Cam basics.docx

8.1 Cams, Cam Types and Classification of Cams

Simple cam mechanisms are 3-link kinematic chains with one cam pair. The other two kinematic pairs can either be revolute or prismatic pair. Hence, we can have 3 kinematic chains with joints R-C-R, R-C-P, P-C-P and 7 different simple cam mechanisms. Both of the kinematic elements that form the cam pair can be of a complex shape, but usually one of the kinematic elements is of a simple shape like a straight line or a circle. In order to reduce the friction and to convert sliding friction to rolling, a roller is attached. The link (which is usually the driven link) containing the simple kinematic element is the follower and the complex curved link (cam) is the driving input link. The resulting mechanisms are schematically shown below

Classification of cams in terms of their kinematic structure as shown was not found sufficient in practice. Several different classifications are used.

According the shape cams are radial, face, wedge, cylindrical, spherical, helical, conical, 3 dimensional ,etc.

Cam pairs are classified as force closed or form closed. Force closed cam pairs are more common since it is expensive to manufacture form closed cams. In form closed cams theoretically the cam and the follower is in contact at two points and there is no need for an external force to keep the surfaces in contact. However, for the system to operate there must be some clearence between the roller and the slot in which it operates. Hence the contact will be at one point and it will change when the direction of force changes.                                 Followers, in general, are classified in one of two ways:1. The construction of the surface in contact (roller, flat faced, cylindrical, spherical) 2. The type of movement (translating or oscillating)The translating follower is also classified as radial or off-set according to the location of movement with respect to the cam rotation axis.

In describing a cam mechanism one must use as many of these classifications as necessary for the clear understanding, such as: force closed radial cam with translating flat faced follower, (.a), form closed wedge cam with translating roller follower (.b) or form closed cylindrical cam with translating follower (c).

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8.2. CAM DESIGN

Kinematic cam design is mainly concerned with the generation of the cam profile. From the standpoint of design, cams can be classified

into two as:

Low-speed cams:For these cams the kinematics is our main concern. The inertia forces may be neglected. These cams exist in sewing machines in toys, in recording instruments. Since the surface quality is not very critical, these types of cams can be produced very cheaply (such as puncing from a sheet metal or molding plastics, etc).  In certain applications they may even be used to reduce the number of parts in the mechanism. From the given kinematic requirements, the cam profile can be layed out without considering the dynamics of the system. However, one must keep in mind the smoothness of the cam surface and the pressure angle even for low speeds.

High-speed cams: Tthe concept of rigidity in case of high speeds will fail. In case of high speeds, low stiffness, large mass or resonance (all of which we shall call high speed cam). Dynamics of the system will be of great concern. For example, one of the application of cams is the internal combustion engine. The cam is used to open and close the intake and exhaust valves of the pistons. The motor speed of 6000 – 8000 rpm is very common. In such a case the follower must be raised (or the valve must be opened) in less than 0.05 seconds. Acceleration of the follower will reach several g’s.

Dynamic design of high speed cams is beyond the topic of this chapter. However, in performing the kinematic design one must take certain precautions, such as the limitation on the acceleration magnitude, so that the cam mechanism obtained can perform reasonably well at low or moderate speeds. For high speed cams certain restrictions must be placed to the input and output relation.

For low speed cams the motion curve which shows the input-output relation theoretically can be of any shape. Usually the motion repeats itself after a full rotation of the cam. Typical application is toys, automatic screw cutting machines, sewing machines, etc. When s = f() (0 <  < 2) function is given. Even for low speed applications, this function must be continuous, and the slope of the curve must not be above a certain value. Otherwise, one may face problems such as excess power requirement, large forces at the bearings, etc. during the operation of the cam.

 Figure 8.9 

 Figure 8.10.

In most cam applications the motion curve for the whole cycle is not exactly defined. What is usually required is in certain parts of the cam rotation the output must remain stationary. This condition is known as the dwell. For example, in internal combustion engines, we want a cam to keep the valve closed for a certain portion of the cycle (0 <  < 1) and then open the valve as fast as possible and keep the valve open for some other portion of the cycle (2 <  < 3).  As another example consider a machine to form plastic cups. The die used to shape the cup must remain (dwell)  at a low position so that the cup can be dispensed for a certain portion of the cycle (0 <  < 1) and then it must move up to give the shape to the plastic and press it (usually heat is applied) open for some other portion of the cycle (2 <  < 3)). The difference between the two applications is the amount of stroke s, the amount of force transmitted and the speed at which the two cams must rotate (In the first case it will be in the range of 3000 - 6000 rpm where as it will be around 30 to 60 rpm in the second example).

