Tooth-meshing-harmonic static-transmission-error amplitudes of helical gears

Mechanical Systems and Signal Processing 98 (2018) 506–533

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Tooth-meshing-harmonic static-transmission-error amplitudes of helical gears William D. Mark Applied Research Laboratory and Graduate Program in Acoustics, The Pennsylvania State University, University Park, PA 16802, USA

a r t i c l e

i n f o

Article history: Received 20 December 2016 Received in revised form 17 April 2017 Accepted 25 April 2017

Keywords: Gear transmission errors Helical gears Tooth-meshing-harmonics Tooth modifications

a b s t r a c t The static transmission errors of meshing gear pairs arise from deviations of loaded tooth working surfaces from equispaced perfect involute surfaces. Such deviations consist of tooth-pair elastic deformations and geometric deviations (modifications) of tooth working surfaces. To a very good approximation, the static-transmission-error tooth-meshingharmonic amplitudes of helical gears are herein expressed by superposition of Fourier transforms of the quantities: (1) the combination of tooth-pair elastic deformations and geometric tooth-pair modifications and (2) fractional mesh-stiffness fluctuations, each quantity (1) and (2) expressed as a function of involute ‘‘roll distance.” Normalization of the total roll-distance single-tooth contact span to unity allows tooth-meshing-harmonic amplitudes to be computed for different shapes of the above-described quantities (1) and (2). Tooth-meshing harmonics p = 1, 2, . . . are shown to occur at Fourier-transform harmonic values of Qp, p = 1, 2, . . ., where Q is the actual (total) contact ratio, thereby verifying its importance in minimizing transmission-error tooth-meshing-harmonic amplitudes. Two individual shapes and two series of shapes of the quantities (1) and (2) are chosen to illustrate a wide variety of shapes. In most cases representative of helical gears, toothmeshing-harmonic values p = 1, 2, . . . are shown to occur in Fourier-transform harmonic regions governed by discontinuities arising from tooth-pair-contact initiation and termination, thereby showing the importance of minimizing such discontinuities. Plots and analytical expressions for all such Fourier transforms are presented, thereby illustrating the effects of various types of tooth-working-surface modifications and tooth-pair stiffnesses on transmission-error generation. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction The static transmission error (STE) is widely recognized to be the dominant source of vibration excitation caused by meshing gear pairs [1–13]. A fictitious pair of meshing gears with equispaced rigid perfect involute tooth-workingsurfaces would yield zero transmission error. But under loading, the teeth of real gears elastically deform [14–20]; moreover, real gear teeth have spacing and other manufacturing errors [21–23]. Therefore, in order to avoid impact loading and transmission-error step (jump) discontinuities, tooth working surfaces entering and leaving the mesh generally are modified by removal of material from otherwise perfect involute surfaces. In the case of helical gears, such modifications take the form of some sort of ‘‘end relief,” ‘‘crowning,” ‘‘generated engagement relief,” or ‘‘bias” modifications, etc. [24–39]. All such

E-mail address: [email protected] http://dx.doi.org/10.1016/j.ymssp.2017.04.039 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.

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modifications and tooth/gearbody elastic deformations generate STE contributions which can serve as excitations in gearing dynamics studies [12,40–42]. Moreover, tooth sliding friction is an additional source of vibration excitation [43–45]. Transmission-error analyses often are carried out in the frequency domain. For accurately manufactured gears, the dominant STE harmonic contributions are the tooth-meshing harmonics, which are caused by tooth/gearbody elastic deformations and geometric deviations from perfect involute surfaces of the averaged working surfaces of the teeth on each of the meshing gears [46, p. 112]. A formulation is provided herein to enable further understanding of the roles of tooth elastic deformations, tooth-working-surface modifications, and mesh stiffness fluctuations on STE tooth-meshing-harmonic contributions of helical gears. In the following pages it is shown that the STE tooth-meshing-harmonic contributions, from a meshing helical gear pair, can be decomposed into two components: (1) the combination of tooth-pair elastic deformations and stiffness-weighted geometric tooth-pair modifications and (2) individual tooth-pair stiffnesses. The Fourier series amplitudes of the STE tooth-meshing harmonics then are shown to be determined by differences of the Fourier transforms of the abovementioned two components. Normalization of the total contact span of individual mating tooth pairs to unity allows different shapes of the above-mentioned components (1) and (2) to be studied, illustrating their STE contributions for any (total) contact ratio. A significant motivation for the study is determining how discontinuities in the above two components (1) and (2), encountered at tooth-pair contact initiation and termination, affect transmission error amplitudes. Two individual models and two series of models, applicable to either of the above-mentioned two components, illustrate a range of discontinuities to accomplish this goal. In particular, it is suggested how different types of tooth-pair modifications are likely to affect such discontinuities, including possible approaches to minimize their transmission error contributions. Apart from the insight provided by these examples, the simplicity of the final analytical formulation, provided in Section 8, should be useful in other studies and in understanding experimental results. Following [4,5], an ‘‘exact” derivation of static-transmission-error contributions is provided in [46], which is the starting point for the work. It is shown that the combined transmission-error contribution from tooth-pair elastic deformations and working-surface deviations can be expressed as a ‘‘rep function” [47, p. 28,46, p. 229] of the individual tooth-pair contributions, thereby allowing the transmission-error Fourier series amplitudes of this combined tooth-pair elastic-deformation/working-sur face-deviation contribution to be expressed by the Fourier transform of this combined contribution evaluated at ‘‘frequency” locations Qp, where Q is the actual (total) contact ratio, and p = 1, 2, . . . are the transmission-error tooth-meshing harmonics, thereby providing a direct relationship between this source of the transmission-error tooth-meshing harmonics and the tooth-meshing-harmonic amplitudes. After normalizing tooth-pair roll-distance spans to unity, two different shapes and two series of shapes of this combined contribution are delineated, and their Fourier transforms computed, thereby showing the relationships between the tooth-working-surface sources of the transmission error and transmission-error tooth-meshingharmonic amplitude contributions from this source. In most of these examples, tooth-meshing-harmonic amplitudes are shown to fall in the harmonic regions controlled by tooth-pair-contact initiation and termination, thereby illustrating the dominant importance of discontinuities at tooth contact initiation and termination, and of actual (total) contact ratios. Following the above-described treatment, it is shown that the mesh-stiffness also can be expressed as a ‘‘rep function” of individual tooth-pair stiffnesses, thereby allowing the fractional mesh-stiffness-fluctuation transmission-error Fourier series contribution to be expressed by the Fourier transform of the stiffness of individual mating tooth pairs evaluated at ‘‘frequency” locations Qp, where Q is the actual (total) contact ratio, and p = 1, 2, . . . are transmission-error tooth-meshing harmonics, as above. Each of the two series of shapes mentioned above also can be suitable for representing the tooth-pair stiffness contributions. The final transmission-error Fourier series representation then is obtained by combining the contributions from the two above-described sources. The resultant model is representative of the physical generation of transmission-error toothmeshing-harmonic contributions of helical gears. 2. Lineal transmission error of parallel-axis gear pairs Fig. 1 illustrates a meshing pair of parallel-axis helical or spur gears with rigid equispaced perfect involute teeth. Such a gear pair would transmit an exactly constant speed ratio. A single independent variable x, roll distance [8,46], can be used to designate the rotational position of the two meshing gears, ð1Þ

ð2Þ

x,Rb hð1Þ ¼ Rb hð2Þ ; ð1Þ

ð1Þ

ð2Þ

where Rb and Rb denote the base-cylinder radii of the two perfect gears (1) and (2), and hð1Þ and hð2Þ denote their instantaneous rotational positions. Let dhð1Þ ðxÞ and dhð2Þ ðxÞ denote the instantaneous rotational deviations of gears (1) and (2), respectively, from the rotational positions hð1Þ and hð2Þ of their perfect involute counterparts. Assume the gear shafts remain parallel and fixed. Define the lineal transmission error, fðxÞ; as the amount the teeth come together in the plane of contact, as in elastic deformations, relative to their perfect involute counterparts. Then [8,46], the lineal transmission error fðxÞ is ð1Þ

ð2Þ

fðxÞ,Rb dhð1Þ ðxÞ  Rb dhð2Þ ðxÞ; where the negative sign in Eq. (2) arises from the sign convention of hð2Þ illustrated in Fig. 1.

ð2Þ

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Fig. 1. Helical or spur gear pair with rigid equispaced perfect involute teeth.

Fig. 2 illustrates the perfect meshing gear pair, where the upper projection shows the lines of tooth contact in the plane of contact (plane of action). These lines of contact, shown dashed, can be envisioned as inscribed on a fictitious belt drive riding on the base cylinders of the two meshing gears. As the gears rotate and independent variable x increases, the lines of tooth contact progress thru the zone of contact. The lines of contact are shown solid when within the nominal zone of contact. Whenever the teeth are in full contact within the zone of contact, it is evident from Eq. (2) and Figs. 1 and 2 that the transmission error arises from the collective deviations of the tooth working surfaces of all teeth in contact within the zone of contact. These deviations consist of geometric deviations of the unloaded tooth working surfaces and tooth elastic deforma-

Fig. 2. Perfect involute helical gear pair. Upper figure shows lines of tooth-pair contract in plane of contact.

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tions. For spur gears, wb ¼ 0 in Fig. 2, and the lines of contact are parallel to the sides of the zone of contact. As the gears rotate and independent variable x increases, the transmission error fðxÞ fluctuates. When operating at constant speed and transmitting constant loading, the transmission error fðxÞ is a periodic function of independent variable x. Unless the stiffness properties of every meshing tooth pair is identical to that of every other meshing tooth pair and the geometric deviations of the individual tooth working surfaces from equispaced perfect involute surfaces on each gear of the pair are identical, there generally are four sets of harmonic contributions arising from a meshing gear pair: (1) generally weak harmonics of the meshing gear pair [48], somewhat stronger rotational-harmonic contributions from each gear of the pair [46, pp. 110–113], and tooth-meshing harmonics of the gear pair. The generally weak harmonics of the gear pair and rotational harmonics from each gear of the pair arise from tooth-to-tooth variations of the elastically deformed working surfaces of each of the two meshing gears, including tooth-to-tooth geometric variations of the working surfaces. The usually dominant tooth-meshing harmonic contributions arise from the mean deviation of the elastically deformed tooth working surfaces from equispaced perfect involute surfaces of each of the two gears of the meshing pair [46, pp. 110–113]. If the gear pair is rotating at a sufficiently slow speed so that inertial effects are negligible, the resultant transmission error is the ‘‘static transmission error,” STE, which is generally accepted to be the dominant source of vibration excitation arising from a meshing gear pair. The tooth-meshing harmonics of the STE, which arise from the mean deviation of the loaded tooth working surfaces, are the harmonic contributions addressed in this paper. An ‘‘exact” expression for the STE is carefully derived in [5,46]. Utilizing the notation of [46, pp. 168–170], let K Tj ðx; yÞ denote the local stiffness per unit length of line of contact of mating tooth-pair j as defined in [46, Appendix 7.A], and

gðÞ j ðx; yÞ denote the geometric deviation from a perfect involute surface of the working surface of tooth j of gear (), () = (1) or (2), of the mating tooth pair j, where the direction of this geometric deviation is ‘‘measured” in the direction defined by the intersection of the plane of contact and transverse plane (plane of the paper of Fig. 2), with positive contributions of ðÞ

deviations gj ðx; yÞ provided by removal of material from involute surfaces as in tooth profile or end reliefs. It follows from Figs. 2 and 3 that for a generic tooth-pair j, specification of coordinate values x, y designates a unique point of contact on the working surfaces of the tooth-pair [46, p. 168]. Denote the integral over the line of contact of the locally-stiffness-weighted working-surface geometric deviation of tooth j of gear () by [46, Eq. (7.11)],

ge ðÞ Kj ðxÞ, sec wb

Z

yBj ðxÞ yAj ðxÞ

ðÞ

K Tj ðx; yÞgj ðx; yÞdy;

