I investigate a new driving mechanism for the cover plate of a flat grinding machine used in carding. The cover plate must move with an approximately constant speed during its working stroke, because this motion directly affects the uniformity of the grinding action and the quality of the carding flats. In the conventional design, the cover plate is driven by a long transmission chain composed of three planar four-bar mechanisms and one cylindrical gear pair with a tooth ratio of 1:2. Although that arrangement can produce a reciprocating motion, it contains many links and joints, occupies a large space, and creates a long power path. I therefore propose a compact combined mechanism that uses a first-order modified eccentric pinion gear, a second-order conjugate non-circular pinion gear, and a concentric crank-slider mechanism connected in series. The eccentric pinion gear pair provides a variable speed ratio, and the crank-slider converts the variable rotation into a reciprocating translation of the cover plate. My goal is to obtain an approximately constant-speed working stroke while reducing the number of components and shortening the transmission route.
The main idea is that the driving pinion gear rotates at a constant input speed, but its pitch curve is not circular. Because the pitch curve radius changes periodically, the driven pinion gear rotates at a non-uniform speed. This non-uniform rotation is then transmitted to the crank of the slider mechanism. By choosing the eccentricity and the modification coefficient of the pinion gear pair properly, I can shape the crank angular velocity so that the slider velocity remains nearly constant over the working stroke. The proposed mechanism can therefore be regarded as a speed-shaping system in which the eccentric pinion gear pair acts as the function generator and the crank-slider acts as the motion converter.
The working principle of the mechanism is as follows. Power enters through the main shaft. A first-order modified eccentric pinion gear is fixed on this shaft and rotates at a constant angular velocity. Through tooth engagement, the motion is transmitted to a second-order conjugate non-circular pinion gear. Because the pitch curve radii of the two pinion gears change with rotation, the driven pinion gear rotates at a variable angular velocity. A crank is fixed on the same shaft as the driven pinion gear, so the crank has exactly the same variable rotation. A connecting rod then transmits the motion to a slider, which represents the cover plate. The slider moves horizontally in a reciprocating manner. The eccentric pinion gear pair is the key component that shapes the velocity, while the concentric crank-slider mechanism converts the variable rotation into an approximately constant-speed translation of the cover plate.
The arrangement of the eccentric pinion gear pair and the crank-slider is shown below.
I establish the mathematical model of the proposed mechanism in several steps. First, I describe the pitch curves of the first-order modified eccentric pinion gear and the second-order conjugate non-circular pinion gear. Second, I derive the center distance from the continuous transmission condition. Third, I obtain the driven pinion gear rotation by numerical integration. Fourth, I formulate the convexity constraints that prevent concave pitch curves. Fifth, I derive the displacement, velocity, and acceleration of the slider from the crank-slider geometry. These equations form the basis of the design and analysis software that I developed.
| Parameter | Meaning | Parameter | Meaning |
|---|---|---|---|
| e | eccentric distance of the pinion gear | R | radius of the eccentric circle |
| m11 | first-segment modification coefficient | m12 | second-segment modification coefficient |
| n1 | order of the driving pinion gear | n2 | order of the driven pinion gear |
| r1 | pitch curve radius of the driving pinion gear | r2 | pitch curve radius of the driven pinion gear |
| phi1 | rotation angle of the driving pinion gear | phi2 | rotation angle of the driven pinion gear |
| omega1 | angular velocity of the driving pinion gear | omega2 | angular velocity of the driven pinion gear |
