Later, through appropriate approximations, the ideal inviscid fluid is introduced. It gives a clear exposition of the fundamentals of fluid dynamics, both viscous and inviscid, including the topic of boundary layer. The text provides introductory concepts of wind tunnel and measurements to give a balanced overview of the subject so that the students are exposed to experiments and laboratory practices at the outset.
This book is primarily intended for the undergraduate students of aeronautical engineering and aerospace engineering. The objective, then and now, was to present the subject of computational? A second edition appeared in with updates to all the chapters and when that printing came to an end, the publisher requested that the editor and authors consider the preparation of a third edition.
Happily, the authors received the request with enthusiasm. The third edition has the goal of presenting additional updates and clari? The book is divided into three parts. John Anderson lays out the subject in Part I by? Methods of discretizing the equations are discussed and transformation techniques and grids are presented. Two examples of numerical methods close out this part of the book: source and vortex panel methods and the explicit method.
Part II is devoted to four self-contained chapters on more advanced material. Roger Grundmann treats the boundary layer equations and methods of solution. Score: 5. It introduces to the rotary-wing aerodynamics, with applications to helicopters, and application of the relevant principles to the aerodynamic design of a helicopter rotor and its blades.
The basic aim of this book is to make a complete text covering both the basic and applied aspects of theory of rotary wing flying machine for students, engineers, and applied physicists. The philosophy followed in this book is that the subject of helicopter aerodynamics is covered combining the theoretical analysis, physical features and the application aspects.
Considerable number of solved examples and exercise problems with answers are coined for this book. Score: 4. New coverage on the Computational Fluid Dynamics CFD and new illustrations to help the students to understand the basic conepts.
More than a dozen "design boxes" are included to help students focus on the practical applications. It examines the challenges of high-speed flight, covers aerodynamic phenomena in supersonic flow and aerodynamic drag in cruising flight, and discusses the advantages and disadvantages of oblique wing aircraft. Essentials of Supersonic Commercial Aircraft Conceptual Design is intended for members of a team producing an initial design concept of an airliner with the capability of making supersonic cruising flights.
It begins with a synopsis of the history of supersonic transport aircraft development and continues with a chapter on the challenges of high-speed flight, which discusses everything from top level requirements and cruise speed requirements to fuel efficiency and cruise altitude. It then covers weight sensitivity; aerodynamic phenomena in supersonic flow; thin wings in two-dimensional flow; flat wings in inviscid supersonic flow; aerodynamic drag in cruising flight, and aerodynamic efficiency of SCV configurations.
The book finishes with a chapter that examines oblique wing aircraft. Provides supersonic aircraft designers with everything they need to know about developing current and future high speed commercial jet planes Examines the many challenges of high-speed flight Covers aerodynamic phenomena in supersonic flow and aerodynamic drag in cruising flight Discusses the advantages and disadvantages of oblique wing aircraft Essentials of Supersonic Commercial Aircraft Conceptual Design is an ideal book for researchers and practitioners in the aerospace industry, as well as for graduate students in aerospace engineering.
The geophysical and astrophysical applications discussed concern e. Non-linear waves solitons are also mentioned. In turn, the book explores problems from everyday life, including the motion of golf balls, life at low Reynolds numbers, the physics of sailing, and the aerodynamics of airplanes and Grand Prix cars. No book on this topic would be complete without a look at chaos and turbulence; here the problems span from Gaussian plumes to chaotic dynamos, to stochastic climate modeling.
Advances in fluid dynamics have produced a wealth of numerical methods and techniques, which are used in many of the applications. The relationship between these energy state functions and the work expression is such that Requirement 2 is satisfied. Thus, the satisfaction of the three requirements during analyses via the indirect approach is somewhat subtle; however, because the indirect approach is a major focus of this book, we shall devote considerable effort to exploring the details by which this is achieved.
The summary comments in Table represent a set of contrasts that are of a general character. As such, itis a table worth referencing as, wwe proceed through the book. In particular, as we shall see, velocities are somewhat easier to obtain than accelerations, and scalars are somewhat easier to manipulate than vectors; thus, the indirect approach is often simpler to use than the direct approach.
On the other hand, the direct approach more readily provides us with the forces that act on and within, the system. Several analyses and calculations were conducted in Example to find the maximum deflection in the spring, Although the primary goal was to illustrate and delineate the di- rect and indirect approaches, as an engineering exercise the example is somewhat deficient.