Usually the dwell periods must be kept as large as possible and the rise and return portions in between the dwells must be as fast as possible. However if the  rise and return portion of the cycle is small, the displacement curve in between the two dwells will get steep hence the velocity and the acceleration will increase (assuming constant velocity of the input, the angular rotation of the cam, , is proportional to the elapsed time). The cam motion curve can have the

following global characteristics:

Dwell-Rise-Dwell (D-R-D): Afeter a certain dwell period the follower rises (or returns) to another dwell period. This is the most frequent cam motion. D-R-D portion of the cam cycle will be followed by a Dwell- Return-Dwell motion which is analysed in a similar manner with D-R-D (a)

Dwell-Rise-Return  (D-R-R): After a certain dwell period the follower rises and returns to the original motion (b)

Rise-Return (R-R): There is no dwell period. For high speed applications one can in most cases instead of using cams one can use a slider-crank or any other mechanism with lower kinematic pairs (c)

(a) (b) (c)

Note that if the global characteristics of the motion curve is as shown in Fig 8.11 a and b, there will be no single function s(q) that defines the motion curve. Instead for each portion of the cycle we will have different functions.

For example for the dwell portions we will have s= 0 or s= H

and  . The motion curves for the rise and return portions has been selected as some basic mathematical functions so that the motion characteristics can be controlled.

CAM LAYOUT AND CAM NOMENCLATURE

 Figure 8.7.

Let us explain the general procedure of graphical determination of the cam profile (generally known as cam lay-out) and explain the nomenclature used by a simple example. Assume a motion curve as shown in Fig. 8.7 is given. We would like to realize this motion curve using a radial cam with an inline translating roller follower. We must first determine the roller radius (rr) and the base circle radius (rb) onto which the cam profile will be applied. The roller radius is usually determined according to the allowable contact stress (known as Hertz stress) after we determine the forces acting at the contact point. The base circle radius is selected so that the cam profile is not very steep or in other words, the force transmission from the cam to the follower is reasonable. This will be explained in section 6.4. Let us assume that we know the roller radius (rr) and base circle radius (rb)  Now, Let us draw a circle (prime circle) of radius rb+rr. The roller centre will be located on this circle when it is at a dwell at the bottom position. Now let us divide the motion curve and the prime circle at equal number of intervals. In figure 8.8 we have 12 equal intervals, corresponding 300 crank rotation each (in a real design case, especially for the rise and return portions, the number of intervals must be quite large to achieve a certain accuracy). In constructing the cam profile we perform kinematic inversion. We keep the cam fixed and release the fixed link and impart a motion to the fixed link such that the relative position of the links in this inverse motion is the same as the relative positions of the original motion. For example, assuming that the cam is rotating counter clockwise 300, the follower will be displaced by a distance s1 relative to the fixed link. When the cam is fixed, for the same relative motion, the fixed link will rotate by 300 clockwise relative to the cam (fixed) and the follower will be displaced by a distance s1 relative to the fixed link which has now rotated 300 clockwise. Hence we measure s1 radially from the prime circle. Thus we can determine the position of the centre of the roller on the follower and we can draw the roller circle for every increment. The cam profile is a smooth curve that is tangent to all these roller circles (Fig.8.8).

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8.3. BASIC CAM MOTION CURVES            In this section we shall discuss the basic philosophy in the selection of motion curves will be discussed and some well known motion curves will be explained. We shall consider the rise portion of the motion curve only. Later we shall see how a full motion curve can be constructed.

Linear motion :

Equation describing a linear motion with respect to time is:           s= a1 t +a0

Assuming constant angular velocity for the input cam (), since t = :          s= a1  +a0

          Let H= Total follower rise (Stroke)

           = angular rotation of the cam corresponding to the total rise of the follower.

Also assume when s = 0   = 0 (rise is to start when t=0). With this assumption when  s = H ,  = .  when these boundary conditions are applied to the linear equation:  a0=0 and a1=H. The linear motion curve is:

and                  

           a = 0

The motion curve and velocity and acceleration curves are as shown. Note that the acceleration is zero for the entire motion (a=0) but is infinite at the ends. Due to infinite accelerations, high inertia forces will be created at the start and at the end even at moderate speeds. The cam profile will be discontinuous. 

One basic rule in cam design is that this motion curve must be continuous and the first and second derivatives (corresponding to the velocity and acceleration of the follower) must be finite even at the transition points.

