**6.1 Definitions**

The concept of transmission error (*TE*) was first introduced by Harris (1958) in relation to the study of gear dynamic tooth forces. He realised that, for high speed applications, the problem was one of continuous vibrations rather than a series of impacts as had been thought before. Harris showed that the measure of departure from perfect motion transfer between two gears (which is the definition of *TE*) was strongly correlated with excitations and dynamic responses. *TE* is classically defined as the deviation in the position of the driven gear (for any given position of the driving gear), relative to the position that the driven gear would occupy if both gears were geometrically perfect and rigid.

NB: *The concept embodies both rigid-body and elastic displacements which can sometimes be confusing.* 

Figure 13 illustrates the concept of transmission error which (either at no-load or under load) can be expressed as angular deviations usually measured (calculated) on the driven member (gear) or as distances on the base plane.

Fig. 13. Concept of transmission error and possible expressions (after Munro, (1989)).

Figures 14 and 15 show typical quasi-static *T.E*. traces for spur and helical gears respectively. The dominant features are a cyclic variation at tooth frequency (mesh frequency) and higher harmonics combined with a longer term error repeating over one revolution of one or both gears.

b. When contact losses occur, response curves exhibit amplitude jumps (sudden

c. Because of a possibly strong sensitivity to initial conditions, several solutions may exist depending on the kinematic conditions i.e., speed is either increased or decreased d. damping reduces the importance of the frequency shift and the magnification at critical

The concept of transmission error (*TE*) was first introduced by Harris (1958) in relation to the study of gear dynamic tooth forces. He realised that, for high speed applications, the problem was one of continuous vibrations rather than a series of impacts as had been thought before. Harris showed that the measure of departure from perfect motion transfer between two gears (which is the definition of *TE*) was strongly correlated with excitations and dynamic responses. *TE* is classically defined as the deviation in the position of the driven gear (for any given position of the driving gear), relative to the position that the

NB: *The concept embodies both rigid-body and elastic displacements which can sometimes be* 

Figure 13 illustrates the concept of transmission error which (either at no-load or under load) can be expressed as angular deviations usually measured (calculated) on the driven

Fig. 13. Concept of transmission error and possible expressions (after Munro, (1989)).

Figures 14 and 15 show typical quasi-static *T.E*. traces for spur and helical gears respectively. The dominant features are a cyclic variation at tooth frequency (mesh frequency) and higher harmonics combined with a longer term error repeating over one

amplitude variations for a small speed variation),

These phenomena are illustrated in the response curves in Figure 12.

driven gear would occupy if both gears were geometrically perfect and rigid.

member (gear) or as distances on the base plane.

revolution of one or both gears.

tooth frequency.

**6. Transmission errors** 

**6.1 Definitions** 

*confusing.* 

Fig. 14. Examples of quasi-static T.E. measurements and simulations – Spur gear (Velex and Maatar, 1996).

Fig. 15. T.E. measurements at various loads – Helical gear example. NASA measurements from www.grc.nasa.gov/WWW/RT2001/5000/5950oswald1.html.

#### **6.2 No-load transmission error (NLTE)**

No-load *T.E*. (*NLTE*) has already been introduced in (2); it can be linked to the results of gear testing equipment (single flank gear tester) and is representative of geometrical deviations. From a mathematical point of view, *NLTE* is derived by integrating (2) and is expressed as:

$$\text{NLLTE} = -\frac{E\_{\text{MAX}}\left(t\right)}{\cos \beta\_b} \tag{45}$$

#### **6.3 Transmission errors under load**

The concept of transmission error under load (TE) is clear when using the classic single degree of freedom torsional model (as Harris did) since it directly relies on the angles of

On the Modelling of Spur and Helical Gear Dynamic Behaviour 99

The theory for 3D models is more complicated mainly because there is no one to one correspondence between transmission error and the degree of freedom vector. It can be demonstrated (Velex and Ajmi, 2006) that, under the same conditions as for the one DOF

