Surging in Coil Springs

Transcription

1 Purdue University Purdue e-pubs nternational Compressor Engineering Conferene Shool of Mehanial Engineering 1996 Surging in Coil Springs R. A. Simmons Purdue University W. Soedel Purdue University Follow this and additional works at: Simmons, R. A. and Soedel, W., "Surging in Coil Springs" (1996). nternational Compressor Engineering Conferene. Paper This doument has been made available through Purdue e-pubs, a servie of the Purdue University Libraries. Please ontat epubs@purdue.edu for additional information. Complete proeedings may be aquired in print and on CD-ROM diretly from the Ray W. Herrik Laboratories at Herrik/Events/orderlit.html

2 SURGNG N COL SPRNGS by R.A. Simmons and W. Soedel Ray W. Herrik Laboratories, Shool of Mehanial Engineering, Purdue University, West Lafayette, N , USA ABSTRACT One potentially important onsideration in ompressor noise ontrol is the effet of surging in oil springs. While studies on spring surging have been pursued in the past [3] this study employs the reeptane tehnique to analyze a mehanial system haraterized by spring surging. 1. NTRODUCTON The objetive of analyzing oil spring surging is in part to understand its ontribution to the response of a larger system. The usual but sometimes not justifiable hypothesis is that springs do not ontribute resonanes of their own to a omposite system. The system was onsiderably simplified to apture the essene of the spring surge problem. The results illustrate phenomena whih are observed by pratial engineers when they try to ahieve noise ontrol by hanging spring rates. For a real ompressor, say a refrigeration ompressor supported inside a hermeti shell by three mounting springs whih may deflet eah in three diretions, the reeptane formulation is only a little more ompliated. The elements of the approah inlude determining the equation of motion for a oil spring as a ontinuous system, the reeptanes for suh a spring, and the total system reeptane expression, for three sub-systems B,C and D, C being the surging spring. One the foundation formulas are obtained, a numerial analysis of a typial system will be performed. This isolation involves the parameters of internal spring damping rate and spring stiffness. The final setion will be devoted to the appliation of the reeptane method to an idealized ompressor shell. 2. SYSTEM ACCOUNTNG FOR SPRNG SURGNG EFFECTS n addition to the oil spring of mass M, stiffness K, and internal damping rate C, designated sub-system C, the system is divided into two other sub-systems, B and D as shown in Figure 1. B is an extremely simplified model of the ompressor body and D is an extremely simplified model of one mode of a ompressor shell. Damping of sub-system C ours in the spring itself; in other words, there is no 'external' damper- the damping of the oil spring ours via material damping or by means of a plasti sleeve strethed around the spring. This will allow for a ontinuous damping effet aross the length L, rather than a net damping effet at the two end points of the spring. 2.1 The Sub-System Band D Reeptanes n general, the reeptane of a system is simply the ratio of harmoni displaement at one point to a harmoni fore at another point. See referenes (1,2] for reeptane definitions. The reeptanes are, for system Band D, 1 1 J3o = J3zz = J3z = (k 2). B, 033 = 044 = 043 = k 2. (1) 1- m1 CO + JCO ( 2 - m 2o ) +]COD 2.2. Derivation of Sub-System C Reeptane A oil spring has mass M, length L, stiffness K, and an internal damping oeffiient C. The displaement along the x diretion is u(x,t). An infinitesimal element dx from the spring length is shown in Figure 2. Thus, the fore F reated in the spring is related to defletion by 721

