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KINEMATICS DYNAMICS AND DESIGN OF MACHINERY 2ND EDITION PDF

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Kinematics, Dynamics, and Design of Machinery, 2nd Edition. Kenneth J. Waldron, G. L. Kinzel. ISBN: Oct pages. Quantity. Kinematics and Dynamics of Machinery - echecs16.info Uploaded by Kinematics, Dynamics, and Design of Machinery 2nd Edition by Waldron Kinzel Chapter 8. Kinematics, Dynamics, and Design of Machinery 2nd Edition by Waldron Kinzel Chapter 8 - Download as PDF File .pdf), Text File .txt) or read online. Solutions.

Problem 8. The amplitude of the follower translation is 3 cm, and the follower radius is 0. The base circle radius is 5 cm, and the offset is 0. The program calls an m-file called follower. The derivatives are not used in this problem, required by the program rf. When this is done, the minimum value in the rise region is 5. Therefore, the minimum base radius to be used is 5.

The machine designer must be aware of this subject and design devices to book. We often see reference to the good or bad ergonomics of an automobile interior or a household appliance. A machine de- signed with poor ergonomics will be uncomfortable and tiring to use and may even be dangerous.

Have you programmed your VCR lately, or set its clock?

There is a wealth of human factors data available in the literature. Some references are noted in the bibliography. The type of information which might be needed for a machine design problem ranges from dimensions of the human body and their distribu- tion among the population by age and gender, to the ability of the human body to with- stand accelerations in various directions, to typical strengths and force generating abili- ty in various positions.

Obviously, if you are designing a device that will be controlled by a human a grass shortener, perhaps , you need to know how much force the user can exert with hands held in various positions, what the user's reach is, and how much noise the ears can stand without damage. If your device will carry the user on it, you need data on the limits of acceleration which the body can tolerate. Data on all these topics exist. Much of it was developed by the government which regularly tests the ability of military personnel to withstand extreme environmental conditions.

Part of the background re- search of any machine design problem should include some investigation of human factors. Many engineering students picture themselves in professional practice spending most of their time doing calculations of a nature similar to those they have done as students.

Fortunately, this is seldom the case, as it would be very boring. Actually, engineers spend the largest percentage of their time communicating with others, either orally or in writ- ing. Engineers write proposals and technical reports, give presentations, and interact with support personnel and managers.

When your design is done, it is usually necessary to present the results to your client, peers, or employer. The usual form of presentation is a formal engineering report. Thus, it is very important for the engineering student to develop his or her communication skills. You may be the cleverest person in the world, but no one will know that if you cannot communicate your ideas clearly and concisely.

In fact, if you cannot explain what you have done, you probably don't understand it your- self. To give you some experience in this important skill, the design project assignments in later chapters are intended to be written up in formal engineering reports. Informa- tion on the writing of engineering reports can be found in the suggested readings in the bibliography at the end of this chapter.

The most common in the United States are the U. The author boldly suggests with tongue only slightly in cheek that this unit of mass in the ips system be called a blob bl to distinguish it more clearly from the slug sl , and to help the student avoid some of the common units errors listed above. Blob does not sound any sillier than slug, is easy to remember, implies mass, and has a convenient abbreviation bl which is an anagram for the abbreviation for pound Ib.

Besides, if you have ever seen a garden slug, you know it looks just like a "little blob. However the student must remember to divide the value of m in Ibm by gc when substituting into this form of Newton's equation.

Thus the Ibm will be divided either by Remember that in round numbers at sea level on earth: This is sometimes also referred to as the mks system.

Force is derived from Newton's law, equation 1. When converting between SI and u. The gravitational constant g in the SI system is ap- proximately 9. The principal system of units used in this textbook will be the U. Most machine design in the United States is still done in this system.

Table shows some of the variables used in this text and their units. The inside front cover contains a table of conversion factors between the U. S, and SI systems. The student is cautioned to always check the units in any equation written for a prob- lem solution, whether in school or in professional practice after graduation. If properly written, an equation should cancel all units across the equal sign. If it does not, then you can be absolutely sure it is incorrect. Unfortunately, a unit balance in an equation does not guarantee that it is correct, as many other errors are possible.

