**2. Nomenclature**

*b* : face width

, *C C m r* : pinion, gear torque

*e M* , *E t MAX* ( ) : composite normal deviation at *M* , maximum of *e M* at time *t* .

*E E*, \*: actual and normalized depth of modification at tooth tips

 *L t* , *t k M e M dM* **<sup>e</sup> q F V M** : time-varying, possibly non-linear forcing term associated

with tooth shape modifications and errors

0 0 *<sup>T</sup>* **G VV**

*H x* : unit Heaviside step function ( *Hx i* 1 1; 0 *f x H x otherwise* )

*mk* , *k t* ,**q** : average and time-varying, non-linear mesh stiffness

 1 *<sup>m</sup> kt k t* , linear time-varying mesh stiffness

<sup>0</sup> *k* : mesh stiffness per unit of contact length

*k M* , mesh stiffness per unit of contact length at *M*

*<sup>p</sup> k* : modal stiffness associated with (*<sup>p</sup>* , **Φ<sup>p</sup>** )

 , *T L t* **<sup>G</sup>** *t kM dM* **q K V M V M** : time-varying, possibly non-linear gear mesh

stiffness matrix

On the Modelling of Spur and Helical Gear Dynamic Behaviour 77

, normalized displacement with respect to the average static mesh deflection

**3. Three-dimensional lumped parameter models of spur and helical gears** 

It is well-known that the speed ratio for a pinion-gear pair with perfect involute spur or helical teeth is constant as long as deflections can be neglected. However, shape errors are present to some extent in all gears as a result of machining inaccuracy, thermal distortions after heat treatment, etc. Having said this, some shape modifications from ideal tooth flanks are often necessary (profile and/or lead modifications, topping) in order to compensate for elastic or thermal distortions, deflections, misalignments, positioning errors, etc. From a simulation point of view, rigid-body rotations will be considered as the references in the vicinity of which, small elastic displacements can be superimposed. It is therefore crucial to characterise rigid-boy motion transfer between a pinion and a gear with tooth errors and/or shape modifications. In what follows, *e*(*M*) represents the equivalent normal deviation at the potential point of contact M (sum of the deviations on the pinion and on the gear) and is conventionally positive for an excess of material and negative when, on the contrary, some material is removed from the ideal geometry. For rigid-body conditions (or alternatively under no-load), contacts will consequently occur at the locations on the contact lines where *e*(*M*) is maximum and the velocity transfer from the pinion to the gear is modified compared

> 11 22 cos

where *E t eM MAX* max ( ) *<sup>M</sup>* with max () *<sup>M</sup>* , maximum over all the potential point of

The difference with respect to ideal motion transfer is often related to the notion of no-load

*<sup>d</sup> dE t NLTE Rb Rb*

Using the Kinetic Energy Theorem, the rigid-body dynamic behaviour for frictionless gears

11 22

*dE t Rb Rb*

*b*

0 *MAX*

*b*

111 222 1 2 *m r J J CC* (3)

*MAX*

1 cos

*dt dt* (2)

*dt* (1)

1 *p pn n* 

> *m*

with ideal gears such that:

transmission error *NLTE* via:

contact at time *t*

is controlled by:

, dimensionless eigenfrequency

**3.1 Rigid-body rotations – State of reference** 

**A** : vector **A** completed by zeros to the total system dimension

, normalized stiffness with respect to the average mesh stiffness

1 2 , : pinion, gear angular velocity

\*

 <sup>ˆ</sup> *mk*

*L t* ,**q** : time-varying, possibly non-linear, contact length

$$L\_w = \varepsilon\_a \frac{b}{\cos \beta\_b} : \text{average contact length}$$

$$I \quad I$$

02 01 2 2 1 02 2 01 *I I m Rb I Rb I* : equivalent mass

*m<sup>p</sup>* : modal mass associated with (*<sup>p</sup>* , **Φ<sup>p</sup>** )

**n1** : outward unit normal vector with respect to pinion flanks

*NLTE* : no-load transmission error

1 2 *O O*, : pinion, gear centre

*Pba* : apparent base pitch

1 2 *Rb Rb* , : base radius of pinion, of gear

**stz** , , : coordinate system attached to the pinion-gear centre line, see Figs. 1&2

*Tm* : mesh period.

*TE , TES* : transmission error, quasi-static transmission error under load

**V M ,V <sup>0</sup>** , structural vector, averaged structural vector

**W** : projection vector for the expression of transmission error, see (44-1)

**XYz** , , : coordinate system associated with the base plane, see Fig. 2

<sup>1</sup> **X KF 0 0** : static solution with averaged mesh stiffness (constant)

**XS** , **X***<sup>D</sup>* **X** : quasi-static, dynamic and total (elastic) displacement vector (time-dependent)

1 2 *Z Z*, : tooth number on pinion, on gear

: small parameter representative of mesh stiffness variations, see (30)

*<sup>p</sup>* : apparent pressure angle

*<sup>b</sup>* : base helix angle

*m*

$$\boldsymbol{\delta}\_{m} = \frac{F\_{\boldsymbol{\delta}}}{k\_{m}} = \mathbf{V}^{\mathrm{T}} \mathbf{X}\_{\boldsymbol{\theta}} \text{ : static mesh deflection with average mesh stiffness } \boldsymbol{\delta}\_{m}$$

*eM E t eM MAX* : instantaneous initial equivalent normal gap at *M*

*M* : mesh deflection at point *M*

: theoretical profile contact ratio

 : overlap contact ratio

1 0 *Cm Rb b k* , deflection of reference

**Φ<sup>p</sup>** : *th p* eigenvector of the system with constant averaged stiffness matrix

*P* : damping factor associated with the *th p* eigenfrequency

: dimensionless extent of profile modification (measured on base plane)

*m t T* , dimensionless time

*<sup>p</sup>* : *th p* eigenfrequency of the system with constant averaged stiffness matrix

$$
\sigma\_{pn} = \frac{\alpha\_p}{n\Omega\_1} \text{ / dimensionless eigenfrequency}
$$

