The output of the cam profile, for the motion curve being simple harmonic motion, for the given specifications of the cam is shown below.
Specifications of the cam:
Base circle radius of the cam: 85
Stroke or lift of the follower: 60
Outward stroke angle: 65 0
First dwell period angle: 75 0
Return stroke angle: 125 0
Second dwell period angle: 95 0
The output of the cam profile, for the motion curve being uniform acceleration, for the given specifications of the cam is shown below.
Specifications of the cam:
Base circle radius of the cam: 90
Stroke or lift of the follower: 40
Outward stroke angle: 90 0
First dwell period angle: 100 0
Return stroke angle: 70 0
Second dwell period angle: 100 0
The output i.e. the displacement diagram and the cam profile are:
The output of the cam profile, for the motion curve being uniform velocity, for the given specifications of the cam is shown below.
Specifications of the cam:
Base circle radius of the cam: 70
Stroke or lift of the follower: 50
Outward stroke angle: 50 0
First dwell period angle: 90 0
Return stroke angle: 120 0
Second dwell period angle: 100 0
The output i.e. the displacement diagram and the cam profile are:
The output of the cam profile, for the motion curve being cycloidal for the given specifications of the cam is shown below.
Specifications of the cam:
Base circle radius of the cam: 100
Stroke or lift of the follower: 50
Outward stroke angle: 100 0
First dwell period angle: 60 0
Return stroke angle: 90 0
Second dwell period angle: 110 0
Offset distance of the follower axis from the cam centre: 30
The output i.e. the displacement diagram and the cam profile are:
The output of the cam profile, for the motion curve being simple harmonic motion, for the given specifications of the cam is shown below.
Specifications of the cam:
Base circle radius of the cam: 95
Stroke or lift of the follower: 70
Outward stroke angle: 1200
First dwell period angle: 500
Return stroke angle: 900
Second dwell period angle: 1000
Offset distance of the follower axis from the cam centre: 40
The output i.e. the displacement diagram and the cam profile are
The output of the cam profile, for the motion curve being uniform acceleration, for the given specifications of the cam is shown below.
Specifications of the cam:
Base circle radius of the cam: 90
Stroke or lift of the follower: 65
Outward stroke angle: 900
First dwell period angle: 700
Return stroke angle: 900
Second dwell period angle: 1100
Offset distance of the follower axis from the cam centre: 30
The output i.e. the displacement diagram and the cam profile are:
In this chapter the output for different cases are discussed i.e. the cases formed when different types of motion curves for the followers (here only knife edge follower) are taken into consideration with the cam being only radial cam. There are even cases of having offset or not. The cases are as follows
CAM-FOLLOWER MOTION WITH OFFSET:
4.2.1. UNIFORM VELOCITY CURVE:
The output of the cam profile, for the motion curve being uniform velocity, for the given specifications of the cam is shown below.