Many such questions would be related to the maximum spring deflection and to the forces that act on the package throughout its transit, including its deceleration toward point D. Note that we emphasize deffection s as well as forces, since the reader should recall that failure of engineering structures is often initiated by too much deformation, even though the forces in a system may not cause yielding or rupture.
Further, in this example the forces, if sufficiently large, could cause damage to the package or the contents of the package. Although we shall defer the calculation of such forees to Chapter 4 and beyond, we briefly consider one of a number of forces encountered by the package in Example In particular, we consider the maximum force applied to the package by the spring in bringing it to a halt.
From Eqs. The result in Eq. For example, is the force sulli- cientiy large to damage the package or its contents or both? Furthermore, from an analytical perspective, will the spring retain its assumed linear characteristics throughout the range of its calculated deformation? If not, appropriate analytical modifications should be consid- ered.
Also, itis clear that Eq. Note that Eq. In summary, we want to emphasize that, whether obvious or obscure, the ultimate goal of all our analyses is the assessment of the performance or reliability of an existing or proposed design. Clearly, the safe delivery of the package including its contents requires that itis capable of withstanding a force equal to three times its weight.
Thus, significant dynamic forces may suddenly appear and then disappear, but nevertheless must be considered in a thorough design analysis. Kinemat- ies is concerned primarily with the time behavior of position, velocity, and acceleration of systems as well as components of systems.
Itis kinematics by which the first of the three requirements in Eq, is satisfied. Although kinematics is conducted without reference to forces, ultimately the calculation of the dynamic forces on elements and systems requites knowledge of kinematics. Thus, a thorough knowledge of kinematics is crucial to acquiring a thorough knowledge of system motions and forces, quantities that are often important inputs to the design process.
It is literally the means by which humankind has kept time throughout history and will no doubt do so for the foreseeable future; itis literally the invention of time as a quantitative scientific concept. It also fixes the invention of kinemat- ics, The surveying by the Ancient Egyptians in order to orient with great accuracy many of their major structures, as well as the invention of astrology, were based upon kinematic astronomical measurements. Following these successes, the development of the lunar cal- endar, the casting of horoscopes, and the cataloging of numerous stars and the motions of Mercury, Venus, Mars, Jupiter, and Saturn by the Mesopotamians were great achievements in kinematics.
These were major kinematic accomplishments that were integrated into the daily lives of the citizens of these two societies; landmarks of human history in the spirit of Ptolemy, Copernicus, Brahe, and Kepler, all of whom would use and build upon them. In the fourteenth century, several scholars at Merton College, Oxford, composed definitions for velocity and acceleration for uniformly accelerated motion.
Between and , the work of Thomas k Bradwardine, William Heytesbury, Richard Swineshead, and John of Dumbleton established t the kinematic analyses that provided the foundations for Galileis studies on falling bodies.
This work was quickly accepted in continental Europe, intially leading most significantly to the analyses of Nicole Oresme ca. This reestablished another useful connection between geometry and the physical work, a tradition the Ancient Egyptians had introduced via their monuments several millenia earlier, and which Ptolemy had notably continued in his astronomy, for example, in his use of epicycles, Galileo Galilei was a pioneer of the modern concept of the testing of a scientific hypothe- sis based on careful experimental design, observation, and measurement, He discovered and expressed the law of falling bodies, as itis often called.
This is not an exclusively kine- matic result, although it is frequently presented as such. His conclusion that in a vacuum all bodies—heavy and light, large and small—fall in the same manner was in part a consequence of meticulous experimentation and brilliant insight.
For in reaching such a conclusion, Galilei had to imagine a vacuum that he could not produce and ignore air resistance that he could not eliminate.
His deduction that an isolated body moves, and will continue to move, with a constant velocity was a great intellectual achievement. The Use and manipulation of quantities that have both magnitude and direction were obviously familiar to the Ancient Egyptians and later the Mesopotamians in their astronomical analy- ses of position, displacement, arid velocity.
Yet, those uses of directed quantities were primarily geometrical and numer- ical, as such use would remain until the rise of postnewtonian mechanics in the eighteenth century.
For example, the velocity of a bus traveling along a highway may be different for observers in different reference frames. A passenger in a cat traveling in the 4 same direction as the bus will observe a different velocity of the bus than that observed by a passenger traveling in a car in the opposite direction.
A pedestrian will observe still another velocity of the bus. Thus, we must be clear in specifying the reference frame used in each kinematic description.
