In the field of mechanical transmission systems, the rack and pinion gear mechanism serves as a critical component for converting rotational motion into linear reciprocating motion, widely employed in vertical transport equipment like construction elevators. The time-varying mesh stiffness (TVMS) generated during the engagement of the rack and pinion is a primary source of internal excitation, significantly influencing system noise and vibration. This study focuses on the rack and pinion system used in construction elevators, where the rack is typically mounted vertically on the guide structure. We develop an analytical model for calculating the TVMS of the rack and pinion gear based on the potential energy method and superposition principle, validated through finite element analysis (FEA). Furthermore, we investigate the effects of key parameters, such as the gear width and meshing center distance, on the TVMS. The results demonstrate that the proposed analytical method accurately computes the TVMS, providing a theoretical foundation for dynamic characteristic analysis in rack and pinion transmission systems.
The rack and pinion gear system is renowned for its efficiency and precision in motion conversion, making it indispensable in applications requiring linear positioning, such as construction elevators. The TVMS arises due to the periodic change in the number of teeth in contact during meshing, leading to fluctuations in stiffness that act as a dynamic激励. Previous research has extensively studied TVMS in gear pairs, but limited work addresses the unique aspects of rack and pinion configurations, especially in vertical orientations. For instance, studies on spur gears have utilized methods like the potential energy approach to incorporate factors like friction and cracks, but these need adaptation for rack and pinion systems. In this paper, we extend these principles to model the rack and pinion gear, considering deformations in the gear tooth, rack tooth, and Hertzian contact, while accounting for the rack’s fixed mounting on the elevator guide.
To compute the TVMS for the rack and pinion, we consider the combined stiffness of the gear, rack, and the contact between them. The overall mesh stiffness \( k_{\text{mesh}} \) for a single tooth pair can be expressed as a series combination of the gear stiffness \( k_L \), rack stiffness \( k_T \), and Hertzian contact stiffness \( k_h \), given by:
$$ \frac{1}{k} = \frac{1}{k_L} + \frac{1}{k_T} + \frac{1}{k_h} $$
For multiple tooth pairs in engagement, the total TVMS is obtained by parallel superposition of individual pair stiffnesses. Given that the rack and pinion gear in construction elevators typically has a contact ratio greater than 1, meshing alternates between single and double tooth pairs. Thus, for double-tooth engagement, the综合 stiffness is the sum of the stiffnesses of the two active pairs.
The gear stiffness \( k_L \) is derived from the bending, shear, axial compression, and fillet foundation stiffnesses of the gear tooth. Using the potential energy method, the gear tooth is modeled as a non-uniform cantilever beam. Under a meshing force \( F \), the stored elastic energies include bending energy \( U_b \), shear energy \( U_s \), and axial compression energy \( U_a \). The relationship between stiffness and energy is defined as \( U = \frac{F^2}{2k} \), leading to the following integrals for a gear tooth:
Bending stiffness \( k_{lb} \):
$$ \frac{1}{k_{lb}} = \int_{\alpha_1}^{\alpha_2} \frac{3 \cos^2 \alpha \{1 + \cos \alpha [(\alpha_2 – \alpha) \sin \alpha – \cos \alpha]\}}{2 E W [(\alpha_2 – \alpha) \cos \alpha + \sin \alpha]^3} d\alpha $$
Shear stiffness \( k_{ls} \):
$$ \frac{1}{k_{ls}} = \int_{\alpha_1}^{\alpha_2} \frac{1.2 (1 + \nu) \cos^2 \alpha (\alpha_2 – \alpha) \cos \alpha}{E W [(\alpha_2 – \alpha) \cos \alpha + \sin \alpha]^2} d\alpha $$
Axial compression stiffness \( k_{la} \):
$$ \frac{1}{k_{la}} = \int_{\alpha_1}^{\alpha_2} \frac{(\alpha_2 – \alpha) \sin \alpha \cos \alpha}{2 E W [(\alpha_2 – \alpha) \cos \alpha + \sin \alpha]^2} d\alpha $$
Here, \( E \) is the Young’s modulus, \( W \) is the gear width, \( \nu \) is Poisson’s ratio, and \( \alpha_1 \), \( \alpha_2 \) are the angular limits of the tooth engagement. The gear foundation stiffness \( k_{lf} \) accounts for deformations in the gear body and is calculated using empirical formulas based on gear geometry, such as:
$$ \frac{1}{k_{lf}} = \frac{\cos^2 \beta}{E L} \left[ L^* M^* + P^* Q^* (1 + \tan^2 \beta) \right] $$
where \( \beta \) is the pressure angle, \( L \) is the tooth width, and \( L^*, M^*, P^*, Q^* \) are constants dependent on gear base and bore radii.
