = \omega _2^2 is the acceleration of point D1 relative to pole D.

The required forces of inertia Fin41 {\vec F_{in41}} and Fin42 {\vec F_{in42}} are determined as: Fin41=m41aD, {\vec F_{in41}} = {m_{41}} \cdot {\vec a_D}, Fin42=m42aD1=m42aD+m42aD1D. {\vec F_{in42}} = {m_{42}} \cdot {\vec a_{{D_1}}} = {m_{42}} \cdot {\vec a_D} + {m_{42}} \cdot {\vec a_{{D_1}D}}.

The notations F1=m42aD {\vec F_1} = {m_{42}} \cdot {\vec a_D} , F2=m42aD1D {\vec F_2} = {m_{42}} \cdot {\vec a_{{D_1}D}} is introduced (Figure 3) and the equilibrium condition is written in the form of a system of three equations – the sum of the projections of all the forces on the x and y axis, and the sum of the moments relative to the point D: R41x(Fin41+F1)sinφ1F2sin(2φ1)+R43cos(γ+αφ3)=0; \matrix{{R_{41}^x} \hfill & {- ({F_{in41}} + {F_1}) \cdot \sin {\varphi _1} - {F_2} \cdot \sin (2{\varphi _1})} \hfill \cr {} \hfill & {+ {R_{43}} \cdot \cos (\gamma + \alpha - {\varphi _3}) = 0;} \hfill \cr} R41y(Fin41+F1)cosφ1F2cos(2φ1)+R43sin(γ+αφ3)=0; \matrix{{R_{41}^y} \hfill & {- ({F_{in41}} + {F_1}) \cdot \cos {\varphi _1} - {F_2} \cdot \cos (2{\varphi _1})} \hfill \cr {} \hfill & {+ {R_{43}} \cdot \sin (\gamma + \alpha - {\varphi _3}) = 0;} \hfill \cr} F1sin(π2φ1)DD1cos(π22φ1)+F1cos(π2φ1)DD1sin(π22φ1)R43xDEcos(π+φ1)R43yDEsin(π+φ1)=0. \matrix{{{F_1}\sin \left({{\pi \over 2} - {\varphi _1}} \right)} \hfill & {\cdot D{D_1} \cdot \cos \left({{\pi \over 2} - 2{\varphi _1}} \right)} \hfill \cr {} \hfill & {+ {F_1}\cos \left({{\pi \over 2} - {\varphi _1}} \right) \cdot D{D_1} \cdot \sin \left({{\pi \over 2} - 2{\varphi _1}} \right)} \hfill \cr {} \hfill & {- R_{43}^x \cdot DE \cdot \cos (\pi + {\varphi _1})} \hfill \cr {} \hfill & {- R_{43}^y \cdot DE \cdot \sin (\pi + {\varphi _1}) = 0.} \hfill \cr}

Considering that R43x=R43cos(γ+αφ3) R_{43}^x = {R_{43}} \cdot \cos (\gamma + \alpha - {\varphi _3}) and R43y=R43sin(γ+αφ3) R_{43}^y = {R_{43}} \cdot \sin (\gamma + \alpha - {\varphi _3}) , from (4), the reaction R43 is determined. Then, from equations (2) and (3), we find the reactions R41x R_{41}^x and R41y R_{41}^y .

Figure 4 shows the kinetostatic equilibrium of link 3. The link makes a rotationally reciprocating motion and is affected by: reaction R30 {\vec R_{30}} in the kinematic pair G, the direction of which is not known in advance; reactions R32 {\vec R_{32}} and R34 {\vec R_{34}} in kinematic pairs F and E directed at an angle α to the tangents O3O4 and O1O2; normal Fin3n \vec F_{in3}^n and tangential Fin3τ \vec F_{in3}^\tau inertia forces applied to the centers of mass of elliptical gears; moment of resistance of the stirred liquid Mr {\vec M_r} and moment of inertia forces Min3 {\vec M_{in3}} .

