fundamentals of engineering supplied-reference handbook - Ventech!

Position Analysis. Given a, b, c, and d, and θ2
⎛ 2 ⎞
− B± B −4AC
θ 4 = 2arctan⎜
⎟
1,2 ⎜
⎝ 2A
⎟
⎠
where A = cos θ2 – K1 – K2 cos θ2 + K3
B = 2sin θ2
C = K1 – (K2 + 1) cos θ2 + K3 , and
K
1
=
d
a
,
K
2
=
d
c
,
K
3
a
=
2
2
− b + c
2ac
2
+ d
In the equation for θ4, using the minus sign in front of the
radical yields the open solution. Using the plus sign yields
the crossed solution.
⎛ 2
− E ± E −4DF
⎞
θ 3 = 2arctan⎜
⎟
1,2 ⎜
⎝ 2 D ⎟
⎠
where D = cos θ2 – K1 + K4 cos θ2 + K5
E = –2sin θ2
F = K1 + (K4 – 1) cos θ2 + K5 , and
K
4
=
d
b
,
K
5
c
=
2
2
− d − a
2 ab
2
− b
In the equation for θ3, using the minus sign in front of the
radical yields the open solution. Using the plus sign yields
the crossed solution.
Velocity Analysis. Given a, b, c, and d, θ2, θ3, θ4, and ω2
aω2
sin ( θ4 −θ2)
ω 3 =
b sin θ −θ
aω
ω 4 =
c
2
sin
sin
( 3 4)
( θ2 −θ3)
( θ −θ )
4 3
V =−aωsin θ , V = a ω cosθ
Ax 2 2 Ay 2 2
V =−bωsin θ , V = b ω cosθ
BAx 3 3 BAy 3 3
V =−cωsin θ , V = c ω cosθ
Bx 4 4 By 4 4
See also Instantaneous Centers of Rotation in the DYNAMICS
section.
Acceleration analysis. Given a, b, c, and d, θ2, θ3, θ4, and
ω2, ω3, ω4, and α2
CD − AF CE −BF
α 3 = , α 4 = , where
AE −BD AE − BD
A= csin θ , B= b sin θ
4 3
2sin 2
2
2cos 2
2
3cos 3
2
4cos 4
C = aα θ + aω θ + bω θ −c ω θ
D= ccos θ , E = b cosθ
4 3
2cos2 2
2sin 2
2
3sin 3
2
4sin 4
F = aα θ −aω θ −bω θ + c ω θ
2
2
207
MECHANICAL ENGINEERING (continued)
Gearing
Involute Gear Tooth Nomenclature
Circular pitch pc = πd/N
Base pitch pb = pc cos φ
Module m = d/N
Center distance C = (d1 + d2)/2
where
N = number of teeth on pinion or gear
d = pitch circle diameter
φ = pressure angle
Gear Trains: Velocity ratio, mv, is the ratio of the output
velocity to the input velocity. Thus, mv = ωout / ωin. For a
two-gear train, mv = –Nin /Nout where Nin is the number of
teeth on the input gear and Nout is the number of teeth on
the output gear. The negative sign indicates that the
output gear rotates in the opposite sense with respect to
the input gear. In a compound gear train, at least one
shaft carries more than one gear (rotating at the same
speed). The velocity ratio for a compound train is:
product of number of teeth on driver gears
m v = ±
product of number of teeth on driven gears
A simple planetary gearset has a sun gear, an arm that
rotates about the sun gear axis, one or more gears
(planets) that rotate about a point on the arm, and a ring
(internal) gear that is concentric with the sun gear. The
planet gear(s) mesh with the sun gear on one side and
with the ring gear on the other. A planetary gearset has
two independent inputs and one output (or two outputs
and one input, as in a differential gearset).
Often, one of the inputs is zero, which is achieved by
grounding either the sun or the ring gear. The velocities
in a planetary set are related by
ω f − ωarm
= ± mv
, where
ω − ω
L
arm
ωf = speed of the first gear in the train,
ωL = speed of the last gear in the train, and
ωarm = speed of the arm.
Neither the first nor the last gear can be one that has
planetary motion. In determining mv, it is helpful to
invert the mechanism by grounding the arm and releasing
any gears that are grounded.
Dynamics of Mechanisms
Gearing
Loading on Straight Spur Gears: The load, W, on
straight spur gears is transmitted along a plane that, in
edge view, is called the line of action. This line makes an
angle with a tangent line to the pitch circle that is called
the pressure angle φ. Thus, the contact force has two
components: one in the tangential direction, Wt, and one
in the radial direction, Wr. These components are related
to the pressure angle by
Wr = Wt tan(φ).