SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
LESSON FIVE FOUNDATIONS OF MACHINE
DESIGN (PART 2)
Action Of a Pair Of Mating Involute Teeth
Let A and B be the base circles of a pair of mating
involute gears,Line CD is a common tangent to the
base circles,while AB is the line of centers,Assume that
CD is part of a string being unwound from A and wound
upon B,while A and B rotate together in such a manner
that string CD remains taut at all times,When A and B
start to rotate,point C on the string will leave circle A
and move towards circle B,thus describing an involute
with respect to circle A.
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
At the same time,however,one can imagine that,with
respect to circle B,point C traces an involute back to its
origin B,The involutes form the actual tooth outline in the
ideal case,It can be proven that
the basic requirement for proper
gear action,namely,that no
changes in speed ratio occur
during the passage of any tooth,
is fulfilled when the normal to
the mating tooth curves at the
point of contact always passes
through the pitch point.
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
Let us investigate whether this condition is satisfied by
the involute,It is clear that since the string is taut at all times,
the path of the point of contact between the two involutes is
a straight line,This line intersects the line of centers (AB) at
P,Also,the involute is by definition normal to its generating
line (i.e.,the string) at all times,since the involute is a
circular arc with everincreasing radius,and a radius is
always perpendicular to its circular arc,Therefore,if we can
prove that point P is the pitch point,we have satisfied all the
above mentioned requirements.
Triangles ACP and BPD are similar since their
corresponding angles are equal,Like the friction drive at the
beginning of this chapter,A and B have the same
circumferential velocity at points C and D,We may
therefore state that
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
BD/AC = nA / nB 6.17
but
BD/AC = BP/AP 6.18
and nA/nB is the speed ratio,Therefore,AP and BP must be
pitch radii and point P must be the pitch point.
Path Of Contact And Contact Ratio
The path of contact is the line described by the point of
contact between two mating teeth during rotation,In
involute gearing,the path of contact coincides with the line
of action,It begins where the addendum circle of the driven
sear intersects the line of action,and ends where the
addendum circle of the driving gear intersects the line of
action,This definition ignores possible interference
conditions with pinions of small tooth numbers,which are
outside the scope of this text.
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
The contact ratio (mc) is a number
indicating the average number of teeth in
contact for a given pair of mating gears,and is
found by dividing the length of the path of
contact Z (the,action”) by the base pitch,or
mc = length of action / base pitch
= Z / Pb 6,19
Cams
A cam and its follower together form a
mechanism that converts rotary motion (often
at constant speed) or oscillating motion,into a
cyclical (repetitive) linear or angular motion.
cam 凸轮
follower 随动件
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
Plate Cams And Cylindrical Cams
Plate cams are made in the shape of a flat plate with
either a contoured periphery or a cylindrical periphery
with a contoured groove in one face,and a hub,They
rotate around an axis,which is perpendicular to the
plane of the plate and they impart mainly radial motion
to their followers.
Cylindrical cams consist of a cylindrical body
designed to rotate around its axis,with either one end,
or a peripheral groove,shaped to impart the desired
motion to a follower,and in a direction mainly parallel
to the cylinder's axis.
There are other types of cams; these are,however,
outside the scope of this chapter.
轮毂
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
Both plate and cylindrical cams have a
wide field of application,and belong to
the most efficient mechanisms known.
Cams of both types are used in both
single and multispindle automatic screw
machines and lathes for actuation of tool
slides,in tool and die design,in internal
combustion engines for valve and
injector actuation,in various indexing
devices,in packaging machinery,and in
many other applications,Two plate cams
acting on a single follower,one in the X
direction,the other in the Y direction,
can be designed to produce almost any
follower path.
multispindle
多轴的
automatic screw
machine
自动切丝机
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
Plate cam followers may be radial,in
which case the centerline of the
follower intersects with the axis of
rotation of the cam; offset when this is
not the case,or swinging,when the
follower is hinged around a point
outside the cam's surface,so that the
point of contact describes a circular arc.
The point of contact between cam and
follower may be in the shape of a point,
a roller,or a flat or rounded surface.
Almost all followers are spring-loaded
to make them follow the cam contour.
offset 偏心
hinge 铰接
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
Sensitivity Of Followers
Not all followers have the same sensitivity to changes in cam
contour,Only the pointed,specially shaped radial follower is able
to follow the sudden change in the particular cam profile shown.
The roller follower is next in sensitivity,while the flat follower is
the least sensitive,Obviously,a cam outline should be based
upon the shape of its follower to obtain the desired displacement,
and only that follower will provide the desired motion.
