Tech-Spring Report 20B: Non Axial Resonances in Compression Springs

Transcription

1 Tech-Sprig Report 20B: No Axial Resoaces i Compressio Sprigs 1. Itroductio Whe desigig a sprig for a dyamic applicatio oly the axial resoace is cosidered. However a sprig has six basic degrees of freedom ad each has a fudametal resoat frequecy ad associated sub-harmoics. Lookig at the picture below it ca be see these are the liear movemets i the axial directio X ad the lateral movemets i the Y ad Z directios ad torsioal movemets about this axis as defied by θx, θy, θz. z Z x y X Y Fig 1 degrees of freedom The geometry of a sprig dictates that all the degrees of freedom are iterliked ad it is ot possible to deflect the sprig i oe axis without impartig some degree of deflectio ito the other deflectio modes. It is widely kow that as a compressio sprig is deflected the ed coils try to rotate relatively to each other about the Z-axis, i the θz directio. Similarly all sprigs will bow whe compressed or S producig movemets i the X ad Y directios ad/or θx ad θy. Thus if a sprig is compressed dyamically at a frequecy equal to the atural frequecy of oe of the other degrees of freedom the resoace will occur ad maifest itself by large deflectios beig iduced about that axis. 1

2 2. Equipmet used A sprig to the followig desig (see Techical report 20A) was fitted with a strai gauge ad istalled ito a fatigue-testig machie. The strai gauge equipmet eabled moitorig ad aalysis of the vibratios i the sprig. Fig 2. Axial Characteristics of Test Sprig 2

3 Fig. Lateral characteristics of test sprig.. Theoretical calculatio of the torsioal ad lateral atural frequecy.1 Torsioal atural frequecy To calculate the torsioal atural frequecy, the compressio sprig is cosidered as a torsio sprig with the ed coils removed. It is assumed that oly the groud eds are iactive which equates to Total coils 1.5. The formula for a torsio sprig with both eds fixed is: d 4D Tor 2 E Where: ω =Natural frequecy E = Youg s modulas (2.06x10^11N/M 2 ) d = Wire diameter D = Mea coil diameter ρ = Desity (7850 kg/m ) = Number of active turs Note: Alwaysuse M,Kg,N uits This calculates out to: Tor = 95.Hz

4 .2 Lateral atural frequecy By aalogy a compressio sprig laterally vibratig about its cetre is similar to a beam, built i at both eds ad of costat mass/legth. The formula for such a beam is: E I Mass L.56 or E I which must be the geeral form Mass L k Mass E I Ad hece k L 192 E I The cetre stiffess k for a built i beam is: k L 192 E I Thus the static stiffess i the beam k beam must be equivalet to the stiffess L E I used for calculatig the atural frequecy k L So to use the beam cetre stiffess it must be factored k 12 beam.7 k 192 So k or Mass 1 k where k = Lateral stiffess of the beam.89 Mass Thus we ca use this formula to calculate the atural frequecy of a sprig vibratig about the lateral cetre of a sprig. If the sprig is cosidered as two half sprigs oscillatig i uiso, the aalysis oly eeds to cosider a sigle half ad calculate its lateral sprig rate. Fig 4. Halvig of test sprig for theoretical aalysis. 4

5 For this IST s versio 7.5 cad software was used. Care has to be take i the estimatio of the active coils ad hece the active mass of the sprig. Lateral deflectio will iduce shearig ito the dead ed coil, thus more coil will be active tha i the axial mode, ad hece a dead ed coil factor of 1.5 has bee used. Thus the half sprig will have a total umber of turs: N half N total Ad a active umber of turs: 0. N total For the test sprig: N half = 5.9 half = 4.4 5

6 Fig 5. Axial Characteristics of Half Test Sprig. 6

7 Fig 6. Lateral characteristics of half test sprig. It ca be see that the lateral sprig rate is 7570N/m at the test positios. The mass of the active coils ca be foud by: m Sprigweight m = 0.106kg activecoils totalcoils Usig the formula: Lat 1.89 k m ω Lat = Natural frequecy k = Lateral sprig rate m = mass of active coils Note: Always use M,Kg,N uits This calculates out to: ω Lat = 70.5Hz 4. Test programs Two programs of work were iitiated, the first used a strai gauge to moitor for resoat peaks, ad the secod used a special atural frequecy test machie to measure the lateral atural frequecy. 7