1 2 , : pinion, gear angular velocity

76 Mechanical Engineering

*L t* ,**q** : time-varying, possibly non-linear, contact length

*<sup>p</sup>* , **Φ<sup>p</sup>** )

**stz** , , : coordinate system attached to the pinion-gear centre line, see Figs. 1&2

**XS** , **X***<sup>D</sup>* **X** : quasi-static, dynamic and total (elastic) displacement vector (time-dependent)

*TE , TES* : transmission error, quasi-static transmission error under load

**W** : projection vector for the expression of transmission error, see (44-1) **XYz** , , : coordinate system associated with the base plane, see Fig. 2 <sup>1</sup> **X KF 0 0** : static solution with averaged mesh stiffness (constant)

: small parameter representative of mesh stiffness variations, see (30)

**<sup>T</sup> V X0** : static mesh deflection with average mesh stiffness

*eM E t eM MAX* : instantaneous initial equivalent normal gap at *M*

**Φ<sup>p</sup>** : *th p* eigenvector of the system with constant averaged stiffness matrix

: dimensionless extent of profile modification (measured on base plane)

*<sup>p</sup>* : *th p* eigenfrequency of the system with constant averaged stiffness matrix

: damping factor associated with the *th p* eigenfrequency

**V M ,V <sup>0</sup>** , structural vector, averaged structural vector

**n1** : outward unit normal vector with respect to pinion flanks

cos *<sup>m</sup>*

*<sup>b</sup> <sup>L</sup>* 

*m*

*m*

 

 

*P* 

*m t T* 

*b*

: average contact length

: equivalent mass

*m<sup>p</sup>* : modal mass associated with (

*NLTE* : no-load transmission error

1 2 *Rb Rb* , : base radius of pinion, of gear

1 2 *Z Z*, : tooth number on pinion, on gear

*<sup>p</sup>* : apparent pressure angle

*M* : mesh deflection at point *M*

: theoretical profile contact ratio

, deflection of reference

: overlap contact ratio

, dimensionless time

1 0 *Cm Rb b k*

*<sup>b</sup>* : base helix angle

*S*

*m F k*

1 2 *O O*, : pinion, gear centre *Pba* : apparent base pitch

02 01 2 2 1 02 2 01 *I I*

*Rb I Rb I*

*Tm* : mesh period.

**A** : vector **A** completed by zeros to the total system dimension

 \* *m* , normalized displacement with respect to the average static mesh deflection

 <sup>ˆ</sup> *mk* , normalized stiffness with respect to the average mesh stiffness

#### **3. Three-dimensional lumped parameter models of spur and helical gears**

#### **3.1 Rigid-body rotations – State of reference**

It is well-known that the speed ratio for a pinion-gear pair with perfect involute spur or helical teeth is constant as long as deflections can be neglected. However, shape errors are present to some extent in all gears as a result of machining inaccuracy, thermal distortions after heat treatment, etc. Having said this, some shape modifications from ideal tooth flanks are often necessary (profile and/or lead modifications, topping) in order to compensate for elastic or thermal distortions, deflections, misalignments, positioning errors, etc. From a simulation point of view, rigid-body rotations will be considered as the references in the vicinity of which, small elastic displacements can be superimposed. It is therefore crucial to characterise rigid-boy motion transfer between a pinion and a gear with tooth errors and/or shape modifications. In what follows, *e*(*M*) represents the equivalent normal deviation at the potential point of contact M (sum of the deviations on the pinion and on the gear) and is conventionally positive for an excess of material and negative when, on the contrary, some material is removed from the ideal geometry. For rigid-body conditions (or alternatively under no-load), contacts will consequently occur at the locations on the contact lines where *e*(*M*) is maximum and the velocity transfer from the pinion to the gear is modified compared with ideal gears such that:

$$\left(Rb\_1\,\Omega\_1 + Rb\_2\,\Omega\_2\right)\cos\beta\_b + \frac{dE\_{MAX}\left(t\right)}{dt} = 0\tag{1}$$

where *E t eM MAX* max ( ) *<sup>M</sup>* with max () *<sup>M</sup>* , maximum over all the potential point of contact at time *t*

The difference with respect to ideal motion transfer is often related to the notion of no-load transmission error *NLTE* via:

$$\frac{d}{dt}\left(\text{NLTE}\right) = Rb\_1\,\Omega\_1 + Rb\_2\,\Omega\_2 = -\frac{1}{\cos\beta\_b}\frac{dE\_{\text{MAX}}\left(t\right)}{dt} \tag{2}$$

Using the Kinetic Energy Theorem, the rigid-body dynamic behaviour for frictionless gears is controlled by:

$$J\_1 \, \Omega\_1 \, \dot{\Omega}\_1 + J\_2 \, \Omega\_2 \, \dot{\Omega}\_2 = \mathcal{C}\_m \, \Omega\_1 + \mathcal{C}\_r \, \Omega\_2 \tag{3}$$

1 2 ,

rotations 1 1

0

*k*

**Y**(1)

**z**

*positive rotation of pinion*)

therefore expressed as:

*p*

 *d*

(pinion) and 2 2

and **t zs** (Fig. 1) or, ii) **XYz** , , attached to the base plane (Fig. 1):

 

**k k k k k k**

*S or* 

where 1 2 *O O*, are the pinion and gear centres respectively

finer line to a negative pinion rotation about axis *O*<sup>1</sup> ,**z** .

**t Y** <sup>2</sup>

*k kk k kk*

**3.3 Deflection at a point of contact – Structural vectors for external gears** 

*t* 

On the Modelling of Spur and Helical Gear Dynamic Behaviour 79

0

(1996), screws of infinitesimal displacements are introduced whose co-ordinates for solid *k* (*conventionally k=1 for the pinion, k=2 for the gear*) can be expressed in two privileged coordinate systems: i) **stz** , , such that **z** is in the shaft axis direction (from the motor to the load machine), **s** is in the centre-line direction from the pinion centre to the gear centre

> *k kk k kk v wu VWu*

Depending on the direction of rotation, the direction of the base plane changes as illustrated in Figure 2 where the thicker line corresponds to a positive rotation of the pinion and the

O1 M O2

Fig. 2. Directions of rotation and planes (lines) of action. (*the thicker line corresponds to a* 

*M* **uMn u Mn 1 12** . . <sup>1</sup>

For a given helical gear, the sign of the helix angle on the base plane depends also on the direction of rotation and, here again; two configurations are possible as shown in Figure 3.