2 2

From (51) and (52), it appears that the excitations in geared systems are mainly controlled by the fluctuations of the quasi-static transmission error and those of the no-load transmission error as long as the contact conditions on the teeth are close to the quasi-static conditions (this hypothesis is not verified in the presence of amplitude jumps and shocks). The typical frequency contents of *NLTE* mostly comprise low-frequency component associated with run-out, eccentricities whose contributions to the second-order time-derivative of NLTE can be neglected. It can therefore be postulated that the mesh excitations are dominated by

*dt* . This point has a considerable practical importance as it shows that reducing the dynamic response amplitudes is, to a certain extent, equivalent to reducing the fluctuations of *TES* . Profile and lead modifications are one way to reach this objective. Equation (50) stresses the fact that, when total displacements have to be determined, the forcing terms are proportional to the product of the mesh stiffness and the difference between *TES* and *NLTE* (and not*TES* !). It has been demonstrated by Velex et al. (2011) that a unique dimensionless equation for quasi-static transmission error independent of the number of

> 

*k t TE t k M e M dM*

Assuming that the mesh stiffness per unit of contact length is approximately constant (see section 2-5), analytical expressions for symmetric profile modifications (identical on pinion and gear tooth tips as defined in Fig. 16) rendering *TE t <sup>S</sup>* constant (hence cancelling most

of the excitations in the gear system) valid for spur and helical gears with 2

,

**X**

**S**

, for any generic variable *A* (normalization with respect to the average

**X** (53)

 

can be

*L t*

\*

**S**

ˆ ˆ cos , <sup>1</sup> \* *b S*

<sup>1</sup> ˆ ˆ , *DD S*

**D KV** , **X XX** *D S* , dynamic displacement vector

2 2 , *DD S d d mx k t x x m TE m NLTE*

model, the corresponding differential system is:

degrees of freedom can be derived under the form:

*m <sup>A</sup> <sup>A</sup>* 

mesh stiffness and the average static deflection).

where <sup>1</sup> <sup>ˆ</sup> cos *m b <sup>k</sup>*

 2 2 *S <sup>d</sup> TE*

with ˆ

*m <sup>A</sup> <sup>A</sup> <sup>k</sup>* , \*

found under the form:

**6.5 Practical consequences** 

 2 2

*dt dt*

*d d <sup>t</sup> TE NLTE*

**MX K X X MD <sup>P</sup> I MD** (52)

(51)

> 2 2 2

*dt Rb dt*

torsion of the pinion and the gear. For other models (even purely torsional ones), the definition of *TE* is ambiguous or at least not intrinsic because it depends on the chosen cross-sections (or nodes) of reference for measuring or calculating deviations between actual and perfect rotation transfers from the pinion to the gear. Following Velex and Ajmi (2006), transmission error can be defined by extrapolating the usual experimental practice based on encoders or accelerometers, i.e., from the actual total angles of rotation, either measured or calculated at one section of reference on the pinion shaft (subscript *I*) and on the gear shaft (subscript *II*). *TE* as a displacement on the base plane reads therefore:

$$TE = Rb\_1 \left[ \int\_0^l \Omega\_1 \, d\xi + \theta\_l \right] + Rb\_2 \left[ \int\_0^l \Omega\_2 \, d\xi + \theta\_{ll} \right] = Rb\_1 \, \theta\_l + Rb\_2 \, \theta\_{ll} + \text{NL.TE} \tag{46}$$

with , a dummy integration variable and , *I II* , the torsional perturbations with respect to rigid-body rotations (degrees of freedom) at node I on the pinion shaft and at node II on the gear shaft.

