The transmission error (TE) of gear is a fundamental concept in the field of gear transmission engineering. It is utilized in guiding high-performance gear design, characterizing gear quality, analyzing gear process errors and predicting dynamic properties of gears (such as vibration and noise). The definition of TE is relatively simple, yet it encompasses a wealth of implications. Understanding the role of TE is essential; however, more importantly, recognizing its limitations is crucial (Ref. 1). Unfortunately, there has been insufficient research on the shortcomings of TE to date. As a result, when applying the concept of TE, its limitations and deficiencies are often overlooked, leading to conclusions that warrant further discussion. In the study of TE, there is a widespread belief in the current literature that research on TE began in the 1950s. In 1958, Harris (Ref. 2) introduced the concept of gear TE while studying gear vibration and noise. His work also laid the theoretical foundation for modern TE research. Undoubtedly, Harris’s contributions to TE research are significant, but the above statement requires further investigation. Upon examining the history of gear technology, it becomes clear that there have always been two forces driving TE research: one group consists of researchers engaged in gear quality control and measurement technology, and the other comprises those involved in gear design and dynamics. As early as the 1930s, to control the quality of gear transmission, researchers obtained the single flank composite error, i.e., TE, of a pair of gears through comparative measurement with standard discs, as shown in Figure 1. This achievement then enabled the control of gear transmission accuracy, and the study of TE originated from this very effort (Ref. 3).

In 1963, Harris introduced the Harris graph, a graphical representation of the relationship between quasi-static load and TE for modified gears (Ref. 4). This graph aimed to provide a theoretical prediction of load-bearing deformation. During the 1960s, R.G. Munro (Ref. 5) developed an optical-grating instrument for single flank testing, which became the first apparatus to utilize a grating technology for TE measurement. This breakthrough marked a significant advancement in the potential for high-precision dynamic measurement of TE. In 1970, Huang Tongnian from China first proposed the concept of Gear Integrated Error (GIE) and developed the measurement technology for GIE (Refs. 6–8). This technology utilizes a specific multi-start worm to implement single-flank testing, as shown in Figure 2. This innovation represented a significant advancement in TE measurement technology and effectively addressed the limitations associated with TE measurement. Concurrently, in the 1980s, the pursuit of smooth gear transmission led Litvin et al. (Ref. 9) to incorporate TE as an objective function in gear design. This approach sparked new developments in the field of gear design.

In 1978, W.D. Mark (Refs. 10–11) introduced the discrete Fourier transform (DFT) method, which allowed the decomposition of TE into elemental deviations of gear. Mark also derived the mathematical expression for TE under low-speed load conditions. Following this, in 1981, Yelle (Ref. 12) developed a mathematical model for cylindrical gear pairs, as depicted in Figure 3, and derived the corresponding mathematical expression for TE. This model accounted for factors such as gear tooth stiffness and pitch deviation. In 1988, J.D. Smith (Ref. 1) identified certain limitations associated with TE. Subsequently, in 2008, as a response to the challenge of TE measurement for fine-pitch gears, Z. Y. Shi proposed the “bidirectional drive synchronous measurement method” for single-flank test of fine-pitch gears (Refs. 13–14).

The development of gears has passed through three stages: geometry, kinematics, and dynamics. The understanding of gears has evolved from “static geometric element” to “moving rigid transmission element,” and then to “dynamically deformed elastic transmission element.” Reviewing the nearly century-long research history of TE, it is found that the understanding of TE also went through three stages: geometric error, kinematic error, and dynamic error. With the widespread application of TE in areas such as gear design, manufacturing error analysis, NVH prediction, and gear pairing, further understanding of TE is particularly urgent. This paper will review the development process, current research status, characteristics, functions, and measurement methods of TE. It will analyze the difficulties and core issues existing in the basic theory of TE, clarify the limitations and deficiencies of TE, and explore ways to overcome the shortcomings of TE.