2. Simple Harmonic Motion (SHM):

Simple harmonic motion curve is widely used since it is simple to design. The curve is the projection of a circle about the cam rotation axis as shown in the figure. The equations relating the follower displacement velocity and acceleration to the cam rotation angle are:

In figure below the displacement, velocity and acceleration curves are shown. The maximum velocity and acceleration values given by equations:

vmax=      ,          amax=  

Note that even though the velocity and acceleration is finite, the maximum acceleration is discontinuous at the start and end of the rise period. Hence the third derivative, jerk, will be infinite at the start and end of the rise portion. This curve will not be suitable for high or moderate speeds.

In cases where the motion curve is composed of rise-return only, if the rise and return takes place for 1800 crank rotation each, simple harmonic motion curve results with a circular cam eccentrically pivoted (eccentricity = H/2, half the rise), as shown.

                                                                                            3. Parabolic or Constant Acceleration Motion Curve:  

Noting that the velocity must be zero at the two ends, we can assume a constant acceleration for the first half and a constant deceleration in the second half of the cycle. The resulting motion curve will be two parabolas. This curve can be graphically drawn by dividing each half displacement into equal number of divisions corresponding to the divisions on the horizontal axis and joining these points with O and O’ for the first and second halves respectively. Point of intersection of these lines with the corresponding vertical lines yield points on the desired curve as shown

The equations relating the follower displacement, velocity and acceleration to the cam rotation angle are:

For the range  0 <  < /2                 For the range  /2 <  < 

In this case the velocity and accelerations will be finite. However the third derivative, jerk, will be infinite at the two ends as in the case of simple harmonic motion. Displacement, velocity and acceleration curves are as shown. This motion curve  has

the lowest possible acceleration.

In the literature, one can also find “skewed constant acceleration”, where the cam rotation angles for the acceleration and deceleration periods are not equal.

Cycloidal Motion Curve:

  If a circle rolls along   a straight without slipping, a point on the circumference traces a curve that is called a cycloid This curve can be drawn by drawing a circle with center C on the line OO’. The circumference of the circle is equal to the total rise; or the diameter is H/ .The circumference is divided into a number of equal parts corresponding to the divisions along the horizontal axis. The points around the circle are first projected to the vertical centreline of the circle and then parallel to OO’ to the corresponding vertical line on the diagram (Fig. 8.18).

Figure 8.18

The equations relating the follower displacement, velocity and acceleration to the cam rotation angle are:

Within the curves we have thus far seen, cycloidal motion curve has the best dynamic characteristics. The acceleration is finite at all times and the starting and ending acceleration is zero. It will yield a cam mechanism with the lowest vibration, stress, noise and shock characteristics. Hence for high speed applications this motion curve is recommended. The maximum velocity and acceleration values are:

Compared to parabolic motion curve the maximum acceleration is 57% larger.

5. Straight line-Circular arc motion curve:

This curve is an improvement to the linear motion curve. To avoid infinite acceleration at the ends of the rise motion, circles are drawn as shown. Although the acceleration is finite, it will be of a high magnitude.

Using the figure it can be shown that (this is left as an exercise):

;      H sin  ;     2=-1

Instead of circular arc the initial and final motions can be simple harmonic or constant acceleration as well as it will be shown in the following example. Straight line motion results in constant velocity. If we are to perform an operation such as cutting during the cam rise, constant velocity is the required motion characteristics.

Example 8.1.

A cam rotates at 50 rpm constant velocity. After a certain dwell period the follower must start to rise at constant acceleration and reach a speed of 200 mm/s and keep this velocity for 600  crank rotation.

The follower must then move by constant decelaration till it has a rise of 60 mm and dwell.

The rise motion is composed of three parts. Within the crank rotation 0<q<q1 we have constant acceleration (or parabolic) motion, within the range  1 <  < 2 we have constant velocity (or straight line) motion and within the range  2 <  <   we have constant deceleration (parabolic) motion curves. Angular velocity of the cam is given as =50*/30 = 5.238 s-1, therefore /3 cam rotation will take place within t= 0.2 s. If the speed of the follower is kept at 200 mm/s during this phase of the motion, then the amount of rise with constant velocity is H’=200*0.2= 40mm. During the acceleration and deceleration periods total rise will be 20 mm. If we assume the rise for each constant acceleration periods is equal than there will be 10 mm rise for each interval. Note that the amount of crank rotation is not known for the constant acceleration periods. Within the range 0 < <1 constant acceleration results with a second order motion curve ( a parabola). This curve can be written as:

           s=c0+c1+c2 2 

The boundary conditions is when =0, s=0 and   (since the motion starts from a dwell period, we want continuity on velocity); and when =1, s=10 mm and   Using the conditions for =0 results c0=c1=0. The condition for =1 results with the equations:

           c212 = H1 = 10mm

           2c21 = 200mm/s

Solving the two equations we obtain:  1 =  /6  (=300 ) and c2=360/2 .