Introducing a projection vector **W** of components *Rb*<sup>1</sup> and *Rb*2 at the positions corresponding to the torsional degrees of freedom at nodes I and II and with zeros elsewhere, transmission error under load can finally be expressed as:

$$TE = \mathbf{W}^T \mathbf{X} + \text{NLE} \tag{47-1}$$

which, for the one DOF model, reduces to:

$$TE = \mathbf{x} + \mathbf{N}LE\mathbf{E}\tag{47.2}$$

#### **6.4 Equations of motion in terms of transmission errors**

For the sake of clarity the developments are conducted on the one-DOF torsional model. Assuming that the dynamic contact conditions are the same as those at very low speed, one obtains from (21) the following equation for quasi-static conditions (i.e., when 1 shrinks to zero):

$$k\left(t,\mathbf{x}\right)\mathbf{x}\_{\mathrm{S}} = \mathrm{F}\_{t} + \zeta \cos \beta\_{b} \int\_{\mathrm{L}\left(t,\mathbf{x}\right)} k\left(M\right) \delta e\left(M\right) dM \tag{48}$$

which, re-injected in the dynamic equation (21), gives:

$$k\hat{m}\ddot{\mathbf{x}} + k\mathbf{(t}\prime\prime\mathbf{x})\mathbf{x} = k\left(t\prime\prime\mathbf{x}\right)\mathbf{x}\_{\rm s} - \kappa\frac{d^2}{dt^2}\left(\text{NLTE}\right) \tag{49}$$

From (47-2), quasi-static transmission error under load can be introduced such that *S S x TE NLTE* and the equation of motion is transformed into:

$$\hat{m}\ddot{\mathbf{x}} + k\mathbf{\bar{x}}(t, \mathbf{x})\mathbf{x} = k\mathbf{\bar{x}}(t, \mathbf{x})[TE\_{\rm S} - NLTE] - \kappa \frac{d^2}{dt^2} \text{(NLTE)}\tag{50}$$

An alternative form of interest can be derived by introducing the dynamic displacement *Dx* defined by *S D xx x* as:

$$
\hat{m}\ddot{\mathbf{x}}\_D + k\left(t, \mathbf{x}\right)\mathbf{x}\_D = -\hat{m}\frac{d^2}{dt^2}\left(\mathbf{T}\mathbf{E}\_S\right) + \left(\hat{m} - \kappa\right)\frac{d^2}{dt^2}\left(\text{NLTE}\right) \tag{51}
$$

The theory for 3D models is more complicated mainly because there is no one to one correspondence between transmission error and the degree of freedom vector. It can be demonstrated (Velex and Ajmi, 2006) that, under the same conditions as for the one DOF model, the corresponding differential system is:

$$\mathbb{E}\left[\mathbf{M}\right]\ddot{\mathbf{X}}\_{\mathcal{D}} + \left[\mathbf{K}(t, \mathbf{X})\right]\mathbf{X}\_{\mathcal{D}} \equiv -\left[\mathbf{M}\right]\hat{\mathbf{D}}\frac{d^2}{dt^2}(TE\_{\mathcal{S}}) + \left[\frac{1}{Rb\_2}\mathbf{I}\_{\mathbf{P}} + \left[\mathbf{M}\right]\hat{\mathbf{D}}\right]\frac{d^2}{dt^2}(NLT\mathbf{E})\tag{52}$$

where <sup>1</sup> <sup>ˆ</sup> cos *m b <sup>k</sup>* **D KV** , **X XX** *D S* , dynamic displacement vector

#### **6.5 Practical consequences**

98 Mechanical Engineering

torsion of the pinion and the gear. For other models (even purely torsional ones), the definition of *TE* is ambiguous or at least not intrinsic because it depends on the chosen cross-sections (or nodes) of reference for measuring or calculating deviations between actual and perfect rotation transfers from the pinion to the gear. Following Velex and Ajmi (2006), transmission error can be defined by extrapolating the usual experimental practice based on encoders or accelerometers, i.e., from the actual total angles of rotation, either measured or calculated at one section of reference on the pinion shaft (subscript *I*) and on the gear shaft