Always double-check your results. You might save a life. Then we will investigate the analysis of those and other mechanisms for their kinematic behavior. Finally, in Part II we will deal with the dynamic analysis of the forces and torques generated by these moving machines. These topics cover the essence of the early stages of a design project. Once the kinematics and kinetics of a design have been determined, most of the concep- tual design will have been accomplished.

What then remains is detailed design-sizing the parts against failure. The topic of detailed design is discussed in other texts such as reference [8]. Machine Design: Upper Saddle River, NJ.

For additional information on the history of kinematics, the following are recommended: Artobolevsky, I. Brown, H. Five Hundred and Seven Mechanical Movements. Erdman, A. Modern Kinematics: Developments in the Last Forty Years. New York. Ferguson, E. Freudenstein, F. Kinematic Synthesis of Linkages. New York, pp. Nolle, H. A Historical Review - II. Developments after A Historical Review -I. Developments up to Spatial Synthesis and Optimization.

Reuleaux, F. The Kinematics of Machinery, A. Kennedy, translator. Dover Publications: Strandh, S. A History of the Machine. For additional information on creativity and the design process, the following are recommended: Alger, J. Creative Synthesis in Design. Allen, M. Morphological Creativity.

Altschuller, G. Creativity as an Exact Science. Gordon and Breach: Buhl, H. Creative Engineering Design. Iowa State University Press: Ames, IA. Field Stone Publishers: Conway, MA. Fey, V. Gordon, W. Haefele, W. Creativity and Innovation. Van Nostrand Reinhold: Harrisberger, L. Monterey, CA. Osborn, A. Applied Imagination. Pleuthner, W. Soh, N. The Principles of Design. Oxford University Press: Taylor, C. Widening Horizons in Creativity. Von Fange, E. Professional Creativity.

For additional information on Human Factors, the following are recommended: Bailey, R. Human Performance Engineering: A Guidefor System Designers. Burgess, W. Designing for Humans: The Human Factor in Engineering. Petrocelli Books. Clark, T. The Ergonomics of Workspaces and Machines. Taylor and Francis. Huchinson, R. New Horizons for Human Factors in Design.

McCormick, D. Human Factors Engineering. Osborne, D. Ergonomics at Work. Pheasant, S. Salvendy, G. Handbook of Human Factors. Sanders, M. Human Factors in Engineering and Design. Woodson, W. Human Factors Design Handbook. For additional information on writing engineering reports, the following are recommended: Barrass, R. Scientists Must Write. Crouch, W. A Guide to Technical Writing. The Ronald Press: Davis, D. Elements of Engineering Reports. Chemical Publishing Co.: Gray, D. Information Resources Press: Washington, D.

Michaelson, H. ISI Press: Philadelphia, PA. Nelson, J. Writing the Technical Report.

Kinematics, Dynamics, and Design of Machinery 2nd edition by Waldron

It will also present some very simple but powerful analysis tools which are useful in the synthesis of mechanisms. The system's DOF is equal to the number of indepen- dent parameters measurements which are needed to uniquely define its position in space at any instant of time. Note that DOF is defined with respect to a selected frame of reference. Figure shows a pencil lying on a flat piece of paper with an x, y coordi- nate system added.

If we constrain this pencil to always remain in the plane of the pa- per, three parameters DOF are required to completely define the position of the pencil on the paper, two linear coordinates x, y to define the position of anyone point on the pencil and one angular coordinate 8 to define the angle of the pencil with respect to the axes. The minimum number of measurements needed to define its position are shown in the figure as x, y, and 8.

This system of the pencil in a plane then has three DOF. Note that the particular parameters chosen to define its position are not unique.