ð3Þ

where differential length along the line of contact is d‘ ¼ sec wb dy, and the limits of integration designate the true endpoints of the line of contact determined by the actual contact region (generally not a perfect rectangle as in Fig. 2). The locally-stiffnessweighted geometric deviations of mating teeth j from both meshing gears (1) and (2) can be combined into a single term

ge Kj ðxÞ, sec wb

Z

yBj ðxÞ yAj ðxÞ

ð1Þ

ð2Þ

K Tj ðx; yÞ½gj ðx; yÞ þ gj ðx; yÞdy:

ð4Þ

The total stiffness of tooth pair j is the integral of the local tooth-pair stiffness K Tj ðx; yÞ over the full line of contact of the tooth pair [46, Eq. (7.10)],

e Tj ðxÞ, sec w K b

Z

yBj ðxÞ

yAj ðxÞ

K Tj ðx; yÞdy:

ð5Þ

The force W j ðxÞ transmitted by tooth pair j is [46, Eq. (7.9)]

e Tj ðxÞ  g e Kj ðxÞ W j ðxÞ ¼ fðxÞ K

ð6Þ

e Tj ðxÞ and g e Kj ðxÞ are given by Eqs. (5) and (4) above. The total mesh stiffness K M ðxÞ is the sum of the local stiffnesses of where K the contacting regions of the individual tooth pairs,

Fig. 3. Axial projection showing lines of tooth-pair contact on tooth working surface.

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K M ðxÞ ¼

X

e Tj ðxÞ: K

ð7Þ

j

The total force transmitted by the mesh is the sum of the forces, Eq. (6), transmitted by the individual tooth pairs,

X

W j ðxÞ ¼ fðxÞ

j

X X K Tj ðxÞ  ge Kj ðxÞ; j

ð8Þ

j

and by substitution of Eq. (7),

X

W j ðxÞ ¼ fðxÞK M ðxÞ 

X

j

ge Kj ðxÞ:

ð9Þ

j

Dividing Eq. (9) by the total mesh stiffness K M ðxÞ and solving for fðxÞ yields an expression [46, Eq. (7.15)] for the static transmission error, fðxÞ,

P fðxÞ ¼

e Kj ðxÞ þg K M ðxÞ

j ½W j ðxÞ

ð10Þ

e Kj ðxÞ and K M ðxÞ are defined by Eqs. (4) and (7), respectively, and the summation over j includes all tooth-pairs j in where g contact. ðÞ e Kj ðxÞ ¼ 0, Eq. (10) If there were no geometric deviations gj ðx; yÞ of the working surfaces of the teeth, and therefore g would attribute the fluctuating transmission error only to tooth elastic deformations. However, positive contributions of working surface geometric deviations, as in tooth tip and/or end reliefs, yield an additional positive contribution to the transe Kj ðxÞ, Eq. (4), in Eq. (10). mission error expressed by the term g 

The dimension of the numerator in Eq. (10) is force, but gKj ðxÞ is not a ‘‘real” force, it is the absence of a working surface force caused by positive geometric working surface deviations, which are positive for removal of material from the involute surfaces. The real tooth forces are described by Eq. (6). As long as good tooth contact is maintained, the representation of the transmission error given by Eq. (10) is essentially exact. 3. Decomposition of transmission-error contributions Gears normally are manufactured with an effort to geometrically modify the working surfaces of all teeth on each gear exactly the same. If this assumption is satisfied for each of gears (1) and (2) of a meshing pair, then the combined modification of each mating tooth pair j in Eq. (4) is the same, enabling the tooth modifications from the two gears to be combined into a single combined modification, ð2Þ gj ðx; yÞ,gð1Þ j ðx; yÞ þ gj ðx; yÞ;

ð11Þ

e Kj ðxÞ in Eq. (4), yielding for g

ge Kj ðxÞ ¼ sec wb

Z

yBj ðxÞ yAj ðxÞ

K Tj ðx; yÞgj ðx; yÞdy;

ð12Þ

which is the same for all mating tooth pairs j. To a very good approximation, the local stiffness of each tooth pair j, K Tj ðx; yÞ, is the same for all tooth pairs. Hence, when the working-surface modification on every tooth of each gear of the meshing pair is the same (and the gears have no manufacturing errors), the meshing action of the gear pair is periodic with period D, the tooth-spacing on the base circle. When these approximations are satisfied, every term in Eq. (10) is periodic with period D, and the resultant harmonic contributions of the transmission error, fðxÞ, are the tooth-meshing harmonics. At this juncture, it is convenient to decompose the mesh stiffness into its mean component K M and its fluctuating component dK M ðxÞ,

K M ðxÞ ¼ K M þ dK M ðxÞ;   dK M ðxÞ : ¼ KM 1 þ KM Then,

( )  1  2  3 1 1 dK M ðxÞ 1 dK M ðxÞ dK M ðxÞ dK M ðxÞ ¼ 1  þ  þ    ; ¼  1þ  M M K M ðxÞ K M KM KM KM K K       dK M ðxÞ     < 1  1 1  dK M ðxÞ ; dK M ðxÞ  1  K  K M  M M M  K K

ð13aÞ ð13bÞ

ð14Þ

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where this last approximation is surprisingly accurate even for spur gears [46, p. 197]. Hence, by using the approximation, Eq. (14), the transmission error, Eq. (10), can be expressed as

fðxÞ  f1 ðxÞ þ dfðxÞ;

ð15Þ

where

P

e Kj ðxÞ þg ;  KM

j ½W j ðxÞ

f1 ðxÞ ¼

ð16Þ

and

dfðxÞ ¼ f1 ðxÞ

dK M ðxÞ KM

:

ð17Þ

The transmission error contribution from Eq. (16) will be treated first, and the contribution from Eq. (17) will be treated later in the paper. 4. Constant-mesh-stiffness transmission-error contribution For the tooth-meshing harmonics, every term in the numerator of Eq. (16) is periodic with period D, the base pitch. Therefore, it can be expressed as

P1

j¼1 ½W j ðx

f1 ðxÞ ¼

e Kj ðx  jDÞ  jDÞ þ g :  KM

ð18Þ

The right-hand side of Eq. (18) has the form of a rept function [47,46, p. 229]. Define the complex Fourier series coefficients of the transmission-error contribution f1 ðxÞ by

ap ½f1 ,

1 D

  i2ppx dx; f1 ðxÞ exp  D D2

Z

D 2

p ¼ 0; 1; 2; . . .

ð19Þ

e Kj ðxÞ by which is periodic with period of the base pitch, and the Fourier transforms of W j ðxÞ and g

Z

Fx ½W j ; g,

1 1

W j ðxÞexpði2pgxÞdx

ð20Þ

g~ Kj ðxÞexpði2pgxÞdx:

ð21Þ

and

~ Kj ; g, Fx ½g

Z

1

1

Then, it follows directly from the rept function, Eq. (18), and [46, p. 229] that the Fourier series coefficients of the e Kj ðxÞ by transmission-error contribution f1 ðxÞ are related to the Fourier transforms of W j ðxÞ and g

ap ½f1  ¼

~ Kj ; p=D Fx ½W j ; p=D þ Fx ½g KMD

;

p ¼ 0; 1; 2; . . . :

ð22Þ

If we further define

ap ½W j ,

Fx ½W j ; p=D ; D

p ¼ 0; 1; 2; . . .

ð23Þ

ap ½g~ Kj ,

~ Kj ; p=D Fx ½g ; D

p ¼ 0; 1; 2; . . . ;

ð24Þ

and

then Eq. (22) can be more concisely written as

ap ½f1  ¼

ap ½W j  þ ap ½ ge Kj  M K

;

p ¼ 0; 1; 2; . . . ;

ð25Þ

which expresses the relationship (22) using Fourier-series-coefficient notation. Explanation of Eq. (22): We have defined the transmission error as positive when the teeth ‘‘come together” relative to their perfect involute counterparts. There are two contributions to the constant-mesh-stiffness transmission-error contribution, f1 ðxÞ : a direct contribution from geometric deviations (modifications) of tooth working surfaces, defined as positive by removal of material, and a second contribution from tooth elastic deformations. The second term in the numerator of Eq. (22) arises from the geometric modification contribution, and the first term arises from the elastic deformation contribution.

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W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533

Notice that the stiffness associated with this elastic deformation contribution is the mean total mesh stiffness, not the individual tooth-pair stiffness. Referring to Eq. (18), f1 ðxÞ expresses the summation of periodic non-negative lagged displacements from these two sources. Because tooth-working-surface modifications control the behavior of tooth-pair loading, and therefore, of elastic deformations, it is sensible to combine these two displacement contributions into a single contribution, while recognizing the two sources of this contribution. Therefore, define

e ðxÞ W ðxÞ g g0j ðxÞ, j þ Kj ; KM KM

ð26Þ

which is the combined contribution from these two displacement sources. Their combined contribution is illustrated in Fig. 4   M denotwhere we have used the abbreviations u ¼ W j ðxÞ=K M denoting the elastic deformation contribution and g ¼ gKj ðxÞ=K ing the geometric modification contribution. The sum of these two contributions is a non-constant function of roll distance x. From Eqs. (19)–(26), the complex Fourier series coefficients of the transmission-error contribution f1 ðxÞ, Eq. (18), are obtained from the Fourier transform of g0j ðxÞ,

Fx ½g0j ; g,

Z

1

1

g0j ðxÞexpði2pgxÞdx

ð27Þ

by

ap ½f1  ¼

Fx ½g0j ; p=D

D

;

p ¼ 0; 1; 2; . . . :

ð28Þ

4.1. Scaling As a gear pair rotates, a single pair of mating teeth remains in contact for an interval of roll-distance x, Eq. (1). That interval of roll-distance is conventionally normalized by the base pitch D, and therefore can be expressed as QD as in Fig. 4, where Q is the actual (total) contact ratio. If the tooth contact region in the plane of contact were a perfect rectangle then Q ¼ Q a þ Q t , which is the sum of the axial Q a and transverse Q t contact ratios [46, p. 115]. Unless the actual contact region in the plane of contact is a perfect rectangle, the actual contact ratio Q is less than Q a þ Q t . We wish to understand how differing shapes of the combined displacement function g0j ðxÞ, Eq. (26), will affect the toothmeshing harmonic transmission-error contributions. This can be accomplished by normalizing to unity the roll-distance interval QD. Therefore, the Fourier transform, Eq. (27), is

Fx ½g0j ; g ¼

Z

QD 2

Q2D

g0j ðxÞexpði2pgxÞdx:

ð29Þ

Define n, QxD ; hence dx ¼ Q Ddn: Then, from Eq. (29),

Fx ½g0j ; g ¼ Q D

Z

1 2

12

g0j ðQ DnÞexpði2pgQ DnÞdn;

ð30Þ

and, from Eq. (28), the tooth-meshing harmonic contributions p ¼ 0; 1; 2;    from the combined displacement, Eq. (26), are