1 1 2 2 1 2

 *I II* 

*TE Rb d Rb d Rb Rb NLTE <sup>I</sup> II <sup>I</sup> II*

rigid-body rotations (degrees of freedom) at node I on the pinion shaft and at node II on the

Introducing a projection vector **W** of components *Rb*<sup>1</sup> and *Rb*2 at the positions corresponding to the torsional degrees of freedom at nodes I and II and with zeros

For the sake of clarity the developments are conducted on the one-DOF torsional model. Assuming that the dynamic contact conditions are the same as those at very low speed, one obtains from (21) the following equation for quasi-static conditions (i.e., when 1 shrinks to

, cos *St b*

*ktxx F*

*S S x TE NLTE* and the equation of motion is transformed into:

which, re-injected in the dynamic equation (21), gives:

*Dx* defined by *S D xx x* as:

,

<sup>2</sup> , , *<sup>S</sup> <sup>d</sup> mx k t x x k t x x NLTE*

From (47-2), quasi-static transmission error under load can be introduced such that

<sup>2</sup> , , *<sup>S</sup> <sup>d</sup> mx k t x x k t x TE NLTE NLTE*

An alternative form of interest can be derived by introducing the dynamic displacement

 

2

(49)

2

*dt*

(50)

*dt*

*Ltx*

 

(46)

 

*<sup>T</sup> TE NLTE* **W X** (47-1)

*TE x NLTE* (47-2)

*k M e M dM* (48)

, the torsional perturbations with respect to

(subscript *II*). *TE* as a displacement on the base plane reads therefore:

0 0

elsewhere, transmission error under load can finally be expressed as:

**6.4 Equations of motion in terms of transmission errors** 

*t t*

 

, a dummy integration variable and ,

which, for the one DOF model, reduces to:

with 

zero):

gear shaft.

From (51) and (52), it appears that the excitations in geared systems are mainly controlled by the fluctuations of the quasi-static transmission error and those of the no-load transmission error as long as the contact conditions on the teeth are close to the quasi-static conditions (this hypothesis is not verified in the presence of amplitude jumps and shocks). The typical frequency contents of *NLTE* mostly comprise low-frequency component associated with run-out, eccentricities whose contributions to the second-order time-derivative of NLTE can be neglected. It can therefore be postulated that the mesh excitations are dominated by 2 2 *S <sup>d</sup> TE dt* . This point has a considerable practical importance as it shows that reducing the dynamic response amplitudes is, to a certain extent, equivalent to reducing the fluctuations of *TES* . Profile and lead modifications are one way to reach this objective. Equation (50) stresses the fact that, when total displacements have to be determined, the forcing terms are proportional to the product of the mesh stiffness and the difference between *TES* and *NLTE* (and not*TES* !). It has been demonstrated by Velex et al. (2011) that a unique dimensionless equation for quasi-static transmission error independent of the number of degrees of freedom can be derived under the form:

$$\cos \beta\_b \,\hat{k}\left(t, \mathbf{X}\_{\mathbf{s}}\right) \, T E\_{\mathbf{s}}^{\dagger}\left(t\right) = 1 - \int\_{\mathbf{L}\left(t, \mathbf{X}\_{\mathbf{s}}\right)} \hat{k}\left(M\right) e^{\mathbf{s}\cdot}\left(M\right) dM \tag{53}$$

with ˆ *m <sup>A</sup> <sup>A</sup> <sup>k</sup>* , \* *m <sup>A</sup> <sup>A</sup>* , for any generic variable *A* (normalization with respect to the average mesh stiffness and the average static deflection).

Assuming that the mesh stiffness per unit of contact length is approximately constant (see section 2-5), analytical expressions for symmetric profile modifications (identical on pinion and gear tooth tips as defined in Fig. 16) rendering *TE t <sup>S</sup>* constant (hence cancelling most of the excitations in the gear system) valid for spur and helical gears with 2 can be found under the form:

On the Modelling of Spur and Helical Gear Dynamic Behaviour 101

0.0

66151

0.19758

0.13186

0.26329

0.19758

0.329

0.26329

0.