| alpha1 | auxiliary angle for the first segment | alpha2 | auxiliary angle for the second segment |
| a | center distance of the pinion gear pair | k | eccentricity ratio, k = e / R |
| l1 | crank length | l2 | connecting rod length |
| i12 | speed ratio of the pinion gear pair | s | slider displacement |
| s_dot | slider velocity | s_ddot | slider acceleration |
For the first-order modified eccentric pinion gear, I use two pitch curve segments. The first segment covers the interval from zero to pi divided by n1 m11. The second segment covers the interval from pi divided by n1 m11 to two pi divided by n1. I define the auxiliary angle for the first segment as alpha1 = n1 m11 phi1. The first pitch curve radius is then
$$ r_{11} = \sqrt{R^2 – e^2 \sin^2 \alpha_1} – e \cos \alpha_1 = R \left( \sqrt{1 – k^2 \sin^2 \alpha_1} – k \cos \alpha_1 \right) $$
For the second segment, I define alpha2 = n1 m12 (2 pi / n1 – phi1), and the second pitch curve radius is
$$ r_{12} = \sqrt{R^2 – e^2 \sin^2 \alpha_2} – e \cos \alpha_2 = R \left( \sqrt{1 – k^2 \sin^2 \alpha_2} – k \cos \alpha_2 \right) $$
The second modification coefficient m12 is not independent. It is related to the first modification coefficient by
$$ m_{12} = \frac{m_{11}}{2 m_{11} – 1} $$
This relation ensures that the two segments of the driving pinion gear pitch curve join smoothly and that the driven pinion gear returns to its initial position after one complete transmission cycle. The eccentricity ratio is defined as
$$ k = \frac{e}{R} $$
For continuous transmission, the driving pinion gear rotates n1 turns while the driven pinion gear rotates n2 turns. This condition can be written as
$$ \frac{2\pi}{n_2} = \int_{0}^{2\pi/n_1} \frac{1}{i_{12}} \, d\phi_1 $$
where the speed ratio of the pinion gear pair is
$$ i_{12} = \frac{\dot{\phi}_2}{\dot{\phi}_1} = \frac{r_1}{a – r_1} $$
Substituting the two pitch curve segments gives the center distance equation
$$ \frac{2\pi}{n_2} = \int_{0}^{\pi/(n_1 m_{11})} \frac{r_{11}}{a – r_{11}} \, d\phi_1 + \int_{\pi/(n_1 m_{11})}^{2\pi/n_1} \frac{r_{12}}{a – r_{12}} \, d\phi_1 $$
I solve this equation numerically. First, I use the advance-retreat method to bracket the center distance. Then I apply the golden-section method to refine the value. Once the center distance is known, the pitch curve of the driven pinion gear follows directly from the conjugate condition
$$ r_2 = a – r_1 $$
The rotation angle of the driven pinion gear is obtained by integrating the reciprocal speed ratio
$$ \phi_2 = \int_{0}^{\phi_1} \frac{1}{i_{12}} \, d\phi_1 = \int_{0}^{\phi_1} \frac{r_1}{a – r_1} \, d\phi_1 $$
I evaluate this integral with the trapezoidal rule because the integrand is available only in discrete form. The angular velocity of the driven pinion gear is then
$$ \dot{\phi}_2 = i_{12} \dot{\phi}_1 = \frac{r_1}{a – r_1} \dot{\phi}_1 $$
For a constant input speed, the angular acceleration of the driven pinion gear is
$$ \ddot{\phi}_2 = \dot{\phi}_1^2 \frac{d}{d\phi_1} \left( \frac{r_1}{a – r_1} \right) $$
These relations show that the driven pinion gear speed is controlled entirely by the pitch curve of the eccentric pinion gear and the center distance. Therefore, the pinion gear geometry is the primary means of shaping the cover plate motion.
To ensure smooth transmission, the pitch curves of both pinion gears must remain convex. A concave pitch curve would cause unfavorable tooth contact, possible undercutting, and unstable motion. The curvature of a polar curve is
$$ \rho = \frac{ \left( r^2 + \left( \frac{dr}{d\phi} \right)^2 \right)^{3/2} }{ r^2 + 2 \left( \frac{dr}{d\phi} \right)^2 – r \frac{d^2 r}{d\phi^2} } $$
For the driving eccentric pinion gear, the no-concavity condition is
$$ k \le \frac{1}{n_1^2 m_{11}^2} $$
$$ k \le \frac{1}{n_1^2 m_{12}^2} $$
For the driven conjugate non-circular pinion gear, the no-concavity conditions are
$$ k \le \frac{1}{n_1^2 m_{11}^2} $$
$$ k \le \frac{1}{n_1^2 m_{12}^2} $$
$$ (1 + k)^2 – \frac{a}{R} \ge 0 $$
$$ k \ge 0 $$
I use these inequalities as constraints in the optimization procedure. They guarantee that the pinion gear pair can operate without concave pitch curves and that the generated tooth profiles remain feasible.