Z

ap ½f1  ¼ Q

1=2

1=2

g0j ðQ DnÞexpði2pQpnÞdn; p ¼ 0; 1; 2;    :

 M . g0 ðxÞ varies with x. e Kj ðxÞ=K Fig. 4. Sketch of combined deviation g0j ðxÞ, Eq. (26), where u ¼ W j ðxÞ=K M and g ¼ g j

ð31Þ

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Hence, when the total contact span QD of mating tooth pairs is normalized to unit span in roll-distance x, the resultant normalized Fourier transform of g0j ðxÞ is to be evaluated at frequencies g = Qp, and its amplitude is to be multiplied by Q. Consequently, the tooth-meshing harmonics p = 1, 2, . . . are to be evaluated at frequencies Q, 2Q, . . ., which are integer multiples of the actual contact ratio Q. Moreover, for a fixed shape of the combined displacement contribution g0j ðxÞ, Eq. (26), with rolldistance span QD normalized to unity as in Eq. (31), the tooth-meshing-harmonic contributions to the transmission-error component f1 ðxÞ will, on the average, become smaller with increasing actual total contact ratio Q, provided that the Fourier transform, Eq. (31), of the normalized combined displacement g0j ðQ DnÞ decays more rapidly than 1/Q, which will be seen next to generally be the case. 4.2. Tooth contact initiation and termination It is known [49,50] that the high-frequency asymptotic behavior of Fourier transforms is governed by the lowest-order discontinuity of the transformed functions. For accurately manufactured gears, such discontinuous behavior normally takes place only at tooth contact initiation and termination, which is controlled by the initiation and termination behavior (ende Kj ðxÞ, Eq. (26), with resultant tooth-meshingpoint behavior) of tooth-pair loading W j ðxÞ and working-surface modifications g harmonic Fourier series coefficients given by Eq. (31). It will be seen that these tooth-meshing-harmonic amplitudes generally occur near or within the high-frequency asymptotic region of the Fourier transforms, Eq. (31), governed by tooth-contact initiation and termination behavior. 5. Tooth-pair working-surface modification examples As a first example, consider tooth-pair modifications from involute surfaces that are constant along each possible line-ofcontract location (Fig. 3), except for minor tip rounding, but with axial crowning at each radial z location. Such axial crowning is necessary to avoid jump (i.e. step) discontinuities at tooth-pair contact initiation and termination. Suppose the resultant tooth-pair crowning is symmetric in the axial y-direction about the centered dashed-line location illustrated in Fig. 3. When such a gear pair is transmitting a non-negligible loading, to a good approximation tooth-pair contact initiation and termination would take place simultaneously along full lines of contact, because tooth-pair modifications have been assumed to be e Kj ðxÞ; Eq. (12), represents an integral over the full line of contact (of the constant on line-of-contact locations. Recalling that g stiffness-weighted tooth-pair modifications), we would expect the combined displacement g0j ðxÞ, Eq. (26), at contact initiation and termination, to initially increase linearly, and terminate linearly, where the slopes of g0j ðxÞ at tooth-pair contact initiation and termination would be dependent on the axial-direction slope of the tooth-pair modification at tooth-pair contact initiations and terminations. 5.1. Half-period cosine-function model Setting the amplitude of g0j ðxÞ, Eq. (26), to be unity at roll-distance x = 0 axially centered on a tooth, a simple function satisfying the above description is the half-period cosine function



g0j ðxÞ ¼ cos

px QD

 ;



QD QD 6x6 ; 2 2

ð32Þ

where QD is the roll-distance tooth-pair contact span, as described above. In Appendix A, the Fourier transform, Eq. (29), of

g0j ðxÞ, Eq. (32), is shown to be

p Q D cosðpQ DgÞ Fx ½g0j ; g ¼ 2 2 ; p  ðpQ DgÞ2

ð33Þ

2

hence, from Eq. (28), the tooth-meshing-harmonic Fourier series coefficients of the transmission-error contribution f1 ðxÞ, Eq. (18), are pQ

cosðpQpÞ

2

 ðpQpÞ2

ap ½f1  ¼ 2p2

;

p ¼ 0; 1; 2; . . . ;

ð34Þ

p ¼ 0; 1; 2; . . .

ð35Þ

and therefore,

ap ½f1  Q

p cosðpQpÞ ¼  22 ; p  ðpQpÞ2 2

which is a function of only a single variable Qp. The absolute value of Eq. (35) is plotted in Fig. 5a as a function of the single variable Qp. At the origin Qp = 0, the value of Eq. (35) is (2/p) = 0.6366. Both the numerator and denominator of Eq. (35) are zero at Qp = ½, yielding the finite value shown at Qp = ½ in Fig. 5a. The remaining zeros of the numerator in Eq. (35) are located at

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W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533

Qp   Fig. 5a. Plot of ap ½f1 =Q , Eq. (35), as a function of Qp for the ‘‘half-period cosine-function” model, where Q is contact ratio and p = 0, ±1, ±2, . . . is toothmeshing harmonic number.

Qp = 3/2, 5/2, 7/2, . . .. Because the numerator in Eq. (35) is bounded by p/2, Eq. (35) asymptotically decays as 1=p2 beyond about Qp = 3/2. Hence, all tooth-meshing harmonics, p = 1, 2, . . . are in the asymptotic attenuation region for actual contact ratios Q larger than about 3/2, as mentioned earlier. For example, if the actual contact ratio Q is Q = 4, then the toothmeshing fundamental harmonic p = 1 would fall at the value of Qp = 4 in Fig. 5a, well into asymptotic decay region, thereby illustrating the very large benefit of large actual contact ratios Q. The asymptotic decay in proportion to 1/p2 is a conse

quence of the discontinuity in slope of g0j ðxÞ ¼ cos QpDx ; Eq. (32), at tooth contact initiation and termination points, i.e., jxj ¼ Q D=2: The value of this slope discontinuity is the slope of g0j ðxÞ at contact initiation and termination. From Eq. (32),

  d 0 p px ; gj ¼  sin dx QD QD

ð36Þ

which evaluated at contact initiation and termination points, jxj ¼ Q D=2, gives

  d 0   g ðxÞ ¼ p=ðQ DÞ; dx j 

ð37Þ

which is reduced by an increasing contact ratio Q. Because the amplitude of g0j ðxÞ, Eq. (32), has been normalized to unity, the actual value of the slope discontinuity would be proportional to the nonnormalized amplitude of g0j ðxÞ: Finally, we note because the left-hand side of Eq. (35) is divided by the actual contact ratio Q, the asymptotic dependence on Q of the tooth-meshing harmonic amplitudes, ap ½f1 , is from Eq. (35), 1/Q, a result consistent with Eq. (37). 5.2. Raised cosine-function model The above-described example was constructed to illustrate the behavior of tooth-meshing-harmonic amplitudes for cases where tooth-pair working-surface modifications from involute surfaces are constant along each possible line of contact. The example showed that the tooth-meshing harmonic amplitudes asymptotically decayed in proportion to 1/p2 with amplitudes proportional to the slope of the combined displacement deviation, Eq. (26), at tooth contact initiation and termination [49,50], where p = ±1, ±2, denotes tooth-meshing-harmonic number. This result suggests that a reduction in tooth-meshingharmonics might be achievable by modifying the working surfaces of mating tooth pairs by providing some sort of ‘‘crowning” along each possible line-of-contact location, thereby minimizing the onset and termination changes in elastic deformation contributions W j ðxÞ=K M in Eqs. (18) and (26), and also minimizing changes in the working-surface-deviation  M in these same two equations, where we note from Eq. (12) that g e Kj ðxÞ=K e Kj ðxÞ denotes an integral over contributions g the (active) line of contact, which would be limited due to crowning. The simplest way to achieve such line-of-contact ‘‘crowning” is to provide straightforward crowning in the axial y-direction (Fig. 3). Contact initiation and termination then would take place on line-of-contact end locations closest to the axial center of the teeth. In the limiting case of sufficient crowning, the dominant discontinuity at contact initiation and termination would be in the rate of change of slope, i.e., in the second derivative with respect to roll-distance x of the combined displacement g0j ðxÞ; Eq. (26). A simple model of g0j ðxÞ meeting this requirement is the ‘‘raised cosine function,”

W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533

g0j ðxÞ; ¼

   1 2p x ; 1 þ cos 2 QD



QD QD 6x6 ; 2 2

515

ð38Þ

which has zero slope at its endpoints, jxj ¼ Q D=2, and is unity at its center location x ¼ 0, axially centered on a tooth, as before. In Appendix B, the Fourier transform, Eq. (29), of g0j ðxÞ; Eq. (38), is shown to be

Fx ½g0j ; g ¼

QD sinðpQ DgÞ : 2pQ Dg ½1  ðQ DgÞ2 

ð39Þ

Hence, from Eq. (28), the Fourier series coefficients of the transmission-error contribution f1 ðxÞ, Eq. (18), for this second example, Eq. (38), are

ap ½f1  ¼

Q sinðpQpÞ ; 2pQp ½1  ðQpÞ2 

p ¼ 0; 1; 2; . . .

ð40Þ

and therefore,

ap ½f1  Q

¼

sinðpQpÞ 2pQp½1  ðQpÞ2 

;

p ¼ 0; 1; 2; . . .

ð41Þ

which, again, is a function of only a single variable Qp. The absolute of Eq. (41) is plotted in Fig 5b as a function of the single variable Qp. It is known that limx!0 sinx x ¼ 1. Hence, Eq. (41) is ½ at Qp = 0, which is shown in Fig. 5b. The numerator and denominator of Eq. (41) both have zeros at Qp = 1, but yielding a finite value of Eq. (41) at Qp = 1 in Fig. 5b. Zero values of Eq. (41) are shown in Fig. 5b at the remaining integer values of Qp = 2, 3, . . .. Because the numerator of Eq. (41) is bounded by unity, beyond about Qp = 2, Eq. (41) decays as 1/p3 due to the fact that the lowest-order discontinuity in Eq. (38), at contact initiation and termination, jxj ¼ Q D=2; is in the second derivative of Eq. (38), [49,50]. Hence, for contact ratios of about Q = 2 or larger, all of the tooth-meshing harmonics p = ±1, ±2, . . . are in the asymptotic decay region of 1/p3. This example again illustrates the importance of the discontinuities of the combined displacement sources g0j ðxÞ, Eq. (26), at tooth-pair contact initiation and termination. This second example, Eq. (38), represents only the limiting case where there is no slope discontinuity in the combined displacement source g0j ðxÞ, Eq. (26), at tooth-pair contact initiation and termination. In reality, there surely would be some contribution from a slope discontinuity. Hence, a better simple model would be a linear superposition of the two models of g0j ðxÞ, Eqs. (32) and (38). The Fourier transform, Eq. (29), of such a superposition is the superposition of the individual Fourier transforms of the two models. In such a superposition of Fourier transforms, the initial asymptotic decay would be somewhere between 1/p2 and 1/p3, while for much higher harmonic values, the asymptotic decay would be as 1/p2. In such a superposition, the zero values of resultant Fourier transforms would no longer be equally spaced as in Figs. 5a and 5b because of the differing asymptotic decay rates of the individual Fourier transforms in these two examples.

  Fig. 5b. Plot of ap ½f1 =Q , Eq. (41), as a function of Qp for the ‘‘raised cosine-function” model, where Q is contact ratio and p = 0, ±1, ±2, . . . is tooth-meshing harmonic number.

