In the realm of precision motion control, harmonic drive gears stand out due to their unique operating principle based on elastic deformation. As a compact and highly efficient reduction device, they offer significant advantages such as high reduction ratios, high torque capacity, multiple simultaneously engaged teeth, minimal wear, and exceptional positioning accuracy. However, one critical parameter that can compromise this precision is the system’s backlash or lost motion. In this article, I will delve into a comprehensive analysis of the sources and quantitative modeling of this lost motion in harmonic drive gears. The phenomenon of lost motion, defined as the maximum angular difference observed at the output shaft when it is subjected to alternating positive and negative load torques, is a primary factor affecting the control accuracy and repeatability of mechanisms employing these reducers. Its origins are multifaceted, stemming from both elastic deformations under load and inherent geometric clearances within the assembly.
The core components of a harmonic drive gear—the wave generator, the flexspline (flexible cup), and the circular spline (rigid ring)—interact in a way that induces controlled elastic deformation in the flexspline. While this principle enables its advantages, it also introduces compliance and clearances that contribute to lost motion. Based on my analysis of the working mechanism, I have identified four primary sources of lost motion: 1) torsional elastic deformation of the output shaft, 2) torsional elastic deformation of the flexspline, 3) the designed flank clearance between the teeth of the flexspline and the circular spline, and 4) the radial clearance within the flexible bearing of the wave generator assembly. The total lost motion, denoted as $\Delta_{bl}$, can be expressed as the sum of the contributions from these individual sources:
$$ \Delta_{bl} = \Delta_{bl1} + \Delta_{bl2} + \Delta_{bl3} + \Delta_{bl4} $$
where $\Delta_{bl1}$ is the lost motion due to output shaft torsion, $\Delta_{bl2}$ is due to flexspline torsion, $\Delta_{bl3}$ is due to tooth flank clearance, and $\Delta_{bl4}$ is due to the flexible bearing’s radial clearance. The first two components are load-dependent elastic effects, while the latter two are primarily geometric, though $\Delta_{bl4}$ manifests as an effective increase in flank clearance.
The image above illustrates a typical harmonic drive gear assembly. The wave generator, comprising an elliptical cam and a thin-walled flexible bearing, is inserted into the flexspline, causing it to deform into an elliptical shape. This deformation engages the teeth of the flexspline with those of the circular spline at two diametrically opposite regions. The relative motion between these components results in speed reduction. However, the clearances and deformations inherent in this system are the focal point of my lost motion analysis.
Elastic Torsional Deformation Contributions
The components transmitting torque, namely the output shaft and the flexspline, are not perfectly rigid. When subjected to a load torque $T_L$, they twist elastically. Upon reversal of the load direction, this stored elastic energy is released, and the components twist back, contributing to the angular lost motion. For a harmonic drive gear system, these components are often in series from a torsional stiffness perspective. The lost motion contributions can be directly related to their respective torsional stiffness values.
The lost motion due to the output shaft’s torsion, $\Delta_{bl1}$, for a bidirectional load of $\pm T_L$, is given by:
$$ \Delta_{bl1} = \frac{2 T_L}{K_e} $$
where $K_e$ is the torsional stiffness of the output shaft (in N·m/rad). Similarly, the lost motion due to the flexspline’s torsion, $\Delta_{bl2}$, is:
$$ \Delta_{bl2} = \frac{2 T_L}{K_b} $$
where $K_b$ is the torsional stiffness of the flexspline. The calculation of $K_e$ and $K_b$ follows from classical torsion theory in mechanics of materials. For a hollow cylindrical shaft (like a typical flexspline or output shaft), the torsional stiffness is:
$$ K = \frac{G J}{L} $$
where $G$ is the shear modulus of the material, $J$ is the polar moment of inertia of the cross-section, and $L$ is the effective length under torsion. For a hollow cylinder with outer radius $r_o$ and inner radius $r_i$, $J = \frac{\pi}{2}(r_o^4 – r_i^4)$. The combined torsional compliance of the output shaft and flexspline in series is $1/K_{total} = 1/K_e + 1/K_b$. Therefore, the total elastic lost motion from these two sources is $\Delta_{bl\_elastic} = 2 T_L / K_{total}$. This component is directly proportional to the applied load and inversely proportional to the system’s torsional stiffness.
