User:Wozmachine

Cam Operated Index Drives

Table of Contents Overview Cam Motion Introduction

Cam Geometry

Slope Angle

Pressure Angle

Radius of Curvature

Parametric Forms

Cam Development

Common Cam Types Introduction

Straight line plate

Swing arm plate

Straight line barrel

Swing arm barrel or Right Angle

Roller Gear

Cam Motions

Constant Velocity

Constant Acceleration

Harmonic

Cycloid

Modified Trapezoid

Modified Sine

3-4-5 Polynomial

Modified Sine 0.33 CV

Modified Sine 0.50 CV

Modified Sine Quick Return

Synthetic Modified Sine Harmonic

Motion Synthesis Techniques

Common Cam Types Geometric Properties 2-D and 3-D

Straight line plate

Swing arm plate

Straight line barrel

Swing arm barrel or Right Angle

Roller Gear

Indexer Rating Forces on Cam Follower

Inertia

Constant

Preload

Cam Stresses

Cam Follower Stresses

Cam Follower B-10 Life

Histograms Torque Calculations

Output Torques

Input Torques

Ind/min, RPM – Type 1 and Type 2 and PPM Hand of Cam and Key Positions

Why 1.3?

Shaft Torques

Vibrations

Overview

A cam is a mechanism used to create motion. Mathematically, a cam is a linear function, i.e. it has a single input and a single output. The input or independent variable is the input axis X or camshaft angle Θ. The output axis of the cam is the dependent variable Y, h or Θout. A common input is a rotation or translation of the cam while the output can also be a translation or rotation. Rotary axes are used most often in industry as they have much less friction than a linear axis. A cam is typically mounted to a camshaft that is supported in rotary bearings and the cam meshes with an output shaft or slide, which is also mounted in bearings. The entire assembly is housed with oil or grease as a lubricant.

For precision positioning applications there must be no backlash or lost motion in the cam mechanism. There are several ways to remove backlash from a cam mechanism. One way is to use a spring to apply a force that pushes the cam follower against the cam

The spring will have to be preload by an amount equal to the maximum of the force that would try to separate the cam follower from the cam track. For the rest of the cycle the spring would exert a force much greater than needed. In addition in order to have the spring last one will have to pick a large or long spring and the mass of this large spring must be added to the mass load of the application.

A better method would be to use a conjugate set of cams and preload the cam followers against the cams or cam tracks by mounting the cam on an adjustable eccentric.

A two dimensional conjugate set of cam is shown in the picture above. The top collection of cam followers mesh with the top cam and the lower set of cam followers mesh with the lower cam. The top and lower cams are conjugates or mirror images of each other, just like your left and right hands.

A three dimensional conjugate roller gear cam oscillator is shown in the picture above. The left cam follower track is said to be the conjugate of the right cam follower track. The left and right cam follower tracks are not mirror images of each other.

The cam follower is a very stiff spring and does not require a large amount of compression to produce a force hat keeps the cam follower loaded against the cam.

A well designed cam mechanism will minimized the size and cost of the cam with respect to the desired maintenance life and accuracy of the application. To minimize the cam size we will need to know more about the geometry of the cam as well as the loads that the cam is expected to carry.

Cam Motion:

The inertia loads on the cam system are a function of the cam motion and rpm of the camshaft. The cam geometry is also a function of the cam motion.

A discussion of cam motions can be found in the latter part of this document. Most applications of a cam require intermittent motion, moving something from rest at position A to rest at position B. The minimum requirement for an acceptable motion is no acceleration discontinuities. The Modified Sine motion is the most common motion used for this purpose. This motion is a combination of a cycloid to a harmonic and back to a cycloid. The harmonic motion uses minimum energy for an object trading kinetic and potential energy. The cycloid is placed before and after the harmonic to eliminate its acceleration discontinuities.

Cam Geometry:

The cam geometry properties that are of interest are the slope angle, pressure angle or transmission angle and the radius of curvature of the cam follower path in the plane of the cam follower.

Slope Angle

The slope angle is the direction the cam follower is moving with respect to the independent reference axis. The cam surface lies perpendicular to the slope angle. This angle is needed to locate the cutting or grinding tool during manufacture.