Since a rigid-body mechanics approach is considered, contact deflections correspond to the interpenetrations of the parts which are deduced from the contributions of the degrees-offreedom and the initial separations both measured in the normal direction with respect to the tooth flanks. Assuming that all the contacts occur in the theoretical base plane (or plane of action), the normal deflection *M* at any point *M* , potential point of contact, is

**<sup>ω</sup> s tz <sup>ω</sup> X Yz** *<sup>k</sup>*=1,2 (4)

**uO s t z uO X Y z**

*t* 

which are small angles associated with deflections superimposed on rigid-body

 *d*

(gear). Following Velex and Maatar

**s**

**X**(2)

*e M* (5)

**X** <sup>1</sup>

with 1 2 *J J*, : the polar moments of inertia of the pinion shaft line and the gear shaft line respectively. , *C C m r* : pinion and gear torques.

The system with 4 unknowns ( 1 2 ,,, *C C m r* ) is characterised by equations (2) - (3) only, and 2 parameters have to be imposed.

### **3.2 Deformed state – Principles**

Modular models based on the definition of gear elements (pinion-gear pairs), shaft elements and lumped parameter elements (mass, inertia, stiffness) have proved to be effective in the simulation of complex gear units (Küçükay, 1987), (Baud & Velex, 2002). In this section, the theoretical foundations upon which classic gear elements are based are presented and the corresponding elemental stiffness and mass matrices along with the possible elemental forcing term vectors are derived and explicitly given. The simplest and most frequently used 3D representation corresponds to the pinion-gear model shown in Figure 1. Assuming that the geometry is not affected by deflections (small displacements hypothesis) and provided that mesh elasticity (and to a certain extent, gear body elasticity) can be transferred onto the base plane, a rigid-body approach can be employed. The pinion and the gear can therefore be assimilated to two rigid cylinders with 6 degrees of freedom each, which are connected by a stiffness element or a distribution of stiffness elements (the discussion of the issues associated with damping and energy dissipation will be dealt with in section 4.3). From a physical point of view, the 12 degrees of freedom of a pair represent the generalised displacements of i) traction: 1 2 *u u*, (axial displacements), ii) bending: 1 12 2 *vwvw* , ,, (translations in two perpendicular directions of the pinion/gear centre), 1 12 2 ,,, (bending rotations which can be assimilated to misalignment angles) and finally, iii) torsion:

Fig. 1. A 3D lumped parameter model of pinion-gear pair.

with 1 2 *J J*, : the polar moments of inertia of the pinion shaft line and the gear shaft line

The system with 4 unknowns ( 1 2 ,,, *C C m r* ) is characterised by equations (2) - (3) only, and

Modular models based on the definition of gear elements (pinion-gear pairs), shaft elements and lumped parameter elements (mass, inertia, stiffness) have proved to be effective in the simulation of complex gear units (Küçükay, 1987), (Baud & Velex, 2002). In this section, the theoretical foundations upon which classic gear elements are based are presented and the corresponding elemental stiffness and mass matrices along with the possible elemental forcing term vectors are derived and explicitly given. The simplest and most frequently used 3D representation corresponds to the pinion-gear model shown in Figure 1. Assuming that the geometry is not affected by deflections (small displacements hypothesis) and provided that mesh elasticity (and to a certain extent, gear body elasticity) can be transferred onto the base plane, a rigid-body approach can be employed. The pinion and the gear can therefore be assimilated to two rigid cylinders with 6 degrees of freedom each, which are connected by a stiffness element or a distribution of stiffness elements (the discussion of the issues associated with damping and energy dissipation will be dealt with in section 4.3). From a physical point of view, the 12 degrees of freedom of a pair represent the generalised displacements of i) traction: 1 2 *u u*, (axial displacements), ii) bending: 1 12 2 *vwvw* , ,, (translations in two perpendicular directions of the pinion/gear centre), 1 12 2

(bending rotations which can be assimilated to misalignment angles) and finally, iii) torsion:

Fig. 1. A 3D lumped parameter model of pinion-gear pair.

 ,,, 

respectively. , *C C m r* : pinion and gear torques.

2 parameters have to be imposed.

**3.2 Deformed state – Principles** 

1 2 , which are small angles associated with deflections superimposed on rigid-body rotations 1 1 0 *t d* (pinion) and 2 2 0 *t d*(gear). Following Velex and Maatar

(1996), screws of infinitesimal displacements are introduced whose co-ordinates for solid *k* (*conventionally k=1 for the pinion, k=2 for the gear*) can be expressed in two privileged coordinate systems: i) **stz** , , such that **z** is in the shaft axis direction (from the motor to the load machine), **s** is in the centre-line direction from the pinion centre to the gear centre and **t zs** (Fig. 1) or, ii) **XYz** , , attached to the base plane (Fig. 1):

$$\{S\_{\boldsymbol{\lambda}}\} \begin{Bmatrix} \mathbf{u}\_{\mathbf{k}} \left( \mathbf{O}\_{\mathbf{k}} \right) = \boldsymbol{\upsilon}\_{\boldsymbol{\lambda}} \mathbf{s} + \boldsymbol{w}\_{\boldsymbol{\lambda}} \mathbf{t} + \boldsymbol{u}\_{\boldsymbol{\lambda}} \mathbf{z} \\ \mathbf{o}\_{\mathbf{k}} = \boldsymbol{\varphi}\_{\boldsymbol{\lambda}} \mathbf{s} + \boldsymbol{\nu}\_{\boldsymbol{\lambda}} \mathbf{t} + \boldsymbol{\theta}\_{\boldsymbol{\lambda}} \mathbf{z} \end{Bmatrix} \text{or} \begin{Bmatrix} \mathbf{u}\_{\mathbf{k}} \left( \mathbf{O}\_{\mathbf{k}} \right) = \boldsymbol{V}\_{\boldsymbol{\lambda}} \mathbf{X} + \boldsymbol{W}\_{\boldsymbol{\lambda}} \mathbf{Y} + \boldsymbol{u}\_{\boldsymbol{\lambda}} \mathbf{z} \\ \boldsymbol{\mathfrak{o}}\_{\mathbf{k}} = \boldsymbol{\mathfrak{o}}\_{\boldsymbol{\lambda}} \mathbf{X} + \boldsymbol{\Psi}\_{\boldsymbol{\lambda}} \mathbf{Y} + \boldsymbol{\theta}\_{\boldsymbol{\lambda}} \mathbf{z} \end{Bmatrix} \text{ &= 1,2. \tag{4}$$

