Kinematic analysis and geometrical parameters justification of a planetary-type mechanism for actuating an inertial vibration exciter

2.1. General design of the laboratory prototype being studied

The laboratory prototype of the single-mass vibratory system (Fig. 1) consists of the planetary-type mechanism 1 driven by the electric motor 2 through the toothed belt transmission 3. The driving pulley 4 is fixed on the motor’s shaft. The ring gear 5 of the mechanism is mounted on the machine’s body 6 suspended from the stationary supporting surface 7 by the set of four coil-type cylindrical springs 8, which perform the vibration isolation function. The arm or, so-called, planet carrier 9 is fixed to the shaft of the driven pulley 10 of the toothed belt transmission 3. Another end of the carrier 9 is hinged to the axle, on which the planet gear 11 is installed. The unbalanced mass 12 is to be connected to the planet gear 11 at different points, and the dependence of the connection point location on the mass 12 trajectory is to be studied in the present paper. The major idea of this research is choosing the appropriate design (geometrical) parameters of the planetary gear train that allow for generating circular, elliptical, and rectilinear motion trajectories of the unbalanced mass. This, in turn, could provide the corresponding vibrations of the machine’s body.

Fig. 1General design of the laboratory prototype being studied

2.2. Kinematic diagram of the planetary-type mechanism of the vibration exciter

The dynamic diagram of the single-mass vibratory system equipped with the planetary-type vibration exciter is presented in Fig. 2(a). The rigid body of the mass $m_1$ can perform planar oscillations along the horizontal ($Ox$) and vertical ($Oy$) axes and is suspended from the stationary base by a set of spring-damper elements. The latter are characterized by the stiffness coefficients $k_x$, $k_y$ and viscous friction factors $c_x$, $c_y$. The system motion is generated due to the action of the inertial forces exerted on the planet gear (mass $m_2$) and unbalanced mass $m_3$ during their uniform rotation around two mutually perpendicular axes located at hinges $O$ and $E$.

Taking into account the main idea of this research mentioned in the previous section, let us consider the corresponding kinematic diagram of the planetary-type mechanism used for actuating the proposed inertial vibration exciter (see Fig. 1(b)). The ring gear of the planetary-type mechanism is supposed to be fixed, while the rotation of the arm (planet carrier) $OE$ is characterized by the angular coordinate ${\phi }_0$. Considering the geometrical parameters of the planetary-type mechanism ($R_0=OK$, $R_1=EK$, $H=OE$, $r=ED$) and assuming the constant angular speed of the arm ${\omega }_0$, let us define the basic relations between the angles ${\phi }_0$ and ${\phi }_1$:

where $k=\left(H+R_1\right)/R_1$.

Fig. 2a) Dynamic diagram of the oscillatory system and b) kinematic diagram of the planetary-type mechanism of the vibration exciter

a)

b)

2.3. Analytical dependencies for determining the kinematic parameters of the mechanism

Considering the coordinate system $xOy$ with its origin at point $O$, let us derive the analytical expression for determining the horizontal and vertical position of the unbalanced mass $D$:

Based on Eqs. (2) and (3), the corresponding formulas for calculating the horizontal and vertical velocities and accelerations of the unbalanced mass can be deduced:

The full speed and acceleration of the unbalanced mass are following:

While performing kinematic analysis, it is more informative to consider the dimensionless analogues of the displacements, speeds, and accelerations instead of their absolute values:

3.1. Simulating the trajectories of the unbalanced mass motion

During the rotation of an arm (planet carrier) at a constant angular speed, it is enough to consider its single revolution to plot the corresponding trajectories of the unbalanced mass in the dimensionless values (see Eq. (9)). The results (Fig. 3) were obtained for the following constant relations between the mechanism geometrical parameters: $H/R_0=$ 0.5 and $R_1/R_0=$ 0.5, whilst the ratio $r/R_1$ and the initial angle of the arm ${\phi }_0$ ($t=$0, ${\phi }_1=$0)$={\mathrm{Ф}}_0$ are considered changeable.