Pressure Angle

The pressure angle is the angle between the direction that the cam pushes against the output and the direction the output is constrained to move. For a straight-line cam the pressure angle and slope angle are the same. For a cam with a swing arm the pressure angle is a function of the slope angle and the output position. The pressure angle influences the magnitude of the force placed on the cam and cam follower and all the bearings supporting the cam and output shafts. For an indexing or oscillating cam with multiple cam followers a large pressure angle creates a thin rib or space between the cam followers and reduces the period of three-follower overlap.

Radius of Curvature:

The curvature of the cam follower path limits the maximum size of the cam follower, maximum torque or force rating and estimated B-10 life of the cam follower. A small curvature produces a large stress on the cam surface and reduces its fatigue life. The first figure below shows a follower moving through a right angle. The edges on either side of the follower are well defined except where the follower turns the corner. The gap at the very corner is a loss of contact created by the sharp corner. The radius or curvature at the corner is undefined. The minimum radius, which does not form a corner, is the same as the cam follower radius. For stress purposes the minimum radius should be greater than the cam follower radius.

The other figures below show undercutting in a three dimensional cam. A sharp corner is also formed at some point and below that point there is a recess or divot similar to a pot hole in a road. The corner and divot can not provide support for the cam follower and the cam follower may bounce or shake as it enters and leaves the divot.

Problems with the pressure angle and curvature can be corrected by increasing the cam size. For a plate cam this means increasing the minimum or base radius and for a cylindrical or barrel cam the diameter would have to be increased. This increased size will increase the expense of the cam and reduce the B -10 or wear rating. A larger cam takes more time to manufacture because the milling cutter and grinding wheel as well as the cam follower must traverse a greater distance.

Undercut 2d and 3d cams

Parametric Form (a simplification):

The cam geometry properties that are of interest are the slope angle, pressure angle or transmission angle and the radius of curvature of the cam follower path in the plane of the cam follower. A strict and rigorous mathematical approach would require writing the location vector of the cam follower position with respect to a reference origin and then differentiating the vector twice. This differentiation can be used to calculate the local slope angle and radius of curvature of a two dimension or plate cam without further correction. When a third dimension is introduced (for a three dimensional cam like a barrel or roller gear cam for example) this method produces the curvature and slope angle. We are interested in the slope angle and curvature in the plane of the cam follower only. We must find the components of the differentiation in the plane of the cam follower. The components are found by a dot product. Is there a simpler method? Yes!

If we define two orthogonal velocities in the plane of the cam follower (a local co-ordinate system) then the cam slope angle and radius of curvature can be easily calculated using the parametric form.

The parametric form of the pressure angle and radius of curvature is as follows

Y = f(Θ) output axis of local co-ordinate system [inches or millimeters]

dY/dΘ = local output velocity [in./rad., mm/rad.]

d2Y/dΘ2 = local output acceleration [in./rad./rad., mm/rad./rad]

X = f(Θ)  input axis of local co-ordinate system [inches or millimeters]

dX/dΘ = local input velocity [in./rad., mm/rad., rad./rad.]

d2X/dΘ2 = local input acceleration [in./rad./rad., mm/rad./rad]

s = invtan (( dY/dΘ) / (dX/dΘ)) where s is the slope angle [radians]

Roc = [ ( dY/dΘ)2 + (dX/dΘ)2 ] 3/2 / [d2Y/dΘ2 * dX/dΘ - d2X/dΘ2 * dY/dΘ]

Cam Development:

Cam development is a two dimensional analytical and graphical technique used to calculated cam geometry. Ordinarily, a cam will rotate about the camshaft axis and move the output axis it is meshed with. The technique of cam development inverts this by making the camshaft axis stationary and rotating the output axis about the cam in equal intervals. The camshaft intervals can be as great as one degree and as fine as 0.125 of a degree. The output axis is rotated in a counter-clock wise sense. The position of the output axis and the cam followers carried by it can be calculated using simple trigonometry.

You will find cam development and geometry easy to understand if you are familiar with triangles and their properties, (law of sines and cosines, etc). A cam mechanism can be analyzed as a collection of triangles with sides or angles that vary with the desired law of motion.

The following is a list of cam developments for five common cam types used in industry.

Straight-line plate

A two dimensional, flat cam, with the cam follower moving in a straight line. The straight line is rotated around the cam axis as the follower is place in its proper position.

Swing Arm plate

A two dimensional, flat cam, with the cam follower mounted on a swinging arm. The center distance between the cam axis and output arm axis is rotated around the cam axis. The output axis arm with the cam follower at its end is rotated relative to the center distance line.

