2.1. Gear Mesh Stiffness Determination
Figure 1 represents the overall flowchart of the TVMS evaluation method proposed in this study. The gears used in this investigation are industrial KHK gears of references: KHK SS2-29/KHK SS2-36 (these two gears were chosen because we have them in our laboratory, so we can verify the extracted geometry by the real gears, but the developed method is applicable for any type of gear). In the initialization part of this method, the gear 3D models downloaded from the constructor website are incorporated in a computer-aided design software to extract the gear teeth geometric features (the tooth profile, the tooth width variation, etc.). These features are used then to reconstruct the gear matrices as explained in
Section 2.1.1. In the calculation part or the dynamic part of this flowchart, these matrices are transformed at each simulation step to adjust the position and the angular rotation of their centers; then, the corresponding LOA equation is defined and swept to detect contact points at every iteration. Finally, these points are used to calculate the stiffness at this iteration using Equations (
1)–(
5) and the angular position of the pinion is incremented for the next iteration. The TVMS evaluation steps are described in more detail in the following subsections.
2.1.1. Gear Representation in the Proposed Method
Each gear is modeled with an
matrix as illustrated in
Figure 2, where
N represents the sampling rate of the chosen gear tooth profile and
Z represents the number of gear teeth. Every cell
of this matrix contains the coordinates of the
nth point of the
zth tooth profile
as shown in
Figure 2. We should note here that only the active side of the gear teeth is considered in this representation.
This representation offers a set of advantages compared to the traditional analytical methods, because it allows one to efficiently consider all types of geometrical modifications or errors of the gear teeth. These modifications include: form errors (profile form error, lead form error, etc.) and location errors (single and multiple pitch errors, runout errors, etc.). Global tooth profile errors/modifications are incorporated directly in the tooth profile vector , whereas local profile errors are assigned to the specific tooth index. Runout error could be considered by performing a periodic translation of teeth profile vectors .
The above mentioned errors could be handled by the two-dimensional representation proposed in this study; other high dimensional modifications such as gear tooth crowning, lead form slope, etc., could also be considered in this method by adding slicing operation on the gear width direction as described in [
26].
As explained in the last paragraph, the first step in building a gear matrix is the determination of the gear tooth profile; three different options could be used to do this: analytically, experimentally or numerically. The analytical method is the most used one in the literature; this method uses the mathematical expression of the involute curve and the tooth root to determine the complete tooth profile curve. The experimental method uses reverse engineering tools and techniques to reconstruct the tooth profile curve from a real gear geometry. Finally, numerical methods use dedicated computer-aided design software to extract the gear tooth profile geometry from the 3D model of the corresponding gear.
The third option is adopted in this study because no explicit mathematical expressions of the teeth root shape are available and experimental methods require sophisticated machines and tools to extract gear teeth profiles from real gears. The CAD models of two commercialized gears were downloaded from the constructor website (KHK-Gears) and the corresponding gear tooth shape was extracted using a CAD software (SOLIDWORKS), as shown in
Figure 3.
The main differences between the extracted gear tooth geometry and that of theoretical one are illustrated in
Figure 3. We can see that the gear addendum diameter is modified, the tooth tip is cornered and the gear tooth width is variable along the tooth length direction. These modifications will affect the gears’ contact ratio and the tooth resistance to the applied contact force and, consequently, the TVMS value of the mating gear pair.
The elementary gear tooth profile and side curve are extracted and saved in two vectors, and L. An adaptive resampling of these vectors is applied when needed to increase the resolution and the calculation precision, but one should note that excessively high resolutions will lead to high calculation time and so will compromise the method’s efficiency. Then, iterative rotational transformations are applied to the curve with respect to each tooth’s angular position and the resulted vector is assigned to the corresponding tooth column in the gear matrix. Other gear parameters such as addendum and root diameters will also be saved to be used in the next steps.
