Dynamics of Mechanical Systems 2009 Part 12 pptx

By inspecting the construction of Figure 16.4.3, we see that given the cam profile it isrelatively easy — at least, in principle — to obtain the follower rise function hθ.. 16.5 Graphica

532 Dynamics of Mechanical Systems

Here we see that both the resultant primary and secondary inertia forces are alsobalanced, leaving the only unbalance with the resultant primary moments Thus, we havestill further improvement in the balance

These examples demonstrate the wide range of possibilities available to the enginedesigner; however, the examples are not meant to be exhaustive Many other practicalconfigurations are possible The examples simply show that the crankshaft configurationcan have a significant effect upon the engine balance In the following section, we willextend these concepts and analyses to eight-cylinder engines

15.10 Eight-Cylinder Engines: The Straight-Eight and the V-8

If we consider engines with eight cylinders the number of options for balancing increasesdramatically The analysis procedure, however, is the same as in the foregoing section Toillustrate the balancing procedure, consider an engine with eight cylinders in a line (the

straight-eight engine) and with crank angles φi for the connecting rods arranged tally at 90° along the shaft Table 15.10.1 provides the listing of the φi for the eight cylinders

incremen-(i = 1,…, 8) together with the trigonometric functions needed to test for balancing A glance

at the table immediately shows that the engine with this incremental 90° crank anglesequence has a moment unbalance

This raises the question as to whether there are crank angle configurations where plete balancing would occur, within the approximations of our analysis To respond to thisquestion, consider again the four-cylinder engine of the previous section For the crankangle configuration of Table 15.9.5, we found that the four-cylinder engine was balancedexcept for the primary moment Hence, it appears that we could balance the eight-cylinderengine by considering it as two four-cylinder engines having reversed crank angle config-urations Specifically, for the four-cylinder engine of Table 15.9.5, the crank angles are 0,

com-90, 270, and 180°; therefore, let the first four cylinders have the crank angles 0, com-90, 270, and180°, and let the second set of four cylinders have the reverse crank angle sequence of 180,

270, 90, and 0° Table 15.10.2 provides a listing of the crank angles φ; together with thetrigonometric functions needed to test for balancing As desired (and expected), the engine

is balanced for both primary and secondary forces and moments

Balancing 533

A practical difficulty with a straight-eight engine, however, is that it is often too long

to conveniently fit into a vehicle engine compartment One approach to solving this

problem is to divide the engine into two parts, into a V-type engine as depicted in Figure 15.10.1 The two sides of the engine are called banks, each containing four cylinders.

Because the total number of cylinders is eight, the engine configuration is commonly

referred to as a V-8.

A disadvantage of this engine configuration, however, is that the engine is no longer inbalance, as compared to the straight-eight engine: To see this, consider again the crankconfiguration of the straight-eight as listed in Table 15.10.2 Taken by themselves, the firstfour cylinders are unbalanced with an unbalanced primary moment perpendicular to theplane of the cylinders as seen in Table 15.9.6; hence, the second set of four cylinders has

an unbalanced primary moment perpendicular to its plane With the cylinder planesthemselves being perpendicular, these unbalanced moments no longer cancel but insteadhave a vertical resultant as represented in Figure 15.10.2 This unbalance will have atendency to cause the engine to oscillate in a yaw mode relative to the engine compartment.This yawing, however, can often be kept small by the use of motor mounts having highdamping characteristics Thus, the moment unbalance is usually an acceptable tradeoff inexchange for obtaining a more compact engine

534 Dynamics of Mechanical Systems

15.11 Closure

Our analysis shows that if a system is out of balance it can create undesirable forces atthe bearings and supports If the balance is relatively small, it can often be significantlyreduced or even eliminated by judicious placing of balancing weights

Perhaps the most widespread application of balancing principles is with the balancing

of internal-combustion engines and with similar large systems Because such systems have

a number of moving parts, complete balance is generally not possible Designers of suchsystems usually attempt to minimize the unbalance while at the same time making com-promises or tradeoffs with other design objectives