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6. Higher-contact-ratio gears In order to utilize the full load-carrying capacity of wider-facewidth helical gears, e.g., Fig. 6, axially centered regions of tooth working surfaces can be left unmodified except for tip/root relief, with modifications required for tooth contact initiation and termination applied to axial ends of the teeth. Two series of models of the combined displacement source, Eq. (26), are described below that are representative of such modifications. As before, the active roll-distance QD of g0j ðxÞ is normalized to unity as in Eqs. (29) and (30), and the amplitude of g0j ðxÞ at x ¼ 0 also is normalized to unity. 6.1. Power-function model The first series of models is, with normalized roll-distance span n ¼ x=Q D, as in Eqs. (29) and (30), the ‘‘power function,”

1 2

1 2

g0j ðnÞ ¼ 1  ð2nÞ2n ;  6 n 6 ; n ¼ 1; 2; . . . :

ð42Þ

Eq. (42) is plotted in Fig. 7 for n = 1, 2, 3, and 4. For increasing values of n, more of the center region of Fig. 7 remains near unity, but with the slopes at the endpoints, jnj ¼ 1=2, increasing with increasing n. For n = 4, Eq. (42) is essentially unity for one-half of the span  12 6 n 6 12, as shown in Fig. 7. Because the lowest-order discontinuity of this model is a slope discontinuity at tooth-pair contact initiation and termination, it can be regarded as representative of tooth-pair modifications that are constant along each possible line-of-contact location (Fig. 3), except for minor tip rounding, as in the half-period cosine model, Eq. (32).

Fig. 6. Single-helical gear with large axial contact ratio.

Fig. 7. Plot of ‘‘power function” model, Eq. (42), for n = 1, 2, 3, 4.

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Fourier transforms ap ½f1 =Q , obtained from Eq. (31), are evaluated in Appendix C for the four functions n = 1, 2, 3, 4 given by Eq. (42). Their absolute values are displayed in Fig. 8(a)–(d) as functions of the single variable Qp, as in Figs. 5a and 5b. Because the shape of Eq. (42) for n = 1, g0j ðnÞ ¼ 1  ð2nÞ2 shown in Fig. 7(a) is very close to the normalized half-cosine function, Eq. (32), with n ¼ x=Q D; i:e: cosðpnÞ, it is not surprising that their Fourier transforms shown in Figs. 5a and 8(a) are almost the same. In Appendix C, the Fourier transform, ap ½f1 =Q , for the case n = 1 of Eq. (42), is shown to be

ap ½f1  Q

¼

2 ðpQpÞ2



cosðpQpÞ 

 sinðpQpÞ ; pQp

p ¼ 0; 1; 2; . . . ;

ð43Þ

which for asymptotically large Qp differs only in the coefficient 2 in Eq. (43) versus the coefficient ðp=2Þ  1:57 in Eq. (35). All four plots, Fig. 8(a)–(d), of Fourier transform absolute values, jap ½f1 =Q j, exhibit very similar behavior, with asymptotic amplitudes increasing with increasing value of n in Eq. (42). These increasing asymptotic amplitudes are a consequence of increasing endpoint slopes shown in Fig. 7. In Appendix C, the asymptotic values of these Fourier transforms are shown to be

ap ½f1  Q



2n ðpQpÞ2

cosðpQpÞ;

p ¼ 0; 1; 2; . . .

ð44Þ

thereby increasing in amplitude with increasing n in proportion to the slope of g0j ðnÞ, Eq. (42), at contact initiation and termination. The absolute value of Eq. (44) is plotted in Fig. 9(a)–(d) for values of n = 1, 2, 3, and 4 respectively, (dashed lines), together with the absolute values of the exact values of jap ½f1 =Q j; as in Fig. 8, but this time plotted only up to Qp = 4. For all four values of n, the envelope of the asymptotic result, Eq. (44), is a good approximation to its exact value beyond about Qp = 4, thereby again illustrating the importance of endpoint slope discontinuities at tooth-pair-contact initiation and termination for actual contact ratios Q 4, for all tooth-meshing harmonics, p = 1, 2, . . .. (Ignore vertical dashed lines at 1 on the abscissa which are an artifact of the plot method.)

(a)

(c)

(b)

Qp

(d)

Qp

Fig. 8. Plot of jap ½f1 =Q j as a function of Qp for the ‘‘power function” model where Q is contact ratio and p = 0, ±1, ±2, . . . is tooth-meshing harmonic number. (a) for n = 1, (b) for n = 2, (c) for n = 3, and (d) for n = 4.

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W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533

6.2. Zero-endpoint-slope model Shown next is a model with zero endpoint slope, as in Eq. (38), but with a constant axially-centered section, as e.g., the case n = 4 in Fig. 7. This is accomplished with the function

(

g0j ðxÞ ¼

1; jxj 6 a

  1 þ cos pb ðx  aÞ ; a 6 jxj 6 ða þ bÞ;

ð45Þ

1 2

which has a total non-zero span of

2ða þ bÞ ¼ Q D:

ð46Þ

Hence, this model is the same as Eq. (38), except that a center section of unit amplitude has been added to the middle of the former model. It therefore is representative of significant crowning along line-of-contact locations, thereby, limiting the span of the integral in Eq. (12) at tooth-pair contact initiation and termination, as in Eq. (38). The span in x of the center section of Eq. (45) is 2a, and the non-zero span of each end is b. In Appendix D, the Fourier transform, Eq. (27), of Eq. (45) is shown to be

Fx ½g0j ; g ¼

Z

aþb

ðaþbÞ

g0j ðxÞexpði2pgxÞdx ¼

b

1

p 2bg½1  ð2bgÞ2 

fsinð2pagÞ þ sin½2pða þ bÞgg:

ð47Þ

With a = 0 and from Eq. (46), 2b = QD, Eq. (47) reduces to the Fourier transform, Eq. (39), of the ‘‘raised cosine function,” Eq. (38), as it must. Incorporation of the constant center section given by the first line of Eq. (45) has affected only the oscillatory term in the brackets of Eq. (47), but has had no effect on the rate of attenuation provided by the first term in Eq. (47).

(a)

(c)

(b)

Qp

(d)

Qp

Fig. 9. Solid curves same as Fig. 8 plotted only to Qp = 4; dashed curves are asymptotic values of jap ½f1 =Q j, Eq. (44), for n = 1 (a), n = 2 (b), n = 3 (c) and n = 4 (d). Ignore vertical dashed line at Qp = 1 which is an artifact of the plot method.

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W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533

6.3. Comparison of zero-endpoint-slope and power-function models From Eq. (42) and Fig. 7 it is apparent that the power-function model of Eq. (42) will have increasing load-carrying capacity with increasing values of n. Similarly, it is apparent from Eqs. (45) and (46) that the zero-endpoint-slope model will have increasing load-carrying capacity with increasing ratios of a/b. Therefore, legitimate comparison of these two models requires equating their load-carrying capacities. This comparison is carried out by determining the ratio a/b in the zeroendpoint-slope model as a function of n in the power-function model that will yield the same load-carrying capacity of the two models. After normalizing the total span, Eq. (46), to unity, each of these models has unity maximum amplitude and unity total span. It then follows from the first term in Eq. (26) that the average force transmitted by a tooth-pair j of each model, as the tooth-pair passes through the zone of contact, is made the same by equating the areas under the function g0j ðxÞ, Eqs. (42) and (45), after normalizing the total span, Eq. (46) of Eq. (45), to unity, thereby equating the load-carrying capacity of the two models. This result is carried out in Appendix E, yielding

a 1 ¼n ; b 2

ð48Þ

and the tabulation in Table 1. With increasing values of n, the fraction of the constant center section of the model, Eq. (45), is seen to be increasing. In Appendix E, the normalized Fourier series coefficients ap ½f1 =Q for the zero-endpoint-slope model, Eq. (45), when expressed as a function of n ¼ ða=bÞ þ 12, is shown to be

  Qp ! cos 2 pnþ ð 12Þ ap ½f1  1 npQp   sin ; ¼ 2 Q pQp n þ 12 1  ðQpÞ1 2 ðnþ2Þ

p ¼ 0; 1; 2;   

ð49Þ

which, again, for any value of n, is a function of only Qp, the product of actual (total) contact ratio Q and tooth-meshingharmonic number p. The absolute value of Eq. (49) is plotted in Fig. 10(a)–(d) for value n = 1, 2, 3, and 4. Because the amplitude of the numerator in Eq. (49) is bounded by unity, the asymptotic value of Eq. (49) is seen to decay in proportion to 1/(Qp)3 because the lowest-order discontinuity of g0j ðxÞ, Eq. (45), is a discontinuity in the second derivative of   g0j ðxÞ at contact initiation and termination. But because of the n þ 12 2 term in the denominator of Eq. (49), for increasing values of n, this decay of 1/(Qp)3 takes place at increasingly larger values of Qp. The independent variable in Figs. 8 and 10 is Qp; hence, the value of the tooth-meshing fundamental p = 1 falls in the figures at values of Q on the abscissas. All tooth-meshing harmonics p = 2, 3, . . . occur at larger values on the abscissas. Comparing Figs. 8(a) and 10(a), we observe better attenuation in Fig. 10(a) beyond about Qp  4; comparing Figs. 8(b) and 10(b), better attenuation in Fig. 10(b) beyond about Qp  6; Figs. 8(c) and 10(c) with better attenuation in Fig. 10(c) likely beyond about Qp8, and Figs. 8(d) and 10(d) with better attenuation in Fig. 10(d) somewhere beyond Qp = 8. In particular, the case (a/b) = 1/2, and therefore, n = 1 in Eq. (49), divides the active roll-distance span x into thirds, with the center 1/3 portion unmodified, yielding from Fig. 10(a) very strong attenuation of all tooth-meshing harmonics for Q > 4. But, for tooth-pair modifications designed to maximize gear-loading capacity, e.g., n 4 or (a/b) 7/2, the zero-endpoint-slope model of Eq. (45) exhibits no transmission-error attenuation advantage over the power-function model of Eq. (42). 6.4. Model limiting behaviors From Eq. (48), for n = 1/2, a = 0, the results from the ‘‘zero-endpoint-slope model”, Eqs. (45) and (46), should reduce to those of the ‘‘raised-cosine-function model.” Setting n = 1/2 in Eq. (49) and using the identity sin h cos h ¼ 12 sinð2hÞ, Eq. (49) is seen to reduce to Eq. (41) for the ‘‘raised-cosine-function model.” Another limiting case of interest is (a/b) ? 1, and therefore from Eq. (48), n ? 1. In this case in Eq. (49),

npQp lim sin n!1 n þ 12

!

