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Mechanics of Machines Dr. Mohammad Kilani Class 5 CAMS Introduction to Cams Cam-follower systems are frequently used in many kinds of machines. Common examples of cam follower systems are the automobile engine valves, which are opened by cams. Machines used in the manufacture of many consumer goods are full of cams. Compared to linkages, cams are easier to design to give a specific output function, but they are much more difficult and expensive to make than a linkage. Cams are a form of degenerate fourbar linkage in which the coupler link has been replaced by a half joint. Cams Terminology Cam-follower systems can be classified in several ways: by type of follower motion, either translating or rotating (oscillating); by type of cam, radial, cylindrical, threedimensional; by type of joint closure, either force- or formclosed; by type of follower, curved or flat, rolling or sliding; by type of motion constraints, critical extreme position (CEP), critical path motion (CPM); by type of motion program, rise-fall (RF), rise-fall-dwell (RFD), rise-dwell-fall-dwell (RDFD). Types of Follower Motion Follower motion can be an oscillation or translation. An oscillating follower rotates around a pivot point. A rotating follower moves in usually rectilinear translation. The choice between these two forms of the cam-follower is usually dictated by the type of output motion desired. Type of Joint Closure Force closure requires an external force be applied to the joint in order to keep the two links, cam and follower, physically in contact. This force is usually provided by a spring. This force, defined as positive in a direction which closes the joint, cannot be allowed to become negative. If it does, the links have lost contact because a forceclosed joint can only push, not pull. Form closure, closes the joint by geometry. No external force is required. There are really two cam surfaces in this arrangement, one surface on each side of the follower. Each surface pushes, in its turn, to drive the follower in both directions. Type of Follower Follower, in this context, refers to that part of the follower link which contacts the cam. Three types are available, flat-faced, mushroom (curved), and roller. The roller follower has the advantage of lower (rolling) friction than the sliding contact of the other two but can be more expensive. Flat-faced followers can package smaller than roller followers and are often favored for that reason as well as cost in automotives. Roller followers are more frequently used in production machinery where their ease of replacement and availability from bearing manufacturers' stock in any quantities are advantages. Type of Cam The direction of the follower's motion relative to the axis of rotation of the cam determines whether it is a radial or axial cam. In radial cams, the follower motion is generally in a radial direction. Open radial cams are also called plate cams. An axial cam is one whose follower moves parallel to the axis of cam rotation. This arrangement is also called a face cam if open (force-closed) and a cylindrical or barrel cam if grooved or ribbed (form-closed). Type of Motion Constraints There are two general categories of motion constraints which determine the shape of the cam, critical extreme position (CEP; also called endpoint specification) and critical path motion (CPM). Critical extreme position refers to the case in which the design specifications define the start and finish positions of the follower (i.e., extreme positions) but do not specify any constraints on the path motion between the extreme positions. This case is the easier of the two to design as the designer has great freedom to choose the cam functions which control the motion between extremes. Critical path motion is a more constrained problem than CEP because the path motion, and/or one or more of its derivatives are defined over all or part of the interval of motion. Dwells and Type of Motion Program A dwell is a period of time for which no change in output motion appears for a changing input motion. Dwells are an important feature of camfollower systems because it is very easy to create exact dwells in these mechanisms. The motion programs rise-fall (RF), rise-fall-dwell (RFD), and