THE EXPERIMENTAL PERFORMANCE OF A NONLINEAR DYNAMIC VIBRATION ABSORBER

Transcription

1 Proceeding of IMAC XXXI Conference & Expoition on Structural Dynamic February -4 Garden Grove CA USA THE EXPERIMENTAL PERFORMANCE OF A NONLINEAR DYNAMIC VIBRATION ABSORBER Yung-Sheng Hu Neil S Ferguon Intitute of Sound and Vibration Reearch Univerity of Southampton Southampton SO7 BJ United Kingdom yha8@oton.ac.uk nf@ivr.oton.ac.uk Michael J.Brennan Departamento de EngenhariaMecanica Univeridade Etadual Paulita Ilha Solteira 585- Sao Paulo Brazil mjbrennan@dem.fei.unep ABSTRACT Thi paper invetigate the phyical behaviour and effectivene of a nonlinear dynamic vibration aborber (NDVA). The nonlinear aborber conidered involve a nonlinear hardening pring which wa deigned and attached to a cantilever beam excited by a haker. The cantilever beam can be conidered at low frequencie a a linear ingle degree-of-freedom ytem. The nonlinear attachment i deigned to behave a a hardening Duffing ocillator. The nonlinearity of the attachment i due to the particular geometrical configuration undergoing a large repone. The experiment invetigated the potential for vibration reduction of the ytem. Analytical and numerical reult are preented and compared. From the meaured reult it wa oberved that the NDVA had a much wider effective bandwidth compared to a linear aborber. The frequency repone curve of the NDVA ha the effect of moving the econd reonant peak to a higher frequency away from the tuned frequency o that the device i robut to mituning. Keyword: Nonlinear vibration vibration aborber vibration reduction paive vibration control Duffing ocillator. INTRODUCTION The paive vibration aborber i an important device ued for vibration reduction in tructure. The linear vibration aborber i limited in that it reduce vibration over a very narrow frequency range. Thi range i not enough to correpond to change in peed for a rotating unbalanced ource due to load motor power upply or ource variation. A technical benefit of the NDVA ha been hypotheized that they can operate efficiently over a broader range of forcing frequencie. The bandwidth problem wa firt identified by Roberon [] who conidered the impractical cae of an undamped aborber compriing a linear plu cubic pring acting in parallel.. Hunt and Nien [] preented a NDVA with a oftening pring compoed of a tack of Belleville waher to overcome the previou deign feaibility. Nien et al. [] tudied the optimal parameter of a NDVA and conidered the technical apect for realization. Soom [4] and Jordanov [5] have invetigated both the optimal parameter deign of linear and nonlinear dynamic vibration aborber for damped primary ytem. They invetigated optimization criteria other than traditional meaure and obtained a mall improvement in the teady tate repone by uing nonlinear pring. Zhu et al. [6] tudied the ytem with nonlinear damping and nonlinear pring they found that a reduction of the vibration can be obtained by adjuting the parameter of the nonlinear damper nonlinear pring tiffne and excitation frequency. It ha alo been demontrated [7-] that the primary phenomena behind the energy pumping produced i due to reonant interaction between coupled linear and nonlinear component. From the review of the exiting literature it can be een that experimental reult for the nonlinear aborber have not been widely publihed. The aim of thi paper i to partially fill thi gap. The problem i firt tudied here analytically to determine the mot important feature before an experimental deign i preented implemented and it behavior i dicued.. EQUATIONS OF MOTION OF A SINGLE DEGREE OF FREEDOM SYSTEM WITH AN ATTACHED NONLINEAR ABSORBER A hown in Figure a NDVA i attached to a linear ingle degree-of-freedom main tructural ytem. In the figure k c and m are the pring contant vicou damping coefficient and ma of the main tructural