After the driven pinion gear rotates, the crank of the concentric crank-slider mechanism rotates with the same angular motion. I place the crank rotation center at the origin of a local coordinate system. The crank has length l1, and the connecting rod has length l2. The crank angle is phi2. The coordinates of the crank pin are
$$ x_A = l_1 \cos \phi_2 $$
$$ y_A = l_1 \sin \phi_2 $$
The slider pin coordinates are
$$ x_B = x_A + l_2 \cos \alpha $$
$$ y_B = y_A + l_2 \sin \alpha $$
Because the crank-slider is concentric, the slider moves along the horizontal axis, so yB = 0. Therefore,
$$ \sin \alpha = \frac{y_B – y_A}{l_2} = – \frac{l_1 \sin \phi_2}{l_2} $$
The slider displacement measured from its inner dead center is
$$ s = x_B – (l_2 – l_1) $$
Substituting the geometric relations gives
$$ s = \sqrt{l_2^2 – l_1^2 \sin^2 \phi_2} + l_1 \cos \phi_2 – (l_2 – l_1) $$
Differentiating with respect to time gives the slider velocity
$$ \dot{s} = – l_1 \left( \sin \phi_2 + \frac{ l_1 \sin(2\phi_2) / l_2 }{ 2 \sqrt{1 – \frac{l_1^2 \sin^2 \phi_2}{l_2^2}} } \right) \dot{\phi}_2 $$
Differentiating again gives the slider acceleration
$$ \ddot{s} = – l_1 \left( \cos \phi_2 + \frac{ l_2^2 \cos(2\phi_2) – \left( l_2^2 – l_1^2 \sin^2 \phi_2 \right) \sin^2 \phi_2 }{ \left( l_2^2 – l_1^2 \sin^2 \phi_2 \right)^{3/2} } \right) \ddot{\phi}_2 $$
These equations complete the kinematic model. They show that the slider velocity depends on the crank angle, the crank length, the connecting rod length, and the angular velocity of the driven pinion gear. Since the driven pinion gear angular velocity is controlled by the eccentric pinion gear pair, the whole mechanism can be tuned by changing the pinion gear parameters.
I developed a design and analysis software tool in Matlab to make the evaluation process efficient. The software interface contains a parameter input area, a calculation button, a simulation button, and a result display area. The user can enter the eccentric distance, the eccentric circle radius, the modification coefficient, the orders of the pinion gears, the crank length, the connecting rod length, and other geometric parameters. The software then calculates the center distance, the pitch curves, the conjugate pitch curve, the driven pinion gear rotation, the speed ratio, the slider displacement, the slider velocity, and the slider acceleration. It also provides an animation of the mechanism so that interference between components can be observed visually. If interference occurs, the corresponding parameter set is rejected. This software is a convenient tool for design analysis and optimization of the eccentric pinion gear driven cover plate mechanism.
I used the software to study how the eccentricity ratio and the modification coefficient affect the cover plate speed. The speed ratio between the output cover plate and the input eccentric pinion gear is an important indicator. If this ratio is nearly constant over the working stroke, the cover plate moves with an approximately constant speed. I first fixed the modification coefficient at m11 = 1 and the eccentric circle radius at R = 40 mm. Then I varied the eccentric distance e. The results are summarized in the following table.
| Eccentric distance e (mm) | Eccentricity ratio k | Cover plate speed range | Motion characteristic |
|---|---|---|---|
| 0 | 0 | [-16.108, 16.108] | Sinusoidal variation, because the pinion gear pair becomes a circular gear pair with ratio 2 |
| 5 | 0.125 | [-13.498, 13.452] | Speed becomes closer to constant over the working stroke |
| 10 | 0.250 | [-15.920, 15.823] | Constant-speed segment begins to show slight concavity |
| 15 | 0.375 | [-20.922, 20.759] | Speed ends increase and a saddle shape appears |
| 20 | 0.500 | [-28.483, 28.757] | Strong saddle shape, poor constant-speed behavior |
The table shows that increasing the eccentricity from zero initially improves the constant-speed behavior. When e = 5 mm, the speed range is narrower and the middle portion of the stroke is flatter. However, if the eccentricity becomes too large, the constant-speed segment develops an inward curvature, and the two ends of the working stroke increase sharply. This produces a saddle-shaped velocity curve. Therefore, for an approximately constant-speed cover plate motion, the eccentricity of the pinion gear must not be too large. The eccentric pinion gear can shape the motion, but excessive eccentricity destroys the desired uniformity.