Any alternate set of three parameters could be used. There is an infinity of sets of parameters possible, but in this case there must be three parameters per set, such as two lengths and an an- gie, to define the system's position because a rigid body in plane motion has three DOF. Hold it above your desktop and move it about. You now will need six parameters to define its six DOF. Any rigid body in three-space has six degrees of freedom.

Try to identify these six DOF by moving your pencil or pen with respect to your desktop. The pencil in these examples represents a rigid body, or link, which for purposes of kinematic analysis we will assume to be incapable of deformation. This is merely a con- venient fiction to allow us to more easily define the gross motions of the body. We can later superpose any deformations due to external or inertial loads onto our kinematic motions to obtain a more complete and accurate picture of the body's behavior.

But re- member, we are typically facing a blank sheet of paper at the beginning stage of the de- sign process. We cannot determine deformations of a body until we define its size, shape, material properties, and loadings. Thus, at this stage we will assume, for purposes of initial kinematic synthesis and analysis, that our kinematic bodies are rigid and massless. In three-dimensional space, there may be rotation about any axis any skew axis or one of the three principal axes and also simultaneous translation which can be resolved into components along three axes.

In a plane, or two-dimensional space, complex motion be- comes a combination of simultaneous rotation about one axis perpendicular to the plane and also translation resolved into components along two axes in the plane.

For simplic- ity, we will limit our present discussions to the case of planar kinematic systems. We will define these terms as follows for our purposes, in planar motion: Pure rotation the body possesses one point center of rotation which has no motion with respect to the "stationary" frame of reference.

All other points on the body describe arcs about that center. A reference line drawn on the body through the center changes only its angular orientation. Pure translation all points on the body describe parallel curvilinear or rectilinear paths.

Complex motion a simultaneous combination of rotation and translation. Any reference line drawn on the body will change both its linear position and its angular orientation. Translation and rotation represent independent motions of the body. Each can ex- ist without the other. If we define a 2-D coordinate system as shown in Figure , the x and y terms represent the translation components of motion, and the e term represents the rotation component. Linkages are the basic building blocks of all mechanisms.

We will show in later chapters that all common forms of mechanisms cams, gears, belts, chains are in fact variations on a common theme of linkages. Linkages are made up of links and joints. A link, as shown in Figure , is an assumed rigid body which possesses at least two nodes which are points for attachment to other links.

Binary link - one with two nodes. Ternary link - one with three nodes. Quaternary link - one with four nodes. A joint is a connection between two or more links at their nodes , which allows some motion, or potential motion, between the connected links.

Joints also called ki- nematic pairs can be classified in several ways: Reuleaux [1] coined the term lower pair to describe joints with surface contact as with a pin surrounded by a hole and the term higher pair to describe joints with point or line contact. However, if there is any clearance between pin and hole as there must be for motion , so-called surface contact in the pin joint actually becomes line contact, as the pin contacts only one "side" of the hole.

Likewise, at a microscopic level, a block sliding on a flat surface actually has contact only at discrete points, which are the tops of the surfaces' asperities. The main practical advantage of lower pairs over higher pairs is their better ability to trap lubricant between their enveloping surfaces.

This is especially true for the rotating pin joint. The lubricant is more easily squeezed out of a higher pair, nonenveloping joint. As a result, the pin joint is preferred for low wear and long life, even over its lower pair cousin, the prismatic or slider joint. Figure a shows the six possible lower pairs, their degrees of freedom, and their one-letter symbols.

The revolute R and the prismatic P pairs are the only lower pairs usable in a planar mechanism. The Rand P pairs are the basic building blocks of all other pairs which are combinations of those two as shown in Table A more useful means to classify joints pairs is by the number of degrees of free- dom that they allow between the two elements joined.

Figure also shows examples of both one- and two-freedom joints commonly found in planar mechanisms. Figure b shows two forms of a planar, one-freedom joint or pair , namely, a rotating pin joint R and a translating slider joint P. These are also referred to as full joints i. These are both contained within and each is a limiting case of another common, one-freedom joint, the screw and nut Figure a.