For  the deceleration period 2 <  <   again we have a second order algebraic curve

s=c0+c1+c2 2 for the motion. When  = 2= /2 s=50 mm,  ; and

when  = ,  s=H=60 mm and   . Using these boundary conditions

          = - 2  =            c0 = H-

/6,

          c1 =     and c2 =  . Now for the whole rise period the motion curves are:

Within 0<  </6    :

           ,    mm/s and   

Within  /6 < </2  

          c0 = H-s=200   ,          and                    Within /2<<2/3 :

            ,    

 and    

The displacement, velocity and acceleration curves are as shown. Note that although the displacement and velocity curves are continuous and the acceleration curve is finite, the third motion derivative (jerk) will be infinite at all the transition points. This motion curve can only be used for low speeds.

6. Trapezoidal Acceleration Curve:

Note that the Straight-line- circular arc curve was an improvement over the straight line motion curve since infinite acceleration was eliminated. Now let us consider constant acceleration motion curve. Although it had the lowest acceleration characteristics, the third derivative was infinite at the start, midpoint and at the end of the rise period since there was a step change on the acceleration curves. To obtain finite third order derivatives the steps in acceleration curve are changed to sloped lines, thus we obtain trapezoids for the acceleration curve as shown. Usually the uniformly accelerated portions of the curve is taken as 1/8 of the cam rotation angle for the total rise, b. In the remaining portions we have constant acceleration motion. Trapezoidal acceleration has the following characteristics.a. It gives finite pulse, limited shock, wear, noise and vibration effects compared to the parabolic motion curve. b. Compared to the third order motion curve (Cubic #1) or cycloidal motion curve, the peak acceleration is lower and the cam size is smaller. c. For the same base circle radius, the transmission angle will be more favourable compared to the third order motion curve.

The curve can further be modified by eliminating the sharp corners of the trapezoids. For example at the ends and at the midpoint instead of uniform acceleration, one can use cycloidal or simple harmonic motion for 1/8 of the cam rotation angle. Trapezoidal motion curves (or modified trapezoidal) are suitable for high speed applications and are used especially in the automotive industry.

7. Cubic or Constant Pulse  (#1):

Cubic #1 curve is the combination of two third order curves. There is no abrupt change at the start or the end of the cycle, but there is an infinite slope on the acceleration curve at the midpoint of the cycle, which is not advantageous. The curve is not very practical. The equations for the displacement velocity and acceleration are:

for  0 << /2                                  for   /2 < <                     

The displacement, velocity and acceleration is plotted as shown.

8. Cubic or Constant Pulse #2 Motion Curve (2-3 polynomial curve): 

The acceleration of cubic#2 curve is a continuous line with a negative slope. There is a finite acceleration at the ends which result with a step change. Jerk will be infinite at these points. Its characteristics is very similar to the harmonic motion curve. This curve is also known as 2-3 polynomial. The displacement, velocity and acceleration equations are  :

9. Double Harmonic motion curve:

The curve is composed as the difference of two harmonics. It is an unsymmetrical curve. The rate of change of acceleration at the start of the rise period is small. It is best suited for D-R-R type of motion. The equations are

There are different methods used in generating motion curves satisfying the certain requirements. Note that for dwell-rise-return cams the main requirement is that there must be continuity of motion and its derivatives (velocity and acceleration). The displacement diagrams of the curves If we compare the displacement of the motion curves shown till now, one can see that there is very little difference . However when we compare the velocity and acceleration of these curves, large differences can be seen. Due to the difference in acceleration, dynamic characteristics will all be different. 

In time, different ways of obtaining cam motion curves were devised. One way is the combination curve in which we use different simple functions within the rise period such as straight-line circular arc or trapezoidal motion curve. Another way of obtaining cam motion curve is to use higher harmonics (Fourier series), the third method is to use higher order polynomials and the fourth is the use of splines. All these methods have found use in cam design. Higher order harmonic motion curves, although they satisfy the boundary condition requirement, create vibration in the system. Therefore they are not preferred. We shall consider higher order polynomials and splines.

