Dynamics of Mechanical Systems 2009 Part 12 pptx

532 Dynamics of Mechanical Systems Here we see that both the resultant primary and secondary inertia forces are also balanced, leaving the only unbalance with the resultant primary moments. Thus, we have still further improvement in the balance. These examples demonstrate the wide range of possibilities available to the engine designer; however, the examples are not meant to be exhaustive. Many other practical configurations are possible. The examples simply show that the crankshaft configuration can have a significant effect upon the engine balance. In the following section, we will extend these concepts and analyses to eight-cylinder engines. 15.10 Eight-Cylinder Engines: The Straight-Eight and the V-8 If we consider engines with eight cylinders the number of options for balancing increases dramatically. The analysis procedure, however, is the same as in the foregoing section. To illustrate the balancing procedure, consider an engine with eight cylinders in a line (the straight-eight engine) and with crank angles φ i for the connecting rods arranged incremen- tally at 90° along the shaft. Table 15.10.1 provides the listing of the φ i for the eight cylinders (i = 1,…, 8) together with the trigonometric functions needed to test for balancing. A glance at the table immediately shows that the engine with this incremental 90° crank angle sequence has a moment unbalance. This raises the question as to whether there are crank angle configurations where com- plete balancing would occur, within the approximations of our analysis. To respond to this question, consider again the four-cylinder engine of the previous section. For the crank angle configuration of Table 15.9.5, we found that the four-cylinder engine was balanced except for the primary moment. Hence, it appears that we could balance the eight-cylinder engine by considering it as two four-cylinder engines having reversed crank angle config- urations. Specifically, for the four-cylinder engine of Table 15.9.5, the crank angles are 0, 90, 270, and 180°; therefore, let the first four cylinders have the crank angles 0, 90, 270, and 180°, and let the second set of four cylinders have the reverse crank angle sequence of 180, 270, 90, and 0°. Table 15.10.2 provides a listing of the crank angles φ; together with the trigonometric functions needed to test for balancing. As desired (and expected), the engine is balanced for both primary and secondary forces and moments. TABLE 15.10.1 Engine Balance: Listing of Terms of Eqs. (15.7.8) to (15.7.11) for the Eight-Cylinder Uniformly Ascending Crank Angle Configuration i φφ φφ i (°) cosφφ φφ i sinφφ φφ i cos2φφ φφ i sin2φφ φφ i (i – 1)cosφφ φφ i (i – 1)sinφφ φφ i (i – 1)cos2φφ φφ i (i – 1)sin2φφ φφ i 1010 10 0 0 0 0 2900 1 –10 0 1 –10 3 180 –10 10 –20 20 4 270 0 –1 –10 0 –3 –30 5 360 1 0 1 0 4 0 4 0 6 450 0 1 –10 0 5 –50 7 540 –10 10 –60 60 8 630 0 –1 –10 0 –7 –70 Totals 0 0 0 0 –4 –4 –40 0593_C15_fm Page 532 Tuesday, May 7, 2002 7:05 AM Balancing 533 A practical difficulty with a straight-eight engine, however, is that it is often too long to conveniently fit into a vehicle engine compartment. One approach to solving this problem is to divide the engine into two parts, into a V-type engine as depicted in Figure 15.10.1. The two sides of the engine are called banks, each containing four cylinders. Because the total number of cylinders is eight, the engine configuration is commonly referred to as a V-8. A disadvantage of this engine configuration, however, is that the engine is no longer in balance, as compared to the straight-eight engine: To see this, consider again the crank configuration of the straight-eight as listed in Table 15.10.2. Taken by themselves, the first four cylinders are unbalanced with an unbalanced primary moment perpendicular to the plane of the cylinders as seen in Table 