Introduction. Mechanisms with non-circular gears have been known for a long time, but their experimental research and practical implementation have long been difficult due to the focused specialization of such mechanisms, the complexity and high cost of their manufacture. Nowadays, there is an increase in the interest of researchers in this topic. At the same time, most scientific papers on designing non-circular gearing are based on the analysis of geometry [1] and kinematics [2] to obtain the required transfer function for practical application [3]. In [4] and [5], mechanisms for synthesizing angular velocity functions were developed and studied. Researchers proposed robotics with non-circular gears, for example, a jumping robot [6], a hexapod robot [7], an exoskeleton mechanism for knee joint rehabilitation [8]. More efficient planting machines for the agricultural industry were studied from the point of view of kinematics [9] and dynamics [10]. Other numerous devices were developed and designed.

A review of sources has shown that the most common gears are those with an elliptical centrode [11][12]. Machines [10] and devices [4] with elliptical gears are created, the geometry [13] is studied, and some design [14] and manufacturing issues are resolved [15].

Elliptical gears are mainly used to transmit rotary movement between parallel fixed axes [16]. However, such use of non-circular gears allows for only one-way nonuniform rotary movement, which limits the scope of their application. Planetary gears have broader capabilities for implementing complex types of movement of working bodies. The creation of drives based on them will provide the development and implementation of more efficient and compact machines for various technological purposes.

Based on the results of the analysis of scientific literature in the field of transmissions with non-circular gears, a planetary mechanism with elliptical wheels is proposed as an object of study. The objective of the work is an experimental kinematic analysis of a planetary transmission. It is justified by the need to verify theoretical provisions for their correct use at the following stages of design — in dynamic, force analysis and strength calculations.

Materials and Methods. The kinematic model of the mechanism is constructed on the basis of the velocity diagram of its links (Fig. 1).

The analog of the angular velocity of the output shaft is determined as follows [17]:

(1)

Segments BD, BC and DE are determined through the polar equation of the ellipse [17]:

(2)

where  — rotation angle, e and a — eccentricity and semi-major axis of an ellipse.

We determine the engagement radii of gears 5 and 6:

(3)

(4)

where e1 and e2 — eccentricities for each pair of elliptical gears, whose rotation angles and  are obtained as follows:

(5)

(6)

Based on the velocity plan and taking into account equations (3), (4), we define the required segments as:

(7)

(8)

(9)

Based on (2)–(9), we obtain an expression for determining the analogue of the angular velocity of the output shaft:

(10)

The rotation angle is determined by integrating equation (10) over joint coordinate :

(11)

Variations of the kinematic scheme of the mechanism include replacing elliptical gears with cylindrical ones in one of the pairs, for the study of which it is required to replace the radius functions with fixed values in the resulting mathematical model.

We consider mechanisms with a pair of elliptical gears 3–6 (e=0.28), and cylindrical wheels 4 and 5 of the following sizes:

Figure 2 shows the graphs of the position functions obtained using (11).

The analysis of the graphs shows (in Fig. 2, numbers indicate studied options) that changing the sizes of the gear wheels allows obtaining different types of output shaft movement: reciprocating-rotational (option 1), discontinuous (option 2) and one-way nonuniform rotational movement (option 3).

Research Results. The kinematic analysis confirmed the implementation of various transfer functions, and changing the sizes of gear wheels allows for the kinematic synthesis of new mechanisms.

The object of the experimental study is a prototype, whose details are shown in Figure 3.

Rotation angles are measured by absolute encoders (Table 1).

The signal is registered by the Arduino controller and processed on the PC (Fig. 4).

The processing and analysis of the measurement results were performed in the MathCAD application package. The schemes of the studied options and the experiment results are shown in Figure 5.

As can be seen from the graphs, the measured values of the output shaft rotation angles are adequate to the constructed model. Let us evaluate the results of the experiment using statistical analysis tools.

We determine the average value of measurement errors [18]:

(12)

where n — number of independent observations qk.

Let us calculate the sample variance:

(13)

An estimate of the dispersion of the mean value is obtained as follows:

(14)

The standard uncertainty of Type A measurements is determined [19]:

(15)

Taking the measurement errors as values qk and processing them according to (12)–(15), we calculate the measurement uncertainty. The results of the statistical analysis are given in Table 2.

The confidence interval is defined as  for a level of certainty of 95% and is –0.16 ± 0.08° (option 1), 0.57 ± 0.22° (option 2), 0.08 ± 0.26° (option 3). Thus, the constructed mathematical model of kinematics is adequate to the physical prototypes.

Discussion and Conclusion. A planetary gear with elliptical wheels is presented, providing the implementation of nonuniform, discontinuous and reciprocating-rotational movement of the output shaft. The type of movement is determined by the parameters of the gear wheels.

A kinematic model of the transmission was constructed, and the law of movement of the output shaft was obtained. The analysis of mechanisms with different parameters of gear wheels showed the feasibility of a wide range of transmission functions and types of movement of the output shaft.

The correspondence of the results of the conducted kinematic analysis to real mechanisms is confirmed by an experimental study of the output shaft position functions for three options of the physical prototype. The performed assessment of the adequacy of the mathematical model provides its using in dynamic and force analysis of machines based on the proposed transmissions. The studied options of the mechanism can be used in drives of compact and easy-to-balance technological machines.