Mathematical Model for Tooth Flank Clearance Contribution
Even in a perfectly manufactured and assembled harmonic drive gear, a small amount of clearance, or backlash, is intentionally designed between the mating tooth flanks of the flexspline and circular spline. This clearance is necessary to prevent binding due to thermal expansion, manufacturing tolerances, and the need for lubrication. However, it directly contributes to lost motion when the direction of torque transmission reverses. This geometric clearance can be expressed in two ways: as normal backlash $j_{bn}$ (perpendicular to the tooth flank) or as circumferential backlash $j_{wt}$ (along the pitch circle). Their relationship for involute gears is:
$$ j_{bn} = j_{wt} \cos \alpha $$
where $\alpha$ is the working pressure angle of the tooth profile. For a standard harmonic drive gear with no profile shift, the pitch circle of the flexspline can be considered coincident with its reference circle. The angular lost motion $\Delta_{bl3}$ caused by this normal backlash can be derived. When the torque reverses, the output shaft can rotate through the equivalent circumferential clearance before the opposite tooth flanks come into contact. The relationship between angular motion and linear circumferential motion at the flexspline’s pitch radius $r_1 = m z_1 / 2$ is used, where $m$ is the module and $z_1$ is the number of teeth on the flexspline. The total circumferential play for bidirectional motion is $2 j_{wt}$. Therefore, the angular lost motion is:
$$ \Delta_{bl3} = \frac{2 j_{wt}}{r_1} = \frac{2 j_{wt}}{(m z_1 / 2)} = \frac{4 j_{wt}}{m z_1} $$
Substituting $j_{wt} = j_{bn} / \cos \alpha$, we get the final expression in terms of the normal backlash:
$$ \Delta_{bl3} = \frac{2 j_{bn}}{m z_1 \cos \alpha} $$
This equation clearly shows that the lost motion from flank clearance is inversely proportional to the module and the number of teeth on the flexspline. For a given design, it is a fixed geometric value, independent of load.
To illustrate the sensitivity, consider a harmonic drive gear with parameters: $m = 0.2$ mm, $z_1 = 200$, $\alpha = 20^\circ$. The relationship between normal backlash $j_{bn}$ and the resulting lost motion $\Delta_{bl3}$ is calculated and presented in Table 1.
| Normal Backlash, $j_{bn}$ (μm) | Lost Motion, $\Delta_{bl3}$ (arc-seconds) |
|---|---|
| 1 | 10.98 |
| 2 | 21.95 |
| 3 | 32.93 |
| 4 | 43.90 |
| 5 | 54.85 |
| 6 | 65.85 |
| 7 | 76.83 |
| 8 | 87.80 |
Mathematical Model for Flexible Bearing Radial Clearance Contribution
This source of lost motion is particularly intriguing and critical for harmonic drive gears. The wave generator’s core is a flexible ball bearing. In an ideal scenario with zero radial clearance, the outer ring of this bearing perfectly conforms to the elliptical shape imposed by the inner ring (cam), transmitting the deformation fully to the flexspline. However, all practical bearings possess internal radial clearance $c$ to allow for assembly, thermal expansion, and lubrication. This clearance causes the outer ring’s deformation to be less pronounced than the inner ring’s; its major and minor axes difference is smaller. Consequently, the flexspline does not deform to its theoretical elliptical shape, meaning its teeth do not engage as deeply with the circular spline’s teeth as designed. This effectively increases the operating flank clearance.
I model this effect by considering it equivalent to a reduction in the operating center distance between the flexspline (modeled as an external gear) and the circular spline (modeled as an internal gear). In the ideal, zero-clearance case, the gears are in mesh with zero backlash at the theoretical center distance $a$. The presence of bearing radial clearance $c$ reduces the effective center distance by an amount $\Delta a$. For a first-order approximation, this reduction can be taken as $\Delta a = c / 2$, as the bearing clearance allows the outer ring to “sink” inward relative to the perfect elliptical contour.