Perhaps the most interesting observation from Figure is that at locations where the point P comes. This fact is likely to surprise many readers. Although we shall not discuss this point further at this time, itis a useful result that we shall use in several analyses, primarily in Chapter 6.
By using Eq. One extremely powerful simplification is afforded by the concept of a rigid body. No real object is rigid, in the sense that it does not deform under load; yet there are many circumstances in which a rigid body can be a useful model for a real system.
This, of course, is a mathematical idealization. In Chapter 6, we shall discuss the concept of a rigid body in more detail. So, in analyses where the deformations of a body are negligible compared with the overall motion of the body, the body is often modeled as a rigid body.
While there may be significant information that is suppressed by such a model most notably, its elastic deformation , the concept of a rigid body is nevertheless a very useful one in dynamics. It is likely that the deformation of the car body will be small compared with the pitching, heaving, and rolling motions of the car; so, adopting the assumption that the car body is rigid will enormously simplify the analysis without significantly affecting the result obtained for the overall motion of the car.
In a subsequent analysis, the assumption of rigidity can be relaxed if it ceases to be useful. For example, once both the motion of the car and the forces applied to the car body by the suspension system have been determined, the assumption that the car is rigid can be relaxed, and a study of the deformation of the nonrigid car traversing the bumpy road can be undertaken, We could model a nonrigid body as a collection of N points, bound by flexible connectors.
In Chapter 4, we shall endow such points with properties. In order to producga useful approximation of the motion of the actual body, NV would have to be large. Assuming each point could move throughout space, each point would require three spatial numbers or c ordinates to locate it completely.
Thus, the system of N points that model the body would require 3N such spatial numbers or coordinates to locate it. If, by contrast, we were able to assume that such a body were rigid, we would find that the entire body would require only six spatial numbers or coordinates to locate it. To see that a rigid body requires only six spatial numbers or coordinates to locate it completely, consider Figure , We can specify the position of the body relative to an OXYZ reference frame by specifying the location of a rigid triangle ABC fixed in the body as follows.
First, specify the three coordinates of point 4; the body is now free to rotate only, but about any axis through A. Second, specify the direction of the line AB; this requires two scalar quantities—say two of the three direction cosines of the line AB or two spherical angles. Third, specify the angle of rotation about AB to locate the plane containing the line AC, using a single coordinate to represent the rotation ofthe body about AB, making six coordinates in all. Other choices of coordinates for locating the body could be made, but any alternative scheme would also require the specification of six independent spatial quantities, From the preceding discussion, we can see that, in general, the kinematic analysis of rigid bodies includes both linear and angular quantities.
In order to establish several impor- tant concepts about the angular or rotational motions of rigid bodies, we shall limit most of our discussion here to the plane kinematics of rigid bodies. This idealization is appropriate for analyzing many rigid body problems in engineering. The plane motion of a rigid body may be divided into several categories, as discussed below.
In translation there is no rotation of amy line in the body at any time. In rectilinear translation, all points in the body move in parallel straight lines see Figure In curvilinear translation, all points move along congruent curves see Figure Clearly, rectilinear translation is a special case of curvilinear translation. Note that in each of the two cases of translation, the motion of the body is completely specified by the motion of any point in the body, since all points undergo the same motion.
Also, note that all ines such as AB in Figures a and remain parallel to their past and future positions. Thus, in rigid body translation, any kinematic describing the motion of a point in the body also completely specify the motion of the body. Ifthe axis of rotation passes through the body, as does the axis through point A in Figure c, points on the axis do not move.
If the axis of rotation does not pass through the body, as does the axis, through point O in Figure d, all points of the body move. In order to proceed quantitatively, however, we must become more precise in our termin- ology. A detailed discussion of rotation of rigid bodies is one of the more challenging aspects of kinematics.
Our tasks will be simplified greatly, however, by limiting our focus on rigid bodies to planar motion. Figure shows a rigid body moving parallel to the XY plane as it undergoes fixed-axis rotation about OZ, where the reference frame OXYZ is fixed in space.
The angular positions of two arbitrary lines in the XY plane, designated as and and fixed to the body, are defined as 0; and 02, respectively, measured from the X axis of the reference frame. Note that lines and do not necessarily extend radially from the origin. The values of 81 and 02 are generally measured in radians, which are dimensionless. The rotation itself can be designated by a directed line segment AO whose length represents the magnitude of the rotation and whose sense is determined by the right-hand rule: When the axis 2 is grasped with the right hand so that the fingers curl in the sense of the rotation A , the right-hand thumb points in the sense of the directed line segment.