2.1.2. The Line of Action Determination
In this section, we will discuss the algorithm used to define the line of action equation for arbitrarily chosen coordinates of pinion and gear centers, respectively. The origin of the global frame used in this study is fixed at the pinion reference center point , so the pinion and gear reference center coordinates are: and .
As shown in
Figure 4, only the gear base circles and centers are required to define the corresponding LOA. Algorithm 1 details the steps followed for the determination of the LOA for varying positions of gear centers:
Algorithm 1 The line of action determination |
1. Define pinion and gear actual positions: |
2. Calculate the pinion-gear center distance: |
3. Determine the intersection point of LOA and the line of center distance:where: |
See Figure 4 for more details. |
4. Calculate the direction of the LOA:where: is the absolute angle of the LOA in the reference frame of study. Not to be confused with the apparent pressure angle as shown in Figure 4. |
5. Define the LOA equation : |
2.1.3. Contact Point Detection
In this section, we will detail the contact point detection algorithm at each iteration. Let be the pinion matrix and the gear matrix; this process is realized following these steps:
Set the pinion angular position at time
t:
where
is the rotation transformation matrix defined as:
Update the position of pinion and gear centers:
Determine the line of action equation:
Use the algorithm described in
Section 2.1.2 to define the line of action equation.
Determine the potential contact points:
Sweep the line of action to determine its intersection points with pinion and gear teeth, respectively. Two sets of points are defined at this step: the first set contains the intersection points of the LOA with the pinion teeth and the second one contains the intersection points of the LOA with the gear teeth.
To optimize the exploration time, this operation is limited to the active portion of the line of action. We can prove geometrically that this part is limited by the two addendum circles of the pinion and gear, respectively.
Rearrange the intersection points sets:
Use the Euclidean distance to rearrange the two sets of intersection points by cross-comparison; note that the two sets are not necessarily of equal size. This step aims to extract the potential contact point pairs.
Rotate the gear matrix until contact between gear teeth occurs:
Use a penalization value
to decide whether the points extracted in the previous step are contact points or not (compare the Euclidean distance of potential contact points to
). This penalization value will depend on the resolution used when building the gear matrices and the line of action. If no contact points are detected, the gear matrix is rotated gradually until contact occurs (the penalization constraint is satisfied) as follows:
repeat until |
where
depending on the nearest intersection point location and
is the number of contact points.
The gear rotation direction
is chosen based on the position of the intersection points determined previously. The gear rotation step
should be refined properly to avoid contact jumping, but very small steps could lead to very high computation time. A sufficient condition to avoid this jumping phenomenon is:
where
is the gear head radius.
Define contact points:
Once contact points at the current iteration are determined, their corresponding indexes, coordinates and the current pressure angle are sent to the mesh stiffness calculation algorithm to determine the corresponding stiffness components.
Update the pinion angular position and return to 1:
Increment the pinion angular position and repeat the above described process to determine the contact points at the next iteration.
The algorithm proposed here to locate contact points is not limited to center distance and eccentricity errors only, but it could be used for any positions of pinion and gear centers. Consequently, this algorithm could be used in an online determination of TVMS of gears when simulating the overall dynamic model of the gear system using the positions of pinion and gear centers determined by the system dynamic model integration.
2.1.4. Gear Mesh Stiffness Components
In this section, the outputs of the contact detection algorithm are used in order to calculate the corresponding components of the gear teeth stiffness. In this calculation, the extracted gear tooth geometry is used for the integration of the different stiffness components.
In the potential energy method, the gear tooth is considered as a non-uniform cantilever beam subjected to the contact force
F as shown in
Figure 5 and the beam theory is applied to calculate the different energies stored in the gear tooth structure. Then, the corresponding stiffnesses are derived from these energies.
The stiffness components considered in this study are the Hertzian stiffness caused by the local deflection at the contact point position. The tooth bending and shearing stiffnesses are caused by the vertical component of the contact force . The tooth axial compression stiffness is caused by the horizontal component of the contact force , and the fillet-foundation stiffness is caused by the gear body torsional deformation.