Straight-line barrel

A three dimensional cylindrical shaped cam where the follower translates along a straight line. The follower line of action and the axis of the cam a parallel to each other. The straight line is rotated around the cam axis at a fixed radius and the follower is place in its proper position.

Swing arm barrel or Right Angle

A three dimensional cylindrical shaped cam where the follower is mounted on a swinging arm. The center distance between the cam axis and output arm axis is rotated around the cam axis. The output axis is at a right angle to the center distance. The output axis arm is in a plane, which is offset at a fixed distance and parallel to the center distance. The cam follower arm is rotated about the output axis. The dwells of these cams generally have parallel and straight side walls.

Roller Gear

A three dimensional concave globoid shaped cam where the follower is mounted on a swinging arm. The center distance between the cam axis and output arm axis is rotated around the cam axis. The output axis is at a right angle to the center distance. The output axis arm is in the same plane as the center distance. The cam follower arm is rotated about the output axis. The dwells of these cams generally have tapered straight side walls.

The swing arm plate, swing arm barrel and roller gear cams can be made with oscillating or indexing motions. They can have one or a multiple number of followers. They can be preloaded using eccentric followers or cams mounted on eccentric camshafts. The swing arm barrel cams with multiple followers are preloaded by manufacturing a slight increase in the space between the followers.

Cam Development and 3-D Surfaces:

The cam development of a 2 dimensional or plate cam is an accurate representation of the cam. The cam development of a 3 dimensional cam resembles that of a tire track through snow or it can be thought to be a mapping or projection of a three-dimensional surface onto a two dimensional surface. Because the cam development of a 3 dimensional cam is a mapping it is not an accurate representation of the cam. The technique of cam development and parametric form of cam geometry simplification can be used to define the correct cam surface of a 3 dimensional cam with some minor modifications to the calculations.

Cam Motions

Included in the reference are source codes for the various motion subroutines along with a compilation of charts and graphs showing the dimensionless motion factors were originally found in the “Moon Book”. These dimensionless factors K, Cv and Ca were used to calculate the displacement, velocity and acceleration of the output before the advent of computers. We have added another factor Cp the power factor to the collection. The power factor is a measure of the power demand for the motion. The graphs of the power factor show the time between the reversing peaks. In order to reduce wear and tear on the drive system (reducer, motor, motor controller, clutch-brakes and couplings) it is important to make these power peaks small and as widely spaced as possible. The following are brief comments about the most common motions along with graphs of their displacement, velocity, acceleration and power.

Constant Velocity

This motion has zero acceleration, and cannot be used for movements as it has velocity discontinuities.

Constant Acceleration

This motion has zero velocity at the beginning and end but it has acceleration discontinuities at the beginning, middle and end. This motion is typical of most servo systems, DC and AC motors and clutch brakes.

Harmonic

This motion has acceleration discontinuities at the beginning and end. This motion has the least amount of energy demand for a system that does not stop or reverses.

Cycloid

This motion has no velocity or acceleration discontinuities and is one of the first motions considered to be smooth and useful.

Modified Trapezoid

This motion is an improvement on the Cycloid but with a lower acceleration present for almost one-half of the cycle.

Modified Sine

This motion is the most common motion used in industry. It is a blend of the cycloid and harmonic. This motion produces a higher B-10 rating for an index drive than the Modified Trapezoid and has the input torque curve peaks spaced further apart.

3-4-5 Polynomial F(Θ) = a Θ 5 + b Θ 4 + c Θ 3 + d Θ 2+ e Θ + f

This motion has similar features as the Modified Sine and is the most common “cam” motion found in servo systems with smooth cam motions.

Motions with constant velocity

Constant velocity is used to reduce the output speed. The constant velocity period is generally placed in the middle of the motion time. For example MSCV 33 means that the first third of the motion time will be modified sine acceleration, the middle third of the motion is at constant velocity, and the last third of the motion time is a modified sine deceleration.

A motion with a constant velocity will reduce the transmission or pressure angle due to the lower speed, but the minimum radius of curvature will be reduced because of the increased acceleration.

For an indexing (where cam followers enter and leave the cam during a cycle) the output speed reduction increases the minimum thickness of material between the cam followers in motion and the amount of distance for a three-follower overlap (one follower enters the cam, the adjacent follower is caught inside the cam and will transfer from one side of the track to the other and the next adjacent follower will leave the cam). However, the acceleration and deceleration values are increased and although the B-10 life rating is increased, the actual application B-10 life hours are reduced due to the increase load demand.

