Integrated Electromechanical Modeling and Dynamic Analysis of a Planetary-Driven Seed-Removing Device for Cotton Gins

1. Introduction

Rotating machine units used in industrial and agricultural equipment increasingly operate under variable technological loads, frequent start-up cycles, and tighter requirements for productivity and energy efficiency. Under these conditions, the motor, transmission, supporting structure, and working members cannot always be treated as dynamically independent components. Elastic deformation, damping, backlash, inertia distribution, and electromagnetic torque variation cause energy exchange between subsystems and may produce transient overloads, torsional oscillations, and non-uniform rotation. These effects are especially important in machines in which the quality and continuity of a technological process depend directly on the rotational stability of several coupled working members.

Recent studies of geared mechanisms have shown that time-varying mesh stiffness, tribological effects, structural flexibility, and load redistribution strongly influence dynamic response [1,2]. Research on controlled transmissions and synchronizing mechanisms has likewise demonstrated that transient motion is governed by the interaction of actuator characteristics, inertia, stiffness, and control laws [3,4,5,6,7]. These findings support the use of system-level models rather than isolated component models when the objective is to predict start-up behaviour, vibration, and power demand.

Integrated modeling of electric motors and mechanical transmissions has become a major direction in rotating machinery research. Electromechanical models of motor-gearbox systems demonstrate that local gear defects, electromagnetic torque pulsations, and mechanical compliance can interact and modify the measured vibration response [8,9,10]. The dynamic response of an induction-motor drive is therefore not fully described by a constant driving torque, particularly during starting and acceleration. Magnetic excitation, voltage distortion, rotor eccentricity, and coupling between lateral and torsional motion may substantially alter the transient load applied to the transmission.

Belt transmissions introduce an additional compliant and dissipative subsystem. Their response depends on belt pretension, nonlinear axial stiffness, transverse vibration, hysteresis, pulley geometry, and operating temperature. Modern analytical, numerical, and experimental studies describe belt drives using distributed-parameter formulations, absolute nodal coordinate methods, viscoelastic models, and data-driven identification [11,12,13,14,15,16,17,18,19,20]. Nevertheless, in many machine-unit models the belt is still reduced to an ideal kinematic ratio or a single linear spring without experimentally supported damping. Such simplification may be acceptable in steady-state kinematic calculations but is insufficient for evaluating start-up oscillations and rotational irregularity.

Planetary transmissions are attractive for compact machines because they provide large speed ratios, coaxial arrangement, and multiple load paths. At the same time, their dynamics are affected by mesh phasing, manufacturing errors, support flexibility, journal-bearing behaviour, and non-uniform load sharing [21,22,23,24,25,26,27,28,29,30,31,32]. Recent models increasingly include flexible components, multi-tooth contact, local faults, and dynamic mesh forces. These works explain the internal behaviour of planetary stages in detail, but most of them do not combine the planetary stage with the induction-motor transient, belt compliance, and a variable technological resistance acting on agricultural working members.

Dynamic models of asynchronous-motor-driven machinery also show that the mechanical response depends on the representation of the motor characteristic. A constant-torque approximation eliminates the most critical part of the transient process: the rapid variation of electromagnetic torque and angular acceleration during starting. Studies of motor vibration, torsional oscillations, tacholess speed estimation, and electromechanical coupling demonstrate the need to include the motor as an active dynamic subsystem rather than as an external constant load [33,34,35,36,37,38,39,40,41,42,43].

The technological object considered in this study is the seed-removing device of a saw-type cotton gin. The device continuously removes seeds from the working chamber and therefore affects process stability, energy demand, and the loading of the main machine. Previous studies of cotton-processing machinery have examined the motion of saw cylinders, distributed-parameter working members, vibration diagnostics, machine-vision inspection, and power consumption [44,45,46,47,48,49,50,51,52,53,54,55,56,57]. Fundamental design and mechanical principles of cotton-processing machines are summarized in classical monographs and dissertations [58,59,60,61]. These studies establish the technological importance of stable rotation, but the drive is often simplified and does not simultaneously include the induction motor, elastic belt, planetary gear, perforated tube, auger, and technological resistance.