¼ sinðpQpÞ;

ð50Þ

!

lim cos

n!1

pQp  ¼ 1; 1



ð51Þ

2 nþ2

Table 1 Equal load-carrying capacities of the two models. n

1

2

3

4

a b

1 2

3 2

5 2

7 2

520

W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533

(a)

(b)

Qp

(c)

(d)

Qp

Fig. 10. Plot of jap ½f1 =Q j, Eq. (49), as a function of Qp for the ‘‘zero-endpoint-slope” model, where Q is contact ratio and p = 0, ±1, ±2, . . . is tooth-meshing harmonic number. Computation carried out for same load-carrying capacities as ‘‘power-function” model for n = 1 (a), n = 2 (b), n = 3 (c) and n = 4 (d). For small b/a, e.g. n = 4, also a suitable model for jap ½dK M =K M j, Eq. (63).

and therefore, from Eq. (49),

lim

n!1

ap ½f1  Q

¼

sinðpQpÞ pQp;

ð52Þ

which is the Fourier transform, Eq. (31), of the ‘rect function” given by the first line of Eq. (45), with b = 0 in Eq. (46). (In the limit, b = 0, the ‘‘zero-endpoint-slope model” has the form of a ‘‘rect” function.) In the limit n ? 1, the ‘‘power-function model,” Eq. (42), also approaches the same ‘‘rect function.” It therefore is of interest to compare the Fourier transforms ap ½f1 =Q of these two models displayed in Figs. 8 and 10 with the Fourier transform shown in Fig. 11 of the ‘‘rect function.” As n gets larger in both Figs. 8 and 10, their behavior approaches that of Fig. 11, but for all values of n, the asymptotic decay is stronger in Figs. 8 and 10 than that in Fig. 11. 6.5. Severe shaft-misalignment model If the amount of shaft misalignment is larger than the combined tooth-pair end/tip relief on one end of a gear, a displacement discontinuity, Eq. (26), will take place at each tooth-pair initiation or termination. A model representative of such behavior, normalized to the roll-distance span, QD, with n ¼ x=Q D, is

g0j ðnÞ ¼

   1 1 1 þ cos pðn  Þ ; 2 2



1 1 6n6 ; 2 2

which is zero at one endpoint, n ¼  12, and is unity at the other endpoint, n ¼ 12.

ð53Þ

W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533

521

Qp Fig. 11. Plot of jap ½f1 =Q j as a function of Qp for the limiting case n ? 1 for ‘‘power function” model and ‘‘zero-endpoint-slope” model. For limit n ? 1, both models ap ½f1 =Q are given by Eq. (52).

In Appendix F, the absolute value of the normalized Fourier series coefficients, ap ½f1 =Q , for the shaft misalignment model, Eq. (53), are shown to be

8 #2 912  2 " jap ½f1 j 1 < sinðpQpÞ 4pQp cosðpQpÞ = ¼ þ ; Q 2: pQp p2  ð2pQpÞ2 ;

p ¼ 0; 1; 2; . . .

ð54Þ

which, again, is a function of Qp. Eq. (54) is plotted in Fig. 12. Because Eq. (53) has a jump (step) discontinuity at the endpoint, n ¼ 12, the asymptotic decay of Eq. (54) is in proportion to 1/(pQ), in contrast to the stronger asymptotic decays of the earlier results. Moreover, in contrast to the earlier results, Fig. 12 is not oscillatory. 7. Mesh-stiffness-variation transmission-error contribution Equation (17) describes the additional contribution to the transmission error that arises from the variation dK M ðxÞ in total mesh stiffness, which also is periodic with tooth-meshing frequencies p = ±1, ±2, . . .. Denote the complex Fourier series coefficients of the fractional-fluctuation in mesh stiffness, dK M ðxÞ=K M in Eq. (13b), by

ap ½dK M =K M ,

1 D

Z

D=2

D=2

dK M ðxÞ expði2ppx=DÞdx; KM

p ¼ 1; 2; . . . :

ð55Þ

Qp Fig. 12. Plot of jap ½f1 =Q j, Eq. (54), as a function of Qp for the ‘‘severe shaft-misalignment” model, where Q is contact ratio and p = 0, ±1, ±2, . . . is toothmeshing harmonic number.

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W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533

Denote the complex Fourier series coefficients of dfðxÞ, Eq. (17), the transmission-error contribution arising from meshstiffness variations, by ap ½df. Then, it follows directly from Eq. (17) and the convolution theorem for Fourier series [46, p. 227, 228] applied to Eqs. (19) and (55) that

ap ½df ¼ 

1 X

app0 ½f1 ap0 ½dK M =K M :

ð56Þ

p0 ¼1 except p0 ¼0

Therefore, from Eq. (56), fluctuations in the total mesh stiffness do not cause any new harmonic contributions, rather, they modify the transmission-error harmonics arising from the constant-mesh-stiffness contribution, Eq. (16). To evaluate Eq. (56), the Fourier series coefficients, Eq. (55), are required. It first is necessary to transform the roll-distance coordinate x to the center of the working surface of each tooth j indicated by the dashed line in Fig. 3. This transformation is described by Eqs. (7.24a)–(7.24c) of [46] applied to Eqs. (7.26a) and (7.26b) of that reference:

e Tj ðxÞ ¼ K T ðx  jDÞ K

ð57Þ



which is the stiffness of a mating tooth pair for line of contact location x ¼ jD centered on a tooth as indicated by the dashed line in Fig. 3. From [46, Eq. (7.97)], the total mesh stiffness is

K M ðxÞ ¼

1 X

K T ðx  jDÞ repD K T ðxÞ

j¼1



ð58Þ



which expresses the total mesh stiffness K M ðxÞ in terms of the stiffness K T ðxÞ of the individual mating tooth pairs, where repD 

is the rep function, [46, p. 229]. If ap ½K M  denotes the Fourier series coefficients of the total mesh stiffness, then from [46, Eqs. (7.C.23)–(7.C.27)],

1 D

ap ½K M  ¼ Fx ½K T ; p=D

ð59Þ

where

Z Fx ½K T ; g ¼ 

QD 2

K T ðxÞ expði2pgxÞdx

ð60Þ

Q2D 

is the Fourier transform of K T ðxÞ, the stiffness of a mating tooth pair at line of contact location x. Using the same transfor

mation as in Eqs. (29) and (30), applied to K T ðxÞ instead of g0j ðxÞ, yields 

Z Fx ½K T ; g ¼ Q D 

1=2

K T ðQ DnÞ expði2pgQ DnÞdn;

ð61Þ

1=2 

and therefore, from Eq. (59),

Z

ap ½K M  ¼ Q

1=2

K T ðQ DnÞ expði2pQpnÞdn; p ¼ 0; 1; 2;   

1=2 

ð62Þ

which are the Fourier series coefficients of the total mesh stiffness KM(x) expressed by the Fourier transform of the individual mating tooth-pair stiffnesses K T ðxÞ; x ¼ Q Dn, with tooth-pair contact span normalized to unit span, where Q is the total 

(actual) contact ratio and D is the base pitch. The Fourier series coefficients of the variation dK M ðxÞ in total mesh stiffness from the mean K M , normalized by the mean, as in Eq. (14), are from Eq. (62),

ap ½dK M =K M  ¼

Q KM

Z

1=2

K T ðQ DnÞ expði2pQpnÞdn;

1=2 

p ¼ 1; 2; . . .

ð63Þ

where K T ðQ DnÞ is the mating tooth-pair stiffness with total contact span x = QD normalized to unity. Because Eq. (63) 

describes the Fourier series coefficients of the ratio dK M =K M , it is independent of any constant amplitude factor of the mating tooth-pair stiffness K T ðxÞ; moreover, it follows directly from Eqs. (62) and (63) evaluated at p = 0 that 

a0 ½dK M =K M  ¼

Q KM

Z

1=2

K T ðQ DnÞdn ¼ 1:

1=2 

ð64Þ

K T ðxÞ describes tooth-pair stiffness as a function of line-of-contact location. Tooth-bending stiffness is the dominant con

tribution to tooth stiffness. Because stiffness contributions of individual teeth of a mating pair are in ‘‘parallel,” a first-order approximation to tooth-pair stiffness is a ‘‘rect function,” given by Eq. (45) with b = 0. However, upon contact initiation, a

W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533

523

mating tooth pair first experiences a softer stiffness that arises from the Hertizan stiffness contribution. Therefore, an improved model of tooth-pair stiffness is likely that described by Eq. (45) with b > 0, thereby providing softer tooth-pair stiffness at contact imitation and termination, perhaps (b/a) = 2/7 as in Table 1. Eq. (63) has the same form as Eq. (31). Hence, the tooth-meshing harmonics of fractional mesh-stiffness fluctuations, Eq. (63), p = ±1, ±2, . . . are to be evaluated at frequencies Q, 2Q, . . . in any of the Fourier transform plots of Eq. (31) that are deemed appropriate for representation of mesh-stiffness fluctuations, especially, Fig. 10(d) for (b/a) = 2/7 in Eq. (45), or possibly Fig. 8(d) for the model, Eq. (42), for n = 4. Let us now turn to evaluating the behavior of Eq. (56). The right-hand side of Eq. (56) consists of products of Fourier transforms of Figs. 10(d) or 8(d), mentioned above, and of Figs. 5a and 5b or 8(a)–(d) or 10(a)–(d). Thus, especially for gears with large contact ratios, Q, all products app0 ½f1 ap0 ½dK M =K M  in Eq. (56) are products of two very small values except for the term p0 ¼ p. Consequently, we have to a good approximation,

ap ½df  a0 ½f1 T ap ½dK M =K M ; p–0;

ð65Þ

where ao ½f1 T denotes the ‘‘true” (not normalized) value of a0 ½f1 : 8. Total tooth-meshing-harmonic contributions Let us first look at the sign in Eq. (65). Transmission error is defined herein as positive when ‘‘teeth come together,” as in removal of material from involute surfaces or in elastic deformations. An increase in mesh stiffness, dK M , works to oppose this; hence, the negative sign in Eq. (65). From Eq. (15) it follows directly that

ap ½f ¼ ap ½f1  þ ap ½df

ð66Þ

which is the sum of Eqs. (31) and (65). But the amplitude of g0j ðxÞ was normalized to be unity at x = 0 in all of our examples, Eqs. (32), (38), (42) and (45). This normalization can be accounted for by multiplying the computed Fourier transforms of the above-mentioned normalized models g0j ðxÞ by the ratio a0 ½f1 T =a0 ½f1  which is the ratio of the true value of a0 ½f1 T to that obtained by the normalized model. Utilizing this adjustment, there follows from Eqs. (65) and (66),



ap ½f  a0 ½f1 T



ap ½f1   a ½dK =K  ; p ¼ 1; 2; . . . : a0 ½f1  p M M

ð67Þ

9. Discussion Eq. (15) expresses the static-transmission-error of a meshing helical gear pair as the superposition of two terms: f1 ðxÞ and dfðxÞ, given by Eqs. (18) and (17) respectively. Eq. (18) is in the form of a rep function of the individual tooth-pair contributions illustrated in Fig. 4, which is superposition of working-surface modifications and elastic deformations of a pair of mating teeth. Eq. (17) is dependent on the mesh stiffness fluctuations, where the mesh stiffness, Eq. (58), also is in the form of a rep function of the individual tooth-pair stiffnesses. The resulting formulation has allowed the Fourier series coefficients of the transmission error, Eq. (66), to be expressed as the superposition of contributions from tooth-pair modifications/elastic deformations and fractional mesh-stiffness variations given by Eq. (67). The first term within the braces in Eq. (67) represents the Fourier series coefficients of the tooth-pair modifications/elastic deformations and the second term within the braces represents the fractional-mesh-stiffness-variation contributions. Thus, upon contact initiation, if the combined tooth-pair modification/elastic deformations were to exactly compensate for fractional increases in mesh stiffness, then there would exist no transmission-error fluctuations; see Eqs. (15)–(17). At this juncture it is sensible to distinguish helical gears with relatively low total (actual) contact ratios from those with significantly larger contact ratios. 9.1. ‘‘Low-contact-ratio” helical gears Except for normalization at the origin, Qp ¼ 0, as given by Eq. (64), it was mentioned above that the Fourier transform amplitude illustrated in Fig. 10(d) is a reasonable first-order approximate representation of the Fourier series coefficients, ap ½dK M =K M , Eq. (63), of the fluctuations in mesh stiffness. According to Eq. (67), ap ½dK M =K M  needs to be compared with ap ½f1 =a0 ½f1 , the Fourier series coefficients arising from tooth-pair elastic deformations/working-surface modifications. For the ‘‘power-function” model, Eq. (42), for n = 2, 3, and 4 in Fig. 7, these Fourier series coefficients are shown in Fig. 8(b)– (d), respectively. When normalized to unity at the origin, all three of these plots are nearly identical. Moreover, for values of Qp of about 4 or less, Fig. 8(b)–(d) are very nearly the same as Fig. 10(d), after normalization of each figure to unity at the origin. From Eq. (67), this comparison suggests that for contact ratios of about Q  4 or smaller, the difference of the two terms within the braces in Eq. (67) will result in significant reduction of the tooth-meshing fundamental harmonic