Figure 4

Calculation scheme of link 3

The kinetostatic balance condition is represented as a system of three equations – the sum of the projections of the all the forces on the x and y axis, and the sum of the moments relative to the point G: R30x+R32cos(γ+αφ3)R34cos(γ+αφ3)=0; \eqalign{& R_{30}^x + {R_{32}} \cdot \cos (\gamma + \alpha - {\varphi _3}) - {R_{34}} \cdot \cos (\gamma + \alpha - {\varphi _3}) \cr & \,\,\,\,\,\,\,\,\, = 0; \cr} R30y+R32sin(γ+αφ3)R34sin(γ+αφ3)=0; \eqalign{& R_{30}^y + {R_{32}} \cdot \sin (\gamma + \alpha - {\varphi _3}) - {R_{34}} \cdot \sin (\gamma + \alpha - {\varphi _3}) \cr & \,\,\,\,\,\,\,\,\, = 0; \cr} MrMin32Fin3τcR32xFGcosφ1R32yFGsinφ1R34xEGcosφ1R34yEGsinφ1=0. \matrix{{{M_r}} \hfill & {- {M_{in3}} - 2F_{in3}^\tau \cdot c - R_{32}^x \cdot FG \cdot \cos {\varphi _1}} \hfill \cr {} \hfill & {- R_{32}^y \cdot FG \cdot \sin {\varphi _1} - R_{34}^x \cdot EG \cdot \cos {\varphi _1}} \hfill \cr {} \hfill & {- R_{34}^y \cdot EG \cdot \sin {\varphi _1} = 0.} \hfill \cr}

Considering that R32x=R32cos(γ+αφ3) R_{32}^x = {R_{32}} \cdot \cos (\gamma + \alpha - {\varphi _3}) , R32y=R32sin(γ+αφ3) R_{32}^y = {R_{32}} \cdot \sin (\gamma + \alpha - {\varphi _3}) , R34x=R34cos(γ+αφ3) R_{34}^x = {R_{34}} \cdot \cos (\gamma + \alpha - {\varphi _3}) , R34y=R34sin(γ+αφ3) R_{34}^y = {R_{34}} \cdot \sin (\gamma + \alpha - {\varphi _3}) , from equation (7), the reaction R32 is determined. Then, from equations (5) and (6), we find the reactions R30x R_{30}^x and R30y R_{30}^y .

Figure 5 presents the kinetostatic equilibrium of link 2, which performs the plane-parallel motion, the instantaneous center of velocity of which is point B. The link is affected by: a reaction R21 {\vec R_{21}} in kinematic pair C, the direction of which is not known in advance; reaction R23 {\vec R_{23}} in kinematic pair F, which is directed opposite to the found reaction R32 {\vec R_{32}} ; reaction R20 {\vec R_{20}} in kinematic pair B, which is directed at an angle α to the tangent O5O6; as well as inertia forces, which are applied to the center of mass of the link. For the convenience of solving the problem, as in the case of link 4, we consider link 2 as the sum of two bodies: body 2.1 (cylindrical wheel and satellite shaft), the inertia force Fin21 {\vec F_{in21}} of which is applied to point C, and body 2.2 (elliptical wheel), the inertia force Fin22 {\vec F_{in22}} of which is attached to the point C1.

Figure 5

Calculation scheme of link 2

The accelerations of points C and C1 are defined as follows. Since ε1 = 0, then aC=aCn=ω12AC {a_C} = a_C^n = \omega _1^2 \cdot AC . Taking point C as a pole, we obtain aC1=aC=aC1 {\vec a_{C1}} = {\vec a_C} = {\vec a_{C1}} , where aC1=aC1n=ω22C1 {a_{C1}} = a_{C1}^n = \omega _2^2 \cdot {C_1} is the acceleration of point C1 relative to pole C.