For the rack and pinion system, the rack stiffness \( k_T \) is similarly composed of bending \( k_{tb} \), shear \( k_{ts} \), axial compression \( k_{ta} \), and foundation stiffness \( k_{tf} \). The rack tooth is analyzed as a beam fixed at the base, with integrals derived along the tooth height. For a rack tooth under meshing force \( F \), the stiffness components are:
Bending stiffness \( k_{tb} \):
$$ \frac{1}{k_{tb}} = \int_{0}^{l} \frac{(x \cos \alpha_1 – h_x \sin \alpha_1)^2}{\frac{2}{3} E W h_x^3} dx $$
Shear stiffness \( k_{ts} \):
$$ \frac{1}{k_{ts}} = \int_{0}^{l} \frac{1.2 (1 + \nu) \cos^2 \alpha_1}{E W h_x} dx $$
Axial compression stiffness \( k_{ta} \):
$$ \frac{1}{k_{ta}} = \int_{0}^{l} \frac{\sin^2 \alpha_1}{2 E W h_x} dx $$
where \( x \) is the distance from the tooth root, \( h_x \) is the semi-tooth thickness at position \( x \), and \( \alpha_1 \) is the rack tooth profile angle. The rack foundation stiffness \( k_{tf} \) is computed using the superposition method, considering the rack as an infinitely long beam fixed at bolt holes. The deflection at the meshing point is evaluated in horizontal and vertical directions, and equivalent stiffness is derived using Hooke’s law. For a force applied at point B on the rack, the displacements are:
Horizontal displacement \( x_B \):
$$ x_B = \frac{F_b L^3}{48 E I} + \frac{F_b a b (L^2 – a^2 – b^2)}{6 E I L} $$
Vertical displacement \( y_B \):
$$ y_B = \frac{F_a a b}{E A L} + \frac{F_a (y_B – a)}{E A} $$
where \( F_b \) and \( F_a \) are the horizontal and vertical components of the meshing force, \( L \) is the rack length, \( I \) is the moment of inertia, \( A \) is the cross-sectional area, and \( a \), \( b \) are distances from the force application point to the supports. The foundation stiffness \( k_{tf} \) is then:
$$ k_{tf} = \frac{F}{x_B \cos \beta + y_B \sin \beta} $$
The Hertzian contact stiffness \( k_h \) for the rack and pinion gear is given by:
$$ k_h = \frac{\pi E W}{4(1 – \nu^2)} $$
For a rack and pinion system with a contact ratio indicating alternating single and double tooth engagement, the total TVMS \( k_{\text{total}} \) over a meshing cycle is the sum of stiffnesses for active tooth pairs. For instance, in double-tooth engagement, \( k_{\text{total}} = k_1 + k_2 \), where \( k_1 \) and \( k_2 \) are the stiffnesses of the two pairs.
To validate the analytical model, we compare it with finite element analysis (FEA) using parameters typical of construction elevator rack and pinion systems. The table below summarizes the key parameters used in the analysis:
| Parameter | Pinion Gear | Rack |
|---|---|---|
| Module (mm) | 8 | 8 |
| Number of Teeth | 15 | 60 |
| Pressure Angle (°) | 20 | 20 |
| Width (mm) | 40 | 40 |
| Material | 20CrMnTi | 60 Steel |
| Young’s Modulus (Pa) | 2.07 × 1011 | 1.90 × 1011 |
| Poisson’s Ratio | 0.25 | 0.3 |
| Density (kg/m³) | 7800 | 7800 |
In the FEA, a 3D model of the rack and pinion gear was created, with the pinion allowed to rotate and the rack fixed at its base. A torque was applied to the pinion center, and strain energy was extracted to compute stiffness. The comparison between analytical and FEA results for TVMS over a meshing cycle shows close agreement, confirming the model’s accuracy. The TVMS curve exhibits periodic fluctuations corresponding to the engagement and disengagement of tooth pairs, with stiffness values ranging from approximately 5 × 107 N/m to 1.5 × 108 N/m.
We further analyze the influence of meshing center distance and gear width on the TVMS of the rack and pinion. The center distance represents the offset between the pinion pitch circle and the rack pitch line, which may vary due to assembly errors. The table below presents TVMS values for different center distances:
| Meshing Center Distance (mm) | Peak TVMS (N/m) | Meshing Cycle Period |
|---|---|---|
| 0 | 1.50 × 108 | Base period |
| 2 | 1.48 × 108 | Reduced by 5% |
| 4 | 1.45 × 108 | Reduced by 10% |
As the center distance increases, the meshing cycle period decreases, but the peak stiffness remains relatively stable, indicating that minor misalignments do not drastically affect stiffness magnitude but alter the engagement timing. This is critical for rack and pinion systems in construction elevators, where precision is vital.
The effect of gear width on TVMS is significant, as width directly influences the length of the contact line. The following table compares TVMS for different widths:
| Gear Width (mm) | Average TVMS (N/m) | Percentage Increase |
|---|---|---|
| 40 | 1.00 × 108 | Base |
| 55 | 1.75 × 108 | 75% |
| 70 | 2.50 × 108 | 150% |
Wider gears result in higher TVMS due to increased cross-sectional area and reduced compliance. This relationship is nearly linear, as stiffness is proportional to width in the analytical model. For rack and pinion applications, selecting an appropriate width can optimize dynamic performance and minimize vibrations.
In conclusion, the analytical model based on the potential energy method effectively computes the time-varying mesh stiffness for rack and pinion gears in construction elevators. The model incorporates gear and rack deformations, Hertzian contact, and foundation effects, providing results consistent with FEA. Parameter studies reveal that meshing center distance shortens the engagement cycle without significantly altering peak stiffness, while gear width has a substantial positive impact on stiffness. These insights aid in the design and optimization of rack and pinion transmission systems, ensuring reliable operation in dynamic environments. Future work could explore the effects of lubrication and wear on TVMS in rack and pinion configurations.