rise-dwell-fall-dwell (RDFD) all refer mainly to the CEP case of motion constraint and in effect define how many dwells are present in the full cycle of motion, either none (RF), one (RFD), or more than one (RDFD) The cam-follower is the design type of choice whenever a dwell is required. Cam-follower systems tend to be more compact than linkages for the same output motion. Dwells and Type of Motion Program If your need is for a rise-fall (RF) CEP motion, with no dwell, then you should really be considering a crank-rocker linkage rather than a cam-follower to obtain all the linkage's advantages over cams of reliability, ease of construction, and lower cost. If your needs for compactness outweigh those considerations, then the choice of a cam-follower in the RF case may be justified. Also, if you have a CPM design specification, and the motion or its derivatives are defined over the interval, then a cam-follower system is the logical choice in the RF case S V A J DIAGRAMS The first task faced by the cam designer is to select the mathematical functions to be used to define the motion of the follower. The easiest approach to this process is to "linearize“ the cam, i.e., "unwrap it" from its circular shape and consider it as a function plotted on Cartesian axes. S V A J DIAGRAMS We plot the displacement functions, its first derivative velocity v, its second derivative acceleration a, and its third derivative jerk}, all on aligned axes as a function of camshaft angle. Note that we can consider the independent variable in these plots to be either time t or shaft angle θ, as we know the constant angular velocity (ω) of the camshaft and can easily convert from angle to time and vice versa. Cam Design Procedure A cam design begins with a definition of the required cam functions and their svaJ diagrams. Functions for the nondwell cam segments should be chosen based on their velocity, acceleration, and jerk characteristics and the relationships at the interfaces between adjacent segments including the dwells.. Double-Dwell Cam Design Choosing S V A J Functions The double-dwell case is quite common design requirement for cams. A double-dwell cam design specifications are often depicted on a timing diagram which is a graphical representation of the specified events in the machine cycle cycle. A machine's cycle is defined as one revolution of its master driveshaft. In a complicated machine. The time relationships among all subassemblies are defined by their timing diagrams which are all drawn on a common time axis. Example 1. Uniform Speed S V A J Diagram Consider the following cam design CEP specifications: dwell rise dwell fall cam speed (ω) at zero displacement for 90 degrees of cam rotation (low dwell) 1 in (25 mm) in 90 degrees of cam rotation at 1 in (25 mm) for 90 degrees of cam rotation (high dwell) 1 in (25 mm) in 90 degrees of cam rotation 1 rev/sec 2π rad/sec Example 1. Uniform Speed S V A J Diagram A uniform velocity cam design merely "connect the dots" on the timing diagram by straight lines to create the displacement diagram. This approach ignores the effect on the higher derivatives of the resulting displacement function. Example 1. Uniform Speed S V A J Diagram The acceleration is zero during the rise and fall intervals, but becomes infinite at the boundaries of the interval, where rise meets low dwell on one side and high dwell on the other. Note that the velocity function is multivalued at these point, creating discontinuities at these boundaries. The effect of these discontinuities is to create a portion of the velocity curve which has infinite slope and zero duration. This results in the infinite spikes of acceleration shown at those points Example 1. Uniform Speed S V A J Diagram Clearly the dynamic forces will be very large at these boundaries and will create high stresses and rapid wear. This is an unacceptable design. In fact, if this cam were built and run at any significant speeds, the sharp comers on the displacement diagram which are creating these theoretical infinite accelerations would be quickly worn to a smoother contour by the unsustainable stresses generated in the materials. The Fundamental law of Cam Design Any cam designed for operation at other than very low speeds must be designed with the following constraints: The cam function