 Figure 8.27

 Figure 8.28

10. Polynomial Motion Curves:

The general expression for a polynomial is given by:           s= c0 + c1 + c22 + c33  +…….+cnn 

where          s= displacement of the follower,           = cam rotation angle           ci = constants (i= 0,,,n)           n= order of the polynomial. For a polynomial of order n we have n+1 unknown constant coefficients. These constant can be determined by considering the end conditions. For cam motion we at least want to have continuity in displacement velocity and acceleration which results with the boundary conditions:,

for =0                                                           for 

s=0                                                                     s=H                                                                                                                                   

Since there are 6 boundary conditions, one can evaluate the value of 6 constants Hence the polynomial must be of fifth order.  The function and its derivatives are:

           s= c0 + c1 + c22 + c33  + c44+c55               c1 + 2c2 + 3c32  + 4c43+5c54

             2c2 + 6c3 + 12c42+20c53

Substituting the boundary conditions we have 6 equations in six unknowns (constants):

          0= c0                                                                      (=0,  s=0)

           H= c0 + c1  + c2 b2 + c3 3  +c4 4+c5 5        (=,  s = 0)

           0= c1                                                                      (=0,  )

           0= c1 + 2c2  + 3c3 2  +4c4 b3 +5c5 4            (,   )

           0=2c2                                                                     (=0,  )

0=2 c2   + 6c3   +12c4 2 +20c5 3                    (,   )

Simultaneous solution of these equations yield: 

           c0=c1 =c2 =0 

             ,         ,   

Hence, the fifth order polynomial is (This polynomial is commonly known as 3-4-5 polynomial) (Figure 8.30):

If we also ask for the third derivative to be zero at the ends (i.e. when q = 0  and when q = b:  ) since there are 8 boundary conditions a seventh order polynomial will be required and we obtain 4-5-6-7 polynomial as:

3-4-5 and 4-5-6-7 polynomials are compared in the above figure. Although 4-5-6-7 polynomial has zero jerk and finite fourth order characteristics. However, note that the maximum velocity and acceleration within the rise period has increased. This must also be taken into account. Using the same procedure one can construct other higher order polynomials as well. However, one must not “overdesign”. Usually the difference between two high order motion curves is very small. If this difference is less that your manufacturing tolerances, even if you use a curve of very good acceleration and jerk characteristics, you will not be able to manufacture such a cam. 

11.  Spline Curves:

In the previous case we were able to fit a single polynomial for the rise. This procedure does not give us flexibility on the motion itself. In order to have some control over the motion the rise period is divided into a number of parts. The end points of these parts are called knots. The first and the last knot is the start and end of the rise and the boundary conditions must be satisfied. At the interior knots, in order to obtain continuity, the two adjacent polynomials must have the same value, slope (first derivative) and curvature (second derivative) etc. For each part a polynomial of order n is written. Using the boundary conditions at the knots the coefficients are solved. Since the mathematics involved is the topic on  numerical analysis, this will be beyond the topic of this book.

Normalization of the motion curves

In order to compare the motion curves that were discussed we let:

w=1 rad/sH= 1 unit

b= 1 radian

This procedure is known as “normalization”. Using this procedure one can then easily compare all these curves with respect to each other.  This comparison is shown in Fig. 8.32 . Cv, Ca ve Cj, are the maximum velocity, acceleration and jerk values for the normalized curves. One can determine the maximum velocity, acceleration and jerk for any . H, w and b  as:

Also the equations given for the normalized motion curves can be converted for any rise H, angular velocity w and crank rotation b by multiplying the equation by H   substituting q/b instead of q. For example the cubic #2 curve is given as:

           s=32+23 

If we multiply the equation by H and substitute q/b instead of q we have:

Or:

Since Cv=1.5, Ca=6, the maximum velocity and acceleration values will be:

vmax=1.5*H/ , amax=6*H()2. 

We have shown the motion curves for the rise and these curves start at  = 0 end at  with a rise s=H. The rise can start at any cam angle and we can have return motion. This can easily be performed by linear transformation of the motion curve. These transformations are shown below. For example in order to have a rise motion starting at an angle , we must substitute () instead of  in any rise motion curve. In order to have a return starting at s=H at an angle , any rise motion curve must first be subtracted from H and instead of ,  () must be substituted (s= f()=H-s() will be the return curve starting at  and s=H). All motion curves except the double harmonic are symmetric motion curves.

 Example 6.2.