I then fixed the eccentricity ratio at k = 5/40 = 0.125 and varied the first modification coefficient m11. The modification coefficient changes the angular position at which the first pitch curve segment reaches its extreme radius. The first segment polar angle becomes pi / m11 instead of pi. The following table summarizes the effect.
| Modification coefficient m11 | First segment polar angle | Effect on cover plate speed |
|---|---|---|
| 0.6 | 5 pi / 3 | Speed extremum appears earlier; motion deviates from constant speed |
| 0.8 | 5 pi / 4 | Speed extremum still appears earlier than the baseline |
| 1.0 | pi | Baseline eccentric pinion gear behavior |
| 1.2 | 5 pi / 6 | Speed extremum appears later; pitch radius variation becomes stronger |
| 1.4 | 5 pi / 7 | Speed extremum appears still later; strong pitch variation |
When m11 is less than one, the first segment of the pitch curve changes more slowly, and the speed extremum occurs earlier. When m11 is greater than one, the first segment changes more rapidly, and the speed extremum occurs later. By adjusting the modification coefficient, I can fine-tune the cover plate speed and lengthen the approximately constant-speed portion of the working stroke. The modification coefficient therefore provides a second design lever in addition to the eccentricity. The eccentric pinion gear and its conjugate non-circular pinion gear can be designed together to produce a favorable speed profile.
I formulated an optimization problem to find the best combination of eccentricity, eccentric circle radius, and modification coefficient. The crank length and connecting rod length were fixed at l1 = 84 mm and l2 = 305 mm based on the existing machine. The order ratio was chosen as n1 : n2 = 1 : 2. This ratio is necessary because if n1 : n2 = 1 : 1, the crank rotates one turn for each input turn, and the minimum speed appears only once per input revolution. To obtain a reciprocating stroke in which the cover plate moves back and forth within one transmission period and maintains an approximately constant speed during the working stroke, the order ratio must be n1 : n2 = 1 : 2. With this ratio, the driven pinion gear completes two revolutions for each revolution of the driving pinion gear, and the crank-slider can produce the required reciprocating motion.
The optimization objective is to minimize the mean square deviation of the speed ratio between the output cover plate and the input eccentric pinion gear over one transmission period. I define the speed ratio at each sample point as
$$ W_i = \frac{\dot{s}_i}{\dot{\phi}_{1,i}} $$
The objective function is then
$$ F(X) = \frac{1}{n} \sum_{i=1}^{n} W_i^2 – \left( \frac{1}{n} \sum_{i=1}^{n} W_i \right)^2 $$
where n is the number of equally spaced points in one transmission period. Minimizing this variance makes the speed ratio as uniform as possible, which in turn makes the cover plate speed approximately constant during the working stroke. I selected the design variables as
$$ X = [e, R, m_{11}] $$
The design variables are subject to the convexity constraints of the pinion gear pair, the geometric bounds of the eccentric circle, and the requirement that the modification coefficient remain positive. In addition, I require that the second modification coefficient m12 be physically meaningful. The genetic algorithm parameters that I used are listed below.
| Genetic algorithm parameter | Value |
|---|---|
| Initial population size | 20 |
| Number of generations | 100 |
| Binary bits per variable | 20 |
| Selection ratio | 0.9 |
| Crossover probability | 0.8 |
| Mutation probability | 0.1 |
After running the genetic algorithm, I selected the result with the smallest deformation of the eccentric pinion gear while still satisfying the constant-speed requirement. I rounded the design parameters to practical values. The optimized parameter set is shown in the following table.
| Parameter | Optimized value |
|---|---|
| Eccentric distance e | 5 mm |
| Eccentric circle radius R | 40 mm |
| Eccentricity ratio k | 0.125 |
| First modification coefficient m11 | 1.05 |
| Second modification coefficient m12 | 1.05 / (2 * 1.05 – 1) = 0.9545 |
| Order of driving pinion gear n1 | 1 |
| Order of driven pinion gear n2 | 2 |
| Crank length l1 | 84 mm |
| Connecting rod length l2 | 305 mm |
Using the optimized parameters, I generated the pitch curves and the tooth profiles. The pitch curve of the first-order modified eccentric pinion gear is a smooth non-circular curve. The conjugate pinion gear pitch curve is obtained from the center distance. I used the generating method to create the tooth profiles, and I checked that the curvature constraints were satisfied everywhere. The resulting pinion gear pair can transmit motion continuously without concave pitch regions. The eccentric pinion gear and the second-order conjugate non-circular pinion gear form a compact speed-varying unit that replaces the long gear-and-linkage chain of the conventional design.