Determination Of The Contour Of a Plate Cam
To design the contour of a plate cam,we must first establish the
motion requirements of the follower,which are most often record
in the form of a displacement diagram,and sometimes as a
displacement table or schedule,Such a diagram shows the desired
displacements of the follower for a number of equal,sequential
angular displacements of the cam,which are usually expressed in
degrees.
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
Furthermore,we must establish the radius of the base circle,
which is numerically equal to the distance from the lowest
point on the cam periphery to the center of rotation of the cam.
The base circle is the basis of the cam outline,to which are
added the sequential distances the point of contact of the
follower must travel,as indicated by the displacement
diagram,The diameter of the base circle must at least be
equal to the diameter of the hub,In the case of cams made
integral with the shaft,the base-circle diameter must be at
least equal to the shaft diameter.
The horizontal line 0-360ois the x axis of the displacement
diagram for certain cam,Imagine that this line represents the
base circle of that cam,flattened to a straight line,Any
movements imparted to the follower are caused by the profile
of the cam material outside the base circle,or,in the diagram,
above the baseline.
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
Assume that we are designing a cam with a radial,pointed
follower,Assume further that the follower must rise
between 0 and 180o,We indicate the starting position of the
follower by A and the position at 180oby B,The question is
now,What path must the follower take from A to B? The
simplest path is,of course,a straight line,Since in this case
the follower moves equal distances in equal angular
displacements,its outward velocity is constant (assuming
constant cam rpm) and the cam constructed from such a
displacement diagram is called a constant velocity cam,It is
clear that at points A and B sudden changes in velocity
occur which may be acceptable for light loads and low rpm
values.
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
To obtain a smoother follower action,we can use either of
two mathematically defined curves,namely,harmonic motion
and constant acceleration and deceleration curves,In harmonic
motion,the shape of the rising and falling portions of the
displacement diagram is that of the curve defining the sine
function (y=sin x); hence,it is also called sinusoidal,Note that
the 30oincrements of the construction circle do not bear any
relation to the increments of the displacement diagram,The
latter should be one-sixth of the total angular displacement of
the cam during the rise or fall of the follower,Also observe that
the total rise or fall of the cam is equal to the diameter of the
construction circle.
It is obvious from inspection of the harmonic motion curve as
compared with the constant-velocity curve that much smoother
motion is imparted to the follower by the former,as velocity
changes at 0 and 180oare gradual rather than sudden.
increment 增量
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
In constant acceleration and deceleration motion,the
distances the follower travels daring rise or fall are
proportional to those given by the equation S = at2 / 2,where
a is the acceleration (in./sec2) and t is the time (sec.),Since
this equation is the mathematical expression of a parabola,
this motion is also called parabolic.
Assume that the entire motion takes 6 sec,During the first
half of the motion (3 sec.),a is an acceleration,so that the
follower velocity increases,Since l/2 a in the equation is a
constant,distance S is proportional to (not equal to) t2,or 12=
l,22= 4,and 32= 9,Successive increments in follower travel
(=S) are then proportional to
S1-S0=1-0=1
S2-S1=-l=3 6.20
S3-S2=9-4=5
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
In the second half of the motion period,the action is
reversed,a becomes a deceleration,and follower velocity
decreases,The increments of S also decrease in reversed
order,so that
S4-S3=9-4=5
S5-S4=4-1=3 6.21
S6-S5=1-0=1
As in the construction for harmonic motion,we thus
again have six equal angular increments providing points
for the displacement curve (at constant cam rpm,equal
time increments mean equal angular increments).
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
To construct the curve on the displacement diagram,we
draw these incremental length (l,3,5,5,3,l),using any
convenient unit of length on a line forming any convenient
angle with the y axis,The total required displacement of the
follower is then marked off on the y axis,and a line is drawn
connecting this mark with the highest mark on the slanting
line.
Lines parallel to this line through the various incremental
points and intersecting with the y axis provide the division of
the total displacement distance into sections of corresponding
ratio,Construction of the displacement curve from then on is
similar to that for the harmonic curve,The motion obtained
by the constant acceleration and deceleration curve is not as
smooth as that provided by harmonic motion.
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
Design a cam for counterclockwise
operation for use with a pointed,radial
follower,Base-circle diameter=4in,The
follower is to rise 0.75in,over 120oof
cam rotation in harmonic motion,then
remain in position,dwell” between
120oand 150o,It must subsequently rise
another 0.25 in,between 150oand 210o
in constant-velocity motion,The
follower must then dwell until 270oand
finally return to its original position at
360oin harmonic motion.
clockwise 顺时针的
counterclockwise
逆时针的
LESSON FIVE FOUNDATIONS OF MACHINE
DESIGN (PART 2)
Action Of a Pair Of Mating Involute Teeth
Let A and B be the base circles of a pair of mating
involute gears,Line CD is a common tangent to the
base circles,while AB is the line of centers,Assume that
CD is part of a string being unwound from A and wound
upon B,while A and B rotate together in such a manner
that string CD remains taut at all times,When A and B
start to rotate,point C on the string will leave circle A
and move towards circle B,thus describing an involute
with respect to circle A.