The required forces of inertia Fin21 {\vec F_{in21}} and Fin22 {\vec F_{in22}} are determined as: Fin21=m21aC,Fin21=m22aC1=m22.aC+m22aC1. \matrix{{{{\vec F}_{in21}} = {m_{21}} \cdot {{\vec a}_C},} \cr {{{\vec F}_{in21}} = {m_{22}} \cdot {{\vec a}_{C1}} = {m_{22}}.{{\vec a}_C} + {m_{22}} \cdot {{\vec a}_{C1}}.} \cr}

The notations F3=m22aC {\vec F_3} = {m_{22}} \cdot {\vec a_C} , F4=m22aC1 {\vec F_4} = {m_{22}} \cdot {\vec a_{C1}} are introduced (Figure 5) and the equilibrium condition is written in the form of a system of three equations – the sum of the projections of all the forces on the x and y axis, and the sum of the moments relative to the point C: R21x+(Fin21+F3)sinφ1+F4sin(2φ1)R23cos(γ+αφ3)+R20cos(αφ1)=0; \matrix{{R_{21}^x} \hfill & {+ ({F_{in21}} + {F_3})\sin {\varphi _1} + {F_4}\sin (2{\varphi _1})} \hfill \cr {} \hfill & {- {R_{23}}\cos (\gamma + \alpha - {\varphi _3}) + {R_{20}}\cos (\alpha - {\varphi _1}) = 0;} \hfill \cr} R21y+(Fin21+F3)cosφ1+F4cos(2φ1)+R23sin(γ+αφ3)+R20sin(αφ1)=0; \matrix{{R_{21}^y} \hfill & {+ ({F_{in21}} + {F_3})\cos {\varphi _1} + {F_4}\cos (2{\varphi _1})} \hfill \cr {} \hfill & {+ {R_{23}}\sin (\gamma + \alpha - {\varphi _3}) + {R_{20}}\sin(\alpha - {\varphi _1}) = 0;} \hfill \cr} F3sin(π2φ1)CC1cos(π22φ1)+F3cos(π2φ1)CC1sin(π22φ1)R23xCFcos(π+φ1)R23yCFsin(π+φ1)+R20xBCcos(π+φ1)R20yBCsin(π+φ1)=0. \matrix{{\,{F_3}\sin \left({{\pi \over 2} - {\varphi _1}} \right)C{C_1}\cos \left({{\pi \over 2} - 2{\varphi _1}} \right)} \hfill \cr {+ {F_3}\cos \left({{\pi \over 2} - {\varphi _1}} \right)C{C_1}\sin \left({{\pi \over 2} - 2{\varphi _1}} \right)} \hfill \cr {- R_{23}^xCF \cdot \cos (\pi + {\varphi _1}) - R_{23}^yCF\sin (\pi + {\varphi _1})} \hfill \cr {+ R_{20}^xBC\cos (\pi + {\varphi _1}) - R_{20}^yBC\sin (\pi + {\varphi _1}) = 0.} \hfill \cr}

Considering that R23x=R23cos(γ+αφ3) R_{23}^x = {R_{23}} \cdot \cos (\gamma + \alpha - {\varphi _3}) , R23y=R23sin(γ+αφ3) R_{23}^y = {R_{23}} \cdot \sin (\gamma + \alpha - {\varphi _3}) , R20.x=R20cos(αφ1) R_{20.}^x = {R_{20}} \cdot \cos (\alpha - {\varphi _1}) , R20.y=R20sin(αφ1) R_{20.}^y = {R_{20}} \cdot \sin (\alpha - {\varphi _1}) , from (10), the reaction R20 is determined. Then, from equations (8) and (9), we find the reactions R21x R_{21}^x and R21y R_{21}^y .

Figure 6 shows the kinetostatic equilibrium of link 1, which performs a rotational motion. The link is affected by: a reaction R10 {\vec R_{10}} in kinematic pair A, the direction and module of which are unknown; a reaction R12 {\vec R_{12}} in kinematic pair C, which is directed opposite to the found reaction R21 {\vec R_{21}} ; reaction R14 {\vec R_{14}} in kinematic pair D, which is directed opposite to the found reaction R41 {\vec R_{41}} ; balancing moment Mb {\vec M_b} .