must be continuous through the first and second derivatives of displacement across the entire interval (360 degrees). Corollary: The jerk function must be finite across the entire interval (360 degrees). The Fundamental law of Cam Design In any but the simplest of cams, the cam motion program cannot be defined by a single mathematical expression, but rather must be defined by several separate functions, each of which defines the follower behavior over one segment, or piece, of the cam. These expressions are sometimes called piecewise functions, and must have third-order continuity at all boundaries. The displacement, velocity and acceleration functions must have no discontinuities. The Fundamental law of Cam Design If any discontinuities exist in the acceleration function, then there will be infinite spikes, or Dirac delta functions, appearing in the derivative of acceleration, jerk. Thus the corollary merely restates the fundamental law of cam design. In the uniform velocity cam design example, the low-degree (linear) polynomial as selected for the displacement function, resulted in discontinuities in the upper derivatives. Simple Harmonic Motion S V A J Diagrams Simple harmonic functions remain continuous throughout any number of differentiations. Differentiation of a harmonic functions only amounts for a phase shift by 90 degrees. The equations of a simple harmonic rise of magnitude h over a cam angle β that starts at a cam angle θi are as follows: s v i h 1 cos 2 h 2 a sin i h i cos 2 j 2 h i sin 2 3 θi θi + β Simple Harmonic Motion S V A J Diagrams s v i h 1 cos 2 h 2 a sin i h i cos 2 j 2 h i sin 2 3 θi θi + β Example 2: Simple Harmonic Motion S V A J Diagrams Consider the cam design CEP specifications of Example 1: dwell rise dwell fall cam speed (ω) at zero displacement for 90 degrees of cam rotation (low dwell) 1 in (25 mm) in 90 degrees of cam rotation at 1 in (25 mm) for 90 degrees of cam rotation (high dwell) 1 in (25 mm) in 90 degrees of cam rotation 1 rev/sec 2π rad/sec Plot the S V A J diagrams resulting from simple harmonic motion for the rise and fall periods. Example 2: Simple Harmonic Motion S V A J Diagrams Consider the cam design CEP specifications of Example 1: dwell rise dwell fall cam speed (ω) at zero displacement for 90 degrees of cam rotation (low dwell) 1 in (25 mm) in 90 degrees of cam rotation at 1 in (25 mm) for 90 degrees of cam rotation (high dwell) 1 in (25 mm) in 90 degrees of cam rotation 1 rev/sec 2π rad/sec Plot the S V A J diagrams resulting from simple harmonic motion for the rise and fall periods. rise: Fall: h s 1 cos h = 1 in., θi = π/2, β = π/2 h = -1 in., θi = 3π/2, β = π/2 i 2 v h 2 a sin i h i cos 2 j 2 h i sin 2 3 s 0 . 5 1 cos 2 2 s 0 . 5 1 cos 2 3 2 v sin 2 2 v sin 2 3 2 a 2 cos 2 2 a 2 cos 2 3 2 j 4 sin 2 2 j 4 sin 2 3 2 Example 2: Simple Harmonic Motion S V A J Diagrams 0 π/2 π 3π/2 2π rise: h = 1 in., θi = π/2, β = π/2 Fall: h = -1 in., θi = 3π/2, β = π/2 s 0 . 5 1 cos 2 2 s 0 . 5 1 cos 2 3 2 v sin 2 2 v sin 2 3 2 a 2 cos 2 2 a 2 cos 2 3 2 j 4 sin 2 2 j 4 sin 2 3 2 Simple Harmonic Motion S V A J Diagrams The velocity function resulting from a SHM is continuous but the acceleration has discontinuities at its starting and ending points. The acceleration is not zero at these points, and thus do not match the zero acceleration of the dwell period. The simple harmonic motion displacement function, when joint with dwells, does not satisfy the fundamental law of cam design. 0 π/2 π 3π/2 2π Simple Harmonic Motion S V A J Diagrams The only case in which the simple harmonic motion displacement functions satisfy the fundamental law of cam design is in which the rise cam angle βr is equal to the fall cam angle βf with no dwells in between. A common example is the nonquick return (equal period) risefall (RF), in which the follower rises in 180° and falls in 180°. βr = βf = 180° in this case. 0 π 2π Cycloidal Cam Motion The cycloidal motion starts with a full wave sine wave defined as the acceleration function for the cam follower through the cam angle β i a C sin 2 , v ( i ) v i 0 , s ( i ) 0 , s i h v a d s v d C using s i 0 , v i 0 , s i h , gives k 1 C i cos 2 2 k 1 k 1 i k 2 4 2 2 i sin 2 i 1 i s h sin 2 2 a 2 h 2 i sin 2 h , k 2 0 , C 2 h i 1 cos 2 , v , j 4 2 i cos 2 h 3 h 2 Example 3: Cycloidal