4.1. GEAR PAIR MESHING IMPACT RESPONSE ANALYSIS 55
When t
oc
< 0, the impact occurs when the subsequent teeth of the driven gear just enter
the meshing state. At this time, the driven gear must maintain the original speed, and the
driving gear should increase its speed, which makes the meshing shock increase. When
t
oc
> 0, the speed of the passive gear slows down due to the inertia of the driven gear,
which result in the impact of both the driving and the driven wheels speed on the meshing
line becomes small.
For these reasons, the impact of the mesh merging is more significant than that of the
mesh-out, so that the following discussion will focus on the impact while the gear mesh
merging.
4.1.1 CALCULATION OF IMPACT ACCELERATION AND IMPACT TIME
Due to the error and elastic deformation of the gear, an impact will occur during the gear mesh-
ing process, and the gear will enter the meshing point outside the meshing line. At this time,
the two tooth profiles have no common normal at the contact point (as shown in Fig. 4.2). e
difference v
a
D v
1a
v
2a
is called the impact velocity, which represents the velocity component
perpendicular to the tooth profile of the driving wheel, v
a
can be calculated by [37]:
v
a
D !
1
r
g1
1 C
1
i
2
4
1
cos
˛
E
0
1
C
1
cos ˛
b
3
5
; (4.11)
where !
1
is the drive wheel angular velocity, i represents the gear ratio, ˛
b
stands for the pressure
angle on the index circle, ˛
E
0
1
C
1
is the argument. When two gears M
1
and M
2
meet with
speed difference, they will hit each other. Its greatest impact is:
F
m
D v
a
s
b
q
E
1
s
J
1
J
2
J
1
r
0
2
g2
C J
2
r
2
g1
: (4.12)
In this formula, q
E
1
is the flexibility for engaging gear pairs, J
1
; J
2
is the moment of
inertia for gears, b is the tooth width, r
g1
is the basic circle radius of the driving gear, r
0
g2
is the
equivalent base circle radius of a passive gear.
If the change of impact force is assumed to be a half-wave sinusoidal pulse, the impact
force of gears 1 and 2 (for convenience, the subscripts 1 and 2 are omitted in the following
expressions, except as specified in particular) [37]:
f
a
.t/ D F
m
sin !
c
t
.
0 t t
c
/
; (4.13)
where !
c
is the frequency of the half wave sine pulse, t
c
is the impact time, then the impact
acceleration can be expressed as:
a
a
.t/ D
F
m
M
sin !
c
t D a
m
sin !
c
t
.
0 t t
c
/
; (4.14)
56 4. TRANSMISSIONSYSTEM PARAMETERSAND MESHING STIFFNESS CALCULATION
C'
E'
1
E
1
C
r'
g2
r'
g1
r
g1
N
2
V
2
V
1
|
V
0
|
N
1
N
1
N
2
ω
2
ω
1
γ
1
α
E'
1
O
2
O
1
r
g2
Figure 4.2: Impact velocity of meshing resulting from gear error.
where a
m
D
F
m
M
.
Because of the errors and elastic deformation of the gears, the gear runs unsteadily, so the
speed v
1
and v
2
of the two gears are unknown at the beginning of the impact, but the impact
speed of the two gears can be obtained from Eq. (4.16). In order to estimate the meshing impact
strength of a pair of gears, the meshing impact system of a pair of gears is simplified as follows.
For the driving gear, the relative velocities of the two teeth along the tooth normal direction at
the meshing point at the beginning and the end of the impact can be described by:
v
a
D v
1a
v
2a
(4.15)
v D v
0
1a
v
0
2a
: (4.16)
e motion of the two gear should satisfy the following momentum conservation equation:
M
1
v
1a
C M
2
v
2a
D M
1
v
0
1a
C M
2
v
0
2a
: (4.17)
According to the impulse theorem, gear 2 satisfies
t
c
Z
0
f
a
.t/dt D M
2
v
0
2a
M
2
v
2a
: (4.18)
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