15.9.6; hence, the second set of four cylinders has an unbalanced primary moment perpendicular to its plane. With the cylinder planes themselves being perpendicular, these unbalanced moments no longer cancel but instead have a vertical resultant as represented in Figure 15.10.2. This unbalance will have a tendency to cause the engine to oscillate in a yaw mode relative to the engine compartment. This yawing, however, can often be kept small by the use of motor mounts having high damping characteristics. Thus, the moment unbalance is usually an acceptable tradeoff in exchange for obtaining a more compact engine. TABLE 15.10.2 Engine Balance: Listing of Terms of Eqs. (15.7.8) to (15.7.11) for the Eight-Cylinder Crank Angle Configuration from Table 15.9.6. i φφ φφ i (°) cosφφ φφ i sinφφ φφ i cos2φφ φφ i sin2φφ φφ i (i – 1)cosφφ φφ i (i – 1)sinφφ φφ i (i – 1)cos2φφ φφ i (i – 1)sin2φφ φφ i 101010 0 0 0 0 2900 1 –10 0 1 –10 3 270 0 –1 –10 0 –2 –20 4 180 –10 10 –30 30 5 180 –10 10 –40 40 6 270 0 –1 –10 0 –5 –50 7900 1 –10 0 6 –60 801010 7 0 7 0 Totals 0 0 0 0 0 0 0 0 FIGURE 15.10.1 A V-type engine. FIGURE 15.10.2 Unbalance primary moments and their resultants. (Resultant) 0593_C15_fm Page 533 Tuesday, May 7, 2002 7:05 AM 534 Dynamics of Mechanical Systems 15.11 Closure Our analysis shows that if a system is out of balance it can create undesirable forces at the bearings and supports. If the balance is relatively small, it can often be significantly reduced or even eliminated by judicious placing of balancing weights. Perhaps the most widespread application of balancing principles is with the balancing of internal-combustion engines and with similar large systems. Because such systems have a number of moving parts, complete balance is generally not possible. Designers of such systems usually attempt to minimize the unbalance while at the same time making com- promises or tradeoffs with other design objectives. We saw an example of such a tradeoff in the balancing of an eight-cylinder engine: the engine could be approximately balanced if the cylinders were all in a line. This arrange- ment, however, creates a relatively long engine, not practical for many engine compart- ments. An alternative is to divide the engine into two banks of four cylinders, inclined relative to each other (the V-8 engine); however, the engine is then out of balance in yaw moments, requiring damping at the engine mounts to reduce harmful vibration. Optimal design of large engines thus generally involves a number of issues that must be resolved for each individual machine. While there are no specific procedures for such optimal design, the procedures outlined herein, together with information available in the references, should enable designers and analysts to reach toward optimal design objectives. References 15.1. Wilson, C. E., Sadler, J. P., and Michaels, W. J., Kinematics and Dynamics of Machinery, Harper & Row, New York, 1983, pp. 609–632. 15.2. Wowk, V., Machine Vibrations, McGraw-Hill, New York, 1991, pp. 128–134. 15.3. Paul, B., Kinematics and Dynamics of Planar Machinery, Prentice Hall, Englewood Cliffs, NJ, 1979, chap. 13. 15.4. Mabie, H. H., and Reinholtz, C. F., Mechanisms and Dynamics of Machinery, Wiley, New York, 1987, chap. 10. 15.5. Sneck, H. J., Machine Dynamics, Prentice Hall, Englewood Cliffs, NJ, 1991, pp. 211–227. 15.6. Swight, H. B., Tables of Integrals and Other Mathematical Data, Macmillan, New York, 1057, p. 1. 15.7. Shigley, J. E., and Uicker, J. J., Jr., Theory of Machines and Mechanisms, McGraw-Hill, New York, 1980, p. 499. 15.8. Martin, G. H., Kinematics and Dynamics of Machines, McGraw-Hill, New York, 1982, p. 419. 