Motion S V A J Diagrams Consider the cam design CEP specifications of Example 1: dwell rise dwell fall cam speed (ω) at zero displacement for 90 degrees of cam rotation (low dwell) 1 in (25 mm) in 90 degrees of cam rotation at 1 in (25 mm) for 90 degrees of cam rotation (high dwell) 1 in (25 mm) in 90 degrees of cam rotation 1 rev/sec 2π rad/sec Plot the S V A J diagrams resulting from simple harmonic motion for the rise and fall periods. rise: rise: h = 1 in., θi = π/2, β = π/2 h = 1 in., θ = π/2, β = π/2 1 s h i sin 2 2 i h i 1 cos 2 v i a 2 2 sin 2 h j 4 2 i cos 2 3 h i s v a j 1 2 2 sin 4 2 2 1 cos 4 8 32 4 2 sin 4 2 cos 4 2 s v a j 1 2 2 sin 4 2 2 1 cos 4 8 32 4 2 sin 4 2 sin 4 2 Polynomial Cam Motion The displacement equation for general polynomial motion can be written as: s C 0 C 1 i C 2 i C N i 2 N s C k i N k k 0 where s is the follower displacement, θ is the angular position of the cam, and θi is the initial cam angle at the beginning of the polynomial motion. The integer N is referred to as the degree of the polynomial The velocity, acceleration and kerk are obtained by successive differentiation as v ds dt ds d d dt C 1 2 C 2 i NC N i N v t kC k i k 1 k 0 a t j t dv dv d dt d dt da da d dt d dt N 2 k k 1 C k i k 2 k 0 N 3 k k 1 k 1 C k k 0 i k 3 N 1 Example 3: Polynomial Motion S V A J Diagrams Determine the polynomial equation that will satisfy the following conditions s C 0 C 1 i C 2 i C 3 i C 4 i C 5 i 2 s = 0 when θ = θi v = 0 when θ = θi a = 0 when θ = θi s = h when θ = θi + β v = 0 when θ = θi + β a = 0 when θ = θi + β 1. 2. 3. 4. 5. 6. C1 0 2 C 2 0 2 a 2 2 2 C 2 3 4 6 C 3 i 12 C 4 i 20 C 5 i 5 C 1 2 C 2 3C 3 4 C 4 5 C 5 2 3 2 C 2 6 C 3 12 C 4 20 C 5 2 Solving the equations produces C0 0 C 3 10 h C1 0 C 4 15 h C2 0 C3 6h 5 3 4 5 3 3 2 2 C 0 C1 C 2 C 3 C 4 C 5 h 2 4 v C 1 2 C 2 i 3C 3 i 4 C 4 i 5 C 5 i Substituting the six required conditions , we get the following set of equations C0 0 3 4 0 0 3 4 The equations for s, v, a, and j are therefore i i s 10 h 15 h 3 30 h v i 60 h 2 4 i 6 h i 60 h 180 h i a 2 2 i 60 h 360 h j 2 3 360 h 3 2 3 i 5 30 h i 4 2 i 3 3 120 h 2 i 2 Example 3: Polynomial Motion S V A J Diagrams Determine the polynomial equation that will satisfy the following conditions 1. 2. 3. 4. 5. 6. s = 0 when θ = θi v = 0 when θ = θi a = 0 when θ = θi s = h when θ = θi + β v = 0 when θ = θi + β a = 0 when θ = θi + β Because of the form of the displacement function, this particular follower motion is referred to as 3-4-5 polynomial motion. i i s 10 h 15 h 3 30 h v i 60 h 2 4 i 6 h i 60 h 180 h i a 2 2 i 60 h 360 h j 2 3 360 h 3 2 3 i 5 30 h i 4 2 i 3 3 120 h 2 i 2 Polynomial Motion S V A J Diagrams There are no jumps in the s, v or a curves, resulting in good dynamic characteristics. The curves are similar in many respected to cycloidal motion, and the two motions are comparable. For the same values of h, ω, and β , cycloidal motion produce higher values of maximum acceleration, but lower values of maximum jerk than does the 3-4-5 polynomial motion. Polynomial Motion S V A J Diagrams Using the same approach, polynomial motions can be derived for a wide range of different boundary conditions for displacement and derivatives. Manufacturing precision requirements, however, increase as the number of boundary conditions increase. Combined Functions Instead of selecting a single function to represent the rise or fall period of the follower, one may specify a number of interconnected segments, each of which is represented by a different function. As seen earlier, the uniform speed rise produced infinite acceleration at the ends of the rise segment. Instead, one can combine a uniformly increasing speed rise for a portion of the rise segment, followed by a uniformly decreasing speed for the other portion. The resulting follower motion is referred to as parabolic motion. Combined Functions – Parabolic Motion Let β be the cam angle turned through which the follower is given a total rise h. When the follower displacement is h/2, the cam has turned by β /2. The constant acceleration motion specifications are: a C , v ( 0 ) 0 , s ( 0 ) 0 v C k 1 , s C 2 k 1 k 2 2 using s 0 0 gives k 2 0 using v 0 0 gives k 1 0 4h h using s gives C 2 2 2 The follower displacement for the first half of the parabolic motion rise is s 2h 2 , v 2 4h 2 , a 4h 2 Combined Functions – Parabolic Motion s During the 2nd half of follower motion, the cam has turned by an additional β /2. The constant acceleration motion specifications are: a C v( 2h ) , s( 2 4h 2 , h ) 2 2 v ( ) 0, s ( ) h Integrating as before, and applying the boundary conditions v 2h 2 k 3 , s using v using s h 2 2 2h 2h 2h k 3 0, k 3 2 k3 k 4 h 2 k4 h h 2h k4 h k 3 k 4 2 2 2 h h 2h 2 , v 2 4h 2 , a 4h 2 Graphical Construction of Radial Cam Profile Knife-Edge Follower with Uniform Speed Rise and Fall A cam is to give the following motion to a knife-edged follower : 1. Outstroke during 60° of cam rotation 2. Dwell for the next 30° of cam rotation 3. Return stroke during next 60° of cam rotation 4. Dwell for the remaining 210° of cam rotation. The stroke is 40 mm and the minimum radius of the cam is 50 mm. The follower moves with uniform velocity during both the outstroke and return strokes. Draw the profile of the cam when (a) the axis of the follower passes through the axis of the cam shaft, and (b) the axis of the follower is offset by 20 mm from the axis of the cam shaft. Graphical Construction of Radial Cam Profile Graphical Construction of Radial Cam Profile Graphical Construction of Radial Cam Profile Knife-Edge Follower with Simple Harmonic Rise and Fall A cam is to be designed for a knife edge follower with the following data : 1. Cam lift = 40 mm during 90° of cam rotation with simple harmonic motion. 2. Dwell for the next 30°. 3. During the next 60° of cam rotation, the follower returns to its original position with simple harmonic motion. 4. Dwell during the remaining 180°. Draw the profile of the cam when (a) the line of stroke of the follower passes through the axis of the cam shaft, and (b) the line of stroke is offset 20 mm from the axis of the cam shaft. The radius of the base circle of the cam is 40 mm. Determine the maximum velocity and acceleration of the follower during its ascent and descent, if the cam rotates at 240 r.p.m. Graphical Construction of Radial Cam Profile Knife-Edge Follower with Simple Harmonic Rise and Fall Graphical Construction of Radial Cam Profile Knife-Edge Follower with Simple Harmonic Rise and Fall Graphical Construction of Radial Cam Profile Knife-Edge Follower with Simple Harmonic Rise and Fall Graphical Construction of Radial Cam Profile Roller Follower with Simple Harmonic Rise and Fall A cam, with a minimum radius of 25 mm, rotating clockwise at a uniform speed is to be designed to give a roller follower, at the end of a valve rod, motion described below : 1. 2. 3. 4. To raise the valve through 50 mm during 120° rotation of the cam ; To keep the valve fully raised through next 30°; To lower the valve during next 60°; and To keep the valve closed during rest of the revolution i.e. 150° ; The diameter of the roller is 20 mm and the diameter of the cam shaft is 25 mm. Draw the profile of the cam when (a) the line of stroke of the valve rod passes through the axis of the cam shaft, and (b) the line of the stroke is offset 15 mm from the axis of the cam shaft. The displacement of the valve, while being raised and lowered, is to take place with simple harmonic motion. Determine the maximum acceleration of the valve rod when the cam shaft rotates at 100 r.p.m. Draw the displacement, the velocity and the acceleration diagrams for one complete revolution of the cam. Graphical Construction of Radial Cam Profile Roller Follower with Simple Harmonic Rise and Fall Graphical Construction of Radial Cam Profile Roller Follower with Simple Harmonic Rise and Fall Graphical Construction of Radial Cam Profile Roller Follower with Simple Harmonic Rise and Fall Graphical Construction of Radial Cam Profile Flat Follower with Simple Harmonic Rise and Fall A cam drives a flat reciprocating follower in the following manner : 1. During first 120° rotation of the cam, follower moves outwards through a distance of 20 mm with simple harmonic motion. 2. The follower dwells during next 30° of cam rotation. 3. During next 120° of cam rotation, the follower moves inwards with simple harmonic motion. 4. The follower dwells for the next 90° of cam rotation. The minimum radius of the cam is 25 mm. Draw the profile of the cam. Graphical Construction of Radial Cam Profile Flat Follower with Simple Harmonic Rise and Fall