Machine Elements Reviewer PDF

**MACHINE ELEMENTS REVIEWER** **CHAPTER 1: INTRODUCTION** **1-1. The science of mechanism** treats the *laws governing the motion of the parts of a machine and the forces transmitted by these parts.* In designing a machine, or in studying the design of an existing machine, two distinct but closely...

**MACHINE ELEMENTS REVIEWER** **CHAPTER 1: INTRODUCTION** **1-1. The science of mechanism** treats the *laws governing the motion of the parts of a machine and the forces transmitted by these parts.* In designing a machine, or in studying the design of an existing machine, two distinct but closely related divisions of the problem present themselves. **First,** *the machine parts must be proportioned and related to one another* so that each has the proper motion. **Second,** *each part must be adapted to withstand the forces imposed upon it.* Thus dividing the science of mechanism into two parts: 1. **Pure mechanism or kinematics of machines**, which treats of the motion and forms of the parts of a machine, and the manner of supporting and guiding them, independent of their strength. 2. **Constructive mechanism or machine design**, which involves the calculation of the forces acting on different parts of the machine; the selection of materials on the basis of strength, durability, and other physical properties in order to withstand these forces, the convenience for repairs and facilities for manufacture also being taken into account. **1-2.** A **machine** *is a combination of resistant bodies so arranged* that by their means the mechanical forces of nature can be compelled to produce some effect or work accompanied by certain determinate motions. In general, it may properly be said that a **machine** *is an assemblage of parts* interposed between the source of power and the work, to adapt one to the other. **1-3.** A **structure** is a *combination of resistant bodies capable of transmitting forces or carrying loads but having no relative motion between parts.* **1-4.** A **mechanism** is a *combination of rigid bodies so arranged that the motion of one compels the motion of the others*, according to a law depending on the nature of the combination. The terms mechanism and machine are often used synonymously, but ***the combination is a mechanism** **when used to transmit or modify*** motion and a ***machine if energy is transferred or work is done.*** Thus***, a machine is a series or train of mechanisms but no mechanism is necessarily a machine.*** **1-5.** The **frame** of a *machine is a structure that supports the moving parts and regulates the path, or kind of motion, of many of the parts*. **1-6.** A **particle** is *an infinitesimal part of a body.* It may be represented on a drawing by a point and is *often referred to as a point*. **1.7.** A **rigid body** is one whose *component particles remain at a constant distance from one another.* For kinematic study, a **line** may be considered as being of *indefinite length* and a **body** of *indefinite magnitude*. **1-8. Driver and Follower**. That ***piece of a mechanism that causes motion is called the driver***, and the ***one whose motion is affected is called the follower.*** **1.9. Modes of Transmission.** If the action of natural forces of attraction and repulsion is not considered, one piece cannot move another unless the two are in contact or are connected by some intervening body that is capable of communicating the motion of one to the other. If an ***intermediate connector is rigid, it is called a link.*** If the ***connector is flexible, it is called a band***, which is supposed to be inextensible and is capable only of transmitting a pull. **1-10. Pairs of Elements.** To compel a body to move in a definite path, it must be paired with another, the shape of which is determined by the nature of the relative motion of the two bodies. **1-11. Closed or Lower Pair.** If one element not only forms the envelope of the other, but also encloses it, the forms of the elements being geometrically identical, ***the one being solid or full, and the other being hollow or open, we have what may be called a closed pair, also called a lower pair.*** Three forms only can exist: 1. a **straight line**, which allows straight; 2. a **circle**, which allows rotation, or revolution and 3. a **helix**, which allows a combination of rotation and straight translation. **1-12. Higher Pairs.** The elementary bodies A and B do not enclose each other in the above sense. Such a pair is called a ***higher pair***, and *the elements are either in point or line contact*. **Ball and roller bearings** are examples of higher pairs. **1.13. Incomplete Pairs of Elements**. Sometimes it is only necessary to prevent forces having a certain definite direction from affecting the pair, and then it is no longer necessary to make the pair complete; one element can be cut away where it is not needed to resist the forces. **1.14. Inversion of Pairs.** In other words, the absolute motion of the moving piece is the same, whichever piece is fixed. This exchange of the fixedness of an element with its partner is called the inversion of the pair, **1-15. Bearings.