15.9. Taylor, C. F., The Internal-Combustion Engine in Theory and Practice, Vol. II: Combustion, Fuels, Materials, Design, MIT Press, Cambridge, MA, 1985, pp. 240–305. Problems Section 15.2 Static Balancing P15.2.1: A 125-lb flywheel in the form of a thin circular disk with radius 1.0 ft and thickness 1.0 in. is mounted on a light (low-weight) shaft which in turn is supported by nearly 0593_C15_fm Page 534 Tuesday, May 7, 2002 7:05 AM Balancing 535 frictionless bearings as represented in Figure P15.2.1. If the flywheel is mounted off-center by 0.25 in., what weight should be placed on the flywheel rim, opposite to the off-center offset, so that the flywheel is statically balanced? P15.2.2: Repeat Problem P15.2.1 if the flywheel mass is 50 kg, with a radius of 30 cm, a thickness of 2.5 cm, and an off-center mounting of 7 mm. P15.2.3: See Problem P15.2.1. Suppose the flywheel shaft has frictionless bearings. What would be the period of small oscillations? P15.2.4: Repeat Problem P15.2.3 for the data of Problem P15.2.2. P15.2.5: See Problem P15.2.3. Suppose a slightly unbalanced disk flywheel, supported in a light shaft with frictionless bearings, is found to oscillate about a static equilibrium position with a period of 7 sec. How far is the flywheel mass center displaced from the shaft axis? Section 15.3 Dynamic Balancing P15.3.1: A shaft with radius r of 3 in. is rotating with angular speed Ω of 1300 rpm. Particles P 1 and P 2 , each with weight w of 2 oz. each, are placed on the surface of the shaft as shown in Figure P15.3.1. If P 1 and P 2 are separated axially by a distance ᐉ of 12 in., determine the magnitude of the dynamic unbalance. P15.3.2: Repeat Problem P15.3.1 if r, w, and ᐉ have the values r = 7 cm, w = 50 g, and ᐉ = 0.333 m. P15.3.3: See Problem P15.3.1. Suppose the dynamically unbalanced shaft is made of steel with a density of 489 lb/ft 3 . Suppose further that it is proposed that the shaft be balanced by removing material by drilling short holes on opposite sides of the shaft. Discuss the feasibility of this suggestion. Specifically, if the holes are to be no more than 0.5 in. in diameter, no more than 0.5 in. deep, and separated axially by no more than 18 in., suggest a drilling procedure to balance the shaft. That is, suggest the number, size, and positioning of the holes. P15.3.4: Repeat Problem P15.3.3 for the shaft unbalance of Problem P15.3.2 if the mass density of the shaft is 7800 kg/m 3 . FIGURE P15.2.1 A flywheel on a light shaft in nearly frictionless bearings. FIGURE P15.3.1 A rotating shaft with unbalance particles. 1 in. Flywheel (125 lb) 12 in. P P Ω ᐉ 1 2 0593_C15_fm Page 535 Tuesday, May 7, 2002 7:05 AM 536 Dynamics of Mechanical Systems Section 15.4 Dynamic Balancing: Arbitrarily Shaped Rotating Bodies P15.4.1: Suppose n 1 , n 2 , and n 3 are mutually perpendicular unit vectors fixed in a body B, and suppose that B is intended to be rotated with a constant speed Ω about an axis X which passes through the mass center G of B and which is parallel to n 1 . Let n 1 , n 2 , and n 3 be nearly parallel to principal inertia directions of B for G so that the components I ij of the inertia dyadic of B for G relative to n 1 , n 2 , and n 3 are: Show that with this configuration and inertia dyadic that B is dynamically out of balance. Next, suppose we intend to balance B by the addition of two 12-oz. weights P and placed opposite one another about the mass center G. Determine the coordinates of P and relative to the X-, Y-, and Z-axes with origin at G and parallel to n 1 , n 2 , and n 3 . P15.4.2: Repeat Problem P15.4.1 if the inertia dyadic components are: and if the masses of P and are each 0.5 kg. P15.4.3: Repeat Problems P15.4.1 and P15.4.2 if B is rotating about the Z-axis instead of the X-axis. Section 15.5 Balancing Reciprocating Machines P15.5.1: Suppose the crank AB of a simple slider/crank mechanism (see Figures 15.5.1 and P15.5.1, below) is modeled as a rod with length of 4 in. and weight of 2 lb. At what distance away from A should a weight of 4 lb be placed to balance AB? P15.5.2: See Problem P15.5.1. Suppose