We saw an example of such a tradeoff in the balancing of an eight-cylinder engine: theengine could be approximately balanced if the cylinders were all in a line This arrange-ment, however, creates a relatively long engine, not practical for many engine compart-ments An alternative is to divide the engine into two banks of four cylinders, inclinedrelative to each other (the V-8 engine); however, the engine is then out of balance in yawmoments, requiring damping at the engine mounts to reduce harmful vibration

Optimal design of large engines thus generally involves a number of issues that must beresolved for each individual machine While there are no specific procedures for suchoptimal design, the procedures outlined herein, together with information available in thereferences, should enable designers and analysts to reach toward optimal design objectives

References

15.1 Wilson, C E., Sadler, J P., and Michaels, W J., Kinematics and Dynamics of Machinery, Harper

& Row, New York, 1983, pp 609–632.

15.2 Wowk, V., Machine Vibrations, McGraw-Hill, New York, 1991, pp 128–134.

15.3 Paul, B., Kinematics and Dynamics of Planar Machinery, Prentice Hall, Englewood Cliffs, NJ,

Section 15.2 Static Balancing

P15.2.1: A 125-lb flywheel in the form of a thin circular disk with radius 1.0 ft and thickness1.0 in is mounted on a light (low-weight) shaft which in turn is supported by nearly

Balancing 535

frictionless bearings as represented in Figure P15.2.1 If the flywheel is mounted off-center

by 0.25 in., what weight should be placed on the flywheel rim, opposite to the off-centeroffset, so that the flywheel is statically balanced?

P15.2.2: Repeat Problem P15.2.1 if the flywheel mass is 50 kg, with a radius of 30 cm, athickness of 2.5 cm, and an off-center mounting of 7 mm

P15.2.3: See Problem P15.2.1 Suppose the flywheel shaft has frictionless bearings Whatwould be the period of small oscillations?

P15.2.4: Repeat Problem P15.2.3 for the data of Problem P15.2.2

P15.2.5: See Problem P15.2.3 Suppose a slightly unbalanced disk flywheel, supported in

a light shaft with frictionless bearings, is found to oscillate about a static equilibriumposition with a period of 7 sec How far is the flywheel mass center displaced from theshaft axis?

Section 15.3 Dynamic Balancing

P15.3.1: A shaft with radius r of 3 in is rotating with angular speed Ω of 1300 rpm Particles

P1 and P2, each with weight w of 2 oz each, are placed on the surface of the shaft as shown

in Figure P15.3.1 If P1 and P2 are separated axially by a distance  of 12 in., determinethe magnitude of the dynamic unbalance

P15.3.2: Repeat Problem P15.3.1 if r, w, and  have the values r = 7 cm, w = 50 g, and

a drilling procedure to balance the shaft That is, suggest the number, size, and positioning

536 Dynamics of Mechanical Systems

Section 15.4 Dynamic Balancing: Arbitrarily Shaped Rotating Bodies

P15.4.1: Suppose n1, n2, and n3 are mutually perpendicular unit vectors fixed in a body B, and suppose that B is intended to be rotated with a constant speed Ω about an axis X

which passes through the mass center G of B and which is parallel to n1 Let n1, n2, and

n3 be nearly parallel to principal inertia directions of B for G so that the components I ij of

the inertia dyadic of B for G relative to n1, n2, and n3 are:

Show that with this configuration and inertia dyadic that B is dynamically out of balance Next, suppose we intend to balance B by the addition of two 12-oz weights P and placed opposite one another about the mass center G Determine the coordinates of P

and relative to the X-, Y-, and Z-axes with origin at G and parallel to n1, n2, and n3

P15.4.2: Repeat Problem P15.4.1 if the inertia dyadic components are:

and if the masses of P and are each 0.5 kg.

P15.4.3: Repeat Problems P15.4.1 and P15.4.2 if B is rotating about the Z-axis instead of the X-axis.

Section 15.5 Balancing Reciprocating Machines

P15.5.1: Suppose the crank AB of a simple slider/crank mechanism (see Figures 15.5.1 and

P15.5.1, below) is modeled as a rod with length of 4 in and weight of 2 lb At what distance

away from A should a weight of 4 lb be placed to balance AB?

P15.5.2: See Problem P15.5.1 Suppose is to be 1.5 in What should be the weight of thebalancing mass ?