Lay out the motion curve for a cam follower that is to have the following motion:Rise 40 mm in  1200 crank rotation (cycloidal motion) Dwell for 300 crank rotation,Return 20 mm in 900 cam rotation  (Simple harmonic motion),Dwell for 300 crank rotation,Return 20 mm in 600 cam rotation  (parabolic motion),The required motion curves as rise motion starting at =00 will be given as:Cycloidal Motion: 

(H=40 mm,/3)               

Simple Harmonic motion

(H=20 mm, /2)

Parabolic Motion

(H=20 mm, /3)  0 < /6

Now, let us transform these equations to the correct cam angles to obtain the equation s for 3600 cam shaft rotation:

0 < 3

  s = 40

When simplified:

 s=20

When simplified:

s=0

Using Excel the cam motion curve is drawn as shown in Figure 8.35. In Figure 8.34 the formula written excel cells are shown. In column A we enter cam angles from 0 (Cell

A3) to 360 (cell A363) in degrees. In column B we convert these angles into degrees and in Column C, in rows 3 to 123 we have the formula for a cycloid, in rows 124 to 152 dwell at s=H= 40 mm etc.

In a similar fashion the derivative ds/dq and   for the motion curves can be evaluated at every cam angle and the required formula can be entered into cells in

Columns D and E respectively (=1   s-1, constant).  These curves are shown below.

Motion curve for the whole cycle can be made much simpler if we write a macro for each type of motion curve as a function routine. This is explained in detail in Appendix. III.

Depending on the application, manufacturing capabilities, etc. only one type of motion curve is usually used for both rise and return portions. In this example different curves are used just as an example.

©es8.4. CAM SIZE DETERMINATION

            Cam size determination is related to the determination of the base circle of the cam. In almost all applications we would like to minimize the size of the cam being used. Large cams are not desired due to the following reasons:

1. More space is required.2. Unbalanced mass increases3. Follower has a longer path to follow for each cycle. Hence the angular velocity of the follower and the surface velocity increases.

However as we decrease the cam size, the following factors take into effect:

1. The force transmission characteristics deteriorate. The cam profile steepness

increases.2.  The curvature of the cam profile decreases (sharp curves)3.  Strength requirements due to the forces and moments acting on the cam.

In practice the cam size is determined by considering two factors: the pressure angle and the minimum radius of curvature..

8.4.1. Pressure angle 

The pressure angle, which is the reciprocal of the transmission angle m (i.e. a=p/2-m ) is defined as:

In cams there is point contact between the two profiles. The force is transmitted along the common normal of the two contacting curves. Pressure angle for oscillating and translating roller follower radial cams are shown below. For flat faced followers the pressure is apparently zero at all times (the normal to the profile is normal to the flat face, which is always constant). Curvature characteristics is used for the determination of cam size.  In force closed cams, the pressure angle is important during the rise portion where cam is driving the follower, in return motion it is the spring force (or any other force used for forced closure) that lowers the follower.

The pressure angle is a function of cam rotation angle. We can express the force ratio expression for the pressure angle using the kinematics of the mechanism.

Consider an inline translating roller follower radial cam . The pressure angle is a function of cam rotation and the amount of rise, s (which is also a function of the cam rotation angle).  At the instant considered, velocity of point B on link 3 (follower) will be vB3 along the slider axis (vertical). The velocity of a point B on link 2 (cam) at this instant is vB2perpendicular to the line OB. These two velocities are related by the equation (see Chapter 2):

          vB3 = vB2 + vB3/B2

In this equation:                                                                           

          vB3= Follower velocity = ds/dt=

vB2 = Velocity of point B on the cam= rw    (r=OB= s+rb+rr.  rb is the base circle radius, rr is the roller radius), The relative velocity direction is along the path tangent. Hence we have:

Here s’ is the derivative of s with respect to q. If the motion curve is known, the

pressure angle variation can be determined (at the dwell periods note that tana=0 or a=0, no motion is transmitted). During the rise or return portion s(q) is monotonically increasing or decreasing. Except the double harmonic, for symmetric curves, s’(q) is maximum at the midpoint (s=H/2, q=b/2). Hence the pressure angle is maximum or minimum at half the rise or return motion for inline roller followers. It will be given by the equation:

Since  , this value can easily be determined for a given motion curve. Usually rr is determined from strength considerations. The designer selects an acceptable maximum pressure angle and solves the above expression for the base circle radius.Maximum pressure angle usually depends on the speed, load and place of application of the cam mechanism. In the literature as a rule of thumb:           If the cam speed is less then 30 rpm:    amax ≤450           If the cam speed is greater then 30 rpm:  amax ≤600           If the follower is eccentric as shown in, Then the pressure angle is given by the equation:

Where e is the eccentricity and c=            or:

In force closed cams the follower is driven by the cam during the rise portion. In the return motion it is the spring force (or weight) that drives the follower towards the cam surface, hence the pressure angle is not that critical. Using eccentricity, for the same can size one can reduce the pressure angle during the rise while there is some increase of pressure angle during the return, or for the same pressure angle a smaller cam size can be used.