I then built a virtual prototype of the complete cover plate driving mechanism in SolidWorks. The model includes the driving eccentric pinion gear, the driven conjugate non-circular pinion gear, the crank, the connecting rod, the slider, and the frame. I assembled the components and applied appropriate mates. The constraints between the components are listed in the following table.
| Constraint or joint | Connected components |
|---|---|
| Fixed joint | Frame and ground |
| Revolute joint with motor drive | First-order eccentric pinion gear and frame |
| Revolute joint | Second-order conjugate non-circular pinion gear and frame |
| Fixed joint | Crank and second-order conjugate non-circular pinion gear |
| Revolute joint | Crank and connecting rod |
| Revolute joint | Connecting rod and slider |
| Prismatic joint | Slider and frame |
I performed a virtual prototype simulation with the Motion add-in of SolidWorks. The simulation settings are summarized below. I used 720 equally spaced points in one transmission period. The motor speed was set to 60 rpm, so the slider completes one reciprocating stroke in 2 s. Because the pitch curve radii of the pinion gear pair vary with rotation, the motion law is not known in advance. I therefore added solid contact between the gear teeth to transmit the torque and to simulate the actual meshing behavior.
| Simulation parameter | Value |
|---|---|
| Number of equally spaced points per cycle | 720 |
| Motor speed | 60 rpm |
| Transmission period | 2 s |
| Gear pair transmission method | Solid contact between teeth |
| Slider motion type | Reciprocating horizontal translation |
The simulation showed that the mechanism runs smoothly. The cover plate moves back and forth without obvious interference. I compared the simulated cover plate speed with the theoretical speed obtained from the mathematical model. The two curves follow the same overall trend. The peaks occur at approximately the same crank angles, and the middle portion of the working stroke is close to constant speed. The agreement between theory and simulation confirms that the mathematical model is correct and that the proposed eccentric pinion gear driven mechanism is feasible.
The two speed curves are not perfectly identical. I identified three main reasons. First, when I calculated the tooth profiles of the first-order modified eccentric pinion gear and the second-order conjugate non-circular pinion gear, the number of calculated points was finite, so the generated tooth profile deviated slightly from the ideal involute-like profile. Second, when I created the three-dimensional gear models, I used spline fitting, and the fitting accuracy was limited. Third, because the pinion gear pair is a variable-speed transmission, I used solid contact to transfer torque in the virtual simulation. Tooth compliance, contact deformation, and numerical contact algorithms introduce small deviations. These effects explain the small differences between the theoretical and simulated speed curves. The overall motion law is nevertheless validated.
The most important result is that the first-order modified eccentric pinion gear and its second-order conjugate non-circular pinion gear can generate a non-uniform rotation that, after the crank-slider, produces an approximately constant-speed reciprocating motion of the cover plate. The eccentric pinion gear pair acts as the speed-shaping element. The crank-slider then converts the rotation into translation. By optimizing the eccentricity and the modification coefficient, I can flatten the velocity curve over the working stroke without adding extra linkages. This is a significant simplification compared with the conventional three-four-bar plus cylindrical gear train.
From a design perspective, the eccentric pinion gear mechanism has several advantages. It has a short transmission path. The number of components is small. The eccentric pinion gear and the conjugate non-circular pinion gear can be manufactured with standard non-circular gear generation methods. The crank-slider is a simple and reliable mechanism. The overall structure is compact and can be integrated into the flat grinding machine more easily than the existing long-chain mechanism. The speed-shaping capability comes from the pinion gear geometry, so no additional cams or linkages are required. The eccentric pinion gear therefore provides a new degree of freedom for optimizing the cover plate motion.
