Charts for the Dynamic Properties of Counterweights

Transcription

1 Purdue University Purdue e-pubs International Compressor Engineering Conference School of Mechanical Engineering 1976 Charts for the Dynamic Properties of Counterweights T. C. Scott G. L. Davis Follow this and additional works at: Scott, T. C. and Davis, G. L., "Charts for the Dynamic Properties of Counterweights" (1976). International Compressor Engineering Conference. Paper This document has been made available through Purdue e-pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information. Complete proceedings may be acquired in print and on CD-ROM directly from the Ray W. Herrick Laboratories at Herrick/Events/orderlit.html

2 CHARTS FOR THE DYNAMIC PROPERTIES OF COUNTERWEIGHTS T. C. SCOTT, SENIOR RESEARCH ENGINEER G. L. DAVIS, MANAGER ADVANCED ENGINEERING DEPARTMENT SUNDSTRAND COMPRESSORS, BRISTOL, VIRGINIA INTRODUCTION Dynamic analyses of reciprocating machines often require that one compute masses and moments of inertia of various shaped counterweights. Relations for the determination of these properties are not generally available in the literature. In this paper, formulas and graphs will be presented for the rapid determination of these properties for several commonly encountered shapes. FUNDAMENTALS Figure (2) illustrates a general counterweight of constant thickness t and having two parallel flat faces of area A oriented perpendicular to the c-c and o-o axes. The polar mass moment of inertia of this element relative to the o-o axis is J, f.?.f"tda = t f f"da (l) Configuration #l Configuration #2 where ;o is the material density. For counterweights, ;o is constant so that it is sufficient for us to determine only the area polar moment of inertia I=-= rda J f2 _r-f: (2) For the dynamic analysis of compressors, one also requires the distance to the center of mass, Rem' and the area, A. The counterweight mass is then Configuration #3 FIG. 1: Three Common Counterweight Designs m =,t>t:a (3) 208

3 For the configurations of Figure (1), time consuming development of equations and their evaluation must be carried out by the designer. In the sections which follow these equations will be presented along with graphs for their rapid evaluation. Details of the derivations will only be presented for configuration #1. CRANK CENTER FIG. 2: General Counterweight FIG. 3: Simple Disk Counterweight For the simple disk counterweight of Figure (3), Rem is easily known and A.,. 1rRZ "' The moment of inertia about the crank centerline is (4) ( 5) For simple configurations such as this the necessary parameters may be determined from tables and formulas in standard texts and handbooks. [ o<.:j 2 A Tr-- R il 2 s FIG. 4: Dimensions for Configuration #1 CONFIGURATION #1 The total area for this shape is and the moment of inertia about the crank center is I = <><R" zrr-«r4 (6) --;r.,+--4-" (7) 209

4 The location of the centroids of the two parts is shown in Figure (4). To find Rem we write ARc.., =A, X, -A_. X.z to obtain For rapid evaluation, charts may be created for the ratio of A to A a circle of radius R 0 for ( 8) ( 9) and the center of mass is at where c [sin z - 3 S.IN,B]- cos(p+7 )j [t+3e1[t-t1 3[: +P z+cj!- z]... - (Ff7) fi= t., -e---j ( 15) (16) (17) the ratio of I to I for a circle of radius R 0 (10)?:o t«.n -I (:r;:::?) There is a limiting value on shape which occurs when the two tangent to the circle of radius This is for this lines ( 18) Rs meet. 1- [I- [ 4!] 31 S"IN F 3{><+ [z.jt-..c][.j} These relations are plotted in Figures (7), (8), and (9). (12) (19) Equations (13) (14), and (15) are plotted in Figures (10), (ll), and (12). H=R cos,.( - j_ 0 2 CRANK CENTER FIG. 5: Dimensions for Configuration #2 CONFIGURATION #2 For this shape the area is the moment of inertia about the crank center is (13) CRANK CENTER CRANK -A, t / -r- bt- _;}- L CENTER A ' I l x, (14) FIG. 6: Dimensions for Configuration #3 210

5 CONFIGURATION #3 For this shape, the properties may be easily determined without the need for charts. For the circular segment the area is the center of mass is located at. X = I 4R 0 SIN 1 ("Vz) 3 [<>(-SIN"'-] (20) (21) and the moment of inertia about the crank center is For the rectangular portion (23) (24) b 3 b [ 3 l] I = -[H+ct] + 3 H +a..j 2 IZ (25) With present electronic pocket calculators, the evaluation of these equations is not difficult except for equation (22). Figure (13) may be of assistance here, being a plot of I;- -.!..cos{)s!n(!!)[i+i!cose('f} (26) R:- f: z c. To locate the counterweight mass center, which is R.:: ern EXAJVIPLE A, X, +Az Xz A, +A.,. f R: SfN"("'Y.e)-+ f [HO?-a.z] 2 zk 0 [o<:-sjno<] + b [a.+h] " To illustrate the use of these charts, consider the problem of finding the thickness t of a counterweight of configuration #2 which will yield (27) (28) when the material is steel with..p= 7.86 gm/cm3, Rs Ro 1.3 em 2.0 em and the angle a< To relate t charts, write is variable. to the parameters on the z mri'. == /'tarz :=..J"t7rR:[AA7[R,] = to.o c,., ern ol "a Inserting the given data then gives o, Oi?S'4 -t = [.J [ J e By taking values from Figures (10) and (12) at various values of <>< along the line R f,3 s - - = 0 "'5" '"' Ro -.i?.o the following table can be quickly created. «(oer;) R"'iR., A/.4., f: (c..,) CONCLUSIONS The graphs presented allow one to quickly evaluate the dynamic properties of several common counterweight shapes. The equations from which the graphs were created may be used directly or programmed for computer solution as a use ul design aid. 211

6 FIG. 7: A/A for --' Configuration #l 0.4 o.c,.: (DEG) 200?50 JOO.%0 FIG. 8: I/I 0 for Configuration #l so /00!SO?00 <>< (DEG) } Jao 0.3 FIG. 9: R /R for em o -- Configuration #l ISO c'oo o( (on;) 300 JoO 212

7 N 0 v.'. i I. ; i i 1. I I!.I' 'I I' I IiI I: I 0 50!00!50,? ]()0 <>< (OEG) FIG. 10: A/A 0 for Configuration #2 50 /00 150,?00 c5o «(DEG') FIG. 11: I/1 0 for Configuration #2 0. f:g H Q 0 I-' I:! [\) H, f-' 0 I=J!:0 Pl..., ct!:0 f-' 0 0 I:! I "'t:: 1-i f\) S::, a""!0 ):) S:::, S:::, "'-... II;! R 'B' '\., <:s CJ '\:, t-' w

8 I /.0 FIG I 4 1 /R for 0 -- Configuration #3 50 /00 ISO coo,?so JOO 360 o< (DEc;) 214