The considered machine unit has a specific coupled architecture. The motor drives the perforated seed-removing tube through a belt transmission. The tube carries the ring gear of a planetary stage. Planet gears rotate on a fixed carrier and drive the sun gear, which is rigidly connected to the auger. Consequently, the tube and auger rotate at different speeds, and their motion is linked by both kinematic constraints and elastic-dissipative interactions. The technological resistance acting on the tube is non-uniform, while the auger is subjected to the torque required to transport seeds. The resulting system contains several inertia, stiffness, damping, and load parameters that influence start-up and steady operation.

The literature review reveals three related gaps. First, detailed models of planetary gears generally focus on internal gear dynamics and do not include the full electromechanical drive and technological working members. Second, cotton-gin studies usually describe individual working bodies or use simplified motor and transmission characteristics. Third, the influence of experimentally identified inertia and transmission parameters on rotational irregularity has not been systematically evaluated for a seed-removing device with a planetary drive.

Accordingly, the objective of this study is to develop and analyze an integrated nonlinear electromechanical model of the machine unit “induction motor–belt transmission–planetary gear–perforated tube–auger” and to determine parameter combinations that reduce transient loading, power consumption, and rotational irregularity.

2. System Description

The investigated machine unit (Figure 1) is installed in the seed-removing section of a saw-type cotton gin. It consists of an induction motor, a belt transmission, a perforated cylindrical tube, a planetary gear stage, and a screw auger. The motor pulley transmits rotation to the pulley rigidly connected to the perforated tube. The tube is rigidly connected to the ring gear (epicycle) of the planetary transmission. Planet gears are mounted on a stationary carrier, and the sun gear is rigidly connected to the auger shaft. This arrangement produces coordinated rotation of the tube and auger while maintaining different angular velocities.

The perforated tube performs two functions: it rotates as a working member interacting with the seed mass and simultaneously acts as the input member of the planetary stage. The auger transports the separated seeds toward the discharge zone. The technological resistance of the tube varies periodically because of non-uniform seed distribution and the repeated interaction of the perforated surface with the material. The auger resistance is determined by seed transport and internal friction.

The belt-drive pulley diameters are $D_m$ = 130 mm on the motor shaft and $D_t$ = 260 mm on the tube shaft. The tooth numbers of the planetary transmission are $Z_1$ = 18 for the sun gear, $Z_2$ = 18 for each planet gear, and $Z_3$ = 54 for the ring gear. The carrier is fixed. The nominal belt ratio and gear relationships are therefore

$$\begin{array}{c}{i}_b={\displaystyle \frac{{\dot{\phi }}_t}{{\dot{\phi }}_m}}={\displaystyle \frac{D_m}{D_t}}={\displaystyle \frac{130}{260}}=0.5,i_{es}={\displaystyle \frac{{\dot{\phi }}_s}{{\dot{\phi }}_t}}=-{\displaystyle \frac{Z_3}{Z_2}},i_{sa}={\displaystyle \frac{{\dot{\phi }}_a}{{\dot{\phi }}_s}}\end{array}$$

(1)

The signs in the gear ratios indicate the relative directions of rotation. In the reduced dynamic model (Figure 2), the actual kinematic relationships are retained through the transmission ratios appearing in the elastic and dissipative deformation terms. The four generalized coordinates are the angular displacements of the motor rotor, tube with ring gear, representative planet gear, and auger with sun gear.

The supports are assumed sufficiently stiff relative to the torsional compliance of the belt and gear contacts. Therefore, support elasticity and housing vibration are not included in the present torsional model. The principal sources of compliance are represented by the belt stiffness $c_b$, ring–planet mesh stiffness $c_{es}$, and planet–sun mesh stiffness $c_{sg}$. Energy dissipation is represented by the corresponding coefficients $b_b$, $b_{es}$, and $b_{sg}$.