Once the axis of rotation has been chosen, the magnitude and the sense of the rotation determine the directed line segment A. However, as we discuss below, for nonplanar rotations, finite rotations are not vectors. The sub- scripts on the differential angles in Eqs. Of the three angular vectors—angular displacement, angular velocity, and angular acceleration—our discussions and analyses will be centered on angular velocity. The angular velocity e» can also be represented by a directed line segment.
Note that OZ does not pass through. All the arguments and all the conclusions made in conjunction with. Figure hold also for Figure ; Figure was simply an intermediate vehicle. Since all line segments in Figure , as well as in Figure , undergo the same angular motions, 4 , w, and could have been sketched any place, as long as they were perpendicular tothe XY plane.
As vectors—which have a magnitude and a direction but not a place—the vectorial representation of these angular motions have no place! Rotating and Nonrotating Disks A larger disk that rotates about its center at a constant angular velocity ay is shown in four successive views in Figures and Actually, the bearings prevent any torque transmission from the larger disk to the smaller disk; but we have no means of knowing this yet.
If the set screw locks the smaller disk to the larger disk, the system behaves as sketched in Figure ; whereas if the set screw is not engaged, the system behaves as sketched in Figure The reference frame OXYZ is fixed. Because the smaller disk can be considered as a rigid body undergoing plane rotation, the decisive issue is whether any line on or in the body undergoes rotation. Note that it does rotate with respect to the larger disk, but not with respect to OXYZ.
Thus, in Figure , the angular velocity of the smaller disk is always zero. Every point in the smaller disk moves with respect to OXYZ with the identical velocity! Conelusions: In Figure , the smaller disk undergoes the same angular rotation as the larger disk.
This case is analogous to Fig. In Figure the smaller disk undergoes only curvilinear translation, as all points within it move in parallel lines and the motion of the entire disk can be completely specified by the motion of any point within the disk. Wrong kinematics guarantees wrong dynamics.
In particular, see Example , which follows, and its reconsideration in Example in Chapter 6. A smaller cylinder of radius r rolls without slip on a larger cylinder of radius R, as sketched in Figure The angle locates the center C of the smaller cylinder with respect to a fixed vertical reference, and point A, which is on the rolling cylinder, and point D, which is, on the stationary cylinder, are coincident when 6 is zero, that is, when the rolling cylinder is in position 1.
Find the angular displacement of the smaller cylinder as a function of 6. Before proceeding with the solution, determine whether you can answer the following little riddle. Stop reading; answer the riddle. Note that by elementary plane geometry, 0 is also the angle between the fixed vertical reference and CB. From Eq. Referring to Figure a and Eq, c , give a physical interpretation to this result. Finally, we note that in this example as well as in our previous and subsequent discus- sions, we have consistently used the word rotation to characterize changes in orientation.
Even more importantly, we emphasize that just as linear displacements have been consistently defined with iespect to a fixed reference, so too an- gular displacements must be defined with respect to a fixed reference. In both cases it is possible to represent displacements, velocities, and accelerations by directed line segments. There is, however, a significant difference between the charac- ter of the directed line segments that represent angular displacements and the directed line segments that represent linear displacements.
This difference emerges when nonplanar motions of finite bodies are analyzed, or equivalently, when we must consider combining a number of angular displacements. Such considerations are usually explored under the cat- egory of finite rotations Ifa point P undergoes several linear displacements in succession, the resulting total linear displacement can be represented by a directed line segment that is the vector sum of, the directed line segments that represent the individual displacements.
However, when a f- nite body undergoes several finite angular displacements about different axes in succession, tte result depends on the order in which the individual angular displacements are taken. The exception to this statement is the case for plane motion, where all the angular displacements are about the same axis.
Under planar motion, the algebra of angular dis- placements does degenerate to vector algebra. A more mathematical presentation of this discussion is given in Appendix A. The desired orientation is achieved and shown at the far right of Figure a. If the order of the instructions to rotate were reversed, as sketched in Figure , the total angular displacement would be entirely different, resulting in a likely unsatisfactory orientation.
Finally, although finite angular displacements, in general, do not combine according to the laws of vector addition, infinitesimal angular displacements do obey the laws of vector addition see Appendix A , So, it follows from the laws of differentiation that angular velocities also must satisfy the laws of vector addition; that is, ifthe infinitesimal 0 satisfies.
