A single-plane rotating imbalance experiment

Transcription

1 A sngle-plane rotatng mbalance experment T. W. NG, Bachelor of Technology Programme, Faculty of Engneerng, Natonal Unversty of Sngapore, 10 Kent Rdge Crescent, Sngapore Receved 10th May 1999 Revsed 2nd March 2001 A smple experment to demonstrate sngle-plane rotatng mbalance s descrbed. Inexpensve and commonly avalable tems were used n the experment to emphasze the concepts presented n an uncluttered manner and to underlne the mportance of smplcty. Students were also requred to assemble the experment by themselves to maxmze experental learnng. The experment was able to provde accurate predctons of the angular poston and magntude that an mbalance mass mposed. Key words: mbalance, balancng, rotatng mass, dynamcs 1. INTRODUCTION Rotatng mbalance s the unequal dstrbuton of mass of a rotor about ts centre-lne. Ths phenomenon s commonly encountered n centrfuges, fans, pumps and other rotatng machnery used n ndustry. The dynamc forces resultng from rotatng mbalance are damagng to the rotor, bearngs and the supportng structures of machnes [1]. A sound knowledge of the mechancs of rotatng mbalance s hence crucal for any asprng mechancal engneer. In dsc-lke rotors, the rotatng masses, and hence mbalance, are concentrated n oneplane. Grndng wheels and celng fans are examples of such systems. A drect approach to demonstrate the effects of sngle-plane rotatng mbalance would be one nvolvng the measurement of dynamc forces [2]. Ths approach, however, has the drawback of requrng the use of sensors and electronc equpment (e.g. osclloscope) for data acquston and analyss. These tems are generally expensve and requre careful handlng. In ths paper, an experment that obvates the use of these tems s presented. The bass of ths approach les wth the complementary satsfacton of condtons between statc and dynamc balancng for rotatng masses n a sngle-plane. At the outset of desgnng the experment, two other desred outcomes were outlned. Frst, the experment should be composed of nexpensve and commonly avalable tems. The purpose of ths was to remove any form of embellshment (e.g. sophstcated nstrumentaton) that would hnder conceptual understandng and to emphasze the advantages of smplcty. Second, the experment would also need to be assembled by students before use. The ratonale behnd such an exercse was to maxmze the level of hands-on experence.

2 166 T. W. Ng 2. THEORY Consder a seres of unbalanced masses located n one plane as shown n Fg. 1. The resultant unbalance can be wrtten as Fg. 1. Descrpton of a seres of sngle-plane masses. mgr cosθ = g m r cosθ (1a) mgrsnθ = g m r snθ (1b) Snce g, the gravtatonal acceleraton s a constant, equaton (1) can be smplfed to mr cosθ = m r cosθ mrsnθ = m r snθ If the radus of all the mbalance masses were equal, equaton (2) smplfes further to mcosθ = m cosθ msnθ = m snθ (2a) (2b) (3a) (3b) Suppose that two masses m 1 and m 2 are placed at angular locatons θ 1 and θ 2 respectvely. Ths allows equaton (3) to be wrtten as mcosθ = cosθ1+ cosθ2 (4a) msnθ = snθ1+ snθ2 (4b) where m s the resultant unbalance mass. Removng m from the nequaltes gves snθ1+ snθ2 tanθ = cosθ1+ cosθ2 (5)