P15.5.3: Repeat Problem P15.5.1 if rod AB has length 10 cm and mass 1 kg.

P15.5.4: Consider again the simple slider/crank mechanism as in Figure P15.5.4, this timewith an objective of eliminating or reducing the primary unbalancing force as developed

in Eq (15.5.23) Specifically, let the length r of the crank arm be 4 in., the length  of the

connecting rod be 9 in., the weight of the piston C be 3.5 lb, and the angular speed w of

Balancing 537

the crank be 1000 rpm Determine the weight (or mass ) of the balancing weight to

eliminate the primary unbalance if the distance h of the weight from the crank axis is 3

in Also, determine the maximum secondary unbalance that remains and the maximum

unbalance in the Y-direction created by the balancing weight.

P15.5.5: Repeat Problem P15.5.4 if the balancing weight compensates for only (a) 1/2 and(b) 2/3 of the primary unbalance

P15.5.6: Repeat Problems P15.5.4 and P15.5.5 if r is 10 cm,  is 25 cm, h is 5 cm, and the mass of C is 2 kg.

Section 15.6 Lanchester Balancing

P15.6.1: Verify Eqs (15.6.1), (15.6.2), and (15.6.3)

P15.6.2: Observe that the geometric parameters of Eq (15.6.5) are such that the Lanchester

balancing mass m need only be a fraction of the piston mass m C Specifically, suppose

that (r/) is 0.5 and (ξ/r) is also 0.5 What, then, is the mass ratio m/m C?

Sections 15.7, 15.8, 15.9 Balancing Multicylinder Engines

P15.7.1: Consider a three-cylinder, four-stroke engine Following the procedures outlined

in Sections 15.7, 15.8, and 15.9, develop a firing order and angular positioning to optimizethe balancing of the engine

16

Mechanical Components: Cams

16.1 Introduction

In this chapter, we consider the design and analysis of cams and cam–follower systems

As before, we will focus upon basic and fundamental concepts Readers interested inmore details than those presented here may want to consult the references at the end

Some cam-pairs simply convert one kind of translation into another kind of translation,

as in Figure 16.1.2 Here, again, the actuator or driving component is the cam and theresponding component is the follower An analogous cam-pair that transforms simplerotation into simple rotation is a pair of meshing gears depicted schematically inFigure 16.1.3 Because gears are used so extensively, they are usually studied separately,which we will do in the next chapter Here, again, however, the actuator gear is called the

driver and the responding gear is called the follower Also, the smaller of the gears is calledthe pinion and the larger is called the gear

To some extent, the study of cams employs different procedures than those used in ourearlier chapters; the analysis of cams is primarily a kinematic analysis Although the study

of forces is important in mechanism analyses, the forces generated between cam–followerpairs are generally easy to determine once the kinematics is known The major focus ofcam analysis is that of cam design Whereas in previous chapters we were generally given

a mechanical system to analyze, with cams we are generally given a desired motion andasked to determine or design a cam–follower pair to produce that motion

In the following sections, we will consider the essential features of cam and followerdesign We will focus our attention upon simple cam pairs as in Figure 16.1.1 Similaranalyses of more complex cam mechanisms can be found in the various references Beforedirecting our attention to simple cam pairs, it may be helpful to briefly review configu-rations of more complex cam mechanisms We do this in the following section and thenconsider simple cam–follower design in the subsequent sections

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540 Dynamics of Mechanical Systems

16.2 A Survey of Cam Pair Types

As noted earlier, the objective of a cam–follower pair is to transform one kind of motioninto another kind of motion As such, cam-pairs are kinematic devices When, in addition

to transforming and transmitting motion, cam-pairs are used to transmit forces, they arecalled transmission devices

There are many ways to transform one kind of motion into another kind of motion.Indeed, the variety of cam–follower pair designs is limited only by one’s imagination Inaddition to those depicted in Figures 16.1.1, 16.1.2, and 16.1.3, Figures 16.2.1, 16.2.2, and16.2.3 depict other common types of cams Figures 16.2.4 and 16.2.5 depict various common

Mechanical Components: Cams 541

follower types Figures 16.2.2 and 16.2.3 show that cam–follower pairs may be used notonly for exchanging types of planar motion but also for converting planar motion intothree-dimensional motion

In the following sections, we will direct our attention to planar rotating cams (or radial cams) with translating followers as in Figure 16.2.4 Initially, we will consider commonnomenclature and terminology for such cams

16.3 Nomenclature and Terminology for Typical Rotating Radial Cams with Translating Followers

Consider a radial cam–follower pair as in Figure 16.3.1 where the axis of the follower shaftintersects the axis of rotation of the cam Let the follower shaft be driven by the cam

Types of rotation followers.