For example, the motion of a turbine rotor on a ship with respect to the shore is conveniently described with respect to an intermediate frame that is fixed in the ship, which itself moves relative to the shore. Before discussing kinematic analysis ulilizing intermediate frames, we consider the problem of relating the time rate of change of a vector as seen by an observer in a moving frame to that seen by an observer in a second reference frame.
The origins 0 and O of the two reference frames are coinci- dent and remain so throughout this discussion. A vector A having a constant. From the second reference frame OXYZ, the vector A appears to be changing its orientation, which is due to «. It can be shown that the rate of change of A observed from the second reference frame OXY is, dA a where the syinbol "x" denotes the vector cross-product.
The derivation of Eq. So, consider in Figure a vector a of constant magnitude that extends from O and is carried around by the rotating frame, which for simplicity is not shown. Since Aq is nearly orthogonal to the plane containing and w, the limiting quantity e is orthogonal to this plane.
Thus, we may write G a da Se wx - a a 13 Finally, by use of Eqs. Find the time derivative of a as seen from oxyz. That is, a vector—characterized by a magnitude and a direction—can be k a NN?
Next, we develop a more general form corresponding to Eqs. Let B be anarbitrary vector defined in oxy, having changing magnitude and changing direction, pos- sibly simultaneously.
Using the notation g to indicate time derivatives as observed from the reference frame OXYZ, the derivative of Eq. The first three terms can be considered to be the rate of change of B as viewed from the ary frame, and the last three terms give the contribution due to the rotation of azyz with respect. The last three terms can be simplified by applying Eqs.
Thus, use of Eqs. Furthermore, if B is parallel to « that is, parallel to the axis of rotation , the cross-product vanishes and the time rates of change of B as viewed from the two frames are identical. So, when the rate of change of w itself is observed from the two reference frames, we note that , 18 a dt Finally, because the vector B in Eq.
Again, we express the fundamental concept that a vector has both magnitude and di- rection; thus a vector can be changed by changing either its magnitude or its direction or both. As we shall illustrate in Example , the first term on the right-hand side of Eq B accounts for the change in scalar components in ozyz of the vector and the second term on the right-hand side of Eq.
Find the first and second time derivatives of w2 with respect to OXYZ. For convenience, oryz is defined as a frame that is fixed in the platform. Note that a2 may be considered simply as an arbitrary vector that is defined in azyz, which itself is rotating at «9, with respect to OXYZ. By noting that a vec- tor has magnitude and direction, the first term on the right-hand side of Eq.
Another application of Eq. Equation b is more difficult to physically interpret than Eq. As sketched in. The intermediate frame is axyz, with unit vectors i, j,k. The frame amry2 translates and rotates with respect to the frame OXYZ. According to Eqs.
An exception to this remark occurs when the intermediate frame is rotating only and ris directed along the axis of rotation. Sey xr Ora ae a aeXr Ry , doy , deo ar PR ox 2 a aw a ae ee Here, again, we emphasize that the left-hand side and middle expressions in Eq. Equation has been introduced into Eq, simply as a matter of convenience. In applying the operational form in Bq. G does not affect it; the second and fourth terms do contain variables defined in aryz, s0 the operator in Eq, must be appropriately applied to them; and Eq.
Thus, Eq. The second term is the relative acceleration of P with respect to axyz. Ulimatey, in the study of hydraulic machines, the French engineer Gustave G. Coriolis in two papers , and more clearly in clesertwl the forces due to this acceleration component that now bears his name. The last term is called the centripetal acceleration.
In fact, Eqs. The reference frame OXYZ does not have to be stationary. Examples through will reinforce this idea; Section will extend it. Accelerations are not easy to visualize.
This is true for each of the terms in Bq. Of the five terms in the acceleration expression in Ea. The example that follows represents an attempt to assist in developing intuition about these two terms. It merits careful study. Anatomy of Centripetal and Cortolis Acceleration A turntable is rotating at a constant angular velocity wo about its fixed center C Figure An ant which we model as a point is walking along a radius of the turntable at a constant veldcity vp, relative to the turntable.
The ant is currently at point 1 and wants to get to the nearby point 2 although it does not know why it wants to get to point2. Also, we see that by the time the ant traverses Ar, point 2 is not where it was when the ant wanted to get there, but is at point 2'. Thus, the approximate acceleration of the ant with respect to the fixed reference frame is the difference between its velocity at points 2' and 1, divided by the time increment At.