Specifications of the cam:
Base circle radius of the cam: 80
Stroke or lift of the follower: 60
Outward stroke angle: 600
First dwell period angle: 450
Return stroke angle: 1200
Second dwell period angle: 1350
Offset distance of the follower axis from the cam centre: 20
The output i.e. the displacement diagram and the cam profile are:
| Curves | Displacement,y | Velocity, v | Acceleration,a |
| Straight Line | hθ/β | ωh/β | 0 |
| Circular Arc | H-[H2-(Rpθ)2]1/2 | ωRp2θ/[H2-(Rpθ)2]1/2 | (ωRpH)2/[H2- (Rpθ)2]3/2 |
| SHM | h/2(1-cos πθ/β) | hπω/2β sinπθ/β | h/2(πω/β)2 cosπθ/β |
| Double harmonic | h/2[(1-cosπθ/β)-1/4(1-cos2πθ/β)] | h/2 πω/β(sin πθ/β- 1/2sin2 πθ/β) | h/2 (πω/β)2(cos πθ/β-cos2πθ/β). |
| Cycloidal | h/π(πθ/β-1/2sin2πθ/β). | hω/β(1-cos2πθ/β) | 2hπω2/β2sin2πθ/β |
| Parabolic θ/β<=0.5 | 2h(θ/β)2 | 4hθω/β2 | 4h(ω/β)2 |
| θ/β>=0.5 | h[1-2(1- θ/β)2] | 4hω/β(1- θ/β). | -4h(ω/β)2 |
| Cubic no.1 θ/β<=0.5 | 4h (θ/β)3 | 12hω/β(θ/β)2 | 24hω2/β2(θ/β) |
| θ/β>=0.5 | h[1-4(1-θ/β)3] | 12hω/β((1-θ/β)2 | -24hω2/β2(1-θ/β) |
| Cubic no.2 | hθ2/β2(3-2θ/β) | 6hωθ/β2(1-θ/β) | 6hω2/β2(1-2θ/β) |
The basic method for the construction of a cam contour is:
The profile is developed by fixing the cam and moving the follower around the cam at its respective relative positions. If we take an infinite number of points, the envelope of the cam contour would be formed. In cam layout, it is necessary to draw enough points for a smooth, reliable cam contour. Accuracy is essential. Also the layout of the roller follower requires plotting the center of the roller and drawing the curve tangent to the rollers.
This curve is similar to the constant acceleration and the pulse no.1 curves. It differs from these , however , in that there is no abrupt change in acceleration at the transition point and also that its acceleration is a continuous curve for the complete rise .similar to the constant acceleration curve , it has the disadvantages of abrupt change in acceleration at the beginning and at the end of the stroke .This cubic curve has characteristics similar to that of the simple harmonic motion curve .It is not often employed but has advantages when used in combination with other curves .No simple construction method is available.
Characteristics:
Displacement y=hθ2/β2(3-2θ/β)
Velocity v=6hωθ/β2(1-θ/β)
Acceleration a=6hω2/β2(1-2θ/β)
Pulse p=-12hω3/β3=constant.
3.8. FOLLOWER CHARACTERISTICS:
In all cams the displacement of the follower is given by the mathematical relationship
y=f(θ)
Where θ=cam angle rotation in radians.
However, since the cam rotates at a angular velocity, the displacement
y=g(t)
and θ=ωt.
Where t=time for cam to rotate through angle θ,
ω=cam angular velocity.
Equations for the follower action are often preferred in the form of the former equation than the latter one, since it is simpler to analyze and use.
The velocity is considered as the instantaneous time rate of change of displacement.
v=dy/dt=slope of the displacement curve at angle θ or time t.
The acceleration being the instantaneous time rate of change of velocity,
a=d2y/dt2=dv/dt
=slope of the velocity curve at angle θ or time t.
This curve of the polynomial family has the property of constant positive and negative accelerations .No other curve will produce a given motion from rest to rest in a given time with so small a maximum acceleration. This is probably the reason that the curve is erroneously known as the best curve .It is in many ways the worst of the all curves
With perfectly rigid members having no backlash or clearance in the system, the constant acceleration curve would give excellent performance .However, all members are somewhat elastic and clearance or backlash always exists, especially in the positive drive-roller groove-type follower. The curve’s abrupt change of acceleration at the dwell ends and the transition point produces noise, vibrations, wear, and requires a large spring size. Thus the parabolic curve should be used only at moderate or lower speeds. One of the reason for the popularity of this curve is the ease of the determining the inertia forces, which are proportional to constant accelerations.