Example  8.3.

Determine the minimum radius of the base circle for the cam motion given in Example 8.2, assuming an inline roller follower radial cam, with maximum permissible pressure angle 260. 

Since the rise is in cycloidal motion, Cv=2. Hence:

            mm

Hence:

          (rb+H/2+rr ) tanamax= 38.20  or  rb+20+rr =38.20 /tan(260)= 78.32 mm

           rb+rr =58.32 mm

If the roller radius is 10 mm (rr=10 mm) then a base circle radius of 50 mm (rb=50 mm) will be an acceptable choice for the cam. In case of oscillating roller followers, The pressure angle is given by:

Where s=+1 or -1 if the cam is rotating away or towards the follower arm pivot. For example, for a counter clockwise rotating cam s=1. If the cam is rotated clockwise then s = -1.

   ( f(q) is the motion curve and  , f0 is the value of the angle y when the roller is on the base circle:

8.4.2. Cam Curvature

In practice for roller followers it is common to determine the cam size using the maximum pressure angle criteria and then check that the cam curvature is satisfactory. In case of flat faced followers, the cam curvature is the determining criteria for the cam size. Graphically when laying out the cam profile, first the successive positions of the follower according to the cam motion curve is drawn while keeping the cam fixed. Consider the case shown below. According to the cam motion requirements, with the selected roller radius and base circle radius the positions of the roller are A,B,C,D and. It is not possible to draw a smooth curve that is tangent to all the roller circles.  With Profile#1, the cam profile is tangent to the circles at positions A,C and E, and in Profile#2 the cam profile is tangent to positions A,B and E of the roller. There is no cam profile that will be tangent to all the positions of the roller. This is known as “undercutting” andoccurs whenever the radius of curvature of the cam profile

is less than the radius of the roller ( ). The only way to eliminate this condition is that one must select a larger base circle radius or (if strength conditions permit) reduce the radius of the roller.

A similar case is shown in case of flat faced followers. The cam profile is not tangent to all the successive positions of the follower.

The radius of curvature of a curve in plane is given in polar coordinates as:

For a radial cam with roller follower the radius of curvature of the pitch curve ( the curve described by the centre of the roller follower, when the cam is fixed) will be given by:

As a rule of thumb for roller followers the following recommendations are made to avoid undercutting:

Use smaller roller diameter (this is limited by the contact stress at the surface)

Utilize a larger cam size (this is usually not desired. It must be applied if necessary)

Employ an internal cam (the curvature is less critical but they are more expensive to manufacture)

For  flat faced follower, the radius of curvature will be given by:

The Location of the contact point P on the follower can be written in complex numbers as:

or

Equating the real and imaginary parts:

               ;    

Noting that the centre of curvature does not change for an infinitesimal motion, the first rates of change of rC and a with respect to q is zero. Taking the derivative of OP with respect to q yields:

or         dX/dq  = -rCsin(a+q)       ds/dq = rCcos(a+q)

Also:

           X= ds/dq

Differentiating:

            = -rCsin(a+q)

Now     or:

From the above equations we can conclude:

Xmax=(ds/dq)max  and Xmin=(ds/dq)min  (maximum velocity during the return motion) Hence the length of the follower L=Xmax-Xmin.

The minimum radius of curvature is when d2s/dq2 is at its minimum (largest negative value). The radius of curvature must always be greater than zero (when r = 0, the curve has a cusp).

©es8.5. CONSTRUCTION OF THE CAM PROFILE: 

After the motion curve is defined and the cam size is determined, the actual cam profile can be found. One can either use geometrical or analytical method for cam profile determination. Since the accuracy of the cam profile is very important, mathematical methods are preferred.Geometrical method for the cam profile determination for an in line roller follower radial cam was shown. In the following figure the cam profile determination for a radial cam with  a flat faced follower is shown. In either case the procedure in generating the cam profile involves kinematic inversion; e.g. cam is fixed and the fixed link is rotated around the cam while the follower is displaced relative to the fixed link by an amount given by the motion curve. Before laying out the cam profile, the base circle radius and the motion curve for the whole cycle must be known. The relative positions of the links are same as that of the original motion. Through this process the cam profile is obtained as the envelope of all the cam profile positions. 

