In the design of mechanical transmission systems, the straight spur gear is one of the most widely used components. Its structural features are highly series-oriented and parameterized. In conventional design, repeated modeling due to parameter changes not only wastes time and effort but also leads to data redundancy. Parameterized design effectively solves this problem by automatically driving the geometry through modification of predefined parameters. Furthermore, the dynamic characteristics of straight spur gear are critical to the reliability and efficiency of the transmission system. By performing finite element modal analysis on the straight spur gear, its natural frequencies and mode shapes can be determined, allowing designers to avoid resonance and harmful vibration patterns during operation. This also helps identify weak areas of the gear, laying the foundation for future modifications, noise control, and optimization. UG (Unigraphics) is a high-end CAD/CAE integrated software that enables parameterized modeling and finite element analysis of straight spur gear within the same platform, eliminating the need for data conversion between different software tools and improving design efficiency while ensuring unified data management.
In this paper, I present my work on the parameterized modeling of straight spur gear based on UG and its finite element modal analysis. The entire process is carried out in a unified environment, demonstrating the effectiveness of the integrated approach. I first derive the involute profile equation based on the generation principle of involute gears. Then, I establish the parametric model of straight spur gear using UG’s expression functionality. Finally, I create the finite element model in UG/Structure and perform modal analysis to extract the first five natural frequencies and mode shapes. The results provide valuable insights for improving the dynamic response and service reliability of straight spur gear.
Parametric Modeling of Straight Spur Gear in UG
Parameterized design embeds design intent into the CAD model by applying various constraints to each feature. The geometric shapes and dimensions of features are represented by variable parameters. When a parameter changes, the geometry updates automatically. Based on the fundamental laws of gear transmission and the generation principle of involute tooth profiles, I built an accurate three-dimensional model of straight spur gear. Using UG’s expression tool, I established the parametric equation of the involute curve, linked the parameters controlling the involute to the basic geometric parameters of straight spur gear, mirrored and trimmed the curve to form a single tooth profile, extruded it into a solid, and then used the associative copy feature to array the teeth around the circumference, achieving a fully parameterized model.
Straight spur gear is defined by five basic parameters and four derived parameters, as summarized in the following table:
| Parameter Type | Symbol | Definition |
|---|---|---|
| Basic | m | Module |
| z | Number of teeth | |
| α | Pressure angle | |
| ha* | Addendum coefficient | |
| c* | Bottom clearance coefficient | |
| Derived | d | Pitch circle diameter: $$d = mz$$ |
| db | Base circle diameter: $$d_b = mz \cos\alpha$$ | |
| da | Addendum circle diameter: $$d_a = d + 2h_a^* m$$ | |
| df | Dedendum circle diameter: $$d_f = d – 2(h_a^* + c^*) m$$ |
The generation of an involute tooth profile starts from the base circle, as illustrated by the generating principle. In polar coordinates, the involute is described by:
$$ r_k = \frac{r_b}{\cos\alpha_k}, \quad \theta_k = \tan\alpha_k – \alpha_k $$
In UG, the default independent variable t ranges from 0 to 1. To draw a 90° involute segment, I define the roll angle S = 90 × t (in degrees). Then:
$$ \tan\alpha_k = \frac{K N}{O N} = \frac{S \cdot r_b}{r_b} = S $$
$$ \theta_k = S – \arctan S $$
Converting to Cartesian coordinates for UG:
$$ x = r_k \cos(180^\circ – \theta_k), \quad y = r_k \sin(180^\circ – \theta_k) $$
Considering the distinction between degrees and radians in UG, the expressions are set accordingly. For a straight spur gear with parameters: m = 2 mm, z = 20, α = 20°, ha* = 1, c* = 0.25, I first generate one side of the involute curve. After mirroring and trimming, I obtain a complete single tooth profile. Using the instance feature (circular pattern), I create the full gear model.