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p = ±1. This observation is a likely explanation of why the strongest tooth-meshing harmonic of low-contact-ratio helical gears often is observed to be p = ±2 or p = ±3 rather than the tooth-meshing fundamental p = ±1. 9.2. Higher-contact-ratio helical gears The above-described observation is a consequence of the similar phase and amplitude behavior of the two terms within the braces in Eq. (67), resulting in partial cancellation of the tooth-meshing fundamental harmonic, p = ±1, at low values of contact ratio Q. For larger contact ratios Q, the tooth-meshing fundamental harmonic p = 1 occurs at larger values of the abscissas Qp, and higher harmonics occur at even larger values of Qp. For the ‘‘power-function” model, Eq. (42), the tooth-meshing harmonics, p = 1, 2, . . ., will always occur in the asymptotic decay region of the abscissa Qp given by Eq. (44), as can be seen from Fig. 9 (a)–(d). In contrast to this behavior, as ða=bÞ ¼ n  12 increases, the asymptotic decay region of the ‘‘zero-endpoint-slope” model, Eq. (45), occurs at increasingly larger values of the abscissas Qp as can be seen from Eqs. (49) and (52) and Figs. 10(a)–(d) and 11. Thus, unless it is possible equate the two terms within the braces in Eq. (67) for the low-order tooth-meshing harmonics, p = ±1, ±2, . . ., i.e.,

ap ½f1  ¼ ap ½dK M =K M ; p ¼ 1; 2; . . . ; ao ½f1 

ð68Þ

thereby matching amplitudes and phases, as mentioned earlier, then one would hope that the tooth-meshing harmonics p = ±1, ±2, . . . especially for ap ½dK M =K M , Eq. (63), can be made to fall in the high-frequency asymptotic region [49,50]. At tooth-pair contact initiation, a more gradual loading increase would imply a longer duration during which the (softer) Hertzian stiffness contribution is operative, and therefore, a larger value of b/a in the ‘‘zero-endpoint-slope” model, Eq. (45), applied to the mating tooth-pair stiffness K T ðxÞ in Eq. (63), as described above. Larger values of b/a (smaller values of n in 

Table 1) move the beginning of the asymptotic decay region to smaller values of the abscissa Qp, as can be seen from Fig. 10(a) and (b), and Fig. 5b for a = 0. The working-surface modification procedure described in Section 5.2 therefore would appear to apply also to minimizing the tooth-meshing-harmonic contributions, Eq. (63), arising from mesh stiffness variations. But because of the dominance of the bending-stiffness contribution that is operative after minimal tooth-pair elastic deformation has taken place, a significant span of very weak tooth-pair loading upon tooth-pair contact initiation and termination would be required for the tooth-meshing fundamental harmonic, p = ±1, to be made to fall within the asymptotic decay region in Eq. (49) and in Fig. 10(a)–(d). If this condition cannot be achieved, the dominant contribution to the lowerorder tooth-meshing harmonics, p = ±1, ±2, will likely be that from the mesh stiffness fluctuations ap ½dK M =K M  in Eq. (67), given by Eq. (63). 9.3. Superposition of models Eq. (68) suggests a means by which STE tooth-meshing harmonic amplitudes can be minimized at one or, possibly, a range of gear-pair loadings. Fig. 16 of [19] displays helical-gear computed tooth-pair stiffnesses for a range of helix angles from zero to 24°. Their shapes are virtually identical to the shapes described by the power-function model, Eq. (42) displayed in Fig. 7, with increasing values of helix angles in [19] corresponding to decreasing values of n in Eq. (42) and Fig. 7. However, Fig. 7 of [14] and Figs. 5 and 6 of [15] show helical-gear tooth-pair stiffnesses with shapes very similar to shapes of the zeroendpoint-slope model, Eq. (45), but with small non-zero slopes at tooth-pair contact initiation and termination. These observations suggest a hybrid model of helical-gear tooth-pair stiffness as a linear superposition of the two models, Eqs. (42) and (45). By allowing a variation in the relative amplitudes of these two models in their superposition, and in model parameters n in Eq. (42) and a and b in Eq. (45), shapes approximately representing all of the above-cited tooth-pair stiffnesses can be achieved. The Fourier transform of any such superposition of the two models is the superposition of their individual Fourier transforms given in Appendices C and D. It is important to recognize that a0 ½dK M =K M  ¼ 1 by definition; see Eq. (64). Hence, the resultant superposition of Fourier transforms, as above, must be divided by its value at zero frequency, to achieve a0 ½dK M =K M  ¼ 1. In an entirely similar manner, a hybrid model of the combined elastic deformation/working-surface modification model, Eq. (26), can be constructed by a linear superposition of any of the models, Eqs. (32), (38), (42), or (45); the Fourier transform of the resultant superposition is superposition of the individual-component Fourier transforms. 10. Summary The static transmission error (STE) is widely recognized to be the principal vibration excitation arising from meshing gear pairs. The tooth-meshing harmonics normally are the dominant STE harmonics of power-transmission gears. They arise from the mean deviation of the tooth working surfaces from equispaced perfect involute surfaces and from tooth/gearbody elastic deformations. Using involute roll-distance as the independent variable, Eq. (1), the tooth-meshing harmonic contributions of the STE are periodic with period equal to the base pitch D of the gear pair (tooth spacing interval).

W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533

525

For helical gears, the ‘‘exact” expression for the STE, Eq. (10), is shown to be well-approximated by the superposition, Eq. (15), of two terms. The first of these terms, Eq. (18), arises from the combined contribution of stiffness-weighted deviations (modifications) of the working surfaces of mating teeth and from tooth-pair elastic deformations. The second of these terms, Eq. (17), arises from stiffness fluctuations of the meshing teeth. As shown by Eq. (18) for the first of these terms, and by Eq. (58) for the second, the dominant contribution of each of these terms is periodic with period D, the base pitch, and therefore is in the form of a ‘‘rep” function. As a consequence of this property, the complex Fourier series coefficients of each of these above-mentioned two terms can be represented in terms of the complex Fourier transform of the repeating function [46, p. 229]. Each mating tooth pair is in contact for a specific roll-distance x. When normalized by the actual (total) contact ratio Q, this contact span is QD. By normalizing this contact span to unit span, two different shapes, Eqs. (32) and (38), and two families of shapes, Eqs. (42) and (45), were utilized to illustrate likely behavior of tooth-meshing harmonic amplitudes of transmission error contributions. As shown by Eqs. (29)–(31) for the first of the above mentioned terms, Eq. (18), the tooth-meshing harmonic contributions occur at Fourier transform harmonic values Qp of the normalized forms, where p = ±1, ±2, . . . are the tooth-meshing harmonics of the transmission-error contribution. In similar manner for the meshstiffness fluctuation term, Eqs. (60)–(62) show that tooth-meshing harmonic contributions also occur at Fourier transform harmonic values Qp of the normalized forms of tooth-pair stiffnesses. Absolute values of Fourier transforms of the abovementioned function shapes are shown in Figs. 5a and 5b or 8(a)–(d) or 10(a)–(d). Each of these Fourier transforms is a monotonically decaying oscillatory function. Consequently, for any tooth-meshing harmonic number p, a larger value of the actual contact ratio Q moves that harmonic p further out on the axis Qp, yielding, on average, a lower transmission-error contribution. These examples therefore illustrate the well-known fact that, on average, larger actual contact ratios of helical gears are associated with lower transmission errors. Gear tooth contact initiation and termination cause discontinuities of various values in transmission errors. Such discontinuous behavior can govern the above-mentioned Fourier transform (high-frequency) behavior Qp where the toothmeshing-harmonic contributions take place. The two families of shapes, Eqs. (42) and (45) were chosen to model such discontinuous behavior. The ‘‘power-function” model, Eq. (42), with normalized roll-distance span of unity, is displayed in Fig. 7 showing differing slope values at contact initiation and termination for n = 1, 2, 3, and 4. The asymptotic values of the Fourier transforms of the four ‘‘power-function” models, n = 1, 2, 3, 4 are given by Eq. (44) and shown by the dashed lines in Fig. 9 (a)–(d), which illustrate that for actual contact ratios Q 4, the Fourier-transform envelope amplitudes are in the asymptotic range controlled by the contact initiation and termination slope discontinuity for all tooth-meshing harmonics, p. Hence, smaller slope discontinuities at contact initiation and termination should result in smaller STE tooth-meshing-harmonic amplitudes. The ‘‘zero-endpoint-slope” model, Eq. (45), is a generalization of the ‘‘raised cosine-function” model, Eq. (38). Its lowestorder discontinuity is in the second derivative, thereby providing a smoother transition than the ‘‘power-function” model, at tooth-contact initiation and termination. As can be seen from Eq. (45), the constant center section of the model is of duration 2a and each end of the model is of duration b. Hence, from Fig. 5b for a = 0 and Fig. 10(a) for (a/b) = ½, as the ratio of a/b increases, the asymptotic decay region of the Fourier transform of Eq. (45) moves to higher frequencies Qp. The limiting value (a/b) ? 1 of the Fourier transform is shown in Fig. 11. The final form of the STE Fourier-series coefficients is given by Eq. (67). The first term within the braces ap ½f1 =a0 ½f1  arises from the stiffness-weighted modification/elastic deformation contribution, Eq. (31), and the second term within the braces ap ½dK M =K M , arises from the mesh-stiffness fluctuations, and is given by Eq. (63). Because of the (softer) Hertzian stiffness contribution encountered upon tooth contact initiation and termination, the ‘‘zero-endpoint-slope” model of tooth-pair stiffness, with small b/a, i.e. large n in Table 1 and in Eq. (49), was suggested as a first-order model for the mesh-stiffness fluctuations dK M =K M . For typical helical gears, the tooth-pair working-surface-modification slope at contact initiation and termination can controlled. Depending on how the working surfaces are modified, either the ‘‘power-function” model, Eq. (42), or the ‘‘zero-end point-slope” model, Eq. (45), or their superposition, might serve as an approximate representation of the stiffness-weighted modification/elastic deformation contribution, Eq. (26), yielding the first term, ap ½f1 =a0 ½f1 , in the expression, Eq. (67), for STE tooth-meshing harmonic amplitudes. Mesh stiffness fluctuations are more difficult to control. As suggested above, a superposition of the ‘‘power-function” and ‘‘zero-envelope-slope” models, Eqs. (42) and (45), respectively, might provide an adequate representation of the fractional mesh stiffness fluctuation, dK M ðxÞ=K M in Eq. (17), yielding the second term, ap ½dK M =K M , in the expression, Eq. (67), for STE tooth-meshing harmonic amplitudes. This reasoning suggests that, in most applications, mesh stiffness fluctuations are likely the stronger contributions to the STE tooth-meshing harmonic amplitudes. It is hoped that the analysis and examples provided herein will be found to be useful in understanding experimental results and in guiding future designs. 10.1. Formulation and solution inherent difficulties In textbook problems, the solution domain of a problem generally is known or given. In gears, this domain is the toothworking-surface contact region which is very strongly dependent on tooth-pair loading. In order to minimize the transmis-

