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If x < 3 then the function is plainly continuous The function is undefined at x = 3 so we may not even speak of continuity at x = 3 The function is also obviously continuous for 3 < x < 4 At x = 4 the limit of g does not exist---it is 1 from the left and 11 from the right So the function is not continuous (we sometimes say that it is discontinuous) at x = 4 By inspection, the function is continuous for x > 4

Let us investigate the automobile camshaft curves applied to the system Characteristic curves of displacement (lift), velocity, acceleration, and jerk will be studied and manipulated to yield the best performance of the follower The total cam curve is generally composed of a ramp and the main event The purpose of the ramp is to minimize the backlash and to control the proper valve seating velocity and seal The ramp events that in turn are blended with sine segments (with no discontinuity) are: No ramp, main event only (not suggested) Constant velocity ramp Constant acceleration ramp Constant jerk ramp

Usually a constant-velocity ramp is chosen The main event designs ( ank of cam), which blend with a ramp, are:

We note that Theorem 21 guarantees that the collection of continuous functions is closed under addition, subtraction, multiplication, division (as long as we do not divide by 0), and scalar multiplication

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Circular-arc see Chap 14 Circular-arc with straight line (tangent cam) see Chap 15 High-order polynomial curve Composite cam designs Polydyne cam (rarely used)

If f g makes sense, if limx c g(x) = , and if lims f (s) = m, then it does not necessarily follow that limx c f g(x) = m [We invite the reader to find an example] One must assume, in addition, that f is continuous at This point will come up from time to time in our later studies

The polynomial equation is the most popular method for blending the ramp and main event The units of the characteristic curve equation are inches and degrees Let us relate these units to the time base employed throughout the book If we let N = cam speed, rpm The follower characteristics at the cam are Displacement ( lift ) y = inches Velocity 1 velocity, in sec = in deg 6N Acceleration y= 1 y= acceleration, in sec 2 = in deg 2 6N Jerk 1 3 = y jerk, in sec = in deg 3 6N In Fig 165 we see the characteristic curves of an automobile camshaft (from software DESINE) with a constant velocity ramp Figure 166 shows diesel intake valve curves In these diesel curves it should be noted that the acceleration and jerk curve shape and maximum values are acceptable since diesels run at about one-half the speed of automobiles

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In this section we investigate the camshaft geometric relationships: Cam pro le Cam pro le curvature Hertz stresses Follower linkage relationship

We further record the important fact that differentiability is a stronger property than continuity If a function f is differentiable at c then it is continuous at c This assertion really follows from Theorem 23: If limh 0 [ f (c + h) f (c)]/ h exists then limh 0 [ f (c + h) f (c)] must be 0 We invite the reader to fill in the details In the next section, we discuss the concept of differentiability in greater detail

The cam pro le or contour must rst be established to determine the curvature of all points for an acceptable minimum radii of curvature for both fabrication and the Hertz stresses In Chap 9 the mathematical relationship for Hertz stresses was presented It is to be noted that the cam and valve have different displacement, velocity, acceleration, and jerk curves (except for direct-acting systems) This phenomenon is because of the linkage angularity and oscillations

Acceleration Velocity Displacement (lift)

Suppose that f is a function whose domain contains the interval (a, b) Let c be a point of (a, b) If the limit lim

Jerk Closing Side Opening Side Angle 140 120 100 80 60 40 20 NOSE 20 40 60 80 100 120 140 60 80 100 120 140 160 180 160 140 120 100 80 60

There is an additional complication as x becomes large, either positive or negative For large x the Fig 2-5 function becomes large number over large number If, however, the fraction is multiplied by 1/ x over 1/ x the limit can be calculated easily:

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