where 1 2 *O O*, are the pinion and gear centres respectively

#### **3.3 Deflection at a point of contact – Structural vectors for external gears**

Depending on the direction of rotation, the direction of the base plane changes as illustrated in Figure 2 where the thicker line corresponds to a positive rotation of the pinion and the finer line to a negative pinion rotation about axis *O*<sup>1</sup> ,**z** .

Fig. 2. Directions of rotation and planes (lines) of action. (*the thicker line corresponds to a positive rotation of pinion*)

For a given helical gear, the sign of the helix angle on the base plane depends also on the direction of rotation and, here again; two configurations are possible as shown in Figure 3.

Since a rigid-body mechanics approach is considered, contact deflections correspond to the interpenetrations of the parts which are deduced from the contributions of the degrees-offreedom and the initial separations both measured in the normal direction with respect to the tooth flanks. Assuming that all the contacts occur in the theoretical base plane (or plane of action), the normal deflection *M* at any point *M* , potential point of contact, is therefore expressed as:

$$
\Delta(M) = \mathbf{u\_1(M).n\_1 - u\_2(M).n\_1 - \delta e(M)}\tag{5}
$$

On the Modelling of Spur and Helical Gear Dynamic Behaviour 81

are defined in Figure 3; 1

An alternative form of interest is obtained when projecting in the **stz** , , frame attached to

cos sin , cos cos , sin , sin sin sin cos

 

1 1 1 1 112 2 2 2 22

sin cos sin sin , cos , cos sin , cos cos ,

*b p b p b bP b p*

 

 

 

 

*bP b p b b p b p*

 

> 

*vwu*

for a positive rotation of the pinion and 1

 

 <sup>2</sup> sin sin , cos 

 

 

For a given direction of rotation, the usual contact conditions in gears correspond to singlesided contacts between the mating flanks which do not account for momentary tooth separations which may appear if dynamic displacements are large (of the same order of magnitude as static displacements). A review of the mesh stiffness models is beyond the scope of this chapter but one usually separates the simulations accounting for elastic convection (i.e., the deflection at one point *M* depends on the entire load distribution on the tooth or all the mating teeth (Seager, 1967)) from the simpler (and classic) thin-slice approach (the deflection at point *M* depends on the load at the same point only). A discussion of the limits of this theory can be found in Haddad (1991), Ajmi & Velex (2005) but it seems that, for solid gears, it is sufficiently accurate as far as dynamic phenomena such as critical speeds are considered as opposed to exact load or stress distributions in the teeth which are more dependent on local conditions. Neglecting contact damping and friction forces compared with the normal elastic components on tooth flanks, the elemental

*Rb p Rb Rb p*

111 2 2

*b p Rb <sup>b</sup>*

*b b p b p*

 

force transmitted from the pinion onto the gear at one point of contact *M* reads:

over the time-varying and possibly deflection-dependent contact length *L t* ,**q** as:

 

**q**

**M O OM n**

,

*L t*

,

**q**

*L t*

The resulting total mesh force and moment at the gear centre *O*2 are deduced by integrating

**1/2 1**

**F n**

**1/2 2 2 1**

*k M M dM*

*k M M dM*

with *k M* : mesh stiffness at point *M* per unit of contact length

*F*

1/2

 

sin , sin sin sin cos

**3.4 Mesh stiffness matrix and forcing terms for external gears** 

  1 1

**dF M 1/2** *k M M dM* **n1** (11)

 

 

*Rb p*

*<sup>b</sup>* is the base helix angle (*always considered as* 

depending on the sign of the

for a negative rotation of

 

 

 

(10)

(12-1)

where 1 2 *Rb Rb* , are the pinion, gear base radii;

2 2

*<sup>T</sup>* **q** *vwu*

*b p*

*Rb p*

sin cos

 

 

 

*positive in this context)*; 1 2 *p p*, ,

the pinion-gear centre line:

 

 

helix angle; 1 

the pinion.

*T*

**V M**

Fig. 3. Helix angles on the base plane.

where *eM eM eM* max ( ) *<sup>M</sup>* is the equivalent initial normal gap at *M* caused by tooth modifications and/or errors for example, **n1** is the outward unit normal vector to pinion tooth flanks (Fig.3)

Using the shifting property of screws, one obtains the expression of *M* in terms of the screw co-ordinates as:

$$\Delta(M) = \mathbf{u}\_1(\mathbf{O}\_1).\mathbf{n}\_1 + (\mathbf{o}\_1 \times \mathbf{O}\_1 \mathbf{M}).\mathbf{n}\_1 - \mathbf{u}\_2(\mathbf{O}\_2).\mathbf{n}\_1 - (\mathbf{o}\_2 \times \mathbf{O}\_2 \mathbf{M}).\mathbf{n}\_1 - \delta c(M) \tag{6}$$

which is finally expressed as:

$$
\Delta(M) = \begin{bmatrix} \mathbf{n\_1} \\ \mathbf{O\_1M} \times \mathbf{n\_1} \\ -\mathbf{n\_1} \\ -\mathbf{O\_2M} \times \mathbf{n\_1} \end{bmatrix}^T \cdot \begin{bmatrix} \mathbf{u\_1(O\_1)} \\ \mathbf{o\_1} \\ \mathbf{u\_2(O\_2)} \\ \mathbf{o\_2} \end{bmatrix} - \delta \varepsilon(M) \tag{7}
$$

or, in a matrix form:

$$\Delta(M) = \mathbf{V}(\mathbf{M})^{\top}\mathbf{q} - \delta\mathbf{e}(M) \tag{8}$$

where **V M** is a structural vector which accounts for gear geometry (Küçükay, 1987) and **q** is the vector of the pinion-gear pair degrees of freedom (*superscript T refers to the transpose of vectors and matrices*)