Gamma

0.05

with

*pn*

*n*

transmission error under load - Spur gear

speeds. It is worth noting that, since

methodology is more suited for helical gears.

 

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

329

0.066151

0.26329

0.19758

0.13186

0.329

E

2 2 1 1 2 2 2 2 <sup>1</sup> 2 2

1 *p pn n* 

1 4 1 4

Equation (56) makes it possible to estimate dynamic tooth loads with minimum computational effort provided that the modal properties of the system with averaged stiffness matrix and the spectrum of *TES* (predominantly) are known. One can notice that the individual contribution of a given mode is directly related to its percentage of strain energy in the meshing teeth and to the ratio of its modal stiffness to the average mesh stiffness. These properties can be used for identifying the usually limited number of critical mode shapes and frequencies with respect to tooth contact loads. They may also serve to test the structural modifications aimed at avoiding critical loading conditions over a range of

*pn p pn pn p pn*

*n pn n p pn n pn n p pn*

*t n t n t*

0.5 1 1.5 2 2.5

*E* \*

Fig. 17. Example of performance diagram: contour lines of the RMS of quasi-static

 1.67.

2 2

*AB BA*

 

\* 1 2\* \* 1 2\*

0.066151

0.13186

0.197

0.26329

Based on 54)

0.329

 

 

sin cos

is supposed to be a small parameter, the proposed

58

$$E = \frac{\Gamma \Lambda}{2\Gamma - 1 + \frac{1}{\mathcal{E}\_a}}\tag{54}$$

submitted to the condition <sup>1</sup> 2 

with *E* : tip relief amplitude; : dimensionless extent of modification (such that the length of modification on the base plane is *Pba* ) and 1 0 *Cm Rb b k* : deflection of reference.

#### Fig. 16. Definition of profile relief parameters

Based on these theoretical results, it can be shown that quasi-static transmission error fluctuations for ideal gears with profile relief depend on a very limited number of parameters: i) the profile and lead contact ratios which account for gear geometry and ii) the normalised depth and extent of modification. These findings, even though approximate, suggest that rather general performance diagrams can be constructed which all exhibit a zone of minimum TE variations defined by (54) as illustrated in Figure 17 (Velex et al., 2011).It is to be noticed that similar results have been obtained by a number of authors using very different models (Velex & Maatar, 1996), (Sundaresan et al., 1991), (Komori et al., 2003), etc.

The dynamic factor defined as the maximum dynamic tooth load to the maximum static tooth load ratio is another important factor in terms of stress and reliability. Here again, an approximate expression can be derived from (51-52) by using the same asymptotic expansion as in (34) and keeping first-order terms only (Velex & Ajmi, 2007). Assuming that *TES* and *NLTE* are periodic functions of a period equal to one pinion revolution; all forcing terms can be decomposed into a Fourier series of the form:

$$-\left[\mathbf{M}\right]\hat{\mathbf{D}}\frac{d^2}{dt^2}(T\mathbf{E}\_s) + \left[\frac{1}{Rb\_2}\mathbf{I}\_\mathbf{p} + \left[\mathbf{M}\right]\hat{\mathbf{D}}\right]\frac{d^2}{dt^2}(NLT\mathbf{E}) = -\Omega\_1^2 \sum\_{n\ge 1} n^2 \left[A^\ast{}\_n \sin n\Omega\_1 t + B^\ast{}\_n \cos n\Omega\_1 t\right] \tag{55}$$

and an approximate expression of the dimensionless dynamic tooth load can be derived under the form:

$$r(t) = \frac{F\_{\mathcal{D}}(t)}{F\_{\mathcal{S}}} \equiv 1 + \sum\_{p} \sqrt{\rho\_{p} \hat{k}\_{\Phi p}} \text{ Y}\_{\mathcal{P}^{\text{w}}}(t) \tag{56}$$