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W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533

sion error, tooth-pair contact initiation and termination discontinuities need to be minimized which results in the tooth contact region, and therefore the actual contact ratio, Q, being very strongly dependent on loading. Because in all of the above example computations shown in Figs. 5 and 8–11, tooth-meshing-harmonic amplitudes are oscillatory and are strongly dependent on actual contact ratios Q, accurate prediction of such amplitudes is virtually impossible, although estimation of envelope values can be achieved. The uncertainty relation for Fourier transforms, e.g., [51, p. 761]], applied to Eqs. (29)–(31) and (60)–(63), shows that it is exceedingly difficult to completely eliminate generation of transmission-error tooth-meshing harmonics over a range of loadings [52]. Appendix A. Fourier transform of ‘‘half-period cosine-function model, Eq. (32) With x ¼ Q Dn as in Eqs. (29) and (30), the Fourier transform, Eq. (30), of Eq. (32) is

Fx ½g0j ; g ¼ Q D

Z

1=2

1=2

cosðpnÞexpði2pgQ DnÞdn:

ðA:1Þ

But,

expði2pgQ DnÞ ¼ cosð2pgQ DnÞ  i sinð2pgQ DnÞ:

ðA:2Þ

The product of two even functions is even and the product of an even function and an odd function is odd. Moreover, the integral over an even range of an odd function is zero, whereas the integral of an even function over an even range is twice the integral over the half range. Therefore,

Fx ½g0j ; g ¼ Q D2

Z

1=2

cosðpnÞ cosð2pQ DgnÞdn;

ðA:3Þ

0

which has the form of a known integral [53, p. 805, #215] that yields

     1=2  sin p2  pQ Dg sin p2 þ pQ Dg sinðp  2pQ DgÞn sinðp þ 2pQ DgÞn : Fx ½g0j ; g ¼ 2Q D ¼ QD þ þ 2ðp  2pQ DgÞ 2ðp þ 2pQ DgÞ 0 ðp  2pQ DgÞ ðp þ 2pQ DgÞ

ðA:4Þ

Using the elementary trigonometric identity,

sinða  bÞ ¼ sin a cos b  cos a sin b;

ðA:5Þ

Eq. (A.4) reduces to

  sin p2 cosðpQ DgÞ  cos p2 sinðpQ DgÞ sin p2 cosðpQ DgÞ þ cos p2 sinðpQ DgÞ þ p  2 p Q Dg p þ 2pQ Dg   cosðpQ DgÞ cosðpQ DgÞ ¼ QD : þ p  2pQ Dg p þ 2pQ Dg

Fx ½g0j ; g ¼ Q D

ðA:6Þ

But, for any real x and y,

1 1 2x þ ¼ ; x  y x þ y x2  y2

ðA:7Þ

therefore, from Eqs. (A.6) and (A.7),

Fx ½g0j ; g ¼

2pQ D cosðpQ DgÞ

p2  ð2pQ DgÞ

2

p Q D cosðpQ DgÞ 2pQ D cosðpQ DgÞ i ¼ 2 2 ¼ h  2 p  ðpQ DgÞ2 4 p2  ðpQ DgÞ2 2

ðA:8Þ

which is Eq. (33). Appendix B. Fourier transform of ‘‘raised cosine-function model, Eq. (38) Derivation of the Fourier transform, Eq. (29) of g0j ðxÞ, Eq. (38), is carried out in Appendix B of [54] using the correspondence of symbols given below. See Eqs. (B.2b) and (B.14) of Appendix B of [54]. Eqs. (29) and (38)

Appendix B of [54]

x

n a

2p QD

g Fx ½g0j ; g

n ND

2N Dcr ðnÞ

W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533

527

Appendix C. Fourier transforms, ap ½f1 =Q , of ‘‘power-function model, Eq. (42), for n = 1, 2, 3, 4 With n ¼ x=Q D, the Fourier transforms ap ½f1 =Q obtained from Eq. (31), for the ‘‘power-function model,” Eq. (42), of the combined displacement, Eq. (26), are

ap ½f1  Q

Z

1=2

¼

½1  ð2nÞ2n expði2pgnÞdn

ðC:1Þ

1=2

with g = Qp, p = 0, ±1, ±2, . . . as in Eq. (31). But ½1  ð2nÞ2n  is an even function of n; hence, as in Eqs. (A.1)–(A.3) of Appendix A, we have for ap ½f1 =Q , Eq. (C.1),

ap ½f1  Q

Z

1=2

¼2

½1  ð2nÞ2n  cosð2pgnÞdn

o

Z ¼ 2½

1=2

Z cosð2pgnÞdn  22n

o

¼

1 sinðpgÞ  22nþ1 pg

Z

1=2

ðC:2aÞ n2n cosð2pgnÞdn

o 1=2

n2n cosð2pgnÞdn:

ðC:2bÞ

o

For the integral in Eq. (C.2b) define x ¼ 2pgn; hence, n ¼ x=2pg and dn ¼ dx=2pg. At n ¼ 12 ; x ¼ pg: Therefore, for the integration in the second term in Eq. (C.2b) we have

Z

1 2

n2n cosð2pgnÞdn ¼

1

Z pg x2n cos xdx:

ðC:3Þ

( ) Z pg 1 1 2n ¼ x cos xdx : sinðpgÞ  pg ðpgÞ2n o

ðC:4Þ

o

ð2pgÞ2nþ1

o

Therefore, from Eqs. (C.2b) and (C.3),

ap ½f1  Q

Fortunately, the integrations in Eq. (C.4) are carried out on p. 101 of [55]. Utilizing Eq. (440.12) of [55] for n = 1,

Z pg

x2 cos xdx ¼ 2pg cosðpgÞ þ ½ðpgÞ2  2 sinðpgÞ:

ðC:5Þ

o

Inserting Eq. (C.5) into Eq. (C.4) and simplifying gives for n = 1,

ap ½f1  Q

¼

2 ðpgÞ3

½ðpgÞ cosðpgÞ  sinðpgÞ

ðC:6Þ

which for g = Qp, as in Eq. (31), yields Eq. (43) in the main text. For n = 2, we obtain from Eq. (440.14) of [55],

Z pg

x4 cos xdx ¼ ½4ðpgÞ3  24ðpgÞ cosðpgÞ þ ½ðpgÞ4  12ðpgÞ2 þ 24 sinðpgÞ:

ðC:7Þ

o

Inserting Eq. (C.7) into Eq. (C.4) and simplifying gives for n = 2,

ap ½f1  Q

¼

4 ðpgÞ

5

n o ½ðpgÞ3  6ðpgÞ cosðpgÞ  ½3ðpgÞ2  6 sinðpgÞ :

ðC:8Þ

For n = 3, we obtain from Eq. (440.16) of [55],

Z pg

x6 cos xdx ¼ ½6ðpgÞ5  120ðpgÞ3 þ 720ðpgÞ cosðpgÞ þ ½ðpgÞ6  30ðpgÞ4 þ 360ðpgÞ2  720 sinðpgÞ:

ðC:9Þ

o

Inserting Eq. (C.9) into Eq. (C.4) and simplifying gives for n = 3,

ap ½f1  Q

¼

6 ðpgÞ

7

n o ½ðpgÞ5  20ðpgÞ3 þ 120ðpgÞ cosðpgÞ  ½5ðpgÞ4  60ðpgÞ2 þ 120 sinðpgÞ :

ðC:10Þ

For n = 4, we obtain from Eq. (440.19) for m = 8, Eq. (430.19) for m = 7 and Eq. (440.16) all of [55],

Z pg

   x8 cos xdx ¼ x8 sin x  8 x7 cos x þ 7 6x5  120x3 þ 720x cos x þ ðx6  30x4 þ 360x2  720Þ sin x

o

which yields, when simplified,

ðC:11Þ

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W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533

Z pg o

n x8 cos xdx ¼ 8 ½ðpgÞ7  42ðpgÞ5 þ 840ðpgÞ3  5040ðpgÞ cosðpgÞ  ½7ðpgÞ6  210ðpgÞ4 o þ 2520ðpgÞ2  5040 sinðpgÞ þ ðpgÞ8 sinðpgÞ:

ðC:12Þ

Combining Eqs. (C.12) and (C.4) yields for n = 4,

ap ½f1  Q

¼

n

8 ðpgÞ

9

o ½ðpgÞ7  42ðpgÞ5 þ 840ðpgÞ3  5040ðpgÞcosðpgÞ  ½7ðpgÞ6  210ðpgÞ4 þ 2520ðpgÞ2  5040 sinðpgÞ : ðC:13Þ

It follows directly from Eqs. (C.6), (C.8), (C.10), and (C.13) that the asymptotic high-frequency values of ap ½f1 =Q are for n = 1, 2, 3, and 4,

ap ½f1 

n ¼ 1;



Q

ap ½f1 

n ¼ 2;



Q

ap ½f1 

n ¼ 3;



Q

ap ½f1 

n ¼ 4;



Q

2 ðpgÞ2 4 ðpgÞ2 6 ðpgÞ2 8 ðpgÞ2

cosðpgÞ

ðC:14Þ

cosðpgÞ

ðC:15Þ

cosðpgÞ

ðC:16Þ

cosðpgÞ;

ðC:17Þ

which for g = Qp yields Eq. (44) of the main text. Appendix D. Fourier transform of ‘‘zero-endpoint-slope model Eq. (45) We require the Fourier transform, Eq. (27), of the zero-endpoint-slope model, Eq. (45). Because g0j ðxÞ described by Eq. (45) is real and an even function of x, as in Appendix A, its Fourier transform, Eq. (27), can be expressed as

Fx ½g0j ; g ¼ 2

Z o

aþb

g0j ðxÞ cosð2pgxÞdx

ðD:1aÞ

¼ I 1 þ I2 ;

