Sandor, Arthur G. Erdman-Advanced Mechanism Design_ Analysis and Synthesis Vol. II () Waldron, Kenneth J. & Gary L. Kinzel - Kinematics, Dynamics and Design of Machinery ()(terney.info) Solution Manual - Machines & Mechanism, 4th Ed.; David terney.info Sandor, arthur g. erdman advanced mechanism design analysis and synthesis vol. ii (). 1. Library Congress Cataloging in Publication. Mechanism Design: Analysis and Synthesis (4th Edition) [Arthur G. Erdman, ANALYSIS AND SYNTHESIS OF MECHANISMS. chpt terney.info chpt 2 - part 1.
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Mechanism Design: Volume 1: Analysis and Synthesis: Vol. 1. Mechanism Design Analysis and Synthesis Forth Edition - Ebook download as PDF File .pdf) . Thank you very much for downloading mechanism design analysis synthesis solution manual. Maybe you have knowledge that, people have look numerous. suitable adjustments compared to cams and geared mechanisms. But at the same time it is usually difficult to change the dimensions of.
Enol silyl ethers with additional common functional groups, such as chlorine, ether, ester, imide and nitrile, were compatible with this protocol and could be converted into the corresponding alkylated enol silyl ethers 3v—3z with equally high efficiency. When the ethyl t-butyl ketone-derived enol silyl ether was subjected to the optimized conditions, the C—C bond formation occurred at the terminal, primary allylic carbon to yield 3Aa. On the other hand, 2-methylcyclohexanone-derived enol silyl ether was alkylated predominantly on the secondary allylic carbon to give 3Ab.
An additional noteworthy aspect of this hybrid catalytic system was that it accommodated complex substrate setting; the allylic alkylation of the estrone derivative featuring a fused polycyclic framework delivered the enol silyl ether 3Ac in excellent yield. Isolated yields are shown. Diastereomeric ratios d. The pKa values of the corresponding radical cations in MeCN were calculated to be 8.
According to the frontier orbital theory, the reactivity of a radical generally depends on both the SOMO level and spin density at the reactive site. The importance of spin localization for ensuring productive bond formation was also corroborated by the similar analysis with 1r in comparison to the outcome with 1s.
This salient structural feature enables diverse transformations to access a variety of complex carbonyl compounds, providing a powerful demonstration of the utility of this method Fig.
On the other hand, selective functionalization of the bis sulfonyl methyl moiety of 3i was feasible by the palladium-catalyzed allylation with allyl carbonate, affording 5b in high yield. Subsequent exposure of 5b to magnesium metal in MeOH effected desulfonylation to give 5c. The malononitrile subunit of the enol silyl ether 3Ab could be selectively converted into imide by reaction with 2-oxazolidone under oxygen atmosphere The resulting 5d underwent intramolecular arylation under palladium catalysis to furnish the fused tricyclic ketone 5e, of which structure is often found in diterpenoids, such as taiwaniaquinol A Under the hybrid catalysis, a series of enol silyl ethers smoothly react with electron-deficient olefins to give the corresponding functionalized enol silyl ethers.
This operationally simple protocol, in concert with the ready availability of enol silyl ethers and their conventional polar reactivity, provides rapid and reliable access to an array of complex carbonyl compounds and will find widespread use among practitioners of organic synthesis. Methods Representative procedure for allylic alkylation To a flame-dried test tube were added 2a The reaction tube was sealed with a rubber septum and then evacuated in vacuo and backfilled with Ar five times.
The slider-crank of Fig. To avoid this, we need to seek another mlkl pair. The question arises: Thus it is seen that there is only one slider dyad, which agrees with the fact that there is only one Ball's point for a set of four ISP, and thus only one finite Ball's point for a set of four FSP. However, the number of pinned dyads is infinite. Thus there will be a single infinity ofsolutions for the four-position motion generator slider-crank.
Each of these has a cognate see Sec. Observe that coupler rotations aj of the first mechanism are the crank rotat ions in the cognate. S is arbitrary, because it determines only the scale and orienta- tion of the linkage. Expanding 3. The Concurrency Point Equation 3. This means that both Z and W rotate with the moving body st, Since W rotates about the unknown fixed center point m, while Z is embedded in and moves with 7T', and because Z and W concur at the unknown circle point k-: Since the tip of Wand the This is illustrated in Fig.