Modified Sine 0.33 CV (33 per cent Constant Velocity)

This motion has Modified Sine acceleration for one-third of its motion time, a constant velocity during the middle third of its motion time and a Modified Sine deceleration during the last third of its motion time.

Modified Sine 0.50 CV (50 per cent Constant Velocity)

This motion has Modified Sine acceleration for one-quarter of its motion time, a constant velocity during the middle half of its motion time and a Modified Sine deceleration during the last quarter of its motion time.

Modified Sine Quick Return (MSQR)

The MSQR motion is used in applications where reciprocating tracking at constant velocity is required. The motion is actually a combination of two motions. The basic motion is a constant velocity for 360 degrees which displaces X amount. A second motion starts with a dwell for time the output is to move at a constant velocity. It is then followed by a modified sine motion which also displaces -X amount of movement. The second motion is added to the basic motion.

In the graph below the red motion (CV) is added to the blue motion (dwell followed by Modified Sine) to create the green motion (MSQR). The Modified Sine motion is preferred but it is also possible to use a Cycloid, Modified Trapezoid or Polynomial instead to create a CYCQR, MTQR or PolyQR.

Synthetic Modified Sine Harmonic (SMSH)

The SMSH is a motion used in oscillating applications (forward and then return) that require a no dwell at one extreme and a dwell at the other extreme. This motion is a blend of a Modified Sine acceleration followed by a full Harmonic motion and finally a Modified Sine deceleration. The time periods for each are equal because the acceleration peaks are to be equal. The Harmonic motion is used during the reversal. The output acceleration has only three peaks.

It is possible (but not recommended) to use a standard motion (Cycloid, MS or MT) for the forward stroke and follow that with another standard motion for the return stroke with no dwell in between. The resulting output of this pairing of motions back to back will have four acceleration peaks instead of three and the power demand peaks will be closer together.

Below for comparison we show the acceleration and power curves for the undesired two back to back Modified Sine motions.

Cam Motions (Subroutines)

The Modified Sine, Harmonic, Modified Trapezoid and Cycloid are common cam motions. The Parabolic motion or constant acceleration is a typical servo motion. The Polynomial motion subroutine most often used is a 3,4,5 polynomial and is close to the modified sine cam motion. The code for this motion is very small and easy to program and several servo suppliers use this as their "cam" motion.

Definition of terms

BR = time period of motion, degrees, radians, seconds

angra = dimensionless time fraction range (0 to 1)

fmsc = fraction of constant velocity range (0 to 1) this constant velocity is centered during the motion interval: for a value of 0 the motion profile is an acceleration to a maximum speed, 1/2 way into the motion time followed by a deceleration to 0 speed. For a value of 0.5 the motion profile would be for the first 1/4 of the motion time there will be an acceleration to maximum speed, followed by 1/2 of the total motion time at maximum speed and then the remaining 1/4 of the motion time is a deceleration.

DF = dimensionless displacement factor (0 to 1)

VF = velocity factor dimensions are 1/BR

AF = acceleration factor dimensions are 1/BR/BR

First Example of a loop in BASIC ( a ' in BASIC prefixes a remark and is ignored by the compiler)

output = 12 ' meters is the total movement

time = 0.999 ' total motion time in seconds

ncal = 999 ' divide the entire motion time into 999 intervals (for this example this interval is exactly one millisecond)

BR= time

fmsc = 0.33333 'one third of the time or 333 milliseconds is acceleration, 333 milliseconds is constant velocity and 333 milliseconds is decel.

for i = 0 to ncal

angra = i/ncal

xtime = angra*time

Call MSCV(BR,angra,fmsc,DF,VF,AF)

displacement = DF * output

velocity = VF * output

acceleration = AF * output

write to file or array or print here ....... xtime,displacement,velocity,acceleration

next i

Second Example

output = 12 ' meters is the total movement

time = 1 ' total motion time in seconds

cam movement is 120 degrees at 60 rpm

ncal = 120 ' we wish to divide the entire motion time into 1 degree intervals

BR= 120 / 6 / 60 ‘ 120 degrees moved at 60 rpm is 1 second

fmsc = 0

For the rest use the same pseudo code as above

Motion Synthesis Techniques

Using the dimensionless motion factors it is possible to create a synthesis or patchwork of motion pieces to create a movement that is a combination of fundamental motions.