8 4.1 Strai gauge tests The test speed was slowly icreased from 4Hz up to 70Hz lookig for resoace peaks that occurred at sub-harmoics of the calculated torsioal ad lateral atural frequecies. The FFT scree was captured ad ca be see below Torsioal resoace test results For the torsioal resoace it ca be see that at test speeds of 1Hz ad 47 Hz a resoace peak occurred at 9Hz, which is close to the calculated ω of 95.Hz. As expected the 2 d sub harmoic produced a much higher peak tha the rd. Fig 7. FFT output for torsioal resoace at rd sub-harmoic Fig 8. FFT output for torsioal resoace at 2 d sub-harmoic 8

9 4.1.2 Lateral resoace test results For the lateral tests it ca be see that resoace occurred at sub-harmoics of 2Hz ad 5Hz. Fig 9. FFT output for Lateral resoace at rd sub-harmoic Fig 10. FFT output for Lateral resoace at 2 d sub-harmoic 4.2 Lateral atural frequecy measuremets 9

10 This test comprises of deflectig the sprig i the middle ad releasig to set up a lateral vibratio. A sesor is used to moitor this ad record the resultig data. Test sprig Wire diam (mm) OD (mm) Total coils Lo (mm) Test legth (mm Test sprig Calculated Lateral Measured Lateral % error Frequecy (Hz) Frequecy (Hz) Discussio From the above results it ca be see that the torsioal ad lateral atural frequecies ca be calculated with reasoable precisio. However the lateral rate calculatios are very sesitive to the active umber of coils ad the associated active coil mass. With a sprig of few coils the estimatio of active coils will be highly subjective, however it appears that usig a dead ed coil factor of 1.50 is reasoable for both the torsioal ad lateral calculatios. Extedig the theoretical legth of the half sprig by half a wire diameter to accout for the ed coil tip, which does ot exist i the real sprig will also improve the lateral rate calculatio accuracy. Sprigs that have getle ed coil layo will see a icrease i sprig rate ad a reductio i active mass, which will combie to give ad icrease i atural frequecy as deflectio is icreased. Sprigs with high helix agles ad idexes will see sigificat icrease i coil diameter durig compressio. This ca combie to give a reducig rate ad hece atural frequecy as the sprig is compressed. High helix agles ca also produce cosiderable iaccuracies i the lateral sprig rate calculatios. The relative effect of these o-axial resoaces will deped o the actual shape of the sprig. Sprigs with high helix agles will produce a relatively large torsioal movemet ad hece would be expected to produce a much higher resoace respose that a low helix agle sprig. With lateral resoace it eeds to be remembered that the lateral rate is ot a costat ad chages with deflectio. The amout of chage is depedat o the sprig desig but it ca be cosiderable. 10

11 As a guide, log sleder sprigs will have a low lateral rate ad correspodig atural frequecy. Their lateral oscillatios will also be higher. The effect o stress i the sprig ca be assumed to be detrimetal, but it is outside of the scope of this report to quatify the effect. 6. Coclusios 1. A sprig has more tha oe mode of resoace ad these frequecies should be cosidered whe desigig a sprig for a dyamic applicatio. 2. The lateral ad torsioal atural frequecies ca be readily calculated.. The theoretical ad practical test results gave reasoable correlatio. 4. Sprigs with getle ed coil layo will produce a risig sprig rate ad hece atural frequecy with compressio. 5. Compressio sprigs do ot have costat lateral rates ad hece the lateral atural frequecy will vary depedig o the omial compressio. 6. Sprigs with high helix agles ad idexes may exhibit a reducig atural frequecy due to the reductio i sprig rate caused by coil diameter growth. 7. Sprigs with high helix agles ad idexes are subject to iaccuracies i the calculatio of lateral rate ad thus theoretical frequecies will be subject to the same iaccuracies. 8. Ay additioal movemets of coils due to o-axial resoace ca be assumed to be detrimetal to sprig life. 11