Along with body rotations a. The unknown vectors Z and Ware always parallel, and they are of infinite length. The unknown circle point k 1 is at infinity in the unknown direction of Z and W in the first position.
The effect of the infinite lengths of W and Z is this: The connection between the unknown center point m and the moving body 7T becomes a turn-slide. This enforces the line of the unknown vector D, which is embedded in 7T, to turn and slide through m.
Hence m is a finite concurrency point. See also Chap. Therefore, we distinguish it with the superscript c, and do likewise for Z and W. The other half is one of a single infinity of pivoted dyads see Fig. Observe that, while the rotations of both D and E are the prescribed aj, E does not change in length, but D does, by the unknown stretch ratios Pj.
We now assume an arbitrary real value for P2, separate the real and imaginary parts of Eq. It is to be noted that the concurrency point mC thus found is unique: It is the unique slider point or finite Ball's point for the inverted motion in which the fixed plane of reference and the moving plane 7T exchange roles.
Thus it is seen that regardless of the arbitrary choice for the value of P2, the end result is the same: There is only one turn-slide dyad that can guide 7T through the four prescribed finite positions. To complete a single-degree-of-freedom four-link mechanism we find a pivoted dyad one of the infinite number available for four finite positions , thus obtaining the mechanism shown in Fig. Furthermore, if design conditions make it desirable to change it to a tumbling block mechanism Fig.
Thanks to Burmester and those who continued his work, we know that there are ideally an infinity of pivoted dyads for any four arbitrarily prescribed positions, and that any two of these can form a four-bar mechanism whose coupler will match these prescribed positions. But can we synthesize four-bar motion generators for five arbitrary positions?
Our first hint toward the answer to this question came when the tabular formula- tion was developed Table 2. Although the table shows that there are no free choices for five prescribed positions, the number of real equations and the number of real unknowns are equal, indicating that these equations can be solved. The second hint will come from further examination of the four-position case and the resulting center- and circle-point curves.
Suppose, as in Fig. In addition, a new set of curves for the same first three positions plus a fifth position Richardson . Two existing five-point dyads are drawn in.
If these curves intersect, a common solution exists and a Burmester pair or dyad has been found that will be able to guide a plane through all five prescribed positions. The circle- and center-point curves can be shown to be cubics [,], so there are a maximum of nine intersections. There are two imaginary intersections at infinity and, discounting the intersections marking the poles Pt 2, Pt 3, and P23, there is a maximum number of four usable real intersections.
Let us see whether this geometric concept can be verified by mathematical methods. Referring again to Fig. Five Positions Li nk Length s: Input Link L,: Ao X,Y - 0. The resulting four-bar is shown in its first position. For four-bar notation, see Fig.
Other precision points are signified by P; and the prescribed angles by line Pja]. For system 3. Thus there are two compatibility equations to be satisfied simulta- neously for five prescribed positions: Thus there are no free choices here Table 2. Then there are up to four usable real intersections of the Burmester curves. This means that there will be up to four dyads that can be used to construct motion generators for the five prescribed positions, for which Z and W can then be found from any two equations of the system of Eq.
The complex conjugates" of these compatibility equations also hold true: Notice that 3. This is begun by multiplying Eqs.
Since these equations have zeros on the right-hand side, the determinant of the coefficients must be zero for the system to yield simultaneous solutions for the four "unknowns. Thus the expansion shows that E is real, so that its imaginary part vanishes identically. Therefore, only the real part of the eliminant is of interest. By way of trigonometric identities, Eq.
Thus Eq. Also, from the determinant form of Eqs. The example in Figs. In general, combining each of four BPPs with every other, we can obtain up to six different such four-bar motion-generator linkages.
See Appendix A3. What about path generation with prescribed timing and function genera- tion with the four-bar? Also, can this theory be extended to other linkage types? Chapter 2 demonstrated that the dyadic standard form equation, Eq. The Roberts-Chebyshev theorem will add more insight to the broad applicability of the dyad form and of the Burmester theory.
Roberts-Chebyshev Theorem An extremely useful property of planar four-bar linkages is revealed in the Roberts- Chebyshev theorem , which states that one point of each of three different but related planar four-bar linkages will trace identical coupler curves.
This means that there will be two additional four-bar linkages associated with each "parent" four- bar linkage which will trace the same path as the parent although the coupler rotations will not be the same.