The simplest expression is that derived in the **XYz** , , coordinate system associated with the base plane leading to:

$$\begin{aligned} \mathbf{V} \begin{pmatrix} \mathbf{M} \end{pmatrix}^{\top} &= \begin{Bmatrix} \cos \beta\_{\flat \prime} & 0, & \varepsilon \sin \beta\_{\flat \prime} & -\zeta \varepsilon \operatorname{Re} b\_{1} \sin \beta\_{\flat}, & \eta - \varepsilon \operatorname{p}\_{1} \sin \beta\_{\flat}, & \zeta \operatorname{R} b\_{1} \cos \beta\_{\flat} \\ -\cos \beta\_{\flat \prime} & 0, & -\varepsilon \sin \beta\_{\flat \prime} & -\zeta \varepsilon \operatorname{R} b\_{2} \sin \beta\_{\flat}, & -\left[\eta + \varepsilon \operatorname{p}\_{2} \sin \beta\_{\flat}\right], & \zeta \operatorname{R} b\_{2} \cos \beta\_{\flat} \end{pmatrix} \\\\ \mathbf{q}^{\top} &= \begin{Bmatrix} V\_{1} & W\_{1} & u\_{1} & \Phi\_{1} & \Psi\_{1} & \Psi\_{1} & V\_{2} & W\_{2} & u\_{2} & \Phi\_{2} & \Psi\_{2} & \theta\_{2} \end{pmatrix} \end{aligned} \tag{9}$$

**z**

*M*

*eM eM eM* max ( ) *<sup>M</sup>* is the equivalent initial normal gap at *M* caused by

.

*e M*

tooth modifications and/or errors for example, **n1** is the outward unit normal vector to

Using the shifting property of screws, one obtains the expression of *M* in terms of the

*T*

**n u O OM n ω n u O OM n ω**

 *<sup>T</sup> M* **VM q**

where **V M** is a structural vector which accounts for gear geometry (Küçükay, 1987) and **q** is the vector of the pinion-gear pair degrees of freedom (*superscript T refers to the transpose* 

The simplest expression is that derived in the **XYz** , , coordinate system associated with

 

 

 

cos , 0, sin , sin , sin , cos , cos , 0, sin , sin , sin , cos

1 11 1 11 2 2 2 2 22

 

 

*bb b b b bb b bb*

**1 1 1 1 1 1 1 2 2 2 1 2**

*M* **uO n 11 1 1 1 1 22 2 2 1** . .. . **ω OM n u O n** <sup>1</sup> **ω OM n**

**X** <sup>1</sup>

*<sup>b</sup> p*<sup>1</sup>

**X** <sup>2</sup>

**n1**

*p*2

(7)

 

> 

*e M* (8)

 1 11 2 22

 

> 

 *VWu* (9)

 

*Rb p Rb Rb p Rb*

  *e M* (6)

*b*

A contact line

*M*

**n1**

**z**

where

pinion tooth flanks (Fig.3)

which is finally expressed as:

screw co-ordinates as:

or, in a matrix form:

*of vectors and matrices*)

**V M**

the base plane leading to:

*T*

 

 

*<sup>T</sup>* **q** *VWu*

Fig. 3. Helix angles on the base plane.

where 1 2 *Rb Rb* , are the pinion, gear base radii; *<sup>b</sup>* is the base helix angle (*always considered as positive in this context)*; 1 2 *p p*, , are defined in Figure 3; 1 depending on the sign of the helix angle; 1 for a positive rotation of the pinion and 1 for a negative rotation of the pinion.

An alternative form of interest is obtained when projecting in the **stz** , , frame attached to the pinion-gear centre line:

 1 1 111 2 2 2 2 cos sin , cos cos , sin , sin sin sin cos sin cos sin sin , cos , cos sin , cos cos , sin , sin sin sin cos sin cos *T bP b p b b p b p b p b p b bP b p b b p b p b p Rb p Rb p Rb Rb p Rb p* **V M** <sup>2</sup> sin sin , cos *b p Rb <sup>b</sup>* 1 1 1 1 112 2 2 2 22 *<sup>T</sup>* **q** *vwu vwu* (10)

#### **3.4 Mesh stiffness matrix and forcing terms for external gears**

For a given direction of rotation, the usual contact conditions in gears correspond to singlesided contacts between the mating flanks which do not account for momentary tooth separations which may appear if dynamic displacements are large (of the same order of magnitude as static displacements). A review of the mesh stiffness models is beyond the scope of this chapter but one usually separates the simulations accounting for elastic convection (i.e., the deflection at one point *M* depends on the entire load distribution on the tooth or all the mating teeth (Seager, 1967)) from the simpler (and classic) thin-slice approach (the deflection at point *M* depends on the load at the same point only). A discussion of the limits of this theory can be found in Haddad (1991), Ajmi & Velex (2005) but it seems that, for solid gears, it is sufficiently accurate as far as dynamic phenomena such as critical speeds are considered as opposed to exact load or stress distributions in the teeth which are more dependent on local conditions. Neglecting contact damping and friction forces compared with the normal elastic components on tooth flanks, the elemental force transmitted from the pinion onto the gear at one point of contact *M* reads:

$$\mathbf{dF\_{1/2}}(\mathbf{M}) = k(M)\Delta(M)dM\,\mathbf{n\_1} \tag{11}$$

with *k M* : mesh stiffness at point *M* per unit of contact length

The resulting total mesh force and moment at the gear centre *O*2 are deduced by integrating over the time-varying and possibly deflection-dependent contact length *L t* ,**q** as:

$$\begin{cases} \left< F\_{1/2} \right> \begin{cases} \mathbf{F\_{1/2}} = \int\_{\iota(t,\mathbf{q})} k(M) \Delta(M) dM \mathbf{n\_1} \\\\ \mathbf{M\_{1/2}}(\mathbf{O\_2}) = \int\_{\iota(t,\mathbf{q})} k(M) \Delta(M) \mathbf{O\_2} \mathbf{M} \times \mathbf{n\_1} dM \end{cases} \end{cases} \tag{12-1}$$