3 au F=KLax. (2) Detennining a onstant mass per unit length m' = dm/dx =ML, the element dx is ated on by fores shown in Figure 3. A fore balane and Newton's 2nd law gives The displaement solution is of the general fonn Applying this to the equation of motion yields a 2 u +. ~ = KL a 2 u (3) at 2 m' at m' ax 2. u(x,t) = (A 1 e-j",x + B 1 ej",x)eiwt (4) 2- KL ' = m'' Using this general expression for the equation of motion of the oil spring, the two boundary ondition of Figure 4 are introdued and the unknown onstants A 1 and B 1 are solved for. When A 1 and B1 are known, the reeptanes at the left hand endpoint y 22 and Y32 are evaluated: u(o, t) ~K,L + e -jk,l u(l, t) 2 Y22 = F _iwr =. KL( jk,l -jk 1 L) ' Y32 = F _iwt =. KL( jk,l -jk 1 L) zc' JK 1 e - e 2C' JK 1 e - e Similarly, reeptanes y 23 and y 33 are evaluated. The boundary onditions, however, are reversed and are shown in Figure 5. ~ U(O,t) _ 2 u(l,t) ~K,L+e-kK,L Yz3- F3eiwt- jkjkl(ej",l-e-j",l)' 'Y33= F3eiwt = jkjkl(ej",l-e-jk,l). Note the symmetry of the sub-system C reeptanes: Y22 = ' 33, ' 32 = Y Reeptane Model of System The sub-systems are generalized as blok diagrams as shown in Figure 6. To obtain the system A (the total system) reeptanes it is neessary to break the system between eah of the arrows and generalize the fore and displaements for eah sub-system. This is shown in Figure 7. Setting up displaement expressions and following the proedure outlined in referenes [ 1,2] yields Beause of damping in some or all parts of the system, a. 31 is a omplex number. To aount for this, the magnitude and phase of the reeptane will be onsidered when response behavior is analyzed. (5) (6,7) (8,9) (10) 3.1. The Effet of Surging on the Response 3. NUMERCAL RESULTS AND DSCUSSON The first observation to be made is the notieable differene in a typial system response (reeptane a 31 ) when spring surging is onsidered, as shown in Figure 8. The natural frequenies of sub-systems Band D were seleted to be 6283 and 2000 rad/s, respetively. The spring dimensions and properties were seleted to be typial for small heat pump ompressors. Several observations an be made. First, notie that the external system spikes (due to the resonanes of systems B and D) at OJ= 6283 rad/s and OJ= 2000 rad/s are magnified when surging ours. Eah of the dashed spikes that do not our at OJ, or 6283 rad/s represent the surge frequenies. 722

4 n addition to the overall inrease at the system resonane, note the near oinidene effet of a spring surge resonane with the sub-system D natural frequeny at 2000 rad/s. The spring surge peak is inreased as a result of this near oinidene. Clearly, surge natural frequeny intervals are a major onsideration in the design of suh a system. The separation between the oil spring resonane will learly ditate whether oinidene is likely to our and thus inrease system response. Furthermore, it may be required to operate at a speifi frequeny or over a range of given frequenies and in these ases, spring surging would redue the operable ranges. f spring surging an be ignored, then a large range between the external sub-system natural frequenies exists in whih to operate. However, if spring surging annot be ignored, one must then onsider how to appropriately spae the intervals so as to avoid detrimental amplifiations. A final note about the system is that spring natural frequeny harmonis will be present at the higher frequenies as well (i.e. those to the right of 6283 rad/s). On the other hand, when surging is ignored the response behavior naturally dereases at these higher frequenies (a false sense of seurity is reated) Use of Coil Spring Damping to Minimize Surge nfluene Of interest here is the extent to whih internal spring damping an redue system response. A plasti sleeve around the oil spring might be one effetive way to reate internal damping. A typial response for various amounts of damping is shown in Figure 9. t is noted that damping of the oil spring appears to be a useful tool in reduing the ontribution of the oil spring to the overall system reeptane. However, while it damps the oil spring resonanes, it does not seem to make a signifiant ontribution to damping the external sub-system ontrolled natural frequenies Variation of Coil Spring Rate, K First onsider a lower value of K. The lassial hypothesis to test here is whether a lower spring rate will always redue the response of system A. f one ignores spring surging, a ase may be made that the most desirable K for response isolation is the lowest one possible. However, in the ase of surging, it will be shown that this is not neessarily always true. Consider the value of a typial spring stiffness. n Figure 10, the solid line demonstrates that the intervals between surge natural frequenies are smaller for K redued by 17% than for the original larger spring rate values (superimposed as the dashed urve). Notie that the surge resonane whih was at approximately 7000 rad/s before has now moved to the left and exhibits a near oinidene behavior of the oil spring natural frequeny with the natural frequeny of subsystem B at 6283 rad/s. The peak has been split in two, eah part of whih is higher than the original system A reeptane. This is one ase whih demonstrates that a lower K is not neessarily advantageous. Next, onsider an even muh lower spring stiffness (K redued by 33% from original value). n Figure 11, the interval between spring surges is muh smaller than before, and oinidene is a great deal more likely. Note that a seond surge peak of the solid line now oinides with the system A resonane at 2000 rad/s. Thus with a muh smaller K, oinidene with other sub-system natural frequenies ours more frequently and refutes the argument that a softer spring rate invariably redues overall system response. (But there is a trend of a lower average off resonane response with dereasing spring rate). Finally onsider a muh higher sub-system C spring rate: K is inreased by a fator of four. This urve is haraterized by a muh higher mean reeptane level, higher peaks (whih are reeptive features), but muh wider surge natural frequeny intervals. As a result of this, oinidene is a great deal less likely. The wider spring resonane intervals an be advantageous if the appliation alls for a system to operate at a speifi frequeny or in a speifi frequeny range. One would simply have to onsider what design spring rate K would have the lowest mean reeptane for the given operation range. The omparison of this larger spring rate (solid line) with the default value (dashed line) is given in Figure 12. As one an observe, the reeptane tehnique is a very appropriate method for understanding how spring surging ontributes to the overall response of a system. An analysis suh as this demonstrate its usefulness when a detailed analysis of spring rate design is important. The onlusion drawn is that a lower oil spring rate does reate a somewhat lower mean system response, but oinidene is more likely to happen due to the tighter intervals of the spring 723