The completed parametric model of straight spur gear allows modifying any basic parameter (e.g., module, number of teeth, pressure angle) through the expression editor, and the geometry updates automatically. For subsequent finite element analysis, cosmetic features such as the central hole and keyway are suppressed to simplify the mesh. This parametric approach ensures that designers do not need to rebuild the model for each design variant, significantly improving efficiency.
Finite Element Modeling in NX Structure
Because UG is a CAD/CAE integrated software, converting the CAD model of straight spur gear into a finite element model does not require data exchange between different programs. This preserves model information integrity and saves time. In the UG/Structure environment, I performed the following three steps for modal analysis of straight spur gear:
- Mesh generation: The straight spur gear is a three-dimensional solid entity. I used Tetra10 elements (10-node tetrahedral) with element size automatically determined by the system.
- Boundary conditions: According to the working condition of straight spur gear, all translational degrees of freedom (X, Y, Z) on the inner cylindrical surface were constrained.
- Material properties: The gear material is isotropic with parameters shown in the table below.
| Property | Value |
|---|---|
| Elastic modulus, E | 2.06e11 Pa |
| Poisson’s ratio, μ | 0.3 |
| Density, ρ | 7800 kg/m³ |
The finite element model of the straight spur gear is then ready for modal analysis.
Modal Analysis of Straight Spur Gear
The natural frequencies and mode shapes of a multi-degree-of-freedom system are obtained by solving the eigenvalue problem. For an undamped n-degree-of-freedom system, the free vibration equation is:
$$ [M]\{\ddot{x}\} + [K]\{x\} = \{0\} $$
Assume a solution of the form:
$$ x_i = A_i^{(j)} \sin(\omega_{nj} t + \varphi_j) $$
Substituting yields the eigenvalue problem:
$$ [K]\{A^{(j)}\} – \omega_{nj}^2 [M]\{A^{(j)}\} = \{0\} $$
For a nontrivial solution, the determinant must vanish:
$$ \det([K] – \omega_{nj}^2 [M]) = 0 $$
This is the characteristic equation of a multi-degree-of-freedom undamped system. Expanding it gives an n-th order polynomial in ωnj², and the natural frequencies satisfy:
$$ 0 < \omega_{n1} < \omega_{n2} < \dots < \omega_{n(n-1)} < \omega_{nn} $$
If the first natural frequency of the straight spur gear is higher than the excitation frequency in the working environment, resonance can be avoided. In this analysis, I extracted the first five modes. The natural frequencies are listed in the following table:
| Mode Order | Natural Frequency (Hz) |
|---|---|
| 1 | 1065 |
| 2 | 1367 |
| 3 | 1936 |
| 4 | 2009 |
| 5 | 2231 |
The corresponding mode shapes and displacement/stress distributions were obtained. For the first mode, the displacement and stress contours indicate a bending-torsional deformation pattern. The second mode exhibits a different nodal diameter pattern. The third, fourth, and fifth modes show increasing complexity with multiple nodal diameters and circles. These mode shapes reveal the dynamic characteristics and potential weak areas of the straight spur gear. For instance, the maximum displacement in the lower modes occurs at the tooth tip or rim regions, indicating that these areas are more susceptible to vibration-induced damage. The stress contours highlight stress concentrations at the tooth root, which aligns with the typical failure location of straight spur gear.
Conclusion
By mathematically deriving the involute tooth profile equation and converting it into UG expressions, I achieved fully parameterized and precise modeling of straight spur gear using UG’s powerful parametric design tools. Simply modifying the basic parameters in the expressions automatically generates a new three-dimensional gear model. This approach eliminates repetitive modeling tasks and greatly enhances design efficiency.
Through the study of the free vibration characteristics of straight spur gear, I obtained its low-order natural frequencies and mode shapes, reflecting the dynamic performance of the gear. The results allow intuitive analysis of the dynamic behavior and identification of weak links, providing a theoretical basis for dynamic experiments, design refinement, and maintenance of straight spur gear. Moreover, the modal data lay the foundation for further dynamic response calculation and analysis of the entire transmission system. The integration of parametric modeling and finite element analysis within the same platform (UG) proves to be a powerful methodology for the efficient design and analysis of straight spur gear.