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
At the same time,however,one can imagine that,with
respect to circle B,point C traces an involute back to its
origin B,The involutes form the actual tooth outline in the
ideal case,It can be proven that
the basic requirement for proper
gear action,namely,that no
changes in speed ratio occur
during the passage of any tooth,
is fulfilled when the normal to
the mating tooth curves at the
point of contact always passes
through the pitch point.
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
Let us investigate whether this condition is satisfied by
the involute,It is clear that since the string is taut at all times,
the path of the point of contact between the two involutes is
a straight line,This line intersects the line of centers (AB) at
P,Also,the involute is by definition normal to its generating
line (i.e.,the string) at all times,since the involute is a
circular arc with everincreasing radius,and a radius is
always perpendicular to its circular arc,Therefore,if we can
prove that point P is the pitch point,we have satisfied all the
above mentioned requirements.
Triangles ACP and BPD are similar since their
corresponding angles are equal,Like the friction drive at the
beginning of this chapter,A and B have the same
circumferential velocity at points C and D,We may
therefore state that
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
BD/AC = nA / nB 6.17
but
BD/AC = BP/AP 6.18
and nA/nB is the speed ratio,Therefore,AP and BP must be
pitch radii and point P must be the pitch point.
Path Of Contact And Contact Ratio
The path of contact is the line described by the point of
contact between two mating teeth during rotation,In
involute gearing,the path of contact coincides with the line
of action,It begins where the addendum circle of the driven
sear intersects the line of action,and ends where the
addendum circle of the driving gear intersects the line of
action,This definition ignores possible interference
conditions with pinions of small tooth numbers,which are
outside the scope of this text.
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
The contact ratio (mc) is a number
indicating the average number of teeth in
contact for a given pair of mating gears,and is
found by dividing the length of the path of
contact Z (the,action”) by the base pitch,or
mc = length of action / base pitch
= Z / Pb 6,19
Cams
A cam and its follower together form a
mechanism that converts rotary motion (often
at constant speed) or oscillating motion,into a
cyclical (repetitive) linear or angular motion.
cam 凸轮
follower 随动件
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
Plate Cams And Cylindrical Cams
Plate cams are made in the shape of a flat plate with
either a contoured periphery or a cylindrical periphery
with a contoured groove in one face,and a hub,They
rotate around an axis,which is perpendicular to the
plane of the plate and they impart mainly radial motion
to their followers.
Cylindrical cams consist of a cylindrical body
designed to rotate around its axis,with either one end,
or a peripheral groove,shaped to impart the desired
motion to a follower,and in a direction mainly parallel
to the cylinder's axis.
There are other types of cams; these are,however,
outside the scope of this chapter.
轮毂
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
Both plate and cylindrical cams have a
wide field of application,and belong to
the most efficient mechanisms known.
Cams of both types are used in both
single and multispindle automatic screw
machines and lathes for actuation of tool
slides,in tool and die design,in internal
combustion engines for valve and
injector actuation,in various indexing
devices,in packaging machinery,and in
many other applications,Two plate cams
acting on a single follower,one in the X
direction,the other in the Y direction,
can be designed to produce almost any
follower path.
multispindle
多轴的
automatic screw
machine
自动切丝机
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
Plate cam followers may be radial,in
which case the centerline of the
follower intersects with the axis of
rotation of the cam; offset when this is
not the case,or swinging,when the
follower is hinged around a point
outside the cam's surface,so that the
point of contact describes a circular arc.
The point of contact between cam and
follower may be in the shape of a point,
a roller,or a flat or rounded surface.
Almost all followers are spring-loaded
to make them follow the cam contour.
offset 偏心
hinge 铰接
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
Sensitivity Of Followers
Not all followers have the same sensitivity to changes in cam
contour,Only the pointed,specially shaped radial follower is able
to follow the sudden change in the particular cam profile shown.
The roller follower is next in sensitivity,while the flat follower is
the least sensitive,Obviously,a cam outline should be based
upon the shape of its follower to obtain the desired displacement,
and only that follower will provide the desired motion.