For a pair of involute gears, a change in center distance alters the operating pressure angle. Let $r_{b1}$ and $r_{b2}$ be the base circle radii of the flexspline and circular spline, respectively. The original center distance is $a = r_2 – r_1$, where $r_1$ and $r_2$ are the reference pitch radii. With the reduced center distance $a’ = a – \Delta a$, the new operating pressure angle $\alpha’$ is given by:
$$ \alpha’ = \arccos\left( \frac{r_{b2} – r_{b1}}{a – \Delta a} \right) = \arccos\left( \frac{(r_2 – r_1) \cos \alpha}{a – \Delta a} \right) $$
This change in center distance and pressure angle introduces an effective normal backlash $j_{bn}’$ between the gear pair, even if they were theoretically zero-backlash gears at the original center distance. To find $j_{bn}’$, I analyze the engagement geometry on the line of action. The normal backlash is the gap between the tooth flanks along the common tangent to the base circles when the gears are positioned at the new center distance. Using fundamental properties of involute geometry, the effective normal backlash can be derived as:
$$ j_{bn}’ = e_{b2} – s_{b1} + 2 (r_{b1} – r_{b2}) \operatorname{inv} \alpha’ $$
where $e_{b2}$ is the base circle tooth space width of the internal gear (circular spline), $s_{b1}$ is the base circle tooth thickness of the external gear (flexspline), and $\operatorname{inv}(x) = \tan x – x$ is the involute function. The base circle tooth dimensions for standard, non-shifted gears are:
$$ s_{b1} = \frac{1}{2} \pi m \cos \alpha + m z_1 \cos \alpha \cdot \operatorname{inv} \alpha $$
$$ e_{b2} = \pi m \cos \alpha – s_{b2} = \pi m \cos \alpha – \left( \frac{1}{2} \pi m \cos \alpha + m z_2 \cos \alpha \cdot \operatorname{inv} \alpha \right) = \frac{1}{2} \pi m \cos \alpha – m z_2 \cos \alpha \cdot \operatorname{inv} \alpha $$
Substituting these into the expression for $j_{bn}’$ and simplifying yields:
$$ j_{bn}’ = m \cos \alpha \left[ \pi – (z_2 – z_1) \operatorname{inv} \alpha \right] – m \cos \alpha \left[ \pi + (z_2 – z_1) \operatorname{inv} \alpha’ \right] + 2 (r_{b1} – r_{b2}) \operatorname{inv} \alpha’ $$
Knowing that $r_{b1} = (m z_1 \cos \alpha)/2$ and $r_{b2} = (m z_2 \cos \alpha)/2$, and that $z_2 – z_1$ is typically 2 for a standard harmonic drive gear (the difference in tooth count between circular spline and flexspline), the expression can be further simplified. The lost motion due to the bearing clearance, $\Delta_{bl4}$, is then the angular equivalent of twice this effective normal backlash (for bidirectional motion), analogous to the formula for $\Delta_{bl3}$:
$$ \Delta_{bl4} = \frac{2 j_{bn}’}{m z_1 \cos \alpha} $$
Combining the equations, a direct relationship between $\Delta_{bl4}$ and the bearing radial clearance $c$ (via $\Delta a = c/2$ and $\alpha’$) is established. This model provides a physical basis for the contribution, moving beyond empirical correction factors.
For a harmonic drive gear with the same parameters as before ($m=0.2$ mm, $z_1=200$, $\alpha=20^\circ$, $z_2=202$) and varying bearing clearance, the calculated lost motion contribution $\Delta_{bl4}$ is shown in Table 2.