is to be 1.5 in. What should be the weight of the balancing mass ? P15.5.3: Repeat Problem P15.5.1 if rod AB has length 10 cm and mass 1 kg. P15.5.4: Consider again the simple slider/crank mechanism as in Figure P15.5.4, this time with an objective of eliminating or reducing the primary unbalancing force as developed in Eq. (15.5.23). Specifically, let the length r of the crank arm be 4 in., the length ᐉ of the connecting rod be 9 in., the weight of the piston C be 3.5 lb, and the angular speed w of FIGURE P15.5.1 A simple slider crank mechanism. I ij = − −− −           18 01 025 01 12 015 0 25 0 15 6 slug ft 2 ˆ P ˆ P I ij = −− − −           30 02 03 02 20 025 03 025 10 kg m 2 ˆ P ˆ r C r m A B ˆ ˆ B ˆ r ˆ m B 0593_C15_fm Page 536 Tuesday, May 7, 2002 7:05 AM Balancing 537 the crank be 1000 rpm. Determine the weight (or mass ) of the balancing weight to eliminate the primary unbalance if the distance h of the weight from the crank axis is 3 in. Also, determine the maximum secondary unbalance that remains and the maximum unbalance in the Y-direction created by the balancing weight. P15.5.5: Repeat Problem P15.5.4 if the balancing weight compensates for only (a) 1/2 and (b) 2/3 of the primary unbalance. P15.5.6: Repeat Problems P15.5.4 and P15.5.5 if r is 10 cm, ᐉ is 25 cm, h is 5 cm, and the mass of C is 2 kg. Section 15.6 Lanchester Balancing P15.6.1: Verify Eqs. (15.6.1), (15.6.2), and (15.6.3). P15.6.2: Observe that the geometric parameters of Eq. (15.6.5) are such that the Lanchester balancing mass m ᐉ need only be a fraction of the piston mass m C . Specifically, suppose that (r/ᐉ) is 0.5 and (ξ/r) is also 0.5. What, then, is the mass ratio m ᐉ /m C ? Sections 15.7, 15.8, 15.9 Balancing Multicylinder Engines P15.7.1: Consider a three-cylinder, four-stroke engine. Following the procedures outlined in Sections 15.7, 15.8, and 15.9, develop a firing order and angular positioning to optimize the balancing of the engine. FIGURE P15.5.4 A slider crank mechanism. A B θ Y r ᐉ C(m ) C X w ˆ m 0593_C15_fm Page 537 Tuesday, May 7, 2002 7:05 AM 0593_C15_fm Page 538 Tuesday, May 7, 2002 7:05 AM 539 16 Mechanical Components: Cams 16.1 Introduction In this chapter, we consider the design and analysis of cams and cam–follower systems. As before, we will focus upon basic and fundamental concepts. Readers interested in more details than those presented here may want to consult the references at the end of the chapter. A cam (or cam-pair ) is a mechanical device intended to transform one kind of motion into another kind of motion (for example, rotation into translation). In this sense, a cam is primarily a kinematic device or mechanism. The most common cam-pairs transform simple uniform (constant speed) rotation into translation (or rectilinear motion). Figure 16.1.1 shows a sketch of such a device. In such mechanisms, the actuator (or active) component (in this case, the rotating disk) is called the cam . The responding (or passive) component is called the follower . Some cam-pairs simply convert one kind of translation into another kind of translation, as in Figure 16.1.2. Here, again, the actuator or driving component is the cam and the responding component is the follower. An analogous cam-pair that transforms simple rotation into simple rotation is a pair of meshing gears depicted schematically in Figure 16.1.3. Because gears are used so extensively, they are usually studied separately, which we will do in the next chapter. Here, again, however, the actuator gear is called the driver and the responding gear is called the follower . Also, the smaller of the gears is called the pinion and the larger is called the gear . To some extent, the study of cams employs different procedures than those used in our earlier chapters; the analysis of cams is primarily a kinematic analysis. Although the study of forces is important in mechanism analyses, the