Determination Of The Contour Of a Plate Cam
To design the contour of a plate cam,we must first establish the
motion requirements of the follower,which are most often record
in the form of a displacement diagram,and sometimes as a
displacement table or schedule,Such a diagram shows the desired
displacements of the follower for a number of equal,sequential
angular displacements of the cam,which are usually expressed in
degrees.
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
Furthermore,we must establish the radius of the base circle,
which is numerically equal to the distance from the lowest
point on the cam periphery to the center of rotation of the cam.
The base circle is the basis of the cam outline,to which are
added the sequential distances the point of contact of the
follower must travel,as indicated by the displacement
diagram,The diameter of the base circle must at least be
equal to the diameter of the hub,In the case of cams made
integral with the shaft,the base-circle diameter must be at
least equal to the shaft diameter.
The horizontal line 0-360ois the x axis of the displacement
diagram for certain cam,Imagine that this line represents the
base circle of that cam,flattened to a straight line,Any
movements imparted to the follower are caused by the profile
of the cam material outside the base circle,or,in the diagram,
above the baseline.
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
Assume that we are designing a cam with a radial,pointed
follower,Assume further that the follower must rise
between 0 and 180o,We indicate the starting position of the
follower by A and the position at 180oby B,The question is
now,What path must the follower take from A to B? The
simplest path is,of course,a straight line,Since in this case
the follower moves equal distances in equal angular
displacements,its outward velocity is constant (assuming
constant cam rpm) and the cam constructed from such a
displacement diagram is called a constant velocity cam,It is
clear that at points A and B sudden changes in velocity
occur which may be acceptable for light loads and low rpm
values.
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
To obtain a smoother follower action,we can use either of
two mathematically defined curves,namely,harmonic motion
and constant acceleration and deceleration curves,In harmonic
motion,the shape of the rising and falling portions of the
displacement diagram is that of the curve defining the sine
function (y=sin x); hence,it is also called sinusoidal,Note that
the 30oincrements of the construction circle do not bear any
relation to the increments of the displacement diagram,The
latter should be one-sixth of the total angular displacement of
the cam during the rise or fall of the follower,Also observe that
the total rise or fall of the cam is equal to the diameter of the
construction circle.
It is obvious from inspection of the harmonic motion curve as
compared with the constant-velocity curve that much smoother
motion is imparted to the follower by the former,as velocity
changes at 0 and 180oare gradual rather than sudden.
increment 增量
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
In constant acceleration and deceleration motion,the
distances the follower travels daring rise or fall are
proportional to those given by the equation S = at2 / 2,where
a is the acceleration (in./sec2) and t is the time (sec.),Since
this equation is the mathematical expression of a parabola,
this motion is also called parabolic.
Assume that the entire motion takes 6 sec,During the first
half of the motion (3 sec.),a is an acceleration,so that the
follower velocity increases,Since l/2 a in the equation is a
constant,distance S is proportional to (not equal to) t2,or 12=
l,22= 4,and 32= 9,Successive increments in follower travel
(=S) are then proportional to
S1-S0=1-0=1
S2-S1=-l=3 6.20
S3-S2=9-4=5
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
In the second half of the motion period,the action is
reversed,a becomes a deceleration,and follower velocity
decreases,The increments of S also decrease in reversed
order,so that
S4-S3=9-4=5
S5-S4=4-1=3 6.21
S6-S5=1-0=1
As in the construction for harmonic motion,we thus
again have six equal angular increments providing points
for the displacement curve (at constant cam rpm,equal
time increments mean equal angular increments).
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
To construct the curve on the displacement diagram,we
draw these incremental length (l,3,5,5,3,l),using any
convenient unit of length on a line forming any convenient
angle with the y axis,The total required displacement of the
follower is then marked off on the y axis,and a line is drawn
connecting this mark with the highest mark on the slanting
line.
Lines parallel to this line through the various incremental
points and intersecting with the y axis provide the division of
the total displacement distance into sections of corresponding
ratio,Construction of the displacement curve from then on is
similar to that for the harmonic curve,The motion obtained
by the constant acceleration and deceleration curve is not as
smooth as that provided by harmonic motion.
SCIENCE AND TECHNOLOGY ENGLISH FOR MECHANICAL ENGINEERING
Design a cam for counterclockwise
operation for use with a pointed,radial
follower,Base-circle diameter=4in,The
follower is to rise 0.75in,over 120oof
cam rotation in harmonic motion,then
remain in position,dwell” between
120oand 150o,It must subsequently rise
another 0.25 in,between 150oand 210o
in constant-velocity motion,The
follower must then dwell until 270oand
finally return to its original position at
360oin harmonic motion.
clockwise 顺时针的
counterclockwise
逆时针的