Cam Follower Track

Cam Follower

Point Follower Plane Follower Roller Follower

Slider Follower Roller Follower

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542 Dynamics of Mechanical Systems

through a follower wheel as shown Figure 16.3.2 shows the commonly employed nology and nomenclature of the radial cam–follower pair [16.1] Brief descriptions of theseitems follow:

termi-• Cam profile — contact boundary of the rotating cam

• Follower wheel — rolling wheel of the follower contacting the cam profile toreduce wear and to provide smooth operation

• Trace point — center of rolling follower wheel; also, point of knife-edge follower

• Pitch curve — path of the trace point

• C — typical location of the trace point on the pitch curve

• θ — pressure angle

Observe the pressure angle in Figure 16.3.2 It is readily seen that the pressure angle is

a measure of the lateral force exerted by the cam on the follower Large lateral forces (asopposed to longitudinal or axial forces) can lead to high system stress and eventuallydeleterious wear A large pressure angle θ can create large lateral forces Specifically, thelateral force is proportional to sinθ

Observe further in Figure 16.3.2 that if the cam profile is circular the pressure angle iszero In this case, no lateral forces are exerted on the follower by the cam However, withthe circular profile, no longitudinal force will be exerted on the follower, so the followerwill remain stationary A stationary follower corresponds to a dwell region for the cam

Cam Normal Follower

Trace Point Follower Wheel Cam Profile

Pitch Curve

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Mechanical Components: Cams 543

These observations show that there is no follower movement without a nonzero pressureangle; hence, a cost of follower movement is the lateral force generated by the pressure angle.Thus, cam designers generally attempt to obtain the desired follower motion with the smallestpossible pressure angle This is usually accomplished by a large cam, as space permits.Finally, observe that the follower wheel simply has the effect of enlarging the cam profilecreating the pitch curve, with the follower motion then defined by the movement of thetrace point

16.4 Graphical Constructions: The Follower Rise Function

The central problem confronting the cam designer is to determine the cam profile thatwill produce a desired follower movement To illustrate these concepts and some graphicaldesign procedures, consider again the simple knife-edge radial follower pair ofFigure 16.4.1 Let the cam profile be represented analytically as (see Figure 16.4.2):

(16.4.1)where r and θ are polar coordinates of a typical point P on the cam profile The angle θmay also be identified with the rotation angle of the cam Thus, given f(θ), we know r as

a function of the rotation angle Then with the knife-edge follower, we can identify r withthe movement of the follower

If the angular speed ω of the cam (ω = ) is constant, the speed v of the follower is:

(16.4.2)and the acceleration a of the follower is:

df d

θ

a r d dt

df d

d dt

df d d

d

df d

d dt

d f d

544 Dynamics of Mechanical Systems

Let r min be the minimum value of r in Eq (16.4.1) Let h be the difference between r and

r min at any cam position: That is, let

(16.4.4)

Then, h(θ) represents the rise (height) of the follower above its lowest position

We can obtain a graphical representation of h(θ) by dividing the cam into equiangularsegments and then by constructing circular arcs from the segment line/cam profile inter-section to the vertical radial line Then, horizontal lines from these intersections determinethe h(θ) profile, as illustrated in Figure 16.4.3

By inspecting the construction of Figure 16.4.3, we see that given the cam profile it isrelatively easy — at least, in principle — to obtain the follower rise function h(θ) Theinverse problem, however, is not as simple; that is, given the follower rise function h(θ),determine the driving cam profile This is a problem commonly facing cam designerswhich we discuss in the following section