Geometrical construction even when we use a CAD package will still require a lot of repetitive work. In some CAM packages there are special units within the package with which you can generate a cam profile.

Analytical method of cam profile determination makes use of envelope theory. A brief discussion of the envelope theory will be given here.An Envelope can be defined as:

If each member of a family of curves is tangent to a certain curve and if each point of this curve at least one member of the family is tangent, the curve is either a part or the whole of the envelope of the family.

A family of curves with one parameter is in the form f(x,y,c)=0. c is the parameter. For each value of c we obtain a member of the family of curves. For example:

           (x-c)2+y2=r2

Is the equation of a circle with radius r (constant) and center (c,0). Since c is the parameter for each value of c we have a different circle. Hence we have a family of circles.

We shall assume that the function f(x,y,c)=0 has as many continuous derivatives with respect to x, y and c as may be required. In addition we shall assume that

any singular point; e.g., points (x,y) satisfying    on any curve having a particular value of c are isolated.The slope of any member in the family is:

Or this slope relation can be written as:

Which may also be written as:

This slope relation is valid for any member within the family. If another curve (the envelope) is tangent to a member of the family at a single point, its slope, likewise, must satisfy the above equation. The total derivative of the function f(x,y,c)=0 is:

or

Since the sum of the first two terms is equal to zero from the slope relationship:

Hence the envelope must satisfy f(x, y, c) = 0 equation and  . Then the envelope of the family of curves f(x,y,c)=0 is obtained by 

eliminating c from f(x,y,c)=0 and the partial derivative of the function with respect

to the parameter c:   

If the family of curves is given in parametric form as:

Where s is the curve parameter (by changing s we obtain x,y values of a member in the family with a value c. For example the family of circles ican be written in parametric form as x=c+rcos(s), y=rsin(s)) The envelope is obtained by the elimination of c from these eqations and 

Example 8.4.

Determine the envelope of the family of circles given by the equation:          f(x,y,c)= (x-c)2+y2-1=0

Using the above equation and its partial derivative with respect to c:            fc=-2(x-c)=0

We can eliminate c from f = 0  and fc =0 to obtain the equation of the envelope:           y=±1 

Thus the horizontal lines y=1 and y=-1 are the envelopes of the family of circles

as shown.

Example 8.5

Determine the envelope generated by the falling of a ladder from a vertical wall while the ends of the ladder are in contact with the wall and the floor as shown.

The parametric equation of the ladder is:           y= -x tang +l sing 

where g is the angle between the horizontal and the ladder and  l is the length of

the ladder. Writing the above equation in the form:           f(x,y,g)= y+ x tan g –l sin g = 0

Partial derivative of the function with respect to g is:

Solving for x and y from these equations:           x = l cos3 g            y = l sin3 g 

Eliminating the parameter g (note that the last two equations are the equations for the envelope in parametric form and it can be used to draw the envelope. For each value of g we obtain a point on the envelope):

Which is the equation of asteroid.

Example 8.6.

Using a gun we shoot a bullet with an initial velocity v at an arbitrary angle a . (Initial velocity v is determined by the type of the bullet, a gunmen cannot control v, he can only control the inclination angle a) We want to determine the area (volume) within which we can shoot an object. We neglect air resistance.

This is the classical projectile motion that is thought in high school physics. The bullet will travel at a constant velocity in x (horizontal) direction and it will be subject to a constant gravitational acceleration in y (vertical) direction. Hence the position of the bullet at any time t is given by:                      x = v cosa t               y = v sin a t –  ½ gt2

We can eliminate the parameter t from the two equations to obtain the trajectory:

          f(x, y, a)=   

This gives us a family of curves with a as the parameter. Each member of the family is a parabola. When we take the derivative of the function with respect to the parameter a:

           fa(x, y, a)= 

when simplified:           x cos 2a + y sin 2a  = 0If we eliminate a from the equations f(x,y,a) ve fa(x,y,a) we obtain the equation of the envelope as: 

The envelope is another parabola. The result is shown below. Note that the bullet can reach any point under the parabola.

A-)   Cam profile for a radial cam with translating roller follower.

B-)   Cam profile for a radial cam with flat faced oscillating follower.

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