On the Modelling of Spur and Helical Gear Dynamic Behaviour 83

along with a forcing term associated with inertial forces (whose expression in **XYz** , , has

sin cos cos sin 0 0 0

sin cos cos sin 0 0 0

Examining the components of the structural vectors in (9) and (10), it can be noticed that most of them are independent of the position of the point of contact M with the exception of

> ,

especially for narrow-faced gears so that the mesh stiffness matrix can be simplified as:

*t k M dM k t* **<sup>G</sup>**

where **V**0 represents an average structural vector and *k t* ,**q** is the time-varying, possibly non-linear, mesh stiffness function (scalar) which plays a fundamental role in gear dynamics.

> 2

Considering the torsional degrees-of-freedom only (Figure 4), the structural vector reads

<sup>1</sup>

*V*2

2

1 2 2 02 2 12 2 2

*Rb Rb* 

> 1 :

*b*

 

, 2 02 2

01 1 2 1 12 1

 

*<sup>I</sup> Rb Rb Rb k t I Rb Rb Rb*

*Cm Rb <sup>I</sup> k M e M dM Cr Rb I*

cos *<sup>b</sup>*

2

cos

**VM V0** (19)

(20)

 

1 01 1

*b*

 

 

,

**q**

*k t* ,**q**

*L t*

(17-2)

0 0

, *<sup>T</sup>*

**K V V q G** (18)

gear

. Their influence is usually discarded

2 2 11 1 1 1 1 11 1 1 1 1 01 1

2 2 22 2 2 2 2 22 2 2 2 2 02 2

*m e m e I*

*t me m e I*

**FG**

the same form on the condition that angles 1,2 are measured from **X** and **Y** ):

those related to bending slopes 1,2 or 1,2 1,2

**3.6.1 Classic one-DOF torsional model** 

Fig. 4. Basic torsional model.

(keeping solely the non-zero components):

1

pinion

and the following differential system is derived <sup>2</sup>

 

0

**q**

*L t*

, , cos <sup>0</sup>

**3.6 Usual simplifications** 

Conversely the mesh force wrench at the pinion centre *O*<sup>1</sup> is:

$$\begin{cases} \left\{ \mathbf{F}\_{2/1} \right\} \\\\ \mathbf{M}\_{2/1} \left( \mathbf{O}\_{1} \right) \end{cases} \begin{aligned} \mathbf{F}\_{2/1} &= -\int\_{\iota(t,\mathbf{q})} k(M) \Delta(M) dM \mathbf{n}\_{1} \\\\ \mathbf{M}\_{2/1} \left( \mathbf{O}\_{1} \right) &= -\int\_{\iota(t,\mathbf{q})} k(M) \Delta(M) \mathbf{O}\_{1} \mathbf{M} \times \mathbf{n}\_{1} dM \end{aligned} \tag{12-2}$$

The mesh inter-force wrench can be deduced in a compact form as:

$$\{F\_{\mathcal{M}}\}\begin{Vmatrix}\{F\_{2/1}\}\\\{F\_{1/2}\}\end{Vmatrix} = -\int\_{L(t,\mathbf{q})} k(\mathcal{M})\Delta(\mathcal{M})\mathbf{V}(\mathbf{M})d\mathcal{M} \tag{13}$$

and introducing the contact normal deflection *<sup>T</sup> M* **VM q** *e M* finally leads to:

$$\mathbb{E}\left\{F\_{\mathcal{M}}\right\} = -\left[\mathbf{K}\_{\mathbf{G}}\left(t\right)\right]\mathbf{q} + \mathbf{F}\_{\mathbf{e}}\left(t\right) \tag{14}$$

where , *T L t* **<sup>G</sup>** *t kM dM* **q K VM VM** is the time-varying gear mesh stiffness matrix

 *L t* , *t k M e M dM* **<sup>e</sup> q F V M** is the excitation vector associated with tooth shape

modifications and errors

#### **3.5 Mass matrix of external gear elements–Additional forcing (inertial) terms**

For solid *k* (pinion or gear), the dynamic sum with respect to the inertial frame can be expressed as:

$$\mathbf{L}\_{\mathbf{k}}^{0} = m\_{\mathbf{k}} \left[ \left( \ddot{w}\_{\mathbf{k}} - e\_{\mathbf{k}} \dot{\Omega}\_{\mathbf{k}} \sin \Theta\_{\mathbf{k}} - e\_{\mathbf{k}} \Omega\_{\mathbf{k}}^{2} \cos \Theta\_{\mathbf{k}} \right) \mathbf{s} + \left( \ddot{w}\_{\mathbf{k}} + e\_{\mathbf{k}} \dot{\Omega}\_{\mathbf{k}} \cos \Theta\_{\mathbf{k}} - e\_{\mathbf{k}} \Omega\_{\mathbf{k}}^{2} \sin \Theta\_{\mathbf{k}} \right) \mathbf{t} + \ddot{w}\_{\mathbf{k}} \mathbf{z} \right] \tag{15}$$

where *mk* and *<sup>k</sup> e* are respectively the mass and the eccentricity of solid *<sup>k</sup>*

A simple expression of the dynamic moment at point *Ok* can be obtained by assuming that *Ok* is the centre of inertia of solid *k* and neglecting gyroscopic components (complementary information can be found in specialised textbooks on rotor dynamics (see for instance (Lalanne & Ferraris, 1998)):

$$\mathbf{S}\_{\mathbf{k}}^{0}\left(\boldsymbol{O}\_{\boldsymbol{k}}\right) \equiv I\_{\boldsymbol{k}}\ddot{\boldsymbol{\phi}}\_{\boldsymbol{k}}\mathbf{s} + I\_{\boldsymbol{k}}\ddot{\boldsymbol{\nu}}\_{\boldsymbol{k}}\mathbf{t} + I\_{\boldsymbol{\alpha}k}\left(\dot{\boldsymbol{\Omega}}\_{\boldsymbol{k}} + \ddot{\boldsymbol{\theta}}\_{\boldsymbol{k}}\right)\mathbf{z} \tag{16}$$

where *<sup>k</sup> I* is the cross section moment of inertia and 0*<sup>k</sup> I* is the polar moment of solid k