16.5 Graphical Constructions: Cam Profiles

If we are given the follower rise function f(θ), we can develop the cam profile by reversingthe graphical construction of the foregoing section To illustrate this, consider a generalfollower rise function as in Figure 16.5.1

Next, we can construct the cam profile by first selecting a minimum cam radius r min.Then, the cam radius at any angular position is defined by Figure 16.5.1 Graphically, thismay be developed by constructing a circle with radius r min and then by scaling points onradial lines according to the follower rise function To illustrate this, let this minimumradius circle (called the base circle) and the radial lines be constructed as in Figure 16.5.2.Next, let the follower rise be plotted on radial lines according to the angular position as

Mechanical Components: Cams 545

in Figure 16.5.3 where, for illustrative purposes, we have used the data of Figure 16.5.1.Then, by fitting a curve through the plotted points on the radial line we obtain the camprofile as in Figure 16.5.3 The precision of this process can be improved by increasing thenumber of radial lines

Although this procedure seems to be simple enough, hidden difficulties may emerge inactual construction, including the effects of differing cam–follower geometries and themore serious problem of obtaining a cam profile that may produce undesirable accelera-tions of the follower We discuss these problems and their solutions in the followingsections

16.6 Graphical Construction: Effects of Cam–Follower Design

In reviewing the procedures of the two foregoing sections, we see and recall from Figure16.4.1 that our discussions assumed a simple knife-edge radial follower If the followerdesign is modified, however, the graphical procedures will generally need to be modified

as well

To illustrate these changes, consider again an elliptical cam profile for which the follower

is no longer radial but instead is offset from the cam rotation axis Also, let the contactsurface between the follower and cam be an extended flat surface as in Figure 16.6.1

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546 Dynamics of Mechanical Systems

If, as before, we attempt to determine the follower rise function (assuming constant camangular speed) by considering equiangular radial lines, we see that, due to the offset, thefollower axis will not coincide with the radial lines Also, due to the flat follower surface,

we see that the base of the follower at the axis of the follower will not, in general, be incontact with the cam profile (see Figure 16.6.2)

A question that arises then is how do these geometrical changes affect the graphicalconstruction of the follower rise profile as in Figure 16.4.3? To answer this question,consider again constructing radial lines on the cams as in Figure 16.4.3 and as shownagain here in Figure 16.6.3

Next, for each radial line let a profile of the follower be superposed on the figure suchthat the axis of the follower is parallel to the radial line but offset from it by the offset ofthe follower axis and the cam axis Let the follower profile be placed so that the flat edge

Mechanical Components: Cams 547

is tangent to the cam profile A representation of this procedure at 45°-angle increments

is shown in Figure 16.6.4

Finally, let the points of tangency between the flat edge of the follower and the camprofile be used to develop the follower rise function using the procedure of Figure 16.4.3and as demonstrated in Figure 16.6.5

The inverse problem of determining the cam profile when given the follower risefunction is a bit more difficult, but it may be solved by following similar procedures tothose of Figures 16.6.3, 16.6.4, and 16.6.5 (As noted earlier, this problem is of greaterinterest to designers, whereas the problem of determining the follower rise function is ofgreater interest to analysts.)

To illustrate the procedure, suppose we are given a follower rise function as in Figure16.5.1 and as shown again in Figure 16.6.6 To accommodate such a function with an offsetflat surface follower as in Figure 16.6.1, we need to slightly modify the procedure ofSection 16.5 in Figures 16.5.2 and 16.5.3 Specifically, instead of constructing radial linesthrough the cam axis as in Figure 16.5.2, we need to account for the offset of the followeraxis This means that, instead of radial lines through the cam axis, we need to constructtangential lines to a circle centered on the cam axis with radius equal to the offset andthus tangent to the follower axis

To see this consider again the representation of the offset flat surface follower ofFigure 16.6.1 and as shown in Figure 16.6.7 Let the offset between the follower and cam

FIGURE 16.6.5

Graphical construction of follower rise profile.

FIGURE 16.6.6

A typical follower rise function.