5 resonanes. Thus, quantifying spring rate S ultimately an issue of the appliation and the range of operating frequenies. 4. COMPRESSOR SHELL RECEPTANCE APPLCATON The preeding system analysis has demonstrated the ease with whih a multiple degree of freedom system an be studied. Clearly the example given has been rather basi in nature, yet it has effetively illustrated how the reeptane method an be applied to a system whih is omprised of several sub-systems of differing design. Beause this study employs the reeptane tehnique, one of its major advantages is that a wide variety of subsystems an be applied to the general expression for a 31. For example, one might wish to substitute a more omplex sub~system into system A in plae of sub-system D. The only requirement beyond what has previously been formulated to make this substitution possible is that the reeptane for sub-system D must be formulated or measured. The example that will be investigated here is a ompressor shell Compressor Shell Reeptane As mentioned, the system A reeptane expression will remain unhanged. n order to substitute a ompressor shell into system A, sub-system D must be replaed by a shell fixed to the spring of sub-system C. n this example, a half shell model, simply supported all around, is used. The numerial values hosen for the system parameters roughly approximate those found in a typial ompressor system: length, L :::: m; radius, a = m: thikness, h = m; and subtended angle, a'= 1t radians (semiirular). The material is steel. Figure 13 illustrates the ompressor half-shell model as it joins with sub-system C: Aording to referene [1] the reeptane for a half-shell at its enter is given by the double sum expression ~: 4 ; ; [ 1. 2 [ m1tx *] 2 [ n7t8*l] u 33 = sm -- sm --, phlaa rn"'l n"'l rornn - ro L a where, at loation 3, (x*,s*) = (L/2,a12) = (0.100, rr./2). The natural frequenies for the (m,n) mode are given by equation (6.12.3), in referene (1]. Notie that the lowest natural frequeny ours in the m=l, n=4 mode and has a value of ro 14 = 3939 rad/s. t is also important to notie that for this partiular half shell example, there are 12 natural frequenies below radls. Only these 12 will ontribute to the system A response for the frequeny range onsidered in this paper (0 to rad/s). The ~ 3 reeptane is evaluated using (10) and is plotted for this range in Figure 14. Next, this modified sub-system D reeptane is substituted into the expression for the system A ross reeptane a3 1, with all other reeptanes remaining the same, to yield the omposite response plot of Figure 15. The response resonanes are labeled C,B, or D if they are primarily due to resonanes of sub-systems C,B or D, respetively. The differene here is that instead of having only one system D resonane as before, the shell introdues numerous resonanes, eah of whih an be in oinidene or near oinidene with a surge frequeny of the spring. Considered mode by mode, however, the system will behave similarly to the simple ases disussed before. (11) 5. CONCLUSON The reeptane method analysis has demonstrated the following: ( 1) Spring surging is often signifiant; it annot ategorially be negleted. (2) nternal damping of the surging oil spring is effetive in damping surge resonane. (3) Surge frequenies should be detuned from other system natural frequenies. (4) Lower oil spring rates redue the average system response, but make oinidene more likely. (5) Higher oil spring rates may beome desirable if the driving frequeny operation range is between surge peaks, provided the driving frequeny is not oinident with other system natural frequenies. 724