Using the same DOF arrangement as for the stiffness matrices, a mass matrix for the piniongear system can be deduced as (note that the same mass matrix is obtained in the **XYz** , , coordinate system):

$$\mathbf{[M\_G]} = \mathbf{diag}\left(m\_1, m\_1, m\_1, l\_1, l\_1, l\_{01}, m\_2, m\_2, l\_2, l\_2, l\_2, l\_{02}\right) \tag{17-1}$$

along with a forcing term associated with inertial forces (whose expression in **XYz** , , has the same form on the condition that angles 1,2 are measured from **X** and **Y** ):

$$\mathbf{F\_G(t)} = \begin{pmatrix} m\_1 e\_1 \left( \dot{\Omega}\_1 \sin \Theta\_1 + \Omega\_1^2 \cos \Theta\_1 \right) & -m\_1 e\_1 \left( \dot{\Omega}\_1 \cos \Theta\_1 - \Omega\_1^2 \sin \Theta\_1 \right) & 0 & 0 & 0 & -I\_{\text{off}} \dot{\Omega}\_1 \\ m\_2 e\_2 \left( \dot{\Omega}\_2 \sin \Theta\_2 + \Omega\_2^2 \cos \Theta\_2 \right) & -m\_2 e\_2 \left( \dot{\Omega}\_2 \cos \Theta\_2 - \Omega\_2^2 \sin \Theta\_2 \right) & 0 & 0 & 0 & -I\_{\text{on}} \dot{\Omega}\_2 \end{pmatrix} \tag{17-2}$$

#### **3.6 Usual simplifications**

82 Mechanical Engineering

 

**q**

**M O OM n**

 

*F k M M dM*

,

*L t*

,

*L t*

**2/1 1**

**F n**

**2/1 1 1 1 q**

*k M M dM*

*k M M dM*

*M* **VM q**

**K VM VM** is the time-varying gear mesh stiffness matrix

**F V M** is the excitation vector associated with tooth shape

For solid *k* (pinion or gear), the dynamic sum with respect to the inertial frame can be

A simple expression of the dynamic moment at point *Ok* can be obtained by assuming that *Ok* is the centre of inertia of solid *k* and neglecting gyroscopic components (complementary information can be found in specialised textbooks on rotor dynamics (see for instance

> *OI I I k kk k k k k k*

Using the same DOF arrangement as for the stiffness matrices, a mass matrix for the piniongear system can be deduced as (note that the same mass matrix is obtained in the

where *<sup>k</sup> I* is the cross section moment of inertia and 0*<sup>k</sup> I* is the polar moment of solid k

 <sup>0</sup> **<sup>0</sup> δ<sup>k</sup> st z** (16)

**M diag <sup>G</sup>** *mmmIII mmmI I I* 1 1 1 1 1 01 2 2 2 2 2 02 , , ,,, , , , ,,, (17-1)

2 2 *mve e k k kk k kk k k kk k kk k k* sin cos *we e u* cos sin **<sup>0</sup> <sup>Σ</sup><sup>k</sup> <sup>s</sup> t z** (15)

*F tt <sup>M</sup>* **K qF G e** (14)

(12-2)

**V M** (13)

*e M* finally leads to:

Conversely the mesh force wrench at the pinion centre *O*<sup>1</sup> is:

*F*

where ,

**q**

**<sup>G</sup>** *t kM dM*

*L t*

*t k M e M dM* 

*L t* ,

expressed as:

modifications and errors

(Lalanne & Ferraris, 1998)):

**XYz** , , coordinate system):

 **<sup>e</sup> q**

2/1

 

The mesh inter-force wrench can be deduced in a compact form as:

*F*

*F*

2/1

and introducing the contact normal deflection *<sup>T</sup>*

1/2 ,

*L t*

*T*

**3.5 Mass matrix of external gear elements–Additional forcing (inertial) terms** 

where *mk* and *<sup>k</sup> e* are respectively the mass and the eccentricity of solid *<sup>k</sup>*

 **q**

 

*M*

Examining the components of the structural vectors in (9) and (10), it can be noticed that most of them are independent of the position of the point of contact M with the exception of those related to bending slopes 1,2 or 1,2 1,2 , . Their influence is usually discarded especially for narrow-faced gears so that the mesh stiffness matrix can be simplified as:

$$\left[\mathbf{K}\_{\mathbf{G}}\left(t\right)\right] \equiv \int\_{\iota\left(t,\mathbf{q}\right)} k\left(M\right) dM \ \mathbf{V}\_{\mathbf{o}} \mathbf{V}\_{\mathbf{o}}^{T} = k\left(t,\mathbf{q}\right) \mathbf{G} \tag{18}$$

where **V**0 represents an average structural vector and *k t* ,**q** is the time-varying, possibly non-linear, mesh stiffness function (scalar) which plays a fundamental role in gear dynamics.

#### **3.6.1 Classic one-DOF torsional model**

Fig. 4. Basic torsional model.