Lanczos and the discussion of his Bq. Expanding Eq. The velocity vp has no effect 0 drag; 9 COUld just as well have been zero. Dividing both sides of Eq. Note that the effect of wo on vo Term 1 is always exactly the same as the effect of vo on wo Term ; or equivalently, the effect of w9 in changing the orientation of vy is exactly the same as the effect of vp in carrying wor to a different radius, changing its magnitude. Finally, although we shall not pursue such detailed study, we note that a better un- derstanding of the kinematics of many problems can be enhanced by studying comparably detailed vector sketches, Several examples will now be presented.
Each of the examples has one or more dis- tinctive features that are intended to amplify some aspect of the use of the kinematic re- sults derived in this chapter. The problems are arranged in order of increasing difficulty, and the reader will likely find the sum of these examples rewarding in future endeavors in kinematics. It can be shown that itis possible to mechanize many kinematics analyses.
The reader is, referred to Section , in which expressions for velocity and acceleration are derived when two intermediate reference frames are used.
The generalizations in Appendix B are presented for future study and are not encouraged during an initial encounter. Similarly, Eqs. Table should be reconsidered after the examples that follow have been studied. A small work piece labeled point P is moving with constant velocity vp along a channel, relative to the channel that is located as shown.
In the problem specification, we are given convenient reference frames. It should be em- phasized that the choice of such reference frames is an important part of the solution. Real objects are not generally encountered with a convenient set of reference axes painted on them. Often, by virtue of symmetry or the definition of the problem itself, some choices of reference frames are to be preferred.
For example, the fixed reference frame OXYZ and the intermediate reference frame axry2 attached to the platform are coincident at their origins. Also, note that the «axis is parallel to the channel. First, we must decide in which frame we want to write our kinematic variables.
By this underlined state- ment, we emphasize that the kinematic variables will be expressed or defined in ory2, and that the velocity and acceleration will be calculated with respect to OXYZ, All unam- biguous kinematic analyses must be accompanied by such a statement!
We must be clear regarding both the reference frame in which the motion is defined and the reference frame with respect to which the motion is calculated. Read this paragraph at least three times! Recalling Eqs. So, let us identify each of the terms in Eqs. There is no derivation here; we must simply observe the problem at hand and use our knowledge of vectors to write each term. Thatis, points o and O are coincident and remain coincident throughout the analysis.
Equations 4 denote the constant angular velocity, and therefore the zero value angular acceleration. In order to find dye and aa, we use Eqs. Now, we have defined all the terms in Eqs. The two terms on the right-hand side of Eq. Also, the centripetal acceleration has been. Itis important to appreciate the fact that although the answers in Eqs.
Indeed, that is precisely the meaning of Eqs. Upon an initial encounter, itis often thought that because the motion in Eqs. This is not the case, as we shall now explain. It is possible to express the answers in terms of the OXYZ unit vectors but—if it were necessary, which in general itis not—we would need more information to do so.
In general, we shall take the forms in Eqs. An arbitrary vec tor A is not affected by choice of reference 2. If we know A in terms of axyz, we know A! We may want to express A in terms of some other reference frame, but that desire or the knowledge itself does not change A. One final point: Now that we have the acceleration in Eq. Well, the sig- nificance of Eqs. For example—though we are jumping ahead of ourselves—in order to apply Newton's second law to the work piece in Figure , we shall need exactly the ac- celeration in Bq, h.
Its the acceleration in Eq, ht —and no other acceleration—for which force equals mass times acceleration. After reading Example , immediately take a peck at Bxample in Chapter 4. An antenna A weight 2 Ib is being deployed from the airplane. This solution also provides a straightfor- ward application of Bqs. Not really. It will be found in Chapter 4 that in situations where the kinematics are completely specified, as in the case here, such quantities as gravity or weight become significant only when we seek forces.
Finally, we emphasize that in this example, as in most of the following examples, the velocity and acceleration that have been found are valid only at the instant shown. At a later instant, the parameters in the problem will be different, requiring another calculation. A series of updated kinematics can be obtained either analytically or numerically by using an electronic computer.
We shall use the results of this example in Chapter 4. At the in- stant shown, a bird of mass m is on a leg of one of the Y's, which is oriented as indicated in Figure Relative to the Y, the bird is running with a velocity v9 and an acceleration ap, at the instant shown, At the same instant, a gust of wind exerts a force F,, on the bird in the X direction Find the acceleration of the bird, which may be modeled as.