Motion of either the nut or the screw with respect to the other results in helical motion. If the helix angle is made zero, the nut rotates without advancing and it becomes the pin joint. If the helix angle is made 90 degrees, the nut will translate along the axis of the screw, and it becomes the slider joint. Figure c shows examples of two-freedom joints h1gherpairs which simultaneously allow two independent, relative motions, namely translation and rotation, between the joined links.

Paradoxically, this two-freedom joint is sometimes referred to as a "half joint," with its two freedoms placed in the denominator. The half joint is also called a roll-slide joint because it allows both rolling and sliding.

A spherical, or ball-and-socket joint Figure a , is an example of a three-freedom joint, which allows three independent angular motions be- tween the two links joined. This ball joint would typically be used in a three-dimensional mechanism, one example being the ball joints in an automotive suspension system.

A joint with more than one freedom may also be a higher pair as shown in Figure c. Note that if you do not allow the two links in Hgore c connected by a roll-slide joint to slide, perhaps by providing a high friction coefficient between them, you can "lock out" the translating At freedom and make it behave as a full joint. This is then called a pure rolling joint and has rotational freedom AD only.

A cornmon example of this type of joint is your automobile tire rolling against die road, as shown in Figure e. In normal use there is pure rolling and no sliding at Ibis joint, unless, of course, you encounter an icy road or become too enthusiastic about accelerating or cornering. If you lock your brakes on ice, this joint converts to a pure sliding one like the slider block in Figure b.

Kinematics, Dynamics, and Design of Machinery 2nd Edition by Waldron Kinzel Chapter 8

Friction determines the actual number of freedoms at this kind of joint. It can be pure roll, pure slide, or roll-slide. To visualize the degree of freedom of a joint in a mechanism, it is helpful to "men- tally disconnect" the two links which create the joint from the rest of the mechanism.

You can then more easily see how many freedoms the two joined links have with respect to one another. Figure c also shows examples of both form-closed and force-closed joints. A form-closed joint is kept together or closed by its geometry. A pin in a hole or a slider in a two-sided slot are form closed. In contrast, a force-closed joint, such as a pin in a half-bearing or a slider on a surface, requires some external force to keep it together or closed. This force could be supplied by gravity, a spring, or any external means.

There can be substantial differences in the behavior of a mechanism due to the choice of force or form closure, as we shall see. The choice should be carefully considered.

In linkag- es, form closure is usually preferred, and it is easy to accomplish. But for cam-follower systems, force closure is often preferred. This topic will be explored further in later chap- ters. Figure d shows examples of joints of various orders, where order is defined as the number of links joined minus one.

It takes two links to make a single joint; thus the simplest joint combination of two links has order one. As additional links are placed on the same joint, the order is increased on a one for one basis. Joint order has significance in the proper determination of overall degree of freedom for the assembly. We gave def- initions for a mechanism and a machine in Chapter 1. With the kinematic elements of links and joints now defined, we can define those devices more carefully based on Reu- leaux's classifications of the kinematic chain, mechanism, and machine.

An assemblage of links and joints, interconnected in a way to provide a controlled out- put motion in response to a supplied input motion. A mechanism is defined as: A kinematic chain in which at least one link has been "grounded," or attached, to the frame of reference which itself may be in motion.

A machine is defined as: A combination of resistant bodies arranged to compel the mechanical forces of nature to do work accompanied by determinate motions. By Reuleaux's definition [1] a machine is a collection of mechanisms arranged to transmit forces and do work. He viewed all energy or force transmitting devices as ma- chines which utilize mechanisms as their building blocks to provide the necessary mo- tion constraints. We will now define a crank as a link which makes a complete revolution and is piv- oted to ground, a rocker as a link which has oscillatory back andforth rotation and is pivoted to ground, and a coupler or connecting rod which has complex motion and is not pivoted to ground.