Based on the Hertzian theory, the potential energy method and the work carried out in [
27], the mathematical expressions of these components are given as follows:
where
are the material Young modulus, shear modulus and Poisson coefficient, respectively, and
are the tooth section area and moment of inertia, respectively, at coordinate
x.
is the gear foundation width and the coefficients
are detailed in [
27].
These stiffness elements are evaluated using the discrete trapezoidal integration method. Then, the total mesh stiffness
at each iteration is determined as follows:
where
is the total number of contact points,
i refers to the contact point index and
j to the pinion and gear.
2.2. Electromechanical Model of a Motor-Gearbox System
To study the effect of assembly errors on the response of a motor-gear system, the electromechanical model of a three-phase induction motor and a one-stage spur gearbox is developed in this section.
A schematic representation of the induction motor is given in
Figure 6. This motor is composed of three phases of windings
on the stator spaced by
and three others on the rotor
. The stator phases are supplied by three-phase sinusoidal voltages at constant frequency and amplitudes, and the rotor windings are short-circuited. The adopted model is based on the following assumptions: proportionality of flux to currents, the influence of skin effect and heating is not taken into account, constant air gap, magneto motor forces are represented with sinusoidal spatial distribution and currents other than in windings are neglected [
5].
The application of the fundamental laws of electromagnetic induction gives the following relationships for all windings [
28,
29]:
With and are the resistance of the stator and rotor winding, respectively. , instantaneous voltages across the stator phases. , instantaneous currents flowing in its phases. the stator and rotor fluxes.
We observe that each flux interacts with the currents of all the phases including its own. If we take the stator flux on phase “a”, the latter can be expressed as follows:
where:
is the maximum mutual inductance between a phase of the stator and a phase of the rotor.
is the angle between the stator frame and the rotor frame. The mutual inductances of the model are dependent on the rotation angle. The Park transformation will be used in order to project the three phases of the windings
of the machine on a frame with two orthogonal two-phase winding
. The purpose of this transformation is to make the mutual inductances independent of the rotation angle. Only the voltage equations on the direct and quadrature axes are used to define the electrical and dynamic model of the induction motor. The equation system constituting the electrical and dynamic model of the induction motor in an equivalent two-phase frame is written as follows:
with:
is the stator self-inductance.
is rotor self-inductance.
is the mutual inductance between stator and rotor.
: the electrical angle between the
d axis and the stator.
: the electrical angle between the
q axis and the rotor. It is therefore possible to estimate from the two preceding equations the stator currents (
) as well as the rotoric fluxes (
), which will subsequently be used for the estimation of the electromagnetic torque:
with
being the stator pulsation and
,
being the total dispersion coefficient;
: rotor time constant and
: the stator time constant.
The electromagnetic torque can then be expressed as follows:
With p being the number of pole pairs.
Then, Newton’s second law is employed to derive the spur gear dynamic model of
Figure 7. For simplification, the LOA direction is assumed to be unchanged in this part of the model, which is justified because we have
in this study. The reference frame of study is chosen such as the
y axis is aligned with the line of action direction as shown in
Figure 7, where
is the rotation center and
is the geometric center of the gear
i.
The equations of motion of the proposed one-stage gearbox are given as follows [
9]:
where
is the centrifugal force and
is the inertial force as represented in
Figure 7. The elastic and damping mesh forces
and
are expressed as follows:
One can see that the coupling between the mechanical and the electrical models is realized through the motor rotation speed and the electromagnetic torque . The proposed model in this study is more precise for simulating the effect of assembly errors because models proposed in the literature often consider only the centrifugal and the inertial efforts induced by the eccentricity and neglect the TVMS deformation or employ approximate formula of this parameter. Additionally, a synchronization is necessary between the centrifugal and inertial fault directions and the TVMS value at each rotation angle; to solve this problem, the TVMS value is injected following the angular positions of the pinion and gear, respectively. This model is general, and one can use the healthy model simply by setting the eccentricities value to zero.