Fundamentals of applied dynamics williams pdf free download

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Fundamentals of Applied Dynamics. 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. The exit veloc- ity and acceleration of the shot relative to the barrel are and F, respectively.

Find the velocity and acceleration of the shot in the barrel with respect to ground when itis a distance r from point O. In Problems through , a set of potentially useful reference frames has been provided.

It may not be necessary to use all the reference frames provided. Alterna- tively, you may choose any other reference frames you prefer, but they must be clearly defined. The mass of each passenger is 75 kg, Note that gravity acts. Link AB is rotating at w1, with respect to the floor, and link BC is rotating at w and dp, both with respect to link AB.

A delicate gem of mass m is being transported; the gems geometric dimensions are negligible compared with! Find the velocity and acceleration of the gem. Problem A carnival ride consists of two plat- forms as sketched in Figure P, The smaller platform of radius rotates at w, with respect to a larger platform of radius R which rotates at «; with respect to ground.

A rider P of mass m is located along the X axis, at the instant shown. Note that grav- ity acts. Find the velocity and acceleration of rider P. Problem A carnival ride consists of two plat- forms of radii r and R rotating at angular velocities ; and wa, respectively, as sketched in Figure P Find the veloc- ity and acceleration of rider P.

A small disk rotates about Q at a constant angular velocity w with respect to the large disk. A point P is located on the perimeter of the small disk such that line PQ is normal to line OQ, as sketched in Figure P, Find the relation between and 0 for which the acceleration of point P would be parallel to line Problem A ride at an amusement park is sketched in Figure P The cockpit.

Point A, which corresponds to an eye of a passenger, is located 0. For the instant shown, find the velocity and acceleration af point A. The pitch curves at a radius,of curvature of ft if viewed from directly above the field, as sketched in Figure P3. Find the acceleration ofa pine tar glob, which weighs 0. Find the velocity and acceleration of point P. The reference frames defined in Figure P are as follows: OXZ is fixed to ground; Oxyyiz1 is fixed to the translating cart and does not rotate; and Oxzy22 is fixed to the pendulum.

At the instant sketched in Figure P, the spoke is in the Z direction and the bead is located at 0. Find the velocity and acceleration of the bead at the instant shown. Arm BC. The lengths of arms AB and BC are 10 mand 3 m, respectively. Find the velocity and ac- celeration of the payload M. Problem An astronaut A weighs W and is on a platform that is rotating at an angular velocity « with respect to ground.

Also, he is walking on a ta- ble which, at the instant shown in Figure P, is in the plane of the platform. His velocity and accel- eration are vp and ao, respectively, both with respect to the rotating table and in a direction that forms an angle 4 with respect to the rotating axis of the plat- form.

The rotating table has an angular velocity wp with respect to the platform. At the instant shown, the astronaut is a radial distance b from the axis of the rotation of the table.

The book is noteworthy in covering both lagrangian dynamics and vibration analysis. The principles covered are relatively few and easy to articulate; the examples are rich and broad. Summary tables, often in the form of flowcharts, appear throughout. End-of-chapter problems begin at an elementary level and become increasingly difficult.

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