** The word bearing is applied, in general, to *the surfaces of contact between two pieces that have relative motion, one of which supports or partially supports the other.* One of the *pieces may be stationary*, in which case the bearing may be called a **stationary bearing**, or both pieces may be moving. **Bearings** may be arranged, according to the relative motions they will allow, in three classes. 1\. **For straight translation**, the bearings must have plane or cylindrical surfaces. ***If one piece is fixed the surfaces of the moving pieces are called slides***; those of the fixed pieces, slides or guides. 2\. **For rotation, or turning,** the bearings must have surfaces of circular cylinders, cones, conoids, or flat disks. ***The surface of the solid or full piece is called a journal, neck, spindle, or pivot;*** that of the hollow or open piece, a bearing, gudgeon, pedestal, plumber-block, pillow-block, bush, or step. 3\. **For translation and rotation combined (helical motion)**, they must have a helical or screw shape. Here, ***the full piece is called a screw, and the open piece is a nut***. **1.16. Collars and Keys.** rings or collars are used to prevent the motion of the pulley along the shaft but allow its free rotation. The **key** may be made fast or integral to either piece, the other having a groove in which it can slide freely. The above arrangement is very common and is called a feather and groove, or spline, or a key and keyway. **1.17. Cranks and Levers**. A **crank** is an arm rotating or oscillating about an axis. When two cranks on the same axis are rigidly connected to each other the name **lever** is often applied to the combination, particularly when the motion is oscillating over a relatively small angle. When the angle between the two arms is less than 90° it is often called a ***bell crank lever***, and when the angle is more than 90° it is often called a ***rocker.*** **1-18. The action of a Crank.** A **crank** may be considered as a rigid piece connecting one member of a pair of cylindrical elements to one member of another pair. The axis of one pair is assumed to be stationary, and the axis of the other pair is constrained by the crank to move in a circular path about the stationary axis. **1-19.** A **link** may be defined as a rigid piece or a non-elastic substance that *serves to transmit force from one piece to another or to cause or control motion*. **1-20.** A **linkage** consists of several pairs of elements connected by links. If the combination is such that the relative motion of the links is possible, and the motion of each piece relative to the others is definite, the linkage becomes a ***kinematic chain***. If one of the links of a kinematic chain is fixed, then the chain becomes a ***mechanism***. **1.21. The four-bar linkage** *consists of two cranks*. The four pieces 1, 2, 3, and 4 are called ***links***. The essential part of a link, from a kinematic standpoint, is its ***centerline*** **1-22. Four-Bar Linkage with a Sliding Member.** In Fig. 1-21, the end of the connecting rod carries a block, pivoted to it at the axis B, which slides back and forth in the circular slot as the crank QA revolves. The center of curvature of the slot is at Q. The center of the crank pin B has the same motion that it would have been it guided by a crank of length QB turning about Q4. The mechanism, therefore, is a four-bar linkage with the lines QA and QB as the center lines of the cranks, Q2Q4 as the line of centers, and AB as the center line of the connecting rod. The mechanism, however, would still be the equivalent of a four-bar linkage where Q₂A is one crank (called the ***finite crank***), the line BQ perpendicular to the slot is the other crank (called the ***infinite crank.*** This mechanism is known as a ***slider-crank mechanism***. **1-23. Machine Analysis**. Pure mechanism \"treats the motions and forms of the parts of a machine,\" which. according to Art. 1-2, is created \"to produce work accompanied with certain definite motions.\" *Motion consists of the three elements*, **displacement, velocity, and acceleration.** The *crank length is often called* **the crank throw**. In reciprocating machinery, the distance between the two extreme positions of the piston travel or the displacement of the piston is called the **stroke**. The *inside diameter of the cylinder* is called the **bore**. **CHAPTER 2: MOTION** **2-1.** **Motion** is the change of position. **2-2. Path.** A point moving in space describes a line called its path, which may be rectilinear or curvilinear. The motion of a body is determined by the paths of three of its points, not by a straight line. If the motion is in a plane, two points suffice, and, if rectilinear, one point suffices, to determine the motion. **2-3. Direction and Sense.** If a point is moving along a straight path the direction of its motion is along the line that constitutes its path; motion toward one end of the line is assumed as having positive direction and indicated by a + sign, the motion toward the other end would be negative and indicated by a sign. Often this is referred to as the sense of the motion. If a point is moving along a curved path, the direction at any instant is along the tangent to the curve and may be indicated as positive or negative, or the sense given, as for rectilinear motion. **2-4. Continuous Motion.