3. Mathematical Model

3.1. Generalized Coordinates and Energy Expressions

The generalized coordinate vector is $q=[{\varphi }_m,{\varphi }_t,{\varphi }_s,{\varphi }_a]T$, where the subscripts m, t, s, and a denote the motor, tube, planet gear, and auger, respectively. The associated moments of inertia are $J_m,J_t,J_s$, and $J_a$. The kinetic energy is

$$\begin{array}{c}T={\displaystyle \frac{1}{2}}{J}_m{\dot{\phi }}_m^2+{\displaystyle \frac{1}{2}}{J}_t{\dot{\phi }}_t^2+{\displaystyle \frac{1}{2}}{J}_s{\dot{\phi }}_s^2+{\displaystyle \frac{1}{2}}{J}_a{\dot{\phi }}_a^2\end{array}$$

(2)

The elastic potential energy is written in terms of relative torsional deformations referred to the corresponding transmission members:

$$\begin{array}{c}V={\displaystyle \frac{1}{2}}{c}_b{({\phi }_m-i_b{\phi }_t)}^2+{\displaystyle \frac{1}{2}}{c}_{es}{({\phi }_t-i_{es}{\phi }_s)}^2+{\displaystyle \frac{1}{2}}{c}_{sg}{({\phi }_s-i_{sg}{\phi }_a)}^2\end{array}$$

(3)

The Rayleigh dissipation function is

$$\begin{array}{c}R={\displaystyle \frac{1}{2}}{b}_b{({\dot{\phi }}_m-i_b{\dot{\phi }}_t)}^2+{\displaystyle \frac{1}{2}}{b}_{es}{({\dot{\phi }}_t-i_{es}{\dot{\phi }}_s)}^2+{\displaystyle \frac{1}{2}}{b}_{sg}{({\dot{\phi }}_s-i_{sg}{\dot{\phi }}_a)}^2\end{array}$$

(4)

3.2. Equations of Motion

Lagrange’s equation of the second kind is applied to each generalized coordinate:

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial T}{\partial {\dot{\phi }}_i}}\right)-{\displaystyle \frac{\partial T}{\partial {\phi }_i}}+{\displaystyle \frac{\partial V}{\partial {\phi }_i}}+{\displaystyle \frac{\partial R}{\partial {\dot{\phi }}_i}}=Q_i,i=m,t,s,a\end{array}$$

(5)

After differentiation, the coupled equations can be written as

$$\begin{array}{c}{J}_m{\ddot{\phi }}_m+b_b({\dot{\phi }}_m-i_b{\dot{\phi }}_t)+c_b({\phi }_m-i_b{\phi }_t)=M_m\end{array}$$

(6)

$$\begin{array}{c}{J}_t{\ddot{\phi }}_t-i_b{b}_b({\dot{\phi }}_m-i_b{\dot{\phi }}_t)-i_b{c}_b({\phi }_m-i_b{\phi }_t)+b_{es}({\dot{\phi }}_t-i_{es}{\dot{\phi }}_s)+c_{es}({\phi }_t-i_{es}{\phi }_s)=-M_t\end{array}$$

(7)

$$\begin{array}{c}{J}_s{\ddot{\phi }}_s-i_{es}{b}_{es}({\dot{\phi }}_t-i_{es}{\dot{\phi }}_s)-i_{es}{c}_{es}({\phi }_t-i_{es}{\phi }_s)+b_{sg}({\dot{\phi }}_s-i_{sg}{\dot{\phi }}_a)+c_{sg}({\phi }_s-i_{sg}{\phi }_a)=-M_s\end{array}$$

(8)

$$\begin{array}{c}{J}_a{\ddot{\phi }}_a-i_{sg}{b}_{sg}({\dot{\phi }}_s-i_{sg}{\dot{\phi }}_a)-i_{sg}{c}_{sg}({\phi }_s-i_{sg}{\phi }_a)=-M_a\end{array}$$

(9)

In matrix form, the system is expressed as $J\ddot{q}+B\dot{q}+Cq=Q(q,\dot{q},t)$. The model is nonlinear because the motor torque depends on rotor speed and the technological resistance may vary with time and angular velocity.