Clearly, the acceleration sought is with respect to OXYZ; thisis the only acceleration which is of ultimate interest. BGT, ao Us a Mobie structure. Again, real bodies do not have convenient reference frames painted on them; the choice of reference frames is an important intellectual part of any analysis, Also, we are aware that the reference frames that have been defined are by no means the only nor perhaps even the best reference frames.
Inthis example, we have multiple rotating frames: rotates at ex; with respect to ground; om-rzyeze Totates at «wy with respect to oyziyi21, ard OXYZ is attached to ground. In this solution, we shall obtain the answer in two steps. Nevertheless, in each step we must be clear regarding the reference frame in which. Motion of bird defined in ony 22 with respect to o. Motion of bird defined in o,x1y with respect to OXYZ.
Once again, the state- ment of the bird's motion as written in the underlined statement is important and should leave no doubt about the intended use of the reference frames for the kinematics. In anticipation of using Eq. As noted in Example , here too we did not need to use the mass m of the bird, the effect of gravity, or the information about the gust of wind. So what; who said that one must use all the information that is provided or that enough information will always be provided?
Each of the links of the robot arm. Atthe instant shown, find the velocity and acceleration of the center of the work piece, labeled point C, as sketched in Figure Reference frame Bayye2? We shall successively apply Eqs. This is so because of the identical orientation of the respective axes in all three frames.
The consistent alignment of the respective axes in multiple rames isa convenient choice whenever the motion at a specified instant is sought. Consider the following important observation. The definition of w: and a 2 here should be contrasted with the definition of w2 in Example , Here w2 and «2 is defined with respect to OXYZ; in Example , «2 was defined with respect to 0, As always, we must be clear regarding the frame in which the motion is defined and the reference frame with respect to which the motion is calculated.
From the problem statement and in anticipation of using Eqs. Also, although it is not needed, we give the value of Ro to encourage the proper interpretation of the various terms for the frames defined in Figure and Figure Substituting Bg. From the problem statement and the results obtained as Eqs.
Substituting Eqs. We shall return to these results in Chapter 6, where we consider the dynamics of rigid bodies. The airplane is moving with velocity vy and acceleration ag, both defined with respect to the aircraft carrier. The airplane is also climbing with angular velocity « 9 and angular acceleration a3, both defined with respect to the aircraft carrier.
Again, we note that the difficult task of selecting a useful set of reference frames has been conducted for the reader. We note that, at the instant shown, i, j, K may be considered as the unit vectors along the respective x, y, z axes in each of the various reference frames.
In this example as in all kinematics examples, the solution technique here is one tech- nique of sequentially using Eqs. The introduction of step 2 into our solution here makes this analysis slightly different from the previous examples. Nevertheless, after completing this example, the reader should reconsider the significance of having introduced this intermediate step. Motion of P defined in os with respect to ox,y In preparation for using Eqs.
Motion of 01 defined in So, by use of Eq. Furthermore, although it would have been slightly inefficient to do so, we could have used Eqs: and to obtain the results in Eqs. In anticipation of the use of Eqs. And, from Figure ,. The use of Eqs. In Chapter 4, we shall return to these results in order to consider the forces on the pilot.
There is no such thing as the technique or method in a kinematic analysis. On the other hand, in Examples and , we suggested that when multiple intermediate reference frames are used, there may be a preferred technique based on the definitions of avg and og in those examples. Now, we explore the generalization of that idea. In this section, we investigate the expressions for velocity and acceleration in the case when two intermediate reference frames are used. Figure shows a point P that.
We consider two cases: the case in which the angular motions of o. Great care must be taken to identify correctly each term in Eqs. In the first step, OXYZ is the inner frame and o,2:y is the outer frame that corresponds to the intermediate frame of Figure and Eqs. Then, for direct use of Eqs. As we have just found in Eqs.
Example and all such problems could have been solved ina single step via Bags. Inthe first step, 0. Thus, in this manner, we successively apply Eqs. From Figure , or the direct use of Bas. Motion of potnt P defined In. Furthermore, it may be of in- terest to note that Eq, is identical to Bq.
B in Appendix B. This approach is not nec- essary see Problem at the end of this chapter , but itis, in general, the easier approach to conceptualize and to compute. Example and all such problems could have been solved in a single step via Bqs. Example could have been solved also in an analogous single step!