Figure 6

Calculation scheme of link 1

The kinetostatic balance condition is represented as a system of three equations – the sum of the projections of all the forces on the x and y axis, and the sum of the moments relative to the point A: R10xR12xR14x=0; R_{10}^x - R_{12}^x - R_{14}^x = 0; R10yR12yR14y=0; R_{10}^y - R_{12}^y - R_{14}^y = 0; My+R12xACcosφ1R12yACsinφ1R14xADcosφ1R14yADsinφ1=0. \matrix{{- {M_y} + R_{12}^x \cdot AC \cdot \cos {\varphi _1} - R_{12}^y \cdot AC \cdot \sin {\varphi _1}} \hfill \cr {- R_{14}^x \cdot AD \cdot \cos {\varphi _1} - R_{14}^y \cdot AD \cdot \sin {\varphi _1} = 0.} \hfill \cr}

The unknown values Mb {\vec M_b} , R10x \vec R_{10}^x , R10y \vec R_{10}^y are found from equations (11) to (13).

Results

Mathematical modeling of kinetostatics is carried out for the rotationally reciprocating stirred tank with planetary actuator, developed on the basis of the Ika Magic Plant laboratory scale process plant (Figure 7). The developed experimental setup contains an electric motor 1, a motor control unit 2, a rack 3, on which the motor is fixed, a planetary actuator 4, an input shaft of an actuator 5, an output shaft of an actuator 6, a coupling 7, a reactor 8, a rack 9, which holds the reactor with the actuator, reactor cover 10, sealing clamp 11, impeller seal 12, impeller 13, product loading fitting 14, outlet valve 15, coolant supply fittings 16.

Figure 7

Ika Magic Plant laboratory scale process plant with rotationally reciprocating actuator

According to the structural scheme in Figure 1b, a planetary actuator is designed and manufactured (Figure 8). The design of the mechanism consists of the rack 1, an input shaft 2, a carrier 3, an output shaft 4, a sun wheel 5, elliptical gears 6 and 7 on the output shaft, located at an angle of 180°, a first satellite consisting of a cylindrical gear 8, an elliptical gear 9 and a shaft 10, a second satellite consisting of a counterweight 11, an elliptical wheel 12 and a shaft 13.

Figure 8

Design of planetary mechanism with elliptical gears

The links of the mechanism have the following inertial characteristics (the numbers of the links correspond to Figure 8): Im = 100 g·cm2 (motor); I2 = 9.8 g·cm2; I3 = 1233 g·cm2; I4 = 30.4 g·cm2; I6 = I7 = 627 g·cm2; I8 = 400 g·cm2, m8 = 0.1 kg; I9 = I12 = 564 g·cm2, m9 = m12 = 0.09 kg; I10 = 19.2 g·cm2, m10 = 0.04 kg; I11 = 350 g·cm2, m11 = 0.09 kg; I13 = 25 g·cm2, m13 = 0.05 kg; Iim = 15 g·cm2 (impeller). By correlating the design of the mechanism (Figure 8) and the calculation schemes in Figures 3 to 6, we obtain the following masses and moments of inertia of the links for substitution in the kinetostatic equations (2) to (13): I1 = Im + I2 + I3 = 1342.8 g·cm2; I3 = I4 + I6 + I7 + Iim = 1299.4 g·cm2; m2.1 = m8 + m10 = 0.14 kg; m2.2 = m9 = 0.09 kg; m4.1 = m11 + m13 = 0.14; m4.2 = m12 = 0.09 kg.

A set of replaceable elliptical gears with eccentricities from 0.28 to 0.6 is manufactured for the planetary mechanism. Kinematic analysis of the mechanism is carried out in [14] by constructing plans of the link velocities for various eccentricities of elliptical gears, while the largest angle of rotation of the output shaft 147° is obtained at e = 0.6. The maximum angular speed of the motor shaft is 62.8 rad/s (n = 600 rpm), which allows to obtain the frequency up to 10 Hz of the rotationally reciprocating motion of impeller. Since the loads on the links of the mechanism increase with an increase in the frequency and angle of oscillation of the impeller, the kinetostatic analysis is carried out for the most loaded mode α = 150°, f = 10 Hz.