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548 Dynamics of Mechanical Systems

rotation axis be ρ as shown Next, construct a circle with radius ρ and with tangent lines

representing the follower axis and placed at equal interval angles around the circle as

shown in Figure 16.6.8

The third step is to select a base circle radius, as before, and to superimpose it over the

offset axes circle (coincident centers) as in Figure 16.6.9 Fourth, using the follower rise

function of Figure 16.6.6, we can use the curve ordinates of the various cam rotation angles

as the measure of the follower displacement above the base circle, along the corresponding

radial lines of Figure 16.6.9 Figure 16.6.10 shows the location of these displacement points

Finally, by sketching the flat surface of the follower placed at these displacement points

we form an envelope [16.2] of the cam profile That is, the flat follower surfaces are tangent

to the cam profile Then, by sketching the curve tangent to these surfaces, we have the

cam profile as shown in Figure 16.6.11

Observe in this process that accuracy is greatly improved by increasing the number of

angle intervals Observe further that, while this process is, at least in principle, the same

as that of the foregoing section (see Figure 16.5.3), the offset of the follower axis and the

flat surface of the follower require additional graphical construction details (One effect

Mechanical Components: Cams 549

of this is that the cam profile can fall inside the base circle as seen in Figure 16.6.11.) Finally,

observe that, although the cam profiles of Figures 16.5.3 and 16.6.11 are not identical, they

are similar

16.7 Comments on Graphical Construction of Cam Profiles

Although the foregoing constructions are conceptually relatively simple, difficulties are

encountered in implementing them First, because they are graphical constructions, their

accuracy is somewhat limited — although emerging computer graphics techniques will

undoubtedly provide a means for improving the accuracy Next, analyses assuming point

contact between the cam and follower (as in Sections 16.4 and 16.5) may produce cam

profiles that are either impractical or incompatible with roller-tipped followers For

exam-ple, for an arbitrary follower rise function, it is possible to obtain cam profiles with cusps

or concavities as in Figure 16.7.1 Finally, if the cam is rotating rapidly (as is often the

FIGURE 16.6.10

Follower displacement points relative to the base circle.

FIGURE 16.6.11

Envelope of follower flat defining the cam profile.

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550 Dynamics of Mechanical Systems

case with modern mechanical systems), the cam profile may create very large follower

accelerations These accelerations in turn may lead to large and potentially destructive

inertia forces In the next section, we consider analytical constructions intended to

over-come these difficulties

16.8 Analytical Construction of Cam Profiles

As with graphical cam profile construction, analytical profile construction has the same

objective; that is, given the follower motion, determine the cam profile that will produce

that motion Because the follower motion may often be represented in terms of elementary

functions, it follows that the cam profile may also often be represented in terms of

ele-mentary functions When this occurs, the analytical method has distinct advantages over

the graphical method Conversely, if the follower motion cannot be represented in terms

of elementary functions, the analytical approach can become unwieldy and impractical,

necessitating the use of numerical procedures and approximations Also, as we will see,

the design of cam profiles with elementary functions may create undesirable inertia forces

with high-speed systems

With these possible deficiencies in mind, we outline the analytical construction in the

following text As with the graphical construction, we develop the fundamentals using

the simple planar cam–follower pair, with the follower axis intersecting the rotation axis

of the cam as in Figure 16.8.1 Also, we will assume the cam rotates with a constant angular

speed ω0

In many applications, the follower is to remain stationary for a fraction of the cam

rotation period After passing through this stationary period, the follower is then generally

required to rise away from this stationary position and then ultimately return back to the

stationary position Figure 16.8.2 shows such a typical follower motion profile The

sta-tionary positions are called periods of dwell for the follower and, hence, also for the cam

The principle of analytical cam profile construction is remarkably simple The follower

rise function h(θ)(as shown in Figure 16.8.2) is directly related to the cam profile function

To see this, note that, because the cam is rotating at a constant angular speed ω0, the

angular displacement θ of the cam is a linear function of time t That is,

(16.8.1)where the initial angle θ may conveniently be taken as zero without loss of generality