** When a point continues to move indefinitely in a given path in the same sense, its motion is said to be continuous. In this case, the path must return on itself, as a circle or other closed curve. **2-5. Reciprocating Motion**. When a point traverses the same path and reverses its motion at the ends of such a path, the motion is said to be reciprocating. **2-6. Oscillation** is a term applied to reciprocating circular motion, such as that of a pendulum. **2-7. Intermittent Motion.** When the motion of a point is interrupted by periods of rest, its motion is said to be intermittent. **2-8. Revolution and Rotation.** A point is said to revolve about an axis when it describes a circle of which the center is in the axis and of which the plane is perpendicular to that axis. When all the points of a body thus move, the body is said to revolve about the axis. If this axis passes through the body, as in a wheel, the word rotation is used synonymously with revolution. The word turn is often used synonymously with revolution and rotation. It frequently occurs that a body not only rotates about an axis passing through itself but also moves in an orbit about another axis. **2-9. An axis of rotation or revolution** is a line whose direction is not changed by the rotation; a fixed axis is one whose position as well as direction remains unchanged. **2-10. A plane of rotation or revolution** is a plane perpendicular to the axis of rotation or revolution. **2-11. The direction of rotation or revolution** is defined by giving the direction of the axis, and the sense is given by stating whether the turning is right-handed (clockwise) or left-handed (counterclockwise), when viewed from a specified side of the plane of motion. **2-12. Coplanar Motion**. A body, or a series of bodies, may be said to have coplanar motion when all their component particles are moving in the same plane or in parallel planes. **2-13. Cycle of Motions.** When a mechanism is set in motion and its parts go through a series of movements that are repeated over and over, the relations between and order of the different divisions of the series being the same for each repetition, one of these series is called a cycle of motions or kinematic cycle **2-14. Period of motion** is the time occupied in completing one cycle. **2-15. Linear speed** is the time rate of motion of a point along its path or the rate at which a point is approaching or receding from another point in its path. If the point to which the motion of the moving point is referred is fixed, the speed is the absolute speed of the point. If the reference point is itself in motion the speed of the point in question is relative. Linear speed is expressed in linear units per unit of time. **2-16. Angular speed** is the time rate of turning of a body about an axis, or the rate at which a line on a revolving body is changing direction, and is expressed in angular units per unit of time. **2-17. Uniform and Variable Speed.** Speed is uniform when equal spaces are passed over in equal times, however small the intervals into which the time is divided. Speed is variable when unequal spaces are passed over in equal intervals of time. **2-18. Velocity** is a word often used synonymously with speed. This is incorrect since velocity includes direction and sense as well as speed. The linear velocity of a point is not fully defined unless the direction and sense in which it is moving and the rate at which it is moving are known. The angular velocity of a line would be defined by stating its angular speed, the direction of the perpendicular to the plane in which the line is turning, and the sense of the motion. **2-19. Linear acceleration** is the time rate of change of linear velocity. Since velocity involves direction as well as the rate of motion, linear acceleration may involve a change in speed or direction or both. Any change in the speed takes place in a direction tangent to the path of the point and is called **tangential acceleration**; a change in direction takes place normal to the path and is called **normal acceleration**. Acceleration may be either positive or negative. If the speed is increasing the acceleration is positive; if the speed is de- creasing the acceleration is negative and is called **retardation or deceleration**. If the speed changes by the same amount during all equal time intervals the acceleration is uniform, but if the speed changes by different amounts during equal intervals of time the acceleration is variable. **2-20. Angular acceleration** is the time rate of change of angular velocity. As in linear acceleration, a change in either speed or direction of rotation, or both, may be involved. Angular acceleration is expressed in angular units change of speed per unit time (such as radians, degrees, or revolutions per minute each minute). **2-21. Translation**. A body is said to have motion of translation when all its component particles have the same velocity, as regards both speed and direction; that is, all points on the body are, for the instant at least, moving in the same direction with equal speeds. If all the particles move in straight lines, as in the piston of an engine, the body has a **rectilinear translation**, and if they move in curved paths, as in the motion of the parallel rod of a locomotive, the body has a **curvilinear translation**. **2-22. Turning Bodies**. All motion consists of translation, turning about an axis, or a combination of the two. It is customary to refer to the motion of turning as revolving or rotating. **2-23. Angular Speed.