3.3. Induction-Motor Model

A dynamic characteristic of the squirrel-cage induction motor is used instead of a constant driving torque. In the adopted representation, the electromagnetic torque $M_m$ and an auxiliary variable $\psi$ evolve according to

$$\begin{array}{c}{\dot{M}}_m=({\omega }_c-P{\dot{\phi }}_m)\psi -{\displaystyle \frac{M_m}{T_e}}\end{array}$$

(10)

$$\begin{array}{c}\dot{\psi }={\displaystyle \frac{2M_k-\psi }{T_e}}-({\omega }_c-P{\dot{\phi }}_m)M_m\end{array}$$

(11)

Here $T_e={({\omega }_c·S_k)}^{-1}$ is the electromagnetic time constant, $S_k$ is the critical slip, P is the number of pole pairs, $M_k$ is the critical torque, and ${\omega }_c=2·\pi ·f_c$ is the electrical angular frequency. The adopted motor is a 4A112M8U3 asynchronous motor with rated power N=3.0 kW, rated speed n=735 rpm, rated torque ${\mathrm{M}}_{\mathrm{n}}={\displaystyle \frac{M_k}{2}}$=38.97672 $\mathrm{N}·\mathrm{m}$, critical torque $M_k$=77.95344 $\mathrm{N}·\mathrm{m}$, line frequency $f_s=50\mathrm{Hz}$, efficiency $\eta =0.83$, power factor $cos\varphi =0.74$, four pole pairs, nominal slip $S_{\mathrm{n}}=({\omega }_{\mathrm{o}}-{\omega }_{\mathrm{n}})/{\omega }_{\mathrm{o}}=0.02$, and critical slip $S_k=0.07464$.

3.4. Technological Resistance

The resistance acting on the perforated tube is represented as the sum of a mean component and a periodic component:

$$\begin{array}{c}{M}_t\left(t\right)=M_{t,\mathrm{av}}+M_{t,0}cos(\pi {\omega }_{s}t+{\theta }_0)\end{array}$$

(12)

For the baseline calculation, $M_{t,av}$ = 35.67 N·m and $M_{t,0}$ = 3.33 N·m. The average auger resistance is estimated from the transported-power requirement as $M_a=76/{\omega }_a$, giving approximately 8.06 N·m at the nominal operating point. The resistance of one planet gear is obtained by distributing the transmitted torque among the three planets and accounting for the relevant ratio.

3.5. Belt and Gear Damping

The equivalent damping coefficients are estimated from the corresponding stiffness and oscillation period. For the belt drive, the adopted relation is

$$\begin{array}{c}{b}_b=0.012665\xi c_b{T}_b\end{array}$$

(13)

where $\xi$ is a dimensionless transmission damping coefficient taken within 0.2–0.6 and $T_b$ is the characteristic period. For the baseline belt section, $\xi =0.5$, $c_b$ = 17.2 N·m/rad, and $T_b$ = 0.803 s, giving $b_b$ = 0.175 N·m·s/rad. Equivalent expressions are used for the two gear meshes.

4. Experimental Methodology

4.1. Identification of Moments of Inertia

The moments of inertia of the motor rotor and the rotating working members were determined experimentally by the acceleration method. Each body was mounted on its operating bearings. A thread was wound around a pulley of known radius, passed over a guide pulley, and loaded by a suspended mass. The load was raised through a known height and released. The descent time was measured from video recordings. Two or more load levels were used to separate the effect of bearing resistance from the inertia term (Figure 3).

For a falling mass m connected to a pulley of radius r, the linear acceleration is calculated from

$$\begin{array}{c}a={\displaystyle \frac{2h}{t^2}}\end{array}$$

(14)

The angular acceleration is $\alpha$ = a/r. Combining the translational equation of the suspended mass with the rotational equation of the tested assembly gives

$$\begin{array}{c}J={\displaystyle \frac{r^2\left[m(g-a)-F_r\right]}{a}}\end{array}$$

(15)

where $F_r$ denotes the equivalent resistance force referred to the thread. When two falling masses are used, the resistance term can be eliminated by solving the two equations simultaneously. The experimentally identified inertias were then introduced into the numerical model without further fitting.

4.2. Numerical Solution

The coupled second-order differential equations and the two first-order motor equations were transformed into a first-order state-space system. The state vector contained four angular displacements, four angular velocities, motor torque, and the auxiliary electromagnetic variable. The system was integrated using the classical fourth-order Runge-Kutta method. The initial angular velocities were zero, and the initial electromagnetic state corresponded to connection of the unloaded motor to the supply.