Mechanical Components: Cams 551

Next, because the follower rise h is often expressed as a function of time t, say h(t), we can solve Eq (16.8.1) for t (t = θ/ω0) and then express the follower displacement as afunction of the rotation angle θ as in Figure 16.8.3 Then, in view of the graphical con-structions of the foregoing sections and in view of Figure 16.8.1, we see that the followerrise function is simply the deviation of the cam profile from its base circle That is, if the

cam profile is expressed in polar coordinates as r( θ), then r(θ) and h(θ) are related as:

(16.8.2)

where r0 is the base circle radius

16.9 Dwell and Linear Rise of the Follower

To illustrate the application of Eq (16.8.2), suppose the follower displacement is in dwell

That is, let the rise function h θ be zero as in Figure 16.9.1 Then, the cam profile r(θ) has

the simple form:

Follower displacement as a function

of the cam rotation angle θ.

r( )θ = +r0 h( )θ

r( )θ =r0 (a constant)

552 Dynamics of Mechanical Systems

Next, suppose the follower rise function is not zero but instead is still a constant (say,

h0) as in Figure 16.9.2 In this case, the follower is still in dwell, but after a displacement

(or rise) of h0 From Eq (16.8.2) the cam profile is then:

Dwell

Follower Rise h( θ)

Cam Rotation θ

Mechanical Components: Cams 553

Hence, from Eq (16.8.2) the cam profile radius r(θ) is:

(16.9.5)

Using Eq (16.9.4) produces a profile as in Figure 16.9.4, where a linearly increasing radius

is shown for 90° of the cam angle

16.10 Use of Singularity Functions

The foregoing analysis may be facilitated by the use of singularity or delta functions To develop this idea, let us first introduce the step function δ1(x) defined as (see References

By integrating, we may generalize the definition of Eq (16.10.1) and thus introduce

higher order step-type functions (or singularity [16.6] functions) as:

554 Dynamics of Mechanical Systems

allow the functions to be continuous at x = 0

Graphically, these functions may be represented as in Figure 16.10.1 Observe that exceptfor δ1(x) all the functions are continuous at x = 0.

Observe further that just as δn+1(x) represents the integration (or antiderivative) of δn(x),

it follows that δn(x) is the derivative of δn+1(x) This raises the question, however, as to the

derivative of δ1(x) From the definition of Eq (16.10.1), we see that, if x ≠ 0, then dδ1(x)/dx

is zero At x = 0, however, the derivative is undefined, representing an infinite change in

the function Thus, we have:

(16.10.10)

δ0(x) is often referred to as Dirac’s delta function — analogous to Kronecker’s delta

function of Section 2.6 The analogy is seen through the “sifting” or “substitution” property

of the functions Recall from Eq (2.6.7) that we have:

(16.10.11)Similarly, it is seen (see References 16.1 to 16.5) that:

Mechanical Components: Cams 555

The functions δ2(x), δ3(x), δ4(x),…, δn+1(x) of Eqs (16.10.5) through (16.10.8), as well as

δ0(x) of Eq (16.10.10), may also be represented with the independent variable shifted as

in Eq (16.10.2) That is,

where δ–1(x) has the form of an impulsive doublet function as in Figure 16.10.3 (Graphical

representations of the higher order derivatives are not as readily obtained, thus the ical representations of the higher order derivatives are not as helpful.)

graph-There has been extensive application of these influence or interval functions in structuralmechanics — particularly in beam theory (see, for example, Reference 16.6) For cam profileanalysis, they may be used to conveniently define the profile, as in Eq (16.10.4) Theprincipal application of the functions in cam analysis, however, is in the differentiation of

556 Dynamics of Mechanical Systems

the follower rise function, providing information about the follower velocity, acceleration,

and rate of change of acceleration (jerk).

To demonstrate this, consider again the piecewise-linear follower rise function of Figure16.9.3 and as expressed in Eq (16.10.4):

(16.10.17)

where as before m is (h2 – h1)/(θ2 – θ1) (see Figure 16.9.3) The velocity v and the acceleration

a of the follower are then:

(16.10.21)

We can readily see that the final terms in both Eqs (16.10.20) and (16.10.21) are zero To

see this, consider first the expression xδ0(x) If x is not zero, then δ0(x) and thus xδ0(x) are

dh d

d dt

dh d

θ

a d h dt

d dt

dh dt

d d

dh dt

d dt

d d

dh d

d h d