## 2 Calculation Method of Gear Backlash Based on Jacobian-Torsor Theory
### Three-Dimensional Tolerance Analysis Parallel Chain Serialization Method  
Two or more assembly positioning nodes exist for inter-part assembly positioning, which constitutes a parallel chain. According to the type of benchmark, parallel chains can be divided into two categories: multiple benchmarks and common benchmarks. Multiple datums and common datums are used as criteria for classifying parallel chains. Multiple datums are based on the spatial degree of freedom restrictions on the part features from mutually independent coordinate systems, while common datums are based on spatial degrees of freedom restrictions on part features from mutually coupled coordinate systems. Multi-basis parallel chains are commonly found in contact pairs formed between two surfaces, where cylindrical nodes combined with planar nodes are the most common type. When a key connection or a clearance fit is used between the gear and the shaft, a chain of parallel tolerances similar to the hole-pin is formed, and the partial parallel chain is calculated with the parallel or intersection operation [31]. The parallel dimensional chain with the geometric tolerance datum in the dimensional chain is the common datum. The tolerance of the two contact surfaces can no longer take the serial chain operation method due to the existence of the leverage effect, and to eliminate the influence of the geometric structure leverage effect on part of the parallel chain, the advantages of the CLIC method and the analytical line method are integrated to transform the parallel chain into the new serial chain tolerance [24]. When a pair of gears meshes, with the active shaft axis as the positioning reference, there is a relative offset between the drive shaft axis and the active shaft axis, and the offset of the center distance tolerance causes the gears on the shaft to also be offset. The offset makes the gears misaligned and there is a local parallel chain between the main serial chain and the center distance tolerance; the dimensional chain is shown in Figure 1. Directly superimposing the center distance tolerance on the gear tolerance will increase the error. The center distance tolerance can be equated to the gear tooth thickness manufacturing deviation through the geometric relationship, which is shown in Figure 1.  
IFE3 in Figure 1 represents the tooth thickness deviation of gear 2, given by  
\[T_{3}=[ \begin{array}{cccc}0&0&w_{3}&\alpha_{3}&\beta_{3}&0\end{array} ] \tag{10}\]  
CFE5 indicates the center distance deviation of the two gears, given by  
\[T_{5}=[ \begin{array}{cccc}u&0&w&\alpha&\beta&0\end{array} ] \tag{11}\]  
In Figure 2, \(t\) is the center distance deviation, \(u\) and \(w\) are the displacement components of the center distance deviation, \(u_{1}\) is the displacement component of the center distance deviation at the point of engagement, \(u_{11}\) and \(w_{11}\) are the displacement components of  
Figure 1: Gear meshing partial parallel dimensional chain diagram. the center distance deviation on the engagement line, and \(\gamma\) and \(\psi\) are the rotational components of the center distance deviation. O\({}_{1}\)O\({}_{3}\) is the length of the pitch circle radius of gear 1, which is denoted by \(r_{a1}\), and O\({}_{2}\)O\({}_{3}\) is the length of the pitch circle radius of gear 2, which is denoted as \(r_{a2}\), the length of O\({}_{1}\)O\({}_{2}\) is the central distance and is denoted as \(q\), and \(\theta\) is the mesh angle of the two gears, which can be expressed as the radius of the base circle \(r_{b1}\) and the radius of the indexing circle \(r_{1}\) of gear 1. The deviation of the center distance of the two gears is in the same direction as the rotational component of the tooth thickness deviation of gear 2 and can be algebraically superimposed. The displacement component of the center distance deviation needs to be translated to the meshing line with the geometric relationship and then algebraically superimposed with the displacement component of the gear 2 tooth thickness deviation.