Ground is defined as any link or links that are fixed nonmov- ing with respect to the reference frame. Note that the reference frame may in fact itself be in motion. We need to be able to quickly determine the DOF of any collection of links and joints which may be suggested as a solution to a problem. Degree of free- dom also called the mobility M of a system can be defined as: Degree of Freedom the number of inputs which need to be provided in order to create a predictable output; also: At the outset of the design process, some general definition of the desired output motion is usually available.

The number of inputs needed to obtain that output mayor may not be specified. Cost is the principal constraint here. Each required input will need some type of actuator, either a human operator or a "slave" in the fonn of a motor, sole- noid, air cylinder, or other energy conversion device.

These devices are discussed in Section 2. These multiple input devices will have to have their actions coordinated by a "controller," which must have some intelligence. This control is now often provid- ed by a computer but can also be mechanically programmed into the mechanism design. There is no requirement that a mechanism have only one DOF, although that is often desirable for simplicity.

Some machines have many DOF. For example, picture the num- ber of control levers or actuating cylinders on a bulldozer or crane. See Figure I-lb p. Kinematic chains or mechanisms may be either open or closed.

Figure shows both open and closed mechanisms. A closed mechanism will have no open attachment points or nodes and may have one or more degrees of freedom. An open mechanism of more than one link will always have more than one degree of freedom, thus requiring as many actuators motors as it has DOF.

A common example of an open mechanism is an industrial robot. An open kinematic chain of two binary links and one joint is called a dyad. The sets of links shown in Figure a and b are dyads. Reuleaux limited his definitions to closed kinematic chains and to mechanisms hav- ing only one DOF, which he called constrained. A multi-DOF mechanism, such as a robot, will be constrained in its motions as long as the necessary number of inputs are supplied to control all its DOF.

Degree of Freedom in Planar Mechanisms To determine the overall DOF of any mechanism, we must account for the number of links and joints, and for the interactions among them. The DOF of any assembly of links can be predicted from an investigation of the Gruebler condition. In Figure c the half joint removes only one DOF from the system because a half joint has two DOF , leaving the system of two links connected by a half joint with a total of five DOF.

In addition, when any link is grounded or attached to the reference frame, all three of its DOF will be removed. This reasoning leads to Gruebler's equation: Thus G is always one, and Gruebler's equation becomes: Multiple joints count as one less than the number oflinks joined at that joint and add to the "full" 11 category.

The DOF of any proposed mechanism can be quickly ascertained from this expression before investing any time in more detailed design. It is interesting to note that this equation has no information in it about link sizes or shapes, only their quantity.

Figure a shows a mechanism with one DOF and only full joints in it. Figure b shows a structure with zero DOF and which contains both half and mul- tiple joints. Note the schematic notation used to show the ground link. The ground link need not be drawn in outline as long as all the grounded joints are identified. Note also the joints labeled "multiple" and "half' in Figure a and b.

As an exercise, compute the DOF of these examples with Kutzbach's equation. Degree of Freedom in Spatial Mechanisms The approach used to determine the mobility of a planar mechanism can be easily ex- tended to three dimensions. Each unconnected link in three-space has 6 DOF, and any one of the six lower pairs can be used to connect them, as can higher pairs with more freedom.

Grounding a link removes 6 DOF. This leads to the Kutzbach mobility equation for spa- tiallinkages: We will limit our study to 2-D mechanisms in this text. There are only three possibilities. If the DOF is positive, it will be a mechanism, and the links will have relative motion.

If the DOF is exactly zero, then it will be a structure, and no motion is possible. If the DOF is negative, then it is a preloaded structure, which means that no motion is possible and some stresses may also be present at the time of assembly. Figure shows examples of these three cases. One link is grounded in each case. Figure a shows four links joined by four full joints which, from the Gruebler equation, gives one DOF. It will move, and only one input is needed to give predictable results.

Figure b shows three links joined by three full joints. It has zero DOF and is thus a structure. Figure c shows two links joined by two full joints.

It has a DOF of minus one, making it a preloaded structure. In order to insert the two pins without straining the links, the center distances of the holes in both links must be exactly the same. Practical- ly speaking, it is impossible to make two parts exactly the same. There will always be some manufacturing error, even if very small.