| Bearing Radial Clearance, $c$ (μm) | Lost Motion, $\Delta_{bl4}$ (arc-seconds) |
|---|---|
| 5 | 17.84 |
| 10 | 33.63 |
| 15 | 46.94 |
| 20 | 57.04 |
Integrated Model and Parameter Sensitivity
The complete model for total lost motion in a harmonic drive gear under a bidirectional load $\pm T_L$ is therefore:
$$ \Delta_{bl}(T_L) = \frac{2 T_L}{K_{total}} + \frac{2 j_{bn}}{m z_1 \cos \alpha} + \frac{2 \left[ e_{b2} – s_{b1} + 2 (r_{b1} – r_{b2}) \operatorname{inv} \alpha’ \right]}{m z_1 \cos \alpha} $$
with $\alpha’ = \arccos\left( \frac{(m(z_2 – z_1)/2) \cos \alpha}{(m(z_2 – z_1)/2) – c/2} \right)$. This equation elegantly separates the load-dependent elastic effects from the geometry-dependent clearance effects. The sensitivity of total lost motion to key design parameters is crucial for optimizing a harmonic drive gear for high-precision applications. I can analyze this by varying one parameter at a time while holding others constant.
First, the effect of load torque $T_L$ is linear for the elastic part but does not affect the geometric parts. Therefore, in high-precision, low-torque applications (e.g., instrument pointing), the geometric terms may dominate. Conversely, in torque-intensive applications, the elastic deformation may be the primary contributor.
Second, the flexspline’s torsional stiffness $K_b$ is a function of its geometry (wall thickness, length, cup diameter) and material (shear modulus $G$). Increasing wall thickness or using a higher shear modulus material (like high-strength steel versus a lighter alloy) increases $K_b$, reducing $\Delta_{bl1}$ and $\Delta_{bl2}$. However, a thicker wall may negatively impact the fatigue life of the flexspline, which undergoes cyclic elastic deformation. This represents a key design trade-off in harmonic drive gear engineering.
Third, the normal backlash $j_{bn}$ is a manufacturing tolerance parameter. Tighter control over gear tooth machining and assembly can minimize $j_{bn}$, directly reducing $\Delta_{bl3}$. However, this increases cost and risk of thermal binding.
Fourth, the flexible bearing radial clearance $c$ is a critical procurement specification. Using a flexible bearing with a “C0” or “C1” clearance class (very small radial clearance) can significantly reduce $\Delta_{bl4}$. However, such bearings are more expensive and may have higher friction torque. The integrated model allows designers to perform a cost-benefit analysis by quantifying the improvement in lost motion per unit reduction in $c$.
To visualize the combined effect, let’s assume a specific harmonic drive gear system with the following nominal parameters: $K_{total} = 1.8 \times 10^4$ N·m/rad, $m=0.2$ mm, $z_1=200$, $\alpha=20^\circ$, $z_2=202$, $j_{bn}=3$ μm, $c=12$ μm, and a test load $T_L = 0.5$ N·m. The calculated contributions are:
– Elastic ($\Delta_{bl1}+\Delta_{bl2}$): $2 \times 0.5 / (1.8 \times 10^4) = 5.556 \times 10^{-5}$ rad ≈ $11.5$ arc-seconds.
– Flank Clearance ($\Delta_{bl3}$): From Table 1, for $j_{bn}=3$ μm, $\approx 32.93$ arc-seconds.
– Bearing Clearance ($\Delta_{bl4}$): For $c=12$ μm, interpolating from Table 2 gives $\approx 40.3$ arc-seconds.
The total predicted lost motion is approximately $11.5 + 32.93 + 40.3 = 84.73$ arc-seconds. This value falls within a predicted range when considering tolerances for $j_{bn}$ and $c$.
Experimental Validation and Discussion
To validate the theoretical model, a lost motion test was conceived and performed. The test setup involves fixing the input shaft (wave generator) of the harmonic drive gear and applying alternating torques of $\pm 0.5$ N·m to the output shaft. A high-resolution encoder measures the output angular position. The lost motion is recorded as the absolute angular difference between the positions reached under positive and negative torque. Multiple units of a 40-size harmonic drive gear were tested. The key geometric parameters of these units were as listed in the previous sections. The test results for four units are summarized in Table 3.