6 Here, the situation to be analyzed was greatly simplified. However, it gave useful answers whih explain what an be seen in engineering pratie. n the future, studies involving more than one spring and surge behavior of the oil springs in more than the axial diretion should be undertaken. REFERENCES 1. W. SOEDEL 1993 Vibrations of Shells and Plates, Seond Edition, New York: Marel Deker. 2. R.E.D. BSHOP and D.C. JOHNSON 1979 The Mehanis of Vibration, New York, Cambridge University Press. 3. AD. DMAROCONAS and S. HADDAD 1992 Vibration for Engineerings, Englewood Cliffs, New Jersey: Prentie Hall. SYsTEM A r SUB-SYSTEM C ' ~l :: , r ~ ~ sub-system B _i u(x,t) : sub-sys't:m D 1: f1(t) :: :-L--+;:. : :. l, , 1 kt :- x,(x,t) : i ~ : i!-- x 2(x,t) Fig. 1. System with surging spring. L x ~rdx Fig. 2. Coil spring with element dx. ~ u(o,t) z: F2d'"'-: x=o: : x=l F~:; CUdx : m'iidx i i--u(x,t) Fig. 3. Fores on element. (JF F+-dx ax Jill;--- F3e Fig. 4. Boundary onditions for left loading. Fig.5. Boundary onditions for right loading. :x=l Fig. 6. System A reeptane blok diagram. Fig. 7. Reeptane omponents. 725

7 -10 j- Unsurged - -Surged -14, ,----r---,----,--,-----,,---,----,---, -16 j , ' - \ '. /' ' \. ' ' ' s,; ;',,, i'. \..,... - '. '. -20 A-22 s Frequei\C) (rad/s) Fig. 8. Surging and nonsurging system response ,----, , "'T'"--, 28 -ooo:;---:1::::o::::oo:;----;;2-;:oo::o:---=3='ooo=--4o="=o-=-o ---=s:-:ooo'='_s_ooo Freaenv (rad/s) ~ = o.J..o-o_a_oo'-o--s-oLoo--, o_jooo Fig. 9. Effet of spring damping on surging. -14r--,--~---~-,--~--~--.---,--~-, _ "28o!--1-'000...,.---,2000-:':-:,--aooo~-= :-:'-:,--:sooo::'::,---::sooo~-7000==---=aooo=~-=9000-:':-:,--,~oooa Frequeny (radls) Fig. 10. System A response when K is redued by 17% (solid line) r--...,...--"T""" , ,----,--~ -28~-~-~~~--~--,~--~-~-~~--~--~ Fruqueny (rad/s) Fig. 11. System A response when K is redued by 33% (solid line) ~ :. 26. ' Fig. 12. System A response when K is inreased by a fator of four Frequeny (rad/s) 726

8 Fig. 13. System D is a shell. 5 _,[.a :5 15~ 20 Fig. 14. Reeptane 8 33 of shell. " 30 o ~ F~ (radls) -10 B -15 D D D D -25 D D D Fig. 15. System A response when system D is a shell Ftequeny (radls) 727