These two additional linkages are called "Roberts-Chebyshev cognates" after their two independent English and Russian discoverers. We can form these cognates geometrically by building on the four-bar linkage shown in full lines in Fig. Find the third fixed point, C, of the Roberts configuration by making triangle The dashed and dot-dashed four-bar mechanisms are the cognates of the basic one. Find one cognate coupler triangle by making I: GPH similar to I: HC is the follower link of the dashed "right cognate" of Fig.
Find the other cognate by making coupler triangle FIP similar to l: Note that ICHP is a parallelogram. Due to the three parallelograms that concur at P, Fig. This property, that every four-bar linkage has two cognate linkages which trace the same path as the parent four-bar, is extremely useful to designers. The cognates are different linkages, even though they share one ground-pivot location with one another. A designer may find that although a particular linkage may trace a desired There are, however, two cognates available which, while they trace the same path, in general will display different kinematic and dynamic characteristics.
It should be mentioned here that cognates are not equivalent linkages. Equivalent linkages are usually employed to duplicate instantaneously the position, velocity, and perhaps acceleration of a direct-contact higher-pair mechanism such as a cam or a noncircular gear by a linkage say, a four-bar.
The dimensions of equivalent linkages are different at various positions of such higher-pair parent mecha- nisms, whereas the link lengths of cognates remain the same for any position of the parent linkage.
Other properties of cognates include the one developed by Cayley : The common coupler-path tracer point and the three instant centers of the three concurrent couplers with respect to ground are collinear at all times and the line containing these points is normal to the coupler curve in every position of the linkage system see points IC! Another observation is: Each grounded link of any of the three FBLs will exhibit the same angular rotations and will rotate at the same angular velocity as one of the grounded links of one of its cognate FBLs and the coupler link of its other cognate FBL, as shown in Fig.
Still another noteworthy fact is that, if the parent linkage and its two cognates were pinned together to form a movable lO-bar linkage, Gruebler's equation Sec. This is an example of an overdosed linkage that has mobility due only to its special geometric properties. Yet another property of the Roberts-Chebyshev configuration is this: In addition to the four rigid similar triangles the three coupler triangles and the triangle formed by the three ground pivots there are also three variable- size triangles, all of which remain always similar to the coupler triangles in the course of motion of the linkage.
These are: The proof may be started as shown in Fig. Proceed similarly with C with respect to m 1. In the resulting stretched-out configuration, in which all link lengths have retained their original lengths, the above-mentioned seven triangles are all clearly similar. The rest of the proof is left to the reader as an exercise. Move C' and m 2 ' toward m , keeping their triangle similar. Four-bar linkages are not the only linkages that have cognates.
The slider- crank a special case of a four-bar ; see Fig. A complex-number proof of the existence of the four-bar cognates, using complex numbers and appropriate rotational operators, can be based on Fig. This is left to the reader as an exercise. For further develop- ment of the above-mentioned properties of cognates and a historical note, refer to Ref.
By employing the Roberts-Chebyshev theorem, path generators with prescribed timing may be obtained from motion-generator four bars. Let us look again at the geometric cognates of Fig. Suppose that the parent four-bar mlktPkim2 is a motion generator that has been synthesized by either the four- or five-precision-point technique.
The rotations of the coupler link aj and the displacements of tracer point P have been prescribed.
According to the Roberts-Chebyshev development, all three cognates trace the identical coupler curves with their common tracer point. What do the individual links of the two other cognates rotate by? Since the originally prescribed rotations aj have been trans- ferred to the grounded links in the cognates, the cognates of a motion generator are path generators with prescribed timing. For every four-bar motion generator there will be two such four-bar path generators.
This development may be utilized to simplify both the four- and five-precision-point synthesis methods, so that the synthesis equations need only be solved once for both tasks: In the second case, the cognates may also be derived via the computer from the parent motion generator. In the five-precision-point case, how many path generators with prescribed timing might we expect?
Since there are either 0, 2, or 4 real roots of the quartic [Eq, 3. Correlation of this procedure with the Roberts-Chebyshev configuration can be observed by referring to points m -, m2 , F and P in both Figs. How many function generators might we expect from the five-precision-point case? One might initially guess that there will be a maximum of 12 function generators if all the roots of the quartic Eq, 3.
Advanced Topics Figure 3. With these, the standard form of the dyadic synthesis equation becomes: This, however, is not the case. First, only half of the four-bar path generator is required to form a function generator, the dyad m 1FP, Figs. Furthermore, since there are only four different dyads that make up the 12 path generators, there are only four different dyad solutions available. Also, there will always be one trivial solution since a circular are, centered at m2 , is being specified as the path of P.