Considering the torsional degrees-of-freedom only (Figure 4), the structural vector reads (keeping solely the non-zero components):

$$\mathbf{V(M)} = \mathbf{V\_0} = \begin{bmatrix} \boldsymbol{\zeta} \ R b\_1 \\ \boldsymbol{\zeta} \ R b\_2 \end{bmatrix} \cos \beta\_b \tag{19}$$

and the following differential system is derived <sup>2</sup> 1 :

$$\begin{aligned} \begin{bmatrix} I\_{01} & 0 \\ 0 & I\_{02} \end{bmatrix} \begin{bmatrix} \ddot{\theta}\_{1} \\ \ddot{\theta}\_{2} \end{bmatrix} + k \begin{Bmatrix} t, \theta\_{1}, \theta\_{2} \end{Bmatrix} \cos^{2} \beta\_{b} \begin{bmatrix} Rb\_{1}^{2} & Rb\_{1}Rb\_{2} \\ Rb\_{1}Rb\_{2} & Rb\_{2}^{2} \end{bmatrix} \begin{bmatrix} \theta\_{1} \\ \theta\_{2} \end{bmatrix} = \\ \begin{bmatrix} Cm \\ Cr \end{bmatrix} + \int\_{l(t,\mathbf{q})} k \begin{Bmatrix} M \end{Bmatrix} \delta e \begin{bmatrix} M \end{bmatrix} dM \begin{bmatrix} \check{\zeta} \ Rb\_{1} \\ \check{\zeta} \ Rb\_{2} \end{bmatrix} \cos \beta\_{b} - \begin{bmatrix} I\_{01}\dot{\Omega}\_{1} \\ I\_{02}\dot{\Omega}\_{2} \end{bmatrix} \end{aligned} \tag{20}$$

On the Modelling of Spur and Helical Gear Dynamic Behaviour 85

*m kt k kt kt kt I Rb kt kt kt t*

*Remark:* The system is ill-conditioned since rigid-body rotations are still possible (no unique static solution). In the context of 3D models with many degrees of freedom, it is not

The problem can be resolved by introducing additional torsional stiffness element(s) which

From the results in section 2-5, it can be observed that, in the context of gear dynamic

This function stems from a 'thin-slice' approach whereby the contact lines between the mating teeth are divided in a number of independent stiffness elements (with the limiting case presented here of an infinite set of non-linear time-varying elemental stiffness elements)

Since the positions of the teeth (and consequently the contact lines) evolve with time (or angular positions), the profiles slide with respect to each other and the stiffness varies because of the contact length and the individual tooth stiffness evolutions. The definition of mesh stiffness has generated considerable interest but mostly with the objective of calculating accurate static tooth load distributions and stress distributions. It has been shown by Ajmi and Velex (2005) that a classic 'thin-slice' model is sufficient for dynamic calculations as long as local disturbances (especially near the tooth edges) can be ignored. In this context, Weber and Banascheck (1953) proposed a analytical method of calculating tooth deflections of spur gears by superimposing displacements which arise from i) the contact

; ()

2 2

**M K** (23-2)

*m kt k kt I Rb k t*

,

,

*L t k t k M dM* **q**

1 1

2 02 2

can represent shafts; couplings etc. thus eliminating rigid-body rotations.

simulations, the mesh stiffness function defined as

2 01 1

interesting to solve for the normal approach *Rb Rb* 11 21

**4. Mesh stiffness models – Parametric excitations** 

Fig. 6. 'Thin-slice' model for time-varying mesh stiffness.

**4.1 Classic thin-slice approaches** 

as schematically represented in Figure 6.

**Contact**

as is done for single DOF models.

**q** plays a key role.

 

Note that the determinant of the stiffness matrix is zero which indicates a rigid-body mode (the mass matrix being diagonal). After multiplying the first line in (20) by *Rb I*1 02 , the second line by *Rb I*2 01 , adding the two equations and dividing all the terms by 2 2 10 2 20 1 *I Rb I Rb* , the semi-definite system (20) is transformed into the differential equation:

$$
\hat{m}\ddot{\mathbf{x}} + k\mathbf{(t,x)}\mathbf{x} = \mathbf{F}\_t + \boldsymbol{\zeta}\cos\beta\_b \int\_{\boldsymbol{\omega}(t,x)} k\left(M\right)\delta e\left(M\right)dM - \kappa \frac{d^2}{dt^2} \left(N\mathbf{L}\,T\mathbf{E}\right) \tag{21}
$$

With 11 22 *x Rb Rb* , relative apparent displacement

02 01 2 2 1 02 2 01 *I I m Rb I Rb I* , equivalent mass

2 02 1 2 2 *I Rb* when the pinion speed 1 and the output torque *Cr* are supposed to be

constant.

#### **3.6.2 A simple torsional-flexural model for spur gears**

The simplest model which accounts for torsion and bending in spur gears is shown in Figure 5. It comprises 4 degrees of freedom, namely: 2 translations in the direction of the line of action 1 2 *V V*, (at pinion and gear centres respectively) and 2 rotations about the pinion and gear axes of rotation 1 2 , . Because of the introduction of bending DOFs, some supports (bearing/shaft equivalent stiffness elements for instance) must be added.

Fig. 5. Simplified torsional-flexural spur gear model.

The general expression of the structural vector **V M** (9) reduces to:

$$\mathbf{V\_{o}}^{\top} = \begin{Bmatrix} 1 & \zeta \ R b\_{1} & -1 & \zeta \ R b\_{2} \end{Bmatrix} \tag{22}$$

Re-writing the degree of freedom vector as 1 11 2 22 \* *<sup>T</sup>* **q** *v Rb v Rb* , the following parametrically excited differential system is obtained for linear free vibrations:

$$\mathbf{M}\ddot{\mathbf{q}}\ ^\ast + \mathbf{K}(t)\mathbf{q}\ ^\ast = \mathbf{0} \tag{23-1}$$

$$\mathbf{M} = \begin{bmatrix} m\_1 \\ & I\_{\text{off}}/Rb\_1^2 \\ & & m\_2 \\ & & & I\_{\text{on}}/Rb\_2^2 \end{bmatrix}; \quad \mathbf{K}(t) = \begin{bmatrix} k(t) + k\_1 & \zeta k(t) & -k(t) & \zeta k(t) \\ & k(t) & -\zeta k(t) & k(t) \\ & & k(t) + k\_2 & -\zeta k(t) \\ & & & k(t) \end{bmatrix} \tag{23-2}$$

*Remark:* The system is ill-conditioned since rigid-body rotations are still possible (no unique static solution). In the context of 3D models with many degrees of freedom, it is not interesting to solve for the normal approach *Rb Rb* 11 21 as is done for single DOF models. The problem can be resolved by introducing additional torsional stiffness element(s) which can represent shafts; couplings etc. thus eliminating rigid-body rotations.