The time step was selected so that further reduction did not materially change peak torque, settling time, or steady-state angular velocities. Simulations were continued until all working members reached a quasi-steady regime. For each case, the following response indicators were calculated: peak electromagnetic torque; maximum angular acceleration; transient duration; mean angular velocity; rotational irregularity; and mean motor power.

$$\begin{array}{c}{\delta }_{\omega }={\displaystyle \frac{{\omega }_{max}-{\omega }_{min}}{{\omega }_{\mathrm{mean}}}}\end{array}$$

(16)

$$\begin{array}{c}{P}_m\left(t\right)=M_m\left(t\right){\dot{\phi }}_m\left(t\right)\end{array}$$

(17)

4.3. Parameter-Variation Plan

A one-factor-at-a-time sensitivity study was performed around the baseline configuration. The effective stiffness and damping of the belt drive, the resistance moments of the tube and auger, and the moments of inertia of the four rotating bodies were varied over the ranges used in the original numerical experiments. The response was evaluated through the auger irregularity, motor power consumption, tube and auger speeds, and the duration of the transient process. The purpose of this study was not global optimization but physical ranking of the parameters and identification of practical adjustment directions.

Table 1. Principal parameters used in the baseline simulation.

Parameter	Symbol	Baseline value	Unit
Motor rated power	N	3.0	kW
Motor rated speed	n	735	rpm
Motor rated torque	$M_n$	38.98	N·m
Critical motor torque	$M_k$	77.95	N·m
Belt pulley diameters	$D_m$/$D_t$	130/260	mm
Sun/planet/ring teeth	$Z_1$/$Z_2$/$Z_3$	18/18/54	–
Mean tube resistance	$M_{t,av}$	35.67	N·m
Tube resistance amplitude	$M_{t,0}$	3.33	N·m
Mean auger resistance	$M_a$	8.06	N·m
Baseline belt stiffness	$c_b$	17.2	N·m/rad
Baseline belt damping	$b_b$	0.175	N·m·s/rad

Table 1. Principal parameters used in the baseline simulation.

Parameter	Symbol	Baseline value	Unit
Motor rated power	N	3.0	kW
Motor rated speed	n	735	rpm
Motor rated torque	$M_n$	38.98	N·m
Critical motor torque	$M_k$	77.95	N·m
Belt pulley diameters	$D_m$/$D_t$	130/260	mm
Sun/planet/ring teeth	$Z_1$/$Z_2$/$Z_3$	18/18/54	–
Mean tube resistance	$M_{t,av}$	35.67	N·m
Tube resistance amplitude	$M_{t,0}$	3.33	N·m
Mean auger resistance	$M_a$	8.06	N·m
Baseline belt stiffness	$c_b$	17.2	N·m/rad
Baseline belt damping	$b_b$	0.175	N·m·s/rad

5. Results

5.1. Motor Start-Up and Transient Response

The simulated motor characteristic exhibits the three expected stages: a rapid torque rise immediately after connection to the supply, acceleration with a gradual torque decrease, and stabilization near the operating point. The electromagnetic torque varies between approximately -79.96 and 101.63 N·m. The positive maximum is 2.61 times the rated torque of 38.98 N·m, confirming that a constant nominal-torque representation would substantially underestimate the transient loading of the belt and planetary transmission.

Figure 4. Electromagnetic torque versus motor angular velocity during start-up.

Figure 4. Electromagnetic torque versus motor angular velocity during start-up.

The principal motor transient lasts approximately 3.5 s. The maximum angular acceleration reaches 2988.6 rad/s² at t = 2.25 s. This short-duration acceleration peak is transmitted through the compliant belt and excites torsional oscillations of the tube, planet gear, and auger. Because the working members have different inertias and are connected through different stiffness and damping parameters, their velocity oscillations are phase-shifted and decay at different rates.

Figure 5. Time histories of the angular velocities of the motor, tube, planet gear, and auger.

Figure 5. Time histories of the angular velocities of the motor, tube, planet gear, and auger.

The perforated tube reaches a maximum angular acceleration of approximately 107.1 rad/s², and its broader mechanical transient persists for about 7 s. The longer settling time of the tube compared with the motor reflects the compliant belt connection and the additional energy exchange with the planetary stage and auger. The calculated mean motor power during operation is approximately 2.5 kW.

Figure 6. Motor power and angular acceleration during the transient process.

Figure 6. Motor power and angular acceleration during the transient process.