V = Cv (Velocity Factor at any point in time) * h ( total displacement ) / B ( total motion time)

Vmax = Cvmax * h / B

The average velocity is h / B, rearranging the above equation gives

h / B = Vmax / Cvmax then Vaverage = Vmax / Cvmax

Example A) - Derivation of Modified Sine with f (the fraction or percent of cv) maximum motion

factors Cvmaxeff and Camaxeff.

Cvmaxeff – Effective Maximum Velocity Factor

Vmax = Cvmaxeff * h ( total displacement ) / B ( total motion time)

f * B = motion time at constant velocity where f is the fraction or percent of cv ( 0.33 or 0.50)

h = Acceleration Displacement + Constant Velocity Displacement+ Deceleration Displacement

h = Vaverage * (B – f * B ) / 2 + Vmax * (f * B) + Vaverage * (B –f * B ) / 2

h = Vaverage * (B –f * B ) + Vmax * (f * B)

h = Vmax / Cvmax * (B –f * B ) + Vmax * (f * B)

h = Vmax * B / Cvmax * (1 + f * (Cvmax – 1))

h / B = Vmax / Cvmax * ( 1 + f * (Cvmax – 1))

Vmax = Cvmaxeff * h / B

h / B = Vmax / Cvmaxeff

Vmax / Cvmaxeff = Vmax / Cvmax * ( 1 + f * (Cvmax – 1))

Cvmaxeff = Cvmax / (1 + f * (Cvmax – 1))

For 33% cv then

Cvmaxeff = 1.7596 / (1 + 0.3333333 * (1.7596– 1)) = 1.404085541 (See above)

Motion Synthesis Techniques (continued)

Camaxeff – Effective Maximum Acceleration Factor

Accmax = Camaxeff * h / B2

Accmax = Camax * 2 * Acceleration Displacement / ( 2 * Acceleration Time ) 2

Accmax = Camax * Vaverage * (B –f * B )/ (B –f * B ) 2

Accmax = Camax * Vmax / Cvmax / (B * (1– f ))

Accmax = Camax * Cvmaxeff * h / B/ Cvmax/ (B * (1–f ))

Accmax = Camax * Cvmax / ( 1 + f * (Cvmax – 1))* h / B/ Cvmax/ (B * (1–f ))

Camaxeff * h / B2 = Camax * Cvmax / ( 1 + f * (Cvmax – 1))* h / B / Cvmax / (B * (1–f ))

Camaxeff = Camax * / ( ( 1 + f * (Cvmax – 1)) * ( (1– f )) )

For 33% cv then

Camaxeff = 5.528 * / (( 1 + 0.3333* (1.7596 – 1)) * ( (1– 0.333))) = 6.6167 (See above)

Motion Synthesis Techniques (continued)

Example B) - Derivation of SMSH time periods a = b

Where a = Modified Sine Acceleration time, b = Harmonic Acceleration time

Modified Sine Velocity = Harmonic Velocity – where they meet velocities must be equal

Vmax = CvmaxMS * hMS / a = CvmaxHarm * hHarm / b

hMS = Vmax * a / CvmaxMS

hHarm = Vmax * b / CvmaxHarm

Modified Sine Acceleration = Harmonic Acceleration - additional requirement for SMSH motion

Modified Sine Acc = Modified Sine Disp* Modified Sine Acc Factor / Modified Sine Time 2

Modified Sine Acceleration = Vmax * a / CvmaxMS * 2 * CamaxMS / ( 2 * a ) 2

Harmonic Acceleration = Harmonic Displacement * Harmonic Acc Factor / Harmonic Time 2

Vmax * b / CvmaxHarm * 2 * CamaxHarm / ( 2 * b ) 2

Vmax * a / CvmaxMS * 2 * CamaxMS / ( 2 * a ) 2 = Vmax * b / CvmaxHarm * 2 * CamaxHarm /  ( 2 * b ) 2

1 / CvmaxMS * CamaxMS / a = 1 / CvmaxHarm * CamaxHarm / b

b = a * CamaxHarm * CvmaxMS / (CvmaxHarm* CamaxMS )

b = a * 4.9348 * 1.7596 / ( (pi /2) )* 5.528 )

b = a