Fig. 17. Example of performance diagram: contour lines of the RMS of quasi-static transmission error under load - Spur gear 1.67.

with

100 Mechanical Engineering

 

<sup>1</sup> 2 1

with *E* : tip relief amplitude; : dimensionless extent of modification (such that the length

 

Based on these theoretical results, it can be shown that quasi-static transmission error fluctuations for ideal gears with profile relief depend on a very limited number of parameters: i) the profile and lead contact ratios which account for gear geometry and ii) the normalised depth and extent of modification. These findings, even though approximate, suggest that rather general performance diagrams can be constructed which all exhibit a zone of minimum TE variations defined by (54) as illustrated in Figure 17 (Velex et al., 2011).It is to be noticed that similar results have been obtained by a number of authors using very different models

The dynamic factor defined as the maximum dynamic tooth load to the maximum static tooth load ratio is another important factor in terms of stress and reliability. Here again, an approximate expression can be derived from (51-52) by using the same asymptotic expansion as in (34) and keeping first-order terms only (Velex & Ajmi, 2007). Assuming that *TES* and *NLTE* are periodic functions of a period equal to one pinion revolution; all forcing

2 2

and an approximate expression of the dimensionless dynamic tooth load can be derived

<sup>1</sup> *<sup>D</sup>*

*r t k t*

*S p*

<sup>1</sup> ˆ ˆ \* sin \* cos *<sup>S</sup> n n*

2 1

*F t*

*F*

2 2 1 11

*d d TE NLTE n A n t B n t*

**M D I MD <sup>P</sup>** (55)

2 2

*n*

*p p pn*

(56)

 

> 1 0 *Cm Rb b k*

: deflection of reference.

Limit of active profile

(54)

Direction of line of action (base plane)

*E*

*Pba* 

 

(Velex & Maatar, 1996), (Sundaresan et al., 1991), (Komori et al., 2003), etc.

terms can be decomposed into a Fourier series of the form:

under the form:

*dt Rb dt*

) and

2 

Extent of modification

Tooth tip

Fig. 16. Definition of profile relief parameters

submitted to the condition <sup>1</sup>

of modification on the base plane is

Tip relief amplitude *E*\*

$$\begin{split} \mathcal{Y}\_{\boldsymbol{\mu}^{n}}(t) = & \sum\_{n \ge 1} \frac{\overline{A}^{\*} \left[ \left( \boldsymbol{\sigma}\_{\boldsymbol{\mu}^{n}} \right)^{2} - 1 \right] + 2 \overline{B}^{\*} \boldsymbol{\omega}\_{\boldsymbol{\mu}} \boldsymbol{\varepsilon}\_{\boldsymbol{\mu}} \left( \boldsymbol{\sigma}\_{\boldsymbol{\mu}^{n}} \right)}{\left[ \left( \boldsymbol{\sigma}\_{\boldsymbol{\mu}^{n}} \right)^{2} - 1 \right]^{2} + 4 \boldsymbol{\varepsilon}\_{p}^{2} \left( \boldsymbol{\sigma}\_{\boldsymbol{\mu}^{n}} \right)^{2}} \sin n\Omega\_{1} t + \frac{\overline{B}^{\*} \boldsymbol{\varepsilon}\_{\boldsymbol{\mu}} \left[ \left( \boldsymbol{\sigma}\_{\boldsymbol{\mu}^{n}} \right)^{2} - 1 \right] - 2 \overline{A}^{\*} \boldsymbol{\varepsilon}\_{\boldsymbol{\mu}} \boldsymbol{\varepsilon}\_{p} \left( \boldsymbol{\sigma}\_{\boldsymbol{\mu}^{n}} \right)}{\left[ \left( \boldsymbol{\sigma}\_{\boldsymbol{\mu}^{n}} \right)^{2} - 1 \right]^{2} + 4 \boldsymbol{\varepsilon}\_{p}^{2} \left( \boldsymbol{\sigma}\_{\boldsymbol{\mu}^{n}} \right)^{2}} \cos n\Omega\_{1} t \\\\ \boldsymbol{\sigma}\_{\boldsymbol{\mu}^{n}} = & \frac{\boldsymbol{\alpha}\_{p}}{n \Omega\_{1}} \end{split}$$