Table 3. TABLE 3. Number of different four bar solutions expected Number of Path generation Function real roots I Motion with generation of the quartic generation prescribed timing See Fig. Some other useful linkages, with more than four links, can be synthesized from these same dyads using simple construction procedures. Suppose that one wishes to obtain a path generator with prescribed timing directly without computing the cognate of the motion generator.
Perhaps the motion generator ground-pivot locations are acceptable but the cognates exhibit an undesirable ground- pivot location. Then either the geared five-bar or parallelogram six-bar path generator with prescribed timing may be useful.
Referring to Fig. Disregarding the parent four-bar, connect the grounded links Z I and Z2 with each other by means of one-to-one gearing using an idler to assure that ZI and Z2 perform identical rotations. Thus a single-degree-of-freedom geared five- bar linkage m I FPGm 2 is obtained which will trace the prescribed path of P with corresponding prescribed input-crank rotations Uj.
For each motion generator there p I Z2 z' I irz: I gear ratio between ZI and Z2 of this single-degree-of-freedom geared five-bar assures that point P, the joint of WI and W2, will trace the same path as the parent linkage, the four-bar motion generator shown in dashed lines.
I velocity ratio between links Zl and Z' of the five-bar path generator. See Fig. Another way to design this linkage is shown in Fig. There is yet another way for converting the two-degree-of-freedom five-bar of Fig. The same objective can be accomplished by adding a parallelogram linkage to the five-bar as shown in Fig.
Notice that the parallelogram connected to Zl and Z2 is not unique. In fact, two parallelograms may be connected together to avoid the dead-center problem Fig.
This is another overdosed linkage whose mobility is assured by its link proportions. The five-bar m 1FPGm2 may be connected by gears of other than I: Six-Bar Parallel Motion Generator An extremely useful linkage is one that will trace a coupler curve while the coupler link undergoes no rotation-a parallel motion curvilinear translation generator. One can easily observe that this is an inappropriate task for a four-bar linkage except in the trivial case of a circular coupler curve of a parallelogram linkage.
The following extension of the Roberts-Chebyshev construction yields a six-bar linkage with one link performing curvilinear translation. P and P' describe identical paths.
If link m2' G' is added, a seven-bar overdosed parallel-motion generator with prescribed timing results because link m 2'G' performs the prescribed rotations aj. Next, draw one of its cognates, say the right dashed cognate m 2GPHC.
Cycloidal-crank and geared linkages are also included. Chapter 5 is a comprehensive treatment of the dynamics of mechanisms. It covers matrix methods, the Lagrangian approach, free and damped vibrations, vibration isolation, rigid-rotor balancing, and linkage balancing for shaking- forces and shaking-moments, all with reference to computer programs.
Also covered is an introduction to kineto-e1astodynamics KED , the study of high-speed mecha- nisms in which the customary rigid-link assumption must be relaxed to account for stresses and strains in elastic links due to inertial forces. Rigid-body kinematics and dynamics are combined with elastic finite-element techniques to help solve this complex problem.
The final chapter of Volume 2, Chapter 6, covers displacement, velocity, and acceleration analysis of three-dimensional spatial mechanisms, including robot manipulators, using matrix methods. It contains an easily teachable, visualizable treatment of Euler-angle rotations. The chapter, and with it the book, closes with an introduction to some of the tools and their applications of spatial kinematic synthe- sis, illustrated by examples.
In viewof the ABET accreditation requirements for increasing the design content of the mechanical engineering curriculum, these books provide an excellent vehicle for studying mechanisms from the design perspective. Many computer programs are either included in the texts as flow charts with example input-output listings or are available through the authors. The complex-number approach in this book is used as the basis for interactive computer programs that utilize graphical output and CRT display terminals.
The designer , without the need for studying the underlying theory, can interface with the computer on a graphics screen and explore literally thousands of possible alterna- tives in search of an optimal solution to a design problem. Thus, while the burden of computation is delegated to the computer, the designer remains in the "loop" at each stage where decisions based on human judgment need to be made. Among xii the latter are Dr. Robert Williams Spatial Mechanisms , and Dr.
Donald R. Riley, who taught from the preliminary versions of the texts and offered numerous suggestions for improvements. Sanjay G.