Studies of fluid dynamics in a reactor with a rotationally reciprocating motion of the impeller were carried out by various authors [35, 36, 37], equations for determining the Reynolds number and the moment of resistance on the working body were obtained. At the same time, researchers agree that in the case of laminar fluid motion, the moment of resistance is proportional to the angular velocity of the impeller, and in turbulent motion, it is proportional to the square of the angular velocity. In the selected operating mode of the stirred tank, a turbulent mode of fluid movement is observed, therefore, according to [37], the following equation is applied to determine the moment of resistance Mr on the impeller: Mr=Bωim2sign(ωim) {M_r} = B \cdot \omega _{im}^2 \cdot sign({\omega _{im}}) where B is the coefficient of resistance square law, ωim is the angular velocity of the impeller. According to the results of calculations for the selected mode, B = 2.67 · 10−6 is obtained and used in further calculations.

After substituting the initial data and solving equations (2) to (13), the reactions in kinematic pairs is determined (Figure 9). Studies have shown that the most loaded are gears (kinematic pairs) B and F, and the rotational pair G.

Figure 9

Graphs of reaction forces Ri(φ1) and balance moment Mb(φ1) in kinematic pairs

Figure 9 presents graphs of the following reaction forces in kinematic pairs:

functions of forces R41x(φ1) R_{41}^x({\varphi _1}) , R41y(φ1) R_{41}^y({\varphi _1}) in kinematic pair D and force R43(φ1) in kinematic pair E are shown in Figure 9a;

functions of forces R30x(φ1) R_{30}^x({\varphi _1}) , R30y(φ1) R_{30}^y({\varphi _1}) in kinematic pair G and force R32(φ1) in kinematic pair F are shown in Figure 9b;

functions of forces R21x(φ1) R_{21}^x({\varphi _1}) , R21y(φ1) R_{21}^y({\varphi _1}) in kinematic pair C and force R20(φ1) in kinematic pair B are shown in Figure 9c;

functions of forces R10x(φ1) R_{10}^x({\varphi _1}) , R10y(φ1) R_{10}^y({\varphi _1}) in kinematic pair A and balance moment Mb(φ1) are shown in Figure 9d.

As it can be seen from the graphs in Figure 9, the maximum values of the forces in the kinematic pairs and the balancing moment are observed during a change in the direction of rotation of the impeller (link 3). Moreover, in most cases, there is a change in the direction of the force and moment vector.

Conclusions

So, in this work, there are obtained systems of equations that allow to perform force analysis of the planetary actuator with elliptical gears and determine the reactions in kinematic pairs, as well as the balancing moment for all the positions of the mechanism. As an example, the calculations of the rotationally reciprocating stirred tank are carried out and the force graphs in kinematic pairs are presented, depending on the position of the mechanism, for the mode f = 10 Hz, α = 150°. The proposed methodology of force analysis can be implemented for other operating modes of the actuator, which allows for extensive studies of kinetostatics, to choose the most rational modes of operation of the stirred tank, as well as to simplify the calculation and design of the future machine. Also, the conducted studies can be applied in the design and analysis of other devices with the proposed scheme of the planetary mechanism.

Figure 1

Rotationally reciprocating stirred tank: a – scheme of the actuator; b – structural scheme of the planetary mechanism
Rotationally reciprocating stirred tank: a – scheme of the actuator; b – structural scheme of the planetary mechanism

Figure 2

Calculation scheme of the mechanism
Calculation scheme of the mechanism

Figure 3

Calculation scheme of link 4
Calculation scheme of link 4

Figure 4

Calculation scheme of link 3
Calculation scheme of link 3

Figure 5

Calculation scheme of link 2
Calculation scheme of link 2

Figure 6

Calculation scheme of link 1
Calculation scheme of link 1

Figure 7

Ika Magic Plant laboratory scale process plant with rotationally reciprocating actuator
Ika Magic Plant laboratory scale process plant with rotationally reciprocating actuator

Figure 8

Design of planetary mechanism with elliptical gears
Design of planetary mechanism with elliptical gears

Figure 9

Graphs of reaction forces Ri(φ1) and balance moment Mb(φ1) in kinematic pairs
Graphs of reaction forces Ri(φ1) and balance moment Mb(φ1) in kinematic pairs

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