Equation (56) makes it possible to estimate dynamic tooth loads with minimum computational effort provided that the modal properties of the system with averaged stiffness matrix and the spectrum of *TES* (predominantly) are known. One can notice that the individual contribution of a given mode is directly related to its percentage of strain energy in the meshing teeth and to the ratio of its modal stiffness to the average mesh stiffness. These properties can be used for identifying the usually limited number of critical mode shapes and frequencies with respect to tooth contact loads. They may also serve to test the structural modifications aimed at avoiding critical loading conditions over a range of speeds. It is worth noting that, since is supposed to be a small parameter, the proposed methodology is more suited for helical gears.

On the Modelling of Spur and Helical Gear Dynamic Behaviour 103

alternative to these time-consuming methods is to use hybrid FE/lumped models as described by Bettaieb et al, (2007). Figure 18 shows an example of such a model which combines i) shaft elements for the pinion shaft and pinion body, ii) lumped parameter elements for the bearings and finally iii) a FE model of the gear + shaft assembly which is sub-structured and connected to the pinion by a time-varying, non-linear Pasternak foundation model for the mesh stiffness. The computational time is reduced but the

A systematic formulation has been presented which leads to the definition of gear elements with all 6 rigid-body degrees-of-freedom and time-varying, possibly non-linear, mesh stiffness functions. Based on some simplifications, a number of original analytical results have been derived which illustrate the basic phenomena encountered in gear dynamics. Such results provide approximate quantitative information on tooth critical frequencies and

Gear vibration analysis may be said to have started in the late 50's and covers a broad range of research topics and applications which cannot all be dealt with in this chapter: multimesh gears, power losses and friction, bearing-shaft-gear interactions, etc. to name but a few. Gearing forms part of traditional mechanics and one obvious drawback of this long standing presence is a definite sense of déjà vu and the consequent temptation to construe that, from a research perspective, gear behaviour is perfectly understood and no longer worthy of study (Velex & Singh, 2010). At the same time, there is general agreement that although gears have been around for centuries, they will undoubtedly survive long into the

Looking into the future of gear dynamics, the characterisation of damping in geared sets is a priority since this controls the dynamic load and stress amplitudes to a considerable extent. Interestingly, the urgent need for a better understanding and modelling of damping in gears was the final conclusion of the classic paper by Gregory et al. (1963-64). Almost half a century later, new findings in this area are very limited with the exception of the results of Li & Kahraman (2011) and this point certainly remains topical. A plethora of dynamic models can be found in the literature often relying on widely different hypotheses. In contrast, experimental results are rather sparse and there is certainly an urgent need for validated models beyond the classic results of Munro (1962), Gregory et al. (1963), Kubo (1978), Küçükay (1984 &87), Choy et al. (1989), Cai & Hayashi (1994), Kahraman & Blankenship (1997), Baud & Velex (2002), Kubur et al. (2004), etc. especially for complex multi-mesh systems. Finally, the study of gear dynamics and noise requires multi-scale, multi-disciplinary approaches embracing non-linear vibrations, tribology, fluid dynamics etc. The implications of this are clear; far greater flexibility will be needed, thus breaking down the traditional boundaries

Ajmi, M. & Velex, P. (2005). A model for simulating the dynamic behaviour of solid wide-

faced spur and helical gear, *Mechanisms and Machine Theory*, vol. 40, n°2, (February

separating mechanical engineering, the science of materials and chemistry.

2005), pp. 173-190. ISSN: 0094 -114X.

modelling issues at the interfaces between the various sub-models are not simple.

mesh excitations held to be useful at the design stage.

21st century in all kinds of machinery and vehicles.

**8. Conclusion** 

**9. References** 