Thus you may have to force the second pin into place, creating some stress in the links. The structure will then be preloaded. You have probably met a similar situation in a course in applied mechanics in the form of an indeterminate beam, one in which there were too many supports or constraints for the equations available. Both structures and preloaded structures are commonly encountered in engineering.

In fact the true structure of zero DOF is rare in engineering practice. Even simple structures like the chair you are sitting in then their interconnection are often preloaded. Since our concern here is with mechanisms, we will concentrate on is impossible. Order in this context refers to the number of nodes perlink, i.

The value of number synthesis is to allow the exhaustive determination of all possible combinations of links which will yield any chosen DOF.

This then equips the designer with a definitive catalog of potential linkages to solve a variety of motion control prob- lems. As an example we will now derive all the possible link combinations for one DOF, including sets of up to eight links, and link orders up to and including hexagonal links.

For simplicity we will assume that the links will be connected with only full rotating joints. We can later introduce half joints, multiple joints, and sliding joints through link- age transformation. First let's look at some interesting attributes of linkages as defined by the above assumption regarding full joints. If all joints are full joints, an odd number of DOFrequires an even number of links and vice versa.

All even integers can be denoted by 2m or by 2n, and all odd integers can be denoted by 2m - I or by 2n - 1, where n and m are any positive integers. The number of joints must be a positive integer. There are other examples of paradoxes which disobey the Gruebler criterion due to their unique geometry.

The designer needs to be alert to these possible inconsistencies. Isomers in chemis- try are compounds that have the same number and type of atoms but which are intercon- nected differently and thus have different physical properties. Figure a shows two hydrocarbon isomers, n-butane and isobutane. Note that each has the same number of carbon and hydrogen atoms C4HlO , but they are differently interconnected and have different properties. Linkage isomers are analogous to these chemical compounds in that the links like atoms have various nodes electrons available to connect to other links' nodes.

The assembled linkage is analogous to the chemical compound. Depending on the particular connections of available links, the assembly will have different motion properties. It will be found that the base circle must be fairly large to avoid cusps.

Similarly, the distance between pivots must be fairly large to avoid interference with the cam and follower pivot. One set of values which will work is a base radius of 3 inches and a distance between pivots of 5 inches. These values and the others given in the problem can be input into the program to determine the cam geometry.

The basic program input is as follows. The displacement diagram does not show the acceleration "spikes" at the beginning and end of the dwell; however, the discontinuities in the radius of curvature are indicated. This cam would perform poorly in a high speed application. The displacement profile along with the first and second derivatives can be computed using the equations in Chapter 8 in a spreadsheet or MATLAB program.

Note: Use the minimum number of Ci possible. If h is 3 cm and is 60, determine the minimum base circle radius to avoid cusps if the cam is used with a flat-faced follower.

Problem S6. S h Dwell Dwell , rad Problem S6. Determine the number of terms required and the corresponding values for the constants Ci if the displacement and velocity are continuous at the end points of the rise points A and B and if the cam velocity is constant.

The cam is to have a translating, flat-faced follower which is offset in the clockwise direction by 0. Assume that the follower is to dwell at zero lift for the first of the motion cycle and to dwell at 1 in lift for cam angles from to The cam will rotate clockwise, and the base circle radius is to be 2 in.

Lay out the cam profile using 20 plotting intervals. In the position to be analyzed, link 2 lies in a plane parallel to the XY plane and points along a line parallel to the Y axis, and link 3 is perpendicular to link 2.

For the position to be analyzed, link 3 is vertical parallel to Z. The joint between link 2 and the frame at A is a cylindrical joint, and that at B is a revolute joint. The gear blank and the worm gear gear 9 are mounted on the same shaft and rotate together.

Kinematics, Dynamics, and Design of Machinery

If the gear blank is to be driven clockwise, determine the hand of the hob. Finally, select gears 3 and 5 which will satisfy the ratio. Gears are available which have all of the tooth numbers from 15 to