** A circular cylinder or wheel, supported on a shaft which in turn is supported in fixed bearings. The speed at which the wheel turns is the rate at which any line on it (radial or otherwise) changes direction. If the wheel makes N complete turns in 1 minute its angular speed is N revolution per minute (written N rpm). In many computations, it is necessary to use as a unit of angular, motion the **radian**, which is the angle subtended by the area of a circle equal in length to its radius. Since the radius is contained in the circumference 2π times there must be 2π radians in 360°, or 1 radian is equal to 57.296°. Hence 1 revolution = 2π radians If N represents the angular speed in revolutions per unit of time and the angular speed in radians per same unit of time then ω = 2πN **2-24. Linear Speed of a Point on a Revolving Body.** The linear speed of a point on the circumference of a revolving wheel is often referred to as the ***periphery speed or surface speed.*** V= 2πR~a~N **2-25. Motion Classified.** Since the motion of a body is determined by the motion of not more than three of its component particles, not lying in a straight line, it is essential before beginning the analysis of the motion of rigid bodies that the laws governing the motion of a particle be fully understood. For this purpose, it is convenient to classify motion as applied to a particle or point according to the kind of acceleration which the moving particle has: 1\. Acceleration zero. 2\. Acceleration constant. 3\. Acceleration variable \(a) According to some simple law which may be expressed in terms of a, u, or t. \(b) In a manner that can be expressed only by a graph or similar means. **2-26. Uniform Motion.** When the acceleration is zero the velocity is constant and the moving particle continues to move in a straight line over equal distances in equal intervals of time. The velocity (or speed) therefore is equal to the length of the path s, in linear units, divided by the time t, in time units, required to traverse the path, or V=s/t **2-27. Uniformly Varying Motion.** In this case, the acceleration is constant; that is, the speed changes by equal amounts in equal intervals of time, like that of a body falling under the action of gravity. \ [\$\$V = \\sqrt{{V\_{o}}\^{2} + 2As}\$\$]{.math.display}\ **2-28. Variable Acceleration**. The acceleration of a moving particle may vary as some function of distance moved, velocity, or time. When this condition exists, definite equations may be written expressing the relations between A, s, V, and t. Three cases will be considered: \(1) Aa function of t; \(2) A- a function of V; and \(3) Aa function of s. \ [*V*^2^ = 2∫Ads]{.math.display}\ **2-29. Semigraphical Methods.** Often no direct relation exists between acceleration, velocity, distance moved, and time that can conveniently be expressed in the form of equations. The data may be obtained by observations or computations at certain frequent intervals during the cycle of motion and the relations worked out on graphs. Example 1. Graphical Differentiation. Let a represent the distance moved from some initial or reference position by a particle having rectilinear motion scales plot a curve with values of t for abscissas and the corresponding values of s as ordinates. This will be called the **space-time curve.** **2-30. Harmonic Motion**. A type of motion in which the acceleration varies directly as the displacement is known as simple harmonic motion. **2-31. Variable and Constant Speed.** Instead of causing a moving piece or particle to travel its entire path with variable motion, it is sometimes desirable to have it travel the major portion of its path with uniform motion, accelerating for a short interval at the beginning, until it has acquired sufficient speed. **CHAPTER 3: VELOCITY ANALYSIS** **3-1. Velocities in Machines.** The fact has been mentioned previously that if the motion of a body is translated, the velocities of all particles composing the body are equal and parallel. If the body has any coplanar motion other than translation, it is necessary to have enough data to determine the velocity of two particles to determine the velocity of any part of the body. The principal cases which occur are the following: \(1) two or more points on the same body; \(2) points on two or more bodies connected by pin joints; \(3) points on bodies in rolling contact, and \(4) points on bodies in sliding contact. There are four commonly used methods for obtaining velocities: 1. resolution and composition; 2. instantaneous axis of velocity; 3. Centro; and 4. relative velocity or velocity polygon. As a general rule, methods 1 and 2 give the quickest solution. Method 2 unamplified version of method 3. Method 4 can be used in the solution of practically all problems and is probably the most desirable method **3-2. Vectors.