| Unit Identifier | Angular Position at +0.5 N·m (arc-seconds) | Angular Position at -0.5 N·m (arc-seconds) | Measured Lost Motion, $\Delta_{bl}$ (arc-seconds) |
|---|---|---|---|
| Unit A | +52 | -34 | 86 |
| Unit B | +54 | -50 | 104 |
| Unit C | +39 | -78 | 117 |
| Unit D | +71 | -21 | 92 |
The measured lost motion values range from 86 to 117 arc-seconds. To compare with theory, I calculate the expected range based on realistic tolerance intervals for the key parameters. The torsional stiffness $K_{total}$ is taken as $1.8 \times 10^4$ N·m/rad, giving an elastic contribution of about 11.5 arc-seconds. For the flank clearance, typical manufacturing tolerances might result in a $j_{bn}$ between 2 and 4 μm, corresponding to $\Delta_{bl3}$ between 22.0 and 43.9 arc-seconds (from Table 1). For the flexible bearing, a common radial clearance range is 10 to 15 μm, corresponding to $\Delta_{bl4}$ between 33.6 and 46.9 arc-seconds (from Table 2). Summing the minimum and maximum possible values from these ranges provides a theoretical lost motion interval:
Minimum predicted: $11.5 + 22.0 + 33.6 = 67.1$ arc-seconds.
Maximum predicted: $11.5 + 43.9 + 46.9 = 102.3$ arc-seconds.
The experimental results (86, 104, 117, 92 arc-seconds) largely fall within or near this predicted range (67.1″ to 102.3″). Unit C’s value of 117″ is slightly above the maximum predicted, which could be attributed to factors not explicitly modeled in this first-order analysis, such as local contact deformations at the tooth interfaces, slight misalignments, or variations in the wave generator’s elliptical profile. Nevertheless, the close agreement between the measured data and the theoretical range validates the integrated mathematical model as a robust engineering tool for predicting and understanding lost motion in harmonic drive gears.
The results underscore several important points. First, for this specific harmonic drive gear size and load condition, the geometric contributions (flank clearance and bearing clearance effects) are significantly larger than the elastic torsional contribution. This highlights that for achieving ultra-low lost motion, meticulous control over gear tooth machining and the specification of low-clearance flexible bearings is paramount. Second, the model successfully decomposes the total lost motion into its physical constituents, providing clear guidance on which design or procurement parameter to focus on for improvement. For instance, if the lost motion needs to be reduced by 20 arc-seconds, the model can estimate whether tightening the tooth flank tolerance or specifying a bearing with 5 μm less radial clearance is more effective or cost-efficient.
Advanced Considerations and Extended Analysis
While the presented model captures the primary sources, the behavior of a harmonic drive gear is complex. To further refine the analysis, additional factors can be considered. One such factor is the hysteresis in the stress-strain curve of the flexspline material. Under cyclic loading, the torsional deformation might not be perfectly linear and reversible, leading to a slight difference between loading and unloading paths. This material hysteresis would manifest as an additional, very small component of lost motion that is also load-dependent.
Another consideration is the compliance of the tooth contact itself. Under load, the teeth in the engagement zones undergo bending and Hertzian contact deformation. This local deformation is nonlinear and depends on the number of teeth in contact. Since a harmonic drive gear has many teeth in simultaneous engagement (often 15-30% of the total teeth), the combined contact compliance can be significant. This can be modeled as an additional torsional spring in series with the flexspline’s bulk torsional stiffness. The stiffness due to tooth contact $K_{tc}$ can be estimated using gear contact mechanics formulas. The total effective stiffness $K_{eff}$ would then be:
$$ \frac{1}{K_{eff}} = \frac{1}{K_e} + \frac{1}{K_b} + \frac{1}{K_{tc}} $$
This would increase the elastic lost motion contribution slightly. A rough estimate for $K_{tc}$ for a typical harmonic drive gear can be derived from the formula for the mesh stiffness of spur gears, scaled by the number of tooth pairs in contact $N_p$:
$$ K_{tc} \approx N_p \cdot c_m \cdot m \cdot b $$
where $c_m$ is a material and geometry-dependent constant (on the order of $10^{10}$ N/m² for steel), and $b$ is the face width. For $N_p=30$, $m=0.0002$ m, $b=0.01$ m, and $c_m=1 \times 10^{10}$ N/m², $K_{tc} \approx 30 \times 1e10 \times 0.0002 \times 0.01 = 600$ N·m/rad. This is much lower than the assumed $K_{total}$ of $1.8 \times 10^4$ N·m/rad, suggesting that in this specific case, tooth contact compliance might be a secondary effect. However, for very high-precision or miniaturized harmonic drive gears, it may become relatively more important.