5.2. Steady-State Speeds and Rotational Irregularity

At steady operation, the model predicts coordinated rotation consistent with the imposed belt and planetary ratios. The auger speed lies near 1466 rpm for the baseline resistance, while the tube speed varies within approximately 355–371 rpm over the investigated parameter range. The rotational irregularity of the auger is close to 0.42–0.44 depending on the selected belt and load parameters.

The calculated response confirms that elastic-dissipative coupling is not merely a numerical correction to the kinematic ratios. It determines the amplitude and decay of speed oscillations and therefore changes both the instantaneous torque demand and the average motor power. The belt drive has the strongest controllable influence because it is the first compliant element excited by the motor start-up torque.

5.3. Sensitivity to Drive and Load Parameters

The resistance moment of the perforated tube and the moment of inertia of the auger have the strongest effect on auger rotational irregularity. The effect of the planet-gear inertia and planet resistance is comparatively small. This ranking is physically reasonable: the tube resistance acts directly at the input member of the planetary stage, whereas the auger inertia is located at the output and stores a substantial part of the oscillatory kinetic energy.

Over the investigated variations, the following changes in the calculated irregularity were obtained: variations associated with the tube resistance produced values around 0.4220–0.4256; changes in motor inertia gave approximately 0.4224–0.4263; changes in tube inertia gave 0.4222–0.4279; changes in planet-gear inertia gave 0.4218–0.4230; and changes in auger inertia gave 0.4216–0.4322. The largest relative change, about 2.46%, corresponded to the auger inertia.

Figure 7. Sensitivity of auger rotational irregularity and motor power to the model parameters.

Figure 7. Sensitivity of auger rotational irregularity and motor power to the model parameters.

As the auger resistance moment increased from the lower bound of the investigated range toward the baseline value, the auger speed decreased from approximately 1479.9 to 1466.3 rpm. Simultaneous variation of the belt properties and the tube and auger resistance moments produced motor power values of approximately 2.45–2.60 kW. The tube speed remained within 355–371 rpm when the belt properties, tube resistance, and tube inertia were varied.

Figure 8. Influence of the resistance moments and inertia parameters on tube and auger speeds.

Figure 8. Influence of the resistance moments and inertia parameters on tube and auger speeds.

5.4. Rational Parameter Adjustment

The calculations indicate that reducing the effective elastic-dissipative parameter assigned to the belt-drive coupling from about 17.7 to 10.3 N·m/rad can reduce auger rotational irregularity from approximately 0.435 to 0.420 while decreasing motor power consumption from about 2.55 to 2.50 kW at an auger speed close to 1466 rpm. This result should be interpreted as a recommendation to tune the effective torsional compliance and damping of the belt transmission through belt selection, pretension, pulley geometry, and operating tension, rather than as a universal stiffness value for all machines.

6. Discussion

6.1. Physical Interpretation of the Integrated Response

The principal advantage of the proposed formulation is that it preserves the causal path from electromagnetic torque generation to the motion of the technological working members. The motor torque peak first excites relative deformation of the belt. The resulting tube acceleration excites the ring–planet mesh, after which the planet–sun mesh transmits the disturbance to the auger. Therefore, the largest transient torque does not occur simultaneously in all components, and a purely kinematic calculation cannot reproduce the phase relationships or decay rates.

The results agree qualitatively with modern studies of integrated motor-gearbox systems [8,9,10], which show that motor dynamics and mechanical compliance must be solved together. They also support the conclusions of belt-drive research [11,12,13,14,15,16,17,18,19,20], where stiffness, pretension, and damping control the magnitude and persistence of transient oscillations. In the planetary stage, the present reduced model does not describe local mesh-force distribution with the detail of flexible multi-mesh models [21,22,23,24,25,26,27,28,29,30,31], but it captures the system-level energy transmission required for machine-unit design.

6.2. Significance of the Parameter Ranking

The sensitivity ranking identifies two distinct engineering targets. The tube resistance is a technological parameter governed by seed density, perforation, friction, and material distribution. Reducing its fluctuations requires process and geometry improvements. The auger inertia is a design parameter governed by shaft dimensions, flight geometry, and attached gear mass. Excessive inertia increases the stored oscillatory energy and can amplify the speed variation after a disturbance. In contrast, moderate changes in planet-gear inertia have little effect on the overall response because the planet gears are relatively small and the load is divided among several identical members.