Fig. 3Dimensionless trajectories of the unbalanced mass motion

a)${\mathrm{\Phi }}_0=$ 0°, $r/R_1=$ 0

b)${\mathrm{\Phi }}_0=$ 0°, $r/R_1=$ 0.5

c)${\mathrm{\Phi }}_0=$ 0°, $r/R_1=$ 1

d)${\mathrm{\Phi }}_0=$ 90°, $r/R_1=$ 0.5

e)${\mathrm{\Phi }}_0=$ 45°, $r/R_1=$ 0.5

f)${\mathrm{\Phi }}_0=$ – 45°, $r/R_1=$ 0.5

g)${\mathrm{\Phi }}_0=$ 90°, $r/R_1=$ 1

h)${\mathrm{\Phi }}_0=$ 45°, $r/R_1=$1

i)${\mathrm{\Phi }}_0=$ – 45°, $r/R_1=$1

The obtained results allow for the conclusion that the circular trajectory of the unbalanced mass can be obtained when the distance between the mass and the planet gear axis is equal to zero ($r/R_1$ = 0). All the other ratios $r/R_1$ within the range of (0…1) provide elliptical paths, whose major axes make the angles ${\mathrm{\Phi }}_0$ with the $Ox$ axis. Similarly, the ratio $r/R_1$ = 1 allows for generating rectilinear motion of the unbalanced mass at different angles ${\mathrm{\Phi }}_0$ with the horizontal axis.

Considering the possible practical implementation of the obtained results, the following conclusions can be drawn. Vertical rectilinear and elliptical trajectories of the unbalanced mass and, as a consequence, of the working member can be used in vibratory compacting machines. Horizontal rectilinear and circular paths can be implemented in screening and sieving equipment. In addition, a circular trajectory of a working member should be provided in vibratory lapping and polishing machines. Inclined rectilinear and elliptical trajectories are preferable for conveying equipment. Various rectilinear paths can be used in vibration-driven locomotion systems.

3.2. Analyzing the speeds and accelerations of the unbalanced mass

In order to perform further analysis, let us consider three basic cases of the unbalanced mass motion (see Figs. 3(a)-(c)): $H/R_0=$ 0.5, $R_1/R_0=$ 0.5, ${\mathrm{\Phi }}_0=$ 0°, at the following changeable ratios $r/R_1=$ 0, 0.5, 1. The plots describing the dependencies of the analogues of speeds and accelerations of the unbalanced mass at different ratios $r/R_1$ are presented in Figs. 4 and 5. Under the conditions of $r=$ 0, the full speed of the unbalanced mass is equal to the speed of the planet gear hinge, while the corresponding horizontal and vertical components periodically reach the maximal values with the phase shift of 90°. Similarly, the full acceleration is equal to the centrifugal acceleration of the unbalanced mass rotating around the crankshaft. The elliptical motion characterized by the horizontal position of the ellipse major axis is also accompanied by the periodical change in the horizontal and vertical speeds and accelerations. In this case, the horizontal quantities take larger peak values than the vertical ones. Considering the rectilinear motion, the vertical components of the speeds and accelerations are equal to zero, while the full speed and acceleration are equal to the horizontal ones and periodically change reaching the maximal values that are twice larger than the ones obtained under the conditions of circular motion. This substantiates the possibilities of implementing the considered planetary-type mechanism with an unbalanced mass for actuating the inertial vibration exciter, which, in turn, can be used for actuating various vibratory equipment, e.g., screens, conveyors, compactors, etc.

Fig. 4Dependencies of analogues of the unbalanced mass speeds on the rotation angle

a)${\mathrm{\Phi }}_0=$ 0°, $r=$ 0

b)${\mathrm{\Phi }}_0=$ 0°, $r=0.5R_1$

c)${\mathrm{\Phi }}_0=$ 0°, $r=R_1$

Fig. 5Dependencies of analogues of the unbalanced mass accelerations on the rotation angle

a)${\mathrm{\Phi }}_0=$ 0°, $r=$ 0

b)${\mathrm{\Phi }}_0=$ 0°, $r=0.5R_1$

c)${\mathrm{\Phi }}_0=$ 0°, $r=R_1$