I also examined the sensitivity of the design to the main parameters. The eccentricity ratio k controls the amplitude of the speed variation. If k is too small, the speed curve approaches the sinusoidal behavior of a circular gear pair. If k is too large, the speed curve develops a saddle shape and the constant-speed segment is lost. The modification coefficient m11 controls the phase and the local slope of the pitch curve. It can shift the speed extremum earlier or later and can slightly lengthen the approximately constant-speed region. The best design lies in a narrow range where the eccentricity is moderate and the modification coefficient is close to unity. This is why the optimized values are e = 5 mm, R = 40 mm, and m11 = 1.05. These values produce a small eccentricity ratio and a mild modification, which together give a smooth and nearly constant speed without sacrificing convexity or manufacturability.
The pinion gear pair must satisfy the continuous transmission condition. The center distance is not arbitrary. It is determined by the integral equation that relates the pitch curve radii to the order ratio. I solve this equation numerically because the pitch curve integrals do not have simple closed-form solutions for all parameter combinations. The advance-retreat method provides a robust bracket, and the golden-section method gives a precise center distance. This numerical procedure is embedded in the Matlab software. Once the center distance is known, the conjugate pitch curve and the driven pinion gear rotation are computed by numerical integration. The trapezoidal rule is sufficiently accurate for the design stage, especially when a large number of sample points is used.
The convexity check is essential for non-circular pinion gears. If the pitch curve becomes concave, the tooth profile generation may produce undercutting, and the contact ratio may become unacceptable. I derived the no-concavity conditions for both the driving eccentric pinion gear and the driven conjugate pinion gear. These conditions are simple inequalities involving the eccentricity ratio k, the order n1, and the modification coefficients m11 and m12. They are included as constraints in the genetic algorithm. The optimization therefore searches only within the feasible region where smooth transmission is possible. This prevents the algorithm from selecting a parameter set that produces a theoretically good speed curve but an impractical pinion gear.
The genetic algorithm is well suited to this problem because the objective function is nonlinear and the design space is constrained. The population size, number of generations, crossover probability, and mutation probability were chosen to balance exploration and convergence. The algorithm evaluates many candidate parameter sets and gradually improves the speed uniformity. The final result is a set of design parameters that satisfies the convexity constraints and minimizes the variance of the speed ratio. I then rounded the parameters to practical values and verified the design with the Matlab software. The rounded design still meets the constant-speed requirement and remains within the feasible region.
The virtual prototype simulation provides an independent check on the mathematical model. The SolidWorks model includes the actual three-dimensional geometry of the pinion gears, the crank, the connecting rod, and the slider. The motion simulation accounts for the contact between the gear teeth and the joints between the links. Although the simulation is not a full physical test, it captures the essential kinematics and some of the contact effects. The fact that the simulated speed curve follows the theoretical curve confirms that the equations are correct and that the mechanism can operate as intended. The small discrepancies are acceptable for a design-stage validation and can be reduced by increasing the number of tooth profile points, improving the spline fitting, and using a more refined contact model.
I conclude that the proposed eccentric pinion gear driven cover plate mechanism is a feasible and promising alternative to the conventional design. The combination of a first-order modified eccentric pinion gear, a second-order conjugate non-circular pinion gear, and a concentric crank-slider mechanism produces an approximately constant-speed working stroke. The eccentric pinion gear pair shapes the speed, and the crank-slider converts the variable rotation into reciprocating translation. The mechanism has a shorter transmission path, fewer components, and a simpler structure than the existing three-four-bar plus cylindrical gear train. By adjusting the eccentricity and the modification coefficient, I can optimize the cover plate speed and lengthen the approximately constant-speed portion of the working stroke. The Matlab design software provides a convenient tool for analysis and optimization. The genetic algorithm successfully finds a practical parameter set, and the virtual prototype simulation validates the feasibility of the design. The mathematical model and the design software are therefore reliable for further development of this type of flat grinding machine cover plate drive.
For future work, I would consider manufacturing a physical prototype and measuring the actual cover plate speed. I would also study the tooth contact stress and the dynamic behavior of the pinion gear pair under load. The effects of manufacturing errors and backlash on the speed uniformity should be investigated. In addition, I would explore other non-circular pitch curves, such as higher-order modified curves, to see whether they can further improve the constant-speed behavior or reduce the maximum acceleration. The eccentric pinion gear concept can also be extended to other textile machines that require a nearly constant-speed reciprocating motion. Overall, the eccentric pinion gear driven cover plate mechanism offers a new and practical direction for the innovative design of flat grinding machines.