** A *scalar quantity* has magnitude only. A *vector quantity* has magnitude, and direction, and represents force, velocity, and acceleration. A **vector** is a line that represents a vector quantity. The length of the line, drawn at any convenient scale, shows the magnitudes of the direction of the line in parallel to the direction in which the quantity acts and an arrowhead or home other suitable convention indicates the sense of the quantity. The initial end of the line is the origin or tail, and the other end is the terminus or head. The sense of the quantity is from the origin tether terminus, and often an arrowhead is placed at the terminus. The sum of the quantities is called their **resultant,** and its vector is the resultant vector. The quantities added together to obtain the resultant are its components, and the corresponding vectors are the component vectors. The sum of two vectors is the closing side of a triangle whose other two sides are formed by using the head of one of the component vectors as the tail for the second. The sense of the resultant vector is toward the head of the second component vector. The process of obtaining the resultant of any number of vectors is called vector composition, and the reversed process of breaking up a vector into components is called vector resolution. **3-3. Scales.** In the graphical solution of problems, it is necessary to draw the machine full scale, to a smaller scale, or a larger scale. This space scale is expressed in three ways: \(1) proportionate size, e.g., one-fourth size (¼ scale) or twice size (double scale); \(2) the number of inches on the drawing equal to 1 foot on the machine, \(3) 1 inch on the drawing equals so many feet. *The **velocity scale*** designated K~v1~, is defined as the linear velocity in distance units per unit of time represented by 1 in. on the drawing. The ***acceleration scale***, designated K~a~, is defined as the linear acceleration in distance units per unit of time per unit of time represented by 1 in, on the drawing. If the linear acceleration of a point is 100 ft/sec and the K. scale is 100, then a line 1 in. long would represent a linear acceleration of 100 ft/see, and would be written K~a~ = 100 ft/sec^2^ **3-4. Rotating and Oscillating Cranks.** The magnitude of the instantaneous linear velocity of a point on a revolving body, rotating crank, or oscillating crank is proportional to the distance of that point from the axis of rotation of the body or crank. **3-5. Resolution and Composition.** If the velocity of one point and the direction of the velocity of any other point on a body are known, the velocity of any other point on that body may be obtained by resolving the known velocity vector into components along and perpendicular to the line joining these points and making one of the components of the velocity of the other point equal to the component along the line. The other component of this velocity will be perpendicular to the line. The validity of this procedure is apparent when it is realized that, in a rigid body, the distance between the two points remains constant and the velocity component along the line joining these points must be the same at each point. In the following discussion, the components will be referred to as the component along the line or link and the component perpendicular to the line or link, or simply along and perpendicular components. **3.7. Instantaneous Axis of Velocity.** Each member of a machine is either rotating about a fixed axis or a moving axis. Instantaneously this moving axis may be thought of as a stationary axis with properties similar to a fixed axis. In other words, the cranks of a machine rotate or oscillate about their respective fixed axes and It should be clearly understood that \(1) there is one instantaneous axis of velocity for each floating link in a machine, \(2) there is not one common instantaneous axis of velocity for all links in a machine, and \(3) the instantaneous axis of velocity changes position as the link moves. **3-11. Centros.** As previously stated, the instantaneous axis of velocity method of obtaining velocities is a simplified version of the Centro method and can be used in obtaining velocities when the instantaneous axis can be located in many mechanisms, the instantaneous axis of rotation cannot be located, since the directions of the motion of two points on the link may not be known. By using the method of centros, velocities in all mechanisms can be obtained. A centro may be defined as \(1) a point common to two bodies having the same velocity in each; \(2) a point in one body about which another body turns; and \(3) a point in one body about which another body tends to turn. The last definition is also the definition of an instantaneous axis of velocity Definitions 2 and 3 satisfy definition 1 in that the velocities are the same, namely, zero. It should be noted that a centro satisfying the second definition is permanently fixed and would be a point in the frame of the machine about which a crank turns. A centro as defined by the first definition may be either a point actually in the two bodies and at the geometric center of the pair of the two bodies and, therefore, a permanent center but movable, or a point in space, not actually in either body, but a point assumed to be in both links and, therefore, movable but not permanent. **3-12. Notation of Centros**. All links, including the frame, are numbered as 1, 2, 3, and so on. The centro has a double number as 12, 13, 23, and so on. The centro 23 (called two-three) is in both links 2 and 3 and maybe notated as 32, but for consistency, the smaller number will be written first. **3-13. Number of Centros.