The belt coupling is the most accessible parameter for commissioning and retrofit. However, the result does not imply that the minimum possible stiffness is always desirable. Insufficient stiffness can increase static angular lag, belt slip, and heat generation. The rational design task is therefore to select a belt and pretension that provide adequate mean torque transmission while avoiding excessive excitation of the downstream torsional modes.

6.3. Engineering Implications

For practical design, the start-up torque ratio of 2.61 should be used when checking belt traction, shaft torque, key and spline strength, and gear-tooth loading. Calculations based only on the rated motor torque may underestimate the short-duration load by more than a factor of two. The approximately 3.5 s motor transient and 7 s tube transient also indicate that repeated start-stop operation should be limited or controlled by a soft starter or variable-frequency drive when process conditions permit.

The model can be used to evaluate alternative pulley diameters, gear ratios, auger geometry, and tube perforation parameters. It is also suitable as a basis for optimization, provided that experimentally measured stiffness, damping, and technological torque histories are supplied. The integration of such data would enable prediction not only of rotational irregularity but also of fatigue-relevant torque cycles.

6.4. Limitations

The present formulation is a reduced torsional model. It neglects lateral and axial vibration, bearing and housing flexibility, belt transverse modes, gear backlash, time-varying mesh stiffness, manufacturing errors, and unequal load sharing among the planet gears. The motor characteristic is represented by a reduced dynamic law rather than a full $d_q$-axis electrical model. The technological resistance is represented by mean and harmonic components and therefore does not reproduce random impacts or strongly non-stationary seed flow.

The experimental stage identifies inertia but does not independently validate every simulated torque and angular-velocity history. Direct validation should include synchronized measurements of motor current, shaft speed, tube speed, auger speed, and torsional vibration during start-up. Accordingly, the numerical results should be regarded as a physically based engineering model and a foundation for expanded validation rather than as a complete digital twin.

6.5. Directions for Further Work

Further research should incorporate time-varying gear-mesh stiffness, backlash, bearing compliance, and distributed belt dynamics. A multi-objective optimization can then minimize motor power, peak torque, and rotational irregularity while satisfying productivity constraints. Experimental work should quantify the technological torque as a function of seed density and throughput and should identify equivalent belt and mesh damping from measured free-decay or operational-response data.

7. Conclusions

1. An integrated electromechanical model of the seed-removing machine unit was developed by coupling a squirrel-cage induction motor, elastic-dissipative belt transmission, planetary gear stage, perforated tube, and auger in a unified Lagrangian formulation.

2. The dynamic induction-motor characteristic is essential for start-up analysis. The calculated maximum torque was 101.63 N·m, which is 2.61 times the rated torque of 38.98 N·m.

3. The principal motor transient lasted approximately 3.5 s, and the maximum motor angular acceleration reached 2988.6 rad/s² at t = 2.25 s. The tube response persisted for approximately 7 s because of energy exchange through the compliant belt and planetary transmission.

4. The acceleration method provided experimentally based moments of inertia for the rotating components and reduced dependence on purely geometric inertia estimates.

5. The resistance moment of the perforated tube and the inertia of the auger were the most influential parameters for auger rotational irregularity. The inertia and resistance of the planet gears had the smallest system-level effect.

6. The auger speed decreased from approximately 1479.9 to 1466.3 rpm as its resistance approached the baseline value, while the perforated-tube speed remained within approximately 355–371 rpm over the investigated variations.

7. Motor power varied within approximately 2.45–2.60 kW in the sensitivity calculations. A rational adjustment of the effective belt-drive coupling reduced the calculated power from about 2.55 to 2.50 kW.

8. Reducing the effective elastic-dissipative belt parameter from about 17.7 to 10.3 N·m/rad reduced the auger rotational irregularity from approximately 0.435 to 0.420 at an auger speed close to 1466 rpm.

9. The model provides a practical basis for checking transient loads, selecting belt-drive parameters, and ranking design changes in planetary-driven cotton-processing machinery.

10. Further experimental validation should include synchronized speed, current, torque, and vibration measurements and an extended model with belt transverse dynamics, time-varying mesh stiffness, backlash, and support flexibility.