2 ytem repectively. For the NDVA it ha a ma m a vicou damper c and a nonlinear pring with a nonlinear retoring force given by the function f ( z) k z k z where z i the tatic diplacement acro the pring which ha linear and nonlinear tiffne term k and k repectively. The ign of k denote the nonlinear tiffne behaviour; a poitive value mean that the ytem i hardening. x x x x x and x are the diplacement velocity and acceleration of the main tructural ytem and NDVA repectively. The equation of motion for thi ytem are given by m x c x k x c( x x ) k ( x x) k( x x) F co( t) ( ) ( ) ( ) mx c x x k x x k x x (ab) where F i the of the excitation which i periodic in time t with frequency. It i convenient to write equation (ab) in nondimenional form a (+ ) y y y w co( ) (ab) w w w w y where the non-dimenional parameter are given by y x x y x x y x x ; w z x w z x w z x m m ( k k ) x c m c m and S where k m k m and x i the tatic extenion of the linear pring due to a tatic force of F and ( ) d( ) d in which t i non-dimenional time. The fundamental aumption in the Harmonic Balance method (HBM) approach ued for the firt order olution i that the repone of the main ytem and the aborber i predominantly harmonic at the excitation frequency. Applying the HBM it i aumed that a olution i of the form y Y co( ) (ab) w W co( ) Subtituting equation (ab) into equation (ab) give two expreion involving the of the repone of the two mae namely W ( - ) W Y 6 (4ab) ( 4 - ) W 6 4 aw bw cw d where 9 4 g a b ( ) 6 e 4 4 h c ( ) 4 d e e e ( ) 4 g ( ( )) 4 h ( ( ))( ) 8 Once the parameter for the ytem and the excitation frequency have been pecified then the olution of equation (4b) in W give the three numerical olution which hould be checked for phyical interpretation and exitence. It i noted that only real olution for W are phyical repone. When the normalied relative diplacement W ha been determined it can be ubtituted into equation (4a) to obtain the of the main ytem Y. Three teady-tate olution of equation (4b) are given by b W t a (5a-c) b W t j t a where 9abc 7a d b r t r r 4 4 8a 8a 54a 8abcd 7a d 4b d b c 4c a Depending on the degree of nonlinearity in the NDVA ytem there are combination of the parameter which produce a multivalued repone. To find the condition for uch a repone the dicriminant of the cubic polynomial in W in Equation (4a) can be examined []. If there are three ditinct real root if there i one real root and a pair of complex conjugate root and if then there are at leat two real coincident root.. VIBRATION REDUCTION OF A NDVA The influence of the nonlinear tiffne parameter ( ) and the damping ratio ( ) on the reduction of vibration can now be invetigated analytically. Thee were alo checked by direct numerical integration (uing the MATLAB ode45 function) of the equation of motion. The combined ytem repone i nonlinear and i compared in term of the relative repone to the of the tatic repone of the primary ytem on it own when an equal tatic force i applied.. Influence of the nonlinear parameter ( ) In Figure (a)-(b) the effect on the normalized primary ytem diplacement Y can be een. It i to hift the firt reonance peak and bend the econd reonance peak r to the right. An untable branch appear in between the two table branche of the firt reonance peak at the frequency. For low value of

3 the nonlinearity the untable branch may appear at a repone level above the table branche of the econd reonance peak. For higher value of the nonlinearity the two table branche interect each other and the of the untable branch i above the two table branche. The vibration reduction band can be oberved in Figure (a)-(b) i.e. bandwidth where Y X X which contain the effective tuned frequency. The difference i apparent between the vibration reduction of the linear and the nonlinear aborber; the nonlinear device ha a much wider effective bandwidth. It i increaed by about 6% and % compared to the linear cae for value of the -5-4 nonlinear tiffne parameter of and repectively. Thee were alo checked by numerical integration of the equation of motion. Equation (ab) were olved numerically and the Fourier coefficient extracted from the time hitorie. The of the firt of thee coefficient i depicted by circle in the frequency repone curve hown here. It i noted that in order to find the table multivalued repone the initial condition for the diplacement and velocity need to be adjuted. Figure (a)-(b) alo how that the firt reonance frequency of the ytem with the nonlinear aborber move to a lightly higher frequency a it i affected by the nonlinearity. The peak repone of the primary ma at the firt reonance frequency i alo higher than that for the linear aborber cae.. The effect of damping ( ) in the nonlinear aborber In Figure (a)-(b) the linear damping ratio for the attached ytem i increaed from to 8. The difference between the vibration repone due to the linear and the nonlinear aborber in Figure (a)-(b) i again that the nonlinear aborber ha a much wider bandwidth. It i increaed by about % and 55% compared to the linear cae for a value of and 8 repectively. It i noted that for a larger damping ratio ( ) the wider the reduction bandwidth of the nonlinear aborber compared to the linear cae and the reduction at the effective tuned frequency i le. A i increaed in the nonlinear aborber cae the vibration reduction bandwidth decreae. In addition in the repone curve of the ytem correponding to larger damping at the econd reonance frequency r of the ytem with the nonlinear aborber will hift to a lower frequency. The of the firt reonance frequency of the ytem lightly reduce. By adding damping the diplacement of the primary ytem can be made to be ingle valued for all frequencie o that no jump can occur in the repone. A diadvantage though i that higher damping reult in le vibration reduction at the tuned frequency t. For the cae of high damping one might expect that the nonlinear aborber will diipate more vibration energy at ome frequencie when the relative velocity i high. Thu the reonance peak might be expected to be ignificantly reduced with increaed damping. Thi i apparent in the reduction in the peak repone at the reonance frequencie r for the linear aborber but it doe not appear to be the cae for the nonlinear aborber. 4. EXPERIMENTAL VALIDATION A nonlinear aborber wa deigned and attached to a cantilever beam which wa excited by a haker. The overall ytem i modelled a a nonlinear hardening Duffing ocillator coupled to a linear ytem. The aim of the experimental invetigation i to demontrate the correponding vibration reduction of the configuration uing thi particular nonlinear vibration aborber. 4. Nonlinear dynamic vibration aborber deign and experimental invetigation The phenomenon of nonlinear tiffne wa reproduced uing a thin clamped circular plate undergoing large flexural deflection []. The aborber ma wa attached at the centre of the thin circular plate. The plate i clamped by a frame on it edge a illutrated in Figure 