Characteristics:
The equation for displacement is
y=2h(θ/β)2
For the velocity and the acceleration, we differentiate, yielding,
v=dy/dt=4hθω/β2
a=dv/dt=4h(ω/β)2
The equation for displacement is
y=h[1-2(1- θ/β)2]
Differentiating we find the velocity
This curve (of the trigonometric family), as the name implies, is basically generated from a cycloid. A cycloid is the locus of a point on a circle which is rolled on a straight line. Applied to cam contours, this line is the y axis, the circumference of the circle is made equal to rise h, and the radius is equal to h/2π .for high speeds, the cycloidal curve is the best of all contours if the accuracy of machining can be maintained at the beginning and at the end of the stroke where other curves exhibit their difficulty.
It has the lowest vibration, wear, stress, noise and shock. The reason for its excellent performance is that there is no change in sudden acceleration at the intersection of the dwell period and the rise curve .It is easy starting , the spring needed to keep the follower on the cam is small , and the side thrust of the translating follower is low.
Characteristics:
The equation for the displacement is
y=h/π (φ-1/2sin2φ)
= h/π (πθ/β-1/2sin2πθ/β).
Differentiating to find the velocity and acceleration yields
v=dy/dt= hω/β (1-cos2πθ/β)
This unsymmetrical curve is composed of the difference between two harmonic motions, one being one-quarter of the amplitude and twice the frequency of the other. It has the advantages of the simple harmonic curve with almost complete elimination of high shock and vibration at the beginning of the stroke .The rate of acceleration change at the beginning of the stroke is small, giving smooth action at that point. However, this slow start requires a larger cam for a minimum cam curvature. The double harmonic curve requires very accurate machining since the errors of cutting its shape (at the start of stroke) usually negate the advantages gained .As the dwell-rise-dwell cam , the limitation of the sudden change in acceleration at the minimum rise point allows only moderate cam speeds . In general, this curve is better applied to dwell-rise-return-dwell cams in which no sudden acceleration for the complete curve.
Characteristics:
The relationship for displacement is
y= h/2[(1-cosφ)-1/4(1-cos2φ)]
= h/2[(1-cosπθ/β)-1/4(1-cos2πθ/β)]
The velocity and acceleration by differentiating are
v=dy/dt=h/2 πω/β(sin πθ/β-1/2sin2 πθ/β)
This curve of the trigonometric family is one of the most popular primarily because of its simplicity in layout and understanding. It provides acceptable performance at moderate speeds. The basis for the harmonic curve is the projection (on a diameter) of the constant angular velocity movement of a point on the circumference of a circle. This circle is called the harmonic circle. The resulting motion of the follower on such a cam is simple harmonic movement similar to that of a swinging pendulum. The simple harmonic curve is a definite improvement over the previous curves. Shock is reduced so that it is no longer serious at moderate speeds.
Characteristics:
The displacement diagram is a cosine curve plotted from points projected from the harmonic circle of radius h/2, giving
y= (h/2) (1-cosφ)
Since the cam rotates β radians while the harmonic circle vector turns through π radians.
Φ/π= θ/β
Solving yields
Φ= πθ/β radians
Where
φ=angle of rotation of harmonic-circle vector, radians.
Substituting yields the displacement
y=h/2(1-cos πθ/β)
Differentiating gives the velocity
V=dy/dt=hπω/2β sin πθ/β.
We observe that the velocity curve is a sine curve with points plotted from a rotating vector hπω/2β in length.
Differentiating again for the acceleration
A=dv/dt=h/2(πω/β)2cos πθ/β.
This member of the trigonometric family is developed from projections of a semi-ellipse .The contour of the elliptical curve and its characteristics depend upon the assumed proportions of the major and minor axes. As the horizontal axis increases the cam becomes larger with the velocities of start and stop slower. In other words, the curve is flatter at the top and the bottom as the ratio of the horizontal axis to the vertical axis is made larger .If the horizontal axis of the ellipse is zero in length, the contour in the displacement is a straight-line curve. A ratio of 2:4 gives a small cam for a given pressure angle .Increasing the ratio further to 11:8 makes the curve approach a parabolic curve .At this ratio , a fair cam-follower performance can be expressed at moderate cam speeds. Further increase in the ratio is not practical, since velocity, acceleration, and cam size becomes prohibitive.