ðD:1bÞ

where from Eq. (45),

Z

a

I1 ¼ 2

ðD:2Þ

cosð2pgxÞdx o

and

Z

aþb

I2 ¼ 2 a

hp io 1n 1 þ cos ðx  aÞ cosð2pgxÞdx: 2 b

ðD:3Þ

I1 is an elementary integral yielding

I1 ¼ 2

1 1 sinð2pgxÞjao ¼ 2 sinð2pagÞ: 2pg 2pg

ðD:4Þ

From Eq. (D.3), I2 can be expressed as

Z

Z

aþb

I2 ¼

aþb

cosð2pgxÞdx þ a

cos a

hp b

i ðx  aÞ cosð2pgxÞdx ¼ I3 þ I4

ðD:5Þ

with obvious definitions of I3 and I4 . For I3 we have

Z

aþb

I3 ¼

cosð2pgxÞdx ¼ a

1 1 ¼ sin ð2pgxÞaþb fsin½2pgða þ bÞ  sinð2pagÞg a 2pg 2p g

ðD:6Þ

From Eqs. (D.4) and (D.6), we have

I1 þ I 3 ¼

1 fsinð2pagÞ þ sin½2pða þ bÞgg: 2pg

Define for I4 , n ¼ x  a, therefore x ¼ n þ a and dx ¼ dn. Therefore, for I4 ,

ðD:7Þ

W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533

Z

b

I4 ¼

cos o

529

Z b p

p

cos n cos½2pgða þ nÞdn ¼ n ½cosð2pagÞ cosð2pgnÞ  sinð2pagÞ sinð2pgnÞdn b b o

¼ cosð2pagÞI5  sinð2pagÞI6

ðD:8Þ

where

Z

cos

p

n cosð2pgnÞdn b

ðD:9Þ

cos

p

n sinð2pgnÞdn: b

ðD:10Þ

b

I5 ¼ o

and

Z

b

I6 ¼ o

Using [53, p. 805, #215],

"     #b   sin 2pg  pb n sin 2pg þ pb n sinð2pbg  pÞ sinð2pbg þ pÞ b sinð2pbgÞ sinð2pbgÞ     I5 ¼ ¼ 2 þ þ 2 ¼ þ p p 2 2pbg  p 2pbg þ p 2 2p g  b 2 2pg þ b ð2pbg  pÞ ð2pbg þ pÞ b b o   b 1 1 : þ ¼  sinð2pbgÞ 2 2pbg  p 2pbg þ p

ðD:11Þ

Using [53, p. 805, #216],

" # "     #b cos 2pg  pb n cos 2pg þ pb n 1 cosð2pbg  pÞ  1 cosð2pbg þ pÞ  1     ¼  þ þ 1 1 2 2 2pg  pb 2 2pg þ pb ð2pbg  pÞ ð2pbg þ pÞ b b o     b  cosð2pbgÞ  1  cosð2pbgÞ  1 b 1 1 ¼ ¼ ½1 þ cosð2pbgÞ : þ þ 2 2pbg  p 2pbg þ p 2 2pbg  p 2pbg þ p

I6 ¼ 

ðD:12Þ

Define

FðbgÞ ¼ ¼

1 1 þ 2pbg  p 2pbg þ p 4pbg 2

ð2pbgÞ  p2 :

ðD:13aÞ ðD:13bÞ

Then, combining Eq. (D.13b) with Eqs. (D.11) and (D.12), and inserting the result into Eq. (D.8) yields for I4 ,

b I4 ¼  fcosð2pagÞ sinð2pbgÞ þ sinð2pagÞ½1 þ cosð2pbgÞgFðbgÞ 2 b ¼  FðbgÞ½sinð2pagÞ cosð2pbgÞ þ cosð2pagÞ sinð2pbgÞ þ sinð2pagÞ 2 b ¼  FðbgÞfsin½2pða þ bÞg þ sinð2pagÞg; 2

ðD:14aÞ

ðD:14bÞ

by using an elementary trigonometric identity. From Eqs. (D.1b) and (D.5),

Fx ½g0j ; g ¼ I1 þ I3 þ I4 ;

ðD:15Þ

and from Eqs. (D.7) and (D.14b)



Fx ½g0j ; g ¼

 b  FðbgÞ fsinð2pagÞ þ sin½2pða þ bÞgg: 2pg 2 1

ðD:16Þ

Moreover, from Eq. (D.13b),

( " # ( ) 2 2 b 1 b 4pbg b 1 4pbg b ð2pbgÞ  p2  4p2 ðbgÞ ¼ ¼  FðbgÞ ¼   2 2 2 2pg 2 2pg 2 ð2pbgÞ  p2 2 pbg ð2pbgÞ  p2 2 pbg½ð2pbgÞ  p2  1

¼

b 2

p2 b 1 ¼ pbg½ð2pbgÞ2  p2  p 2bg½1  ð2bgÞ2 :

ðD:17Þ

Combining Eqs. (D.16) and (D.17) yields the final result,

Fx ½g0j ; g ¼

b

1

p 2bg½1  ð2bgÞ2 

which is Eq. (47) of the main text.

fsinð2pagÞ þ sin½2pða þ bÞgg

ðD:18Þ

530

W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533

Appendix E. Equating load-carrying capacities of ‘‘power-function and ‘‘zero-endpoint-slope models, and Fourier transform, ap ½f1 =Q ; of ‘‘zero-endpoint-slope model The total roll-distance span of the zero-end-point-slope model, Eq. (45), is 2(a + b) = QD, Eq. (46). By normalizing this span to unity, as in the power-function model, Eq. (42), the zero-endpoint-slope model is reduced below to dependence on only a single parameter b/a or its reciprocal a/b. Then, by equating the load-carrying capacities of the two models, quantity a/b is obtained as a function of n in the power-function model, thereby allowing a direct comparison of the Fourier transforms of the two models for equal load-carrying capacities. The area under the central span, jxj  a of Eq. (45) is 2a. The area under each end a  x  (a + b) of Eq. (45) is b/2. Therefore, the total area under Eq. (45) is 2a + b. The average height of the model, Eq. (45), therefore is

1 þ 12 ba 2a þ b ¼ : 2a þ 2b 1 þ ba

ðE:1Þ

To equate load-carrying capacities of the two models, this average height must be equated to the average height of the power-function model, Eq. (42). The integral under Eq. (42) is

Z

1=2

½1  ð2nÞ2n dn ¼ 2

1=2

Z

1=2

½1  ð2nÞ2n dn ¼ 1  2

Z

0

1=2

ð2nÞ2n dn ¼ 1  22nþ1

0

" # ð1=2Þ2nþ1 1 ¼1 ; ¼ 1  22nþ1 2n þ 1 2n þ 1

Z

1=2

1=2

n2n dn ¼ 1  22nþ1

0

n2nþ1 2n þ 10

ðE:2Þ

which is the average height of the model Eq. (42) because its span in n is unity. Putting Eq. (E.2) under a common denominator gives

1

1 2n þ 1  1 2n ¼ ¼ : 2n þ 1 2n þ 1 2n þ 1

ðE:3Þ

Equating this average height of the power-function model to that of the zero-endpoint-slope model, Eq. (E.1), gives

1 þ 12 ba 2n ¼ 2n þ 1 1 þ ba

ðE:4Þ

which can be solved in straight-forward fashion to yield

b 1 ¼ a n  12

ðE:5Þ

a 1 ¼n b 2

ðE:6Þ

or

which is Eq. (48) of the main text. Equation (E.6) equates the load carrying capacities of the two models. We now need to utilize Eqs. (47) and (E.6) to obtain an expression for the normalized Fourier-series tooth-meshing harmonic amplitudes ap ½f1 =Q as a function of the contact ratio Q and power-parameter n for the ‘‘zero-endpoint-slope” model, in order to compare this model’s and the ‘‘power-function” model’s harmonic amplitudes for the same load-carrying capacity. Using an elementary trigonometric relation, we have for the sinusoidal terms in Eq. (47),

1 1 sinð2pagÞ þ sin½2pða þ bÞg ¼ 2 sin ½2pag þ 2pða þ bÞg cos ½2pða þ bÞg  2pag 2 2 ¼ 2 sin½pð2a þ bÞg cosðpbgÞ:

ðE:7Þ

But, from Eq. (E.6), 2a = (2n  1)b and 2a + b = 2nb; therefore, the expression (E.7) becomes 2 sinð2pnbgÞ cosðpbgÞ, and from Eq. (47), we have

Fx ½g0j ; g ¼

2b cosð2bg p=2Þ sinðnp2bgÞ

p2bg½1  ð2bgÞ2 

:

ðE:8Þ

Moreover, from Eq. (E.6), a ¼ nb  12 b; a þ b ¼ ðn þ 12Þb; and 2ða þ bÞ ¼ 2bðn þ 12Þ; therefore, using Eq. (46),

2bg ¼

2ða þ bÞg Q Dg ¼ : n þ 12 n þ 12

Substituting Eq. (E.9) into Eq. (E.8), and applying Eq. (28), yields the final result,

ðE:9Þ

W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533



! pQp 1 cos 2ðnþ12Þ npQp   sin ¼ ; p ¼ 0; 1; 2;    2 pQp n þ 12 1  ðQpÞ1 2

ap ½f1  Q

531

ðE:10Þ

ðnþ2Þ

which is a function of Qp and n. This expression enables the zero-endpoint-slope model Fourier series coefficients to be compared with those of the power-function model for the same load-carrying capacities. Equation (E.10) is Eq. (49) of the main text. Appendix F. Fourier transform, ap ½f1 =Q ; of ‘‘severe shaft-misalignment model, Eq. (53) With n ¼ x=Q D, the Fourier transform ap ½f1 =Q , obtained from Eq. (31), for the ‘‘severe shaft-misalignment” model, Eq. (53), is

Z

ap ½f1 

1=2

¼

Q

1=2

Z

1=2

¼ 1=2

    1 1 1 þ cos p n  expði2pgnÞdn 2 2     1 1 1 þ cos p n  ½cosð2pgnÞ  i sinð2pgnÞdn 2 2

¼ I1 þ I 2 ;

ðF:1aÞ ðF:1bÞ ðF:1cÞ

where

I1 ¼ ¼

Z

1 2

1=2

½cosð2pgnÞ  i sinð2pgnÞdn

ðF:2aÞ

1=2

Z

1 2 Z

1=2

cosð2pgnÞdn

ðF:2bÞ

1=2 1=2

¼

cosð2pgnÞdn

ðF:2cÞ

0

¼

1 sinðpgÞ; 2pg

ðF:2dÞ

by using even and odd function properties of cosine and sine functions, and where

I2 ¼

Z

1 2

1=2

cos 1=2

   1 ½cosð2pgnÞ  i sinð2pgnÞdn: p n 2

ðF:3Þ

But, using an elementary trigonometric identity,

 

cos

p n

 1 ¼ sinðpnÞ: 2

ðF:4Þ

Therefore, by using even and odd function properties of Eqs. (F.3) and (F.4), I2 becomes

I2 ¼ i

1 2 Z

¼ i

Z

1=2

sinðpnÞ sinð2pgnÞdn

ðF:5aÞ

1=2 1=2

sinðpnÞ sinð2pgnÞdn:

ðF:5bÞ

0

This integral can be evaluated using [53, p. 804, #197],

 1 sinðp  2pgÞn sinðp þ 2pgÞn 2 I2 ¼ i :  2ðp  2pgÞ 2ðp þ 2pgÞ 0

ðF:6Þ

Using elementary trigonometric relations gives

I2 ¼ 

i 4pg cosðpgÞ : 2 p2  ð2pgÞ2

ðF:7Þ

Therefore, combining Eqs. (F.1c), (F.2d), and (F.7), there follows

ap ½f1  Q

" # 1 sinðpgÞ 4pg cosðpgÞ : ¼ i 2 pg p2  ð2pgÞ2

ðF:8Þ

We require the absolute value of ap ½f1 =Q . Multiplying Eq. (F.8) by its complex conjugate, then forming its square root gives

532

W.D. Mark / Mechanical Systems and Signal Processing 98 (2018) 506–533

8 " #2 91=2     = ap ½f1  1 < sinðpgÞ 2 4 p g cosð p gÞ   þ :  Q  ¼ 2: 2 pg p2  ð2pgÞ ;

ðF:9Þ

Substituting g = pQ, p = 0, ±1, ±2, . . ., as in Eq. (31), yields Eq. (54). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44]

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