Their value lies in the fact that they represent complete expressions that may be implemented without further differentiation. How much time does the pi lot have to make a correction if the plane is to avoid fiying into the ground? Problem Sketch a graph that represents the acceleration of the object as a function of time. Make rough sketches of velocity versus time and accelera- tion versus time for this motion. Category Il: Intermediate Problem 3 Aparticle moves in a straight line with a constant acceleration a and initial velocity vp.
Given the same constant acceleration, what would be the stopping distance if the initial speed were 75 mishe? Problem Two cars travel along a straight section of highway at the same constant speed Vo. After a 0. The acceleration versus time graphs for both cars are sketched in Figures Pa and b, respectively. Geometrical constraints: Problems through Problem Explain how every point on the smaller disk in Figure can have the same velocity despite the fact that some points are farther from the axis of rotation than others.
Assuming the trip from location 1 back to location 1 takes seconds, find the average velocity and the average angular velocity of the automobile. Problem Explain intuitively why the answer is, 4in Example Problem Assume that the nonstipping disk in Figure rolls inside the stationary disk as sketched in Figure P How many complete ro- tations does the smaller disk make in one complete transit around the inside of the larger disk?
Problem Gears are important mechanisins for transferring rotational motion from one axis to an- other. Show that the ratio of the angular velocity of a pair of gears is the reciprocal of the ratio of their radii called the gear ratio.
Use the result to deter- mine the speed at which the weight W in the system sketched in Figure P is being lifted when the mo- tor turns at 50 revis.
Kinematically, the difference in times is caused by the use of two different reference frames with respect to which the Earth's rotation is measured. Such a difference is illustrated in Figure P Given that a solar year is approximately solar days and that a solar day is defined to be 24 hours, determine the length of a sidereal day. Also, how many sidereal days are there in a solar year? Shaft AB is mounted on a cone-shaped base of conical angle Fin De argu aceon a ess Goth spect to ground.
The rod is in a horizontal plane and the disk isin a vertical plane. The wheel is rolling without slip on the horizontal supporting sur- face, as sketched in Figure P Find the angular velocity and angular acceleration of the wheel with respect to ground Use of Eqs.
To encourage this, approach, a set of potentially useful reference frames has been provided in each problem. Of course, read- ers may choose any other reference frames they pre- fer, but such frames must be clearly defined.
Problem A boy is spinning a bucket of water, such that the bucket moves in a circle of radius R in the vertical plane at a constant speed v, as sketched in Figure P Note that gravity acts. Determine the acceleration of the bucket with respect to ground at locations A and B; that is, the highest and lowest points of the path.
Problem An amusement park ride consists of swinging arms only two of which are shown pinned to a rotating column, as sketched in Figure P Bach swinging arm has an effective length of 50 ft.
At the instant shown, the columns rotating at 0. Note that gravity aets. If in a particular test, starting from rest, the driving motor supplies a con- stant angular acceleration of 0.
Intermsof andits derivatives, find thevelocity andaccelerationofpoints4,B,andContheladder with respectto ground asit falls while remaining in contact with the wall and the floor. Problem A commuter train is moving toward the right along a straight rail at the speed w and accel- eration ap.
A boy in the train runs toward the forward end of the car, along the center aisle at the speed and acceleration a , both with respect to the train, as sketched in Figure P Find the velocity and accel- eration of the boy with respect to ground. Anywhere on the other bank but should take the minimum time to cross the river.
For each of the four destinations, determine the time required to complete the crossing. Problem Rotor blades ofa helicopter are of ra- dius 5. Problem A test chamber sketched in Figure 3. A passenger's head of mass m is located at point, P. At the instant shown, the chamber is moving out- ward at speed 9 and acceleration ap, with respect to the rotating base, which is rotating at angular veloc- ity w and angular acceleration j, both with respect to ground.
Find the velocity and acceleration of the passenger's head with respect to ground. Problem A skater is spinning on the ice about the vertical axis of her body with angular velocity , with respect to ground. At the instant shown, her arms are nearly fully extended, and she is moving her hands radically in- ward toward her body. At the instant shown, each ofher arms extends to a radius of about the axis of her body.
Find the velocity and acceleration of each weight with respect to ground. Problem A sportsman is skeet shooting. As sketched in Figure P, as the shot of mass 7m is exiting the barrel of the rife, the sportsman is rotat- ing the rifle about a vertical axis that passes through point O, at an angular velocity 6 and angular acceler- ation 0, both with respect to ground.
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