This curve composed of two circular arcs tangent to each other is only acceptable insofar as it has some improvement over the infinite acceleration, straight-line curve. Although its acceleration is finite at all times, the curve gives large follower accelerations and excessive velocities. Therefore it is used for low speeds only, if at all.
Characteristics:
Let H=radii of circular arcs.
Rp=radius of pitch circle.
l=developed length of cam for angle θ.
The displacement of the follower from 0 to β/2 is
y=H-[H2-L2]1/2
But we know for radial cams
l=Rpθ
Substituting gives displacement
y=H-[H2-(Rpθ)2]1/2
Differentiating gives velocity and acceleration
v=dy/dt=ωRp2θ/[H2-(Rpθ)2]1/2
a=dv/dt=(ωRpH)2/[H2-(Rpθ)2]3/2
Characteristics:
Let h=maximum displacement of follower.
β=cam angle rotation for rise h, radians.
The displacement of follower
y=hθ/β
Differentiating in the range of the curve for the velocity and acceleration, we find
v=dy/dt=hω/β=a constant.
a=dv/dt=0.
The displacement equations of simple polynomial curves are of the form
Y=Cθn
Where n=any number.
C= a constant.
In this polynomial family, we have the following popular curves with integer powers:
Straight line, n=1;
Parabolic or constant acceleration, n=2;
Cubic or constant pulse, n=3.
Trigonometric curves:
When a cam turns through one motion cycle the follower executes a series of events consists of rise, dwells and falls. The following three events of a follower’s motion are generally used in cam profile design:
a) DRDF(dwell-rise-dwell-fall)
b) DRFD(dwell-rise-fall-dwell)
c) RF(rise-fall)
The position of the follower from a specific zero or rest position (usually it’s the position when the follower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam is the follower displacement.
Pressure angle:
A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower is called trace point. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller.
The path generated by the trace point at the follower is rotated about a stationary cam is called pitch curve.
Pitch circle:
Prime circle (reference circle):
Base circle:
Stroke or throw:
The cam profile is the actual working surface contour of the cam. It is the surface in contact with the knife-edge, roller surface, or flat-faced follower. The cam profile may be any shape whatsoever, external or internal, single or multi-lobe, etc.
Displacement diagram:
The displacement diagram is a rectangular coordinate layout of the follower motion in one cycle of cam operation. The rise of the follower is shown as the ordinate with the length of the abscissa arbitrarily chosen. The abscissa is divided into equal cam angles or equal time divisions since the cam usually rotates at a constant speed. The displacement diagram is generally drawn or sketched as the first step in the development of the cam profile.
The transition point is the point on the cam at which the follower has its maximum velocity. In the displacement diagram, the transition point (point of inflection) is located at the maximum cam slope.
The follower velocity and acceleration are of pertinent concern in most machines since proper design requires accurate analysis and control of the velocity and acceleration curves. In general, the higher the speed, the more critical becomes the investigation. This is especially true of the acceleration curve which is the determining factor of the dynamic loads and vibrations of a cam-follower system. For years mathematically related basic curves have been found convenient .These curves have been proved popular from a standpoint of ease of layout, reproduction analysis and control.
There are following points of interest while designing a cam profile:
i) The contour which is the rise portion of the cam, should not be too steep since this will produce jamming of the translating follower on the sides. This curve slope can be reduced by using a longer cam length for the same rise of the follower .A larger cam results.
ii) The contour should be smooth, free of sharp changes or “bumps”.
iii) The speed of the wedge cam is pertinent to later investigation of cam-follower action-wear, shock ,spring size ,dynamic loads ,vibration and lubrication all being affected .
The follower is known as a radial follower if the line of movement of the follower passes through the centre of the rotation of the cam.
In this type, as the cam rotates, the follower reciprocates or translates in the guides.