forces generated between cam–follower pairs are generally easy to determine once the kinematics is known. The major focus of cam analysis is that of cam design. Whereas in previous chapters we were generally given a mechanical system to analyze, with cams we are generally given a desired motion and asked to determine or design a cam–follower pair to produce that motion. In the following sections, we will consider the essential features of cam and follower design. We will focus our attention upon simple cam pairs as in Figure 16.1.1. Similar analyses of more complex cam mechanisms can be found in the various references. Before directing our attention to simple cam pairs, it may be helpful to briefly review configu- rations of more complex cam mechanisms. We do this in the following section and then consider simple cam–follower design in the subsequent sections. 0593_C16_fm Page 539 Tuesday, May 7, 2002 7:06 AM 540 Dynamics of Mechanical Systems 16.2 A Survey of Cam Pair Types As noted earlier, the objective of a cam–follower pair is to transform one kind of motion into another kind of motion. As such, cam-pairs are kinematic devices. When, in addition to transforming and transmitting motion, cam-pairs are used to transmit forces, they are called transmission devices. There are many ways to transform one kind of motion into another kind of motion. Indeed, the variety of cam–follower pair designs is limited only by one’s imagination. In addition to those depicted in Figures 16.1.1, 16.1.2, and 16.1.3, Figures 16.2.1, 16.2.2, and 16.2.3 depict other common types of cams. Figures 16.2.4 and 16.2.5 depict various common FIGURE 16.1.1 A simple cam–follower pair. FIGURE 16.1.2 A translation to translation cam–follower pair. FIGURE 16.1.3 A gear pair. FIGURE 16.2.1 Cam–follower with cam–follower track. FIGURE 16.2.2 Cylindrical cam with cam–follower track. Follower Cam Follower Cam Driver Follower Cam Follower Cam Follower Track Cam Rotation Rotation Cam Follower Track Cam Follower 0593_C16_fm Page 540 Tuesday, May 7, 2002 7:06 AM Mechanical Components: Cams 541 follower types. Figures 16.2.2 and 16.2.3 show that cam–follower pairs may be used not only for exchanging types of planar motion but also for converting planar motion into three-dimensional motion. In the following sections, we will direct our attention to planar rotating cams (or radial cams ) with translating followers as in Figure 16.2.4. Initially, we will consider common nomenclature and terminology for such cams. 16.3 Nomenclature and Terminology for Typical Rotating Radial Cams with Translating Followers Consider a radial cam–follower pair as in Figure 16.3.1 where the axis of the follower shaft intersects the axis of rotation of the cam. Let the follower shaft be driven by the cam FIGURE 16.2.3 Conical cam with cam–follower. FIGURE 16.2.4 Types of translation followers. FIGURE 16.2.5 Types of rotation followers. Cam Follower Track Cam Follower Point Follower Plane Follower Roller Follower Slider Follower Roller Follower 0593_C16_fm Page 541 Tuesday, May 7, 2002 7:06 AM [...]... equal to the offset and thus tangent to the follower axis To see this consider again the representation of the offset flat surface follower of Figure 16.6.1 and as shown in Figure 16.6.7 Let the offset between the follower and cam 0593_C16_fm Page 548 Tuesday, May 7, 2002 7:06 AM 548 Dynamics of Mechanical Systems ρ FIGURE 16.6.7 Offset flat surface follower of Figure 16.6.1 FIGURE 16.6.8 Offset axes... AM 544 Dynamics of Mechanical Systems FIGURE 16.4.3 Graphical construction of follower rise profile Let rmin be the minimum value of r in Eq (16.4.1) Let h be the difference between r and rmin at any cam position: That is, let h = r − rmin = f (θ) − rmin = h(θ) (16.4.4) Then, h(θ) represents the rise (height) of the follower above its lowest position We can obtain a graphical representation of