** The number of centros in a mechanism is the number of possible combinations of the links taken two at a time. It may be obtained by the equation Number of centros= N(N-1)/2 where N is the number of links, including the frame, in the mechanism. **3-14. Location of Centros.** Centros are located by \(1) observation and \(2) the application of Kennedy\'s theorem, which states that any three bodies having plane motion relative to each other have only three centros that lie along the same straight line. In other words, the three centros that are akin to each other lie along the same straight line. **3-18. Relative Velocity.** All motions, strictly speaking, are relative motions in that some arbitrary set of axes or planes must be established in order for the motion may be defined. It is customary to assume that the earth is a fixed reference plane when analyzing the velocities and motions of machine members and to refer to such motions as absolute motions. **CHAPTER 7: CAMS** **7-1. A cam** is a plate, cylinder, or other solid with a surface of contact so designed as to cause or modify the motion of a second piece, or of the cam itself. Either the cam or the other piece or both may be moving. The most common case is that of a plate, cylinder, or other solid having a curved outline or a curved groove, which rotates about a fixed axis and, by its rotation, imparts motion to a piece in contact with it, known as the **follower.** The latter type of follower is known as an offset follower. A cylinder containing an irregular groove and known as a cylindrical cam. Many machines, particularly automatic machines, depend largely upon cams, properly designed and properly timed, to give motion to the various parts. Usually, a cam is designed for the special purpose for which it is to be used. Ordinarily, in practice, the condition to be fulfilled in designing a cam does not directly involve the speed ratio, but it assigns a certain series of definite positions that the follower is to assume while the driver occupies a corresponding series of definite positions. The relations between the successive positions of the driver and follower in a cam motion may be represented by means of a displacement diagram, whose abscissas are linear distances arbitrarily chosen. **7.2. Motion of Follower.** It often happens that a cam is required to give a definite displacement to the follower in a short interval of time, the nature of the motion not being fixed. The velocity of the follower increases from zero to a maximum from O to m. This type of motion is called ***accelerated harmonic motion.*** From m to n the velocity of the follower decreases. This motion is called ***decelerated or retarded harmonic motion.*** ***Gravitational Motion.*** A body dropped from the hand has no initial velocity at the start, but has a uniformly increasing velocity, under the action of gravity, until it reaches the ground. The motion of the follower may obey this same law of gravity and have a uniformly accelerated motion until the middle of its path is reached, then a uniformly retarded motion to the end of its path. This type of motion is called \(1) gravitational motion, \(2) parabolic motion, or \(3) uniformly accelerated and retarded motion. The follower may be caused to move during a time interval according to either uniformly accelerated motion, uniformly retarded motion, or both uniformly accelerated and retarded motion. **7-3. Plate Cams.** A plate cam imparts motion to a follower guided so that it is constrained to move in a plane that is perpendicular to the axis about which the cam rotates, that is, in a plane coincident with or parallel to the plane in which the cam itself lies. The nature of the motion given to the follower depends upon the shape of the cam. The follower may move continuously or intermittently; it may move with uniform speed or variable speed; or it may have uniform speed part of the time and variable speed part of the time. that center and the pitch profile. The pitch profile or pitch line is the path traveled by the reference point on the following during one revolution of the cam. **7-4. Positive Motion Plate Cams.** It will be noticed that, in each of the cams that have been discussed, the follower must be held in contact with the surface of the cam by some external force such as gravity, or a spring. The cam can only force the follower away from the camshaft; some outside force must bring it back. If it is desired to make the cam positive in its action in either direction without depending upon external force, the cam must be so constructed as to act on both sides of the follower\'s roller, or there must be two rollers, one on either side of the cam. In Fig. 7-19, the pitch line of the cam is made the center line of a groove of a width slightly greater than the roller diameter, thus enabling the cam to move the roller in either direction