Temperature effects also play a role. Changes in temperature affect material properties (shear modulus $G$ decreases with increasing temperature, reducing stiffness) and cause thermal expansion, altering clearances. The radial clearance $c$ of the flexible bearing, for example, will change with temperature. The coefficients of thermal expansion (CTE) of the bearing steel, flexspline, and housing must be considered for applications operating over a wide temperature range, such as in aerospace. The lost motion model can be extended to include temperature $T$ as a variable:
$$ \Delta_{bl}(T_L, T) = \frac{2 T_L}{K_{total}(T)} + \frac{2 j_{bn}(T)}{m z_1 \cos \alpha} + \frac{2 j_{bn}'(c(T), \alpha'(T))}{m z_1 \cos \alpha} $$
where $K_{total}(T) \propto G(T)$, $j_{bn}(T)$ changes due to differential expansion of gear bodies, and $c(T)$ changes due to bearing ring expansion.
Furthermore, the model for the bearing clearance effect assumes a simple radial shift. In reality, the thin-walled flexible bearing’s behavior under an elliptical load is complex. The outer ring’s deformation might not be a perfect ellipse with uniformly reduced curvature. Finite Element Analysis (FEA) could be used to obtain a more precise relationship between the applied elliptical inner ring profile, the bearing’s internal clearance, and the resulting outer ring profile. This profile would then determine the exact engagement condition of each tooth pair around the circumference, potentially leading to a non-uniform effective backlash that varies with the rotation angle. However, for the purpose of estimating the average or maximum lost motion, the center-distance reduction model provides a sufficiently accurate and analytically tractable solution.
The performance of a harmonic drive gear over its lifetime is also critical. As the device undergoes millions of cycles, wear occurs on the tooth flanks and in the flexible bearing. Wear typically increases clearances. Therefore, the geometric lost motion components $\Delta_{bl3}$ and $\Delta_{bl4}$ are not static but will gradually increase over time. Predictive maintenance schedules for precision mechanisms can be informed by monitoring the growth of lost motion, which serves as a health indicator for the harmonic drive gear. The initial lost motion measurement establishes a baseline, and subsequent periodic tests can track its degradation.
Conclusion
In this detailed analysis, I have systematically investigated the phenomenon of lost motion in harmonic drive gears. By breaking down the problem into fundamental physical sources—elastic torsion of structural components, designed tooth flank clearance, and the amplifying effect of flexible bearing radial clearance—I have developed an integrated mathematical model that quantifies each contribution. The model is grounded in theories of mechanics of materials, involute gear geometry, and bearing mechanics. The formulas derived, particularly for the bearing clearance effect, provide a clear physical understanding that moves beyond empirical corrections.
The experimental validation on multiple units of a 40-size harmonic drive gear showed that the measured lost motion values correlated well with the theoretical range predicted by the model, confirming its practical utility. The analysis revealed that for the tested conditions, the geometric clearance sources contributed more significantly to the total lost motion than the elastic deformation from the applied load. This insight directs attention to manufacturing tolerances and component specifications as primary levers for enhancing the precision of a harmonic drive gear.
For engineers designing or selecting harmonic drive gears for high-precision applications, this model serves as a valuable tool. It enables predictive performance analysis, sensitivity studies to guide design trade-offs (e.g., stiffness vs. weight, clearance vs. cost), and root-cause analysis of excessive lost motion in existing systems. By understanding and accurately modeling the sources of lost motion, one can better harness the exceptional capabilities of harmonic drive gear technology, pushing the boundaries of accuracy in robotics, aerospace instrumentation, satellite pointing mechanisms, and other advanced fields where precise motion control is paramount. Future work could focus on integrating the effects of temperature, detailed contact mechanics, and wear progression into an even more comprehensive lifetime performance model for the harmonic drive gear.