h(θ)... application of these influence or interval functions in structural mechanics — particularly in beam theory (see, for example, Reference 16.6) For cam profile analysis, they may be used to conveniently define the profile, as in Eq (16.10.4) The principal application of the functions in cam analysis, however, is in the differentiation of 0593_C16_fm Page 556 Tuesday, May 7, 2002 7:06 AM 556 Dynamics of Mechanical Systems. .. pair [16.1] Brief descriptions of these items follow: • Cam profile — contact boundary of the rotating cam • Follower wheel — rolling wheel of the follower contacting the cam profile to reduce wear and to provide smooth operation • Trace point — center of rolling follower wheel; also, point of knife-edge follower • Pitch curve — path of the trace point • C — typical location of the trace point on the pitch... function of Figure 16.6.6, we can use the curve ordinates of the various cam rotation angles as the measure of the follower displacement above the base circle, along the corresponding radial lines of Figure 16.6.9 Figure 16.6.10 shows the location of these displacement points Finally, by sketching the flat surface of the follower placed at these displacement points we form an envelope [16.2] of the cam... profiles with cusps or concavities as in Figure 16.7.1 Finally, if the cam is rotating rapidly (as is often the 0593_C16_fm Page 550 Tuesday, May 7, 2002 7:06 AM 550 Dynamics of Mechanical Systems (a) (b) FIGURE 16.7.1 Incompatible and impractical cam profiles for a roller follower case with modern mechanical systems) , the cam profile may create very large follower accelerations These accelerations in turn... Tuesday, May 7, 2002 7:06 AM 542 Dynamics of Mechanical Systems FIGURE 16.3.1 A radial cam–follower pair Cam Normal Direction of Follower Motion Follower Trace Point Follower Wheel Cam Profile Pitch Curve FIGURE 16.3.2 Radial cam–follower pair terminology and nomenclature through a follower wheel as shown Figure 16.3.2 shows the commonly employed terminology and nomenclature of the radial cam–follower pair... inserting the rising segment of the cosine function as shown in Figure 16 .12. 2 By comparing Figures 16 .12. 1 and 16 .12. 2, we see that to obtain the desired cosine segment fit between (h1, θ1) and (h2, θ2), the amplitude A must be: A = ( h2 − h1 ) 2 (16 .12. 1) In like manner, the phase and period of the cosine function must be adjusted so that when θ has the values θ1 and θ2, the argument ψ of the cosine function... θ 1 ) (16 .12. 2) Then, it is readily seen that the desired cosine function for the rise segment of Figure 16 .12. 2 may be represented by the function φ(θ) given by: φ= FIGURE 16 .12. 1 Cosine function  h1 + h2   h2 − h1  + cos ψ  2   2  (16 .12. 3) 0593_C16_fm Page 561 Tuesday, May 7, 2002 7:06 AM Mechanical Components: Cams 561 Follower Rise h(θ) h2 cosine Rise Segment h1 θ FIGURE 16 .12. 2 Fitting... where, from Eqs (16 .12. 2) and (16 .12. 3), dφ/dθ, d2φ/dθ2, and d3φ/dθ3 are: dφ  h − h  π  =− 2 1  sin ψ  2   θ 2 − θ1  dθ  (16 .12. 8) 2 d 2φ  h − h  π  =− 2 1  cos ψ 2  2   θ 2 − θ1  dθ  (16 .12. 9) and 3 d 3φ  h2 − h1   π  = sin ψ dθ 3  2   θ 2 − θ 1    (16 .12. 10) From Eq (16 .12. 2) we see that when θ is θ1, then ψ is π, and when θ is θ2, ψ is 2π Hence, from Eq (16 .12. 3), we have: . AM 540 Dynamics of Mechanical Systems 16.2 A Survey of Cam Pair Types As noted earlier, the objective of a cam–follower pair is to transform one kind of motion into another kind of motion rotating shaft with unbalance particles. 1 in. Flywheel (125 lb) 12 in. P P Ω ᐉ 1 2 0593_C15_fm Page 535 Tuesday, May 7, 2002 7:05 AM 536 Dynamics of Mechanical Systems Section 15.4 Dynamic. internal-combustion engines and with similar large systems. Because such systems have a number of moving parts, complete balance is generally not possible. Designers of such systems usually attempt to minimize