The follower is pivoted at a suitable point on the frame and oscillates as the cam makes the rotatory motion.
It is quite simple in construction. However its use is limited as it produces a great wear of the surface at the point of contact.
It is a widely used cam follower and has a cylindrical roller free to rotate about a pin joint, at low speeds, the follower has a pure rolling action, but a high speeds, some sliding also occurs.
In case of steep rise, a roller follower jams the cam and, therefore, is not preferred.
1. Shape
2. Movement, and
3. Location of line and movement
To reproduce exactly the motion of the cam to the follower, it is necessary that the two remain in touch at all speeds at all times. The cams can be classified according to the manner in which this is achieved.
A pre loaded compression spring is used for the purpose of keeping the contact between the cam and follower.
In this type, constant touch between the cam and the follower is maintained by a roller follower operating in the groove of a cam. The follower cannot go out of this groove under the normal working operations. A constrained or positive drive is also obtained by the use of a conjugate cam.
If the rise of the cam is achieved by rising surface of the cam and return by the force of gravity or due to the weight of the cam, the cam is known as gravity cam. However, these cams are not preferred due to their uncertain behaviour.
The motions of the followers are distinguished from each other by the dwells they have. A dwell is the zero displacement or the absence of motion of the follower during the motion of the cam.
Cams are classified according to the motions of the followers in the following ways:
In this there is alternate rise and return of the follower with no periods of dwells [Fig. (a)]. Its use is very limited in the industry. The follower has a linear or an angular displacement.
In such a type of cam, there is rise and return of the follower after the dwell
[Fig. (b)]. This type is used more frequently than the R-R-R type of cam.
It is the most widely used type of cam. The dwelling of the cam is followed by rise and dwell and subsequently by return and dwell. In case the return of the follower is by a fall. The motion may be known as Dwell-Rise-Dwell (D-R-D).
Cams are classified according to:
1. Shape
2. Follower movement, and
3. Manner of constraint of the follower.
A wedge cam has a wedge W which, in general, has a translation motion. The follower F can either translate or oscillate. A spring is, usually used to maintain the constant between the cam and the follower. The cam is stationary and the follower constraint or guide G causes the relative motion of the cam and the follower.
Instead of using a wedge, a flat plate with a groove can also be used. In the groove, the follower is used. Thus, a positive drive is achieved without the use of a spring.
A cam in which the follower moves radially from the centre of rotation of the cam is known as a radial or disc cam. Radial cams are very popular due to their simplicity and compactness.
A spiral cam is a face cam in which a groove is cut in the form of a spiral. The spiral groove consists of teeth which mesh with a pin gear follower. The velocity of the follower is proportional to the radial distance of the groove from the axis of the cam.
The use of such cam is limited as the cam has to reverse the direction to reset the position of the follower. It finds its use in computers.
In a cylindrical cam, a cylinder which has circumference counter cut in the surface rotates about its axis. The follower motion can be of two types as follows:
In the first type, a groove is cut on the surface of the cam and a roller follower has constrained (or positive) oscillating motion. A spring loaded follower translates along or parallel to the axis of the rotating cylinder.
Cylinder cams are also known as barrel or drum cams.
A conjugate cam is a double disc cam, the two discs being keyed together and is in constant touch to the two rollers of a follower. Thus, the follower has a positive constraint. Such type of cam is preferred when the
requirements are low wear, low noise, better control of the follower, high
A globoidal cam can have two types of surfaces, convex and concave. A circumferential contour is cut on the surface of rotation of the cam to impart motion to the follower which has an oscillary motion. The application of such cams is limited to moderate speeds and where the angle of oscillation of the follower is large.
In a spherical cam, the follower oscillates about an axis perpendicular to the rotation of the cam. Note that in a disc cam, the follower oscillates about an axis parallel to the axis of rotation of cam.
A spherical cam is in the form of a spherical surface which transmits motion to the follower.