3 A sngle-plane rotatng mbalance experment 167 It s possble to rewrte equaton (5) as snθ1 cosθ1tanθ = cosθ2 tanθ snθ2 (6) Suppose that the mass 2 causes the mbalance n a sngle plane rotatng system. In order to cancel ths mbalance, t would be necessary to determne m 2 and θ 2. It s possble to determne the values of θ by locatng a known balance mass m 1 at dfferent values of θ 1. By nspectng equaton (6), t s clear that a lnear relatonshp y = kx + c exsts, where y = snθ1 cosθ1tanθ k = cosθ2 x = tanθ c = snθ2 (7) If a graph of y aganst x s plotted, t should be possble to determne m 2 and θ 2 from the slope and y-axs ntercept. It s mportant to note that the sgns of the sne and cosne of θ 2 must be used to determne the quadrant that θ 2 belongs to. Table 1 gves a descrpton of how ths can be sorted out. Once the values of m 2 and θ 2 are determned, t s possble to fnd the resultant mbalance mass for balance m 1 located at dfferent values of θ 1 (from equaton (4)) usng m= ( snθ1+ sn θ2) 2 + ( cosθ1+ cos θ 2) 2 (8) Table 1. Procedure to determne the actual angle of the mbalance mass Condton sn θ 2 + ve and cos θ 2 + ve sn θ 2 + ve and cos θ 2 ve sn θ 2 ve and cos θ 2 ve sn θ 2 ve and cos θ 2 + ve Actual angle θ 2 =θ 2 θ 2 = 180 θ 2 θ 2 = θ 2 θ 2 = 360 θ 2 3. EXPERIMENTAL EQUIPMENT AND PROCEDURE The tems used n the experment were a C-clamp, a battery, two approxmately equal masses and a sngle-plane motorzed rotor mounted on a cantlever. The student was requred to assemble the components as shown n Fg. 2. The rotor has to be spun a few tmes wthout any mass attached to ensure that t dd not return to any fxed poston usng the poston ndcator shown n Fg. 3. The mbalance mass m 2 was then attached to an arbtrary locaton and spun a few tmes. Despte the number of spns, the rotor should nevtably settle an angle poston θ. The known balance mass m 1 was then placed at varous angular postons θ 1. For each spn of the rotor, the angular poston of θ was recorded. A graph based on the values of y

4 168 T. W. Ng Fg. 2. General vew of the expermental set-up. Fg. 3. Front vew of the rotor n the expermental set-up. aganst x usng equaton (7) was next plotted. From ths graph t was possble to determne m 2 and θ 2. When the values of m 2 and θ 2 were known, a plot of the resultant unbalance mass m for

5 A sngle-plane rotatng mbalance experment 169 values of θ 1 from 0 to 360 usng equaton (8) was made. From the second graph, t was possble to determne the angular poston of θ 1 wheren the resultant mbalance mass was mnmal. After the balance mass m 1 was placed at ths angular poston, the wres of the motor were connected to the ends of the battery to rotate the rotor. Before ths porton of the experment was carred out, students were remnded to stay a safe dstance and to ensure that the amount of clampng was suffcent. Students were requred to observe the magntude of ths vbraton and to compare t wth the alternatve cases where (a) m 1 was placed at an adjacent locaton and (b) m 1 was removed to leave the mbalance mass m 2 alone. 4. EXPERIMENTAL RESULTS AND DISCUSSION As antcpated students found the experment easy to understand and conduct. The mbalance mass was 11.9 g and located at angular poston 240. Students were encouraged to use a spreadsheet program lke Excel to plot the graphs. From the graph n Fg. 4 t was found that the lnear equaton (generated usng the regresson analyss tool found n the spreadsheet program) correspondng to the plot was y = 0.456x Snce m 1 and m 2 must be postve, ths could only mean that both sn θ 2 and cos θ 2 were negatve from equaton (7). From Table 1, t was clear that θ 2 must then le n the thrd quadrant wth a value of tan 1 ( ) = Based on ths plot, the value of m 2 was found to be 11.6 g. The angular poston and magntude values therefore, were only 1.11% and 2.52% respectvely off n predcton. Fg. 4. Graph of y aganst x usng sample results from the experment. Fg. 5 gave the resultant mbalance masses calculated for dfferent angular postons θ 1. The resultant mbalance mass was found to be mnmum when the balance mass was located at angular poston θ 1 =60. The dynamc test mentoned n the prevous secton confrmed ths. As an avenue for further dscusson, students were asked to theoretcally verfy the

6 170 T. W. Ng observaton that masses should always be placed opposte each other n a two-mass mbalance stuaton. Fg. 5. Graph of resultant unbalance mass aganst angular poston of the balancng mass usng sample results from the experment. 5. CONCLUSIONS An experment to llustrate sngle-plane rotatng mbalance was descrbed n ths work. The experment dd not requre the use of any sensors nor electronc equpment. It was successful n presentng the subject matter concsely, encouraged smplcty, and provded experental learnng. Despte the somewhat rustc nature of the set-up, accurate predcton of the angular poston and magntude of the mbalance mass was acheved. REFERENCES [1] K. J. Waldon and G. L. Knzel, Knematcs, Dynamcs and Desgn of Machnery, John Wley, New York, (1999). [2] L. Ercol and S. La Malfa, Analytcal and expermental model for teachng the sngle-plane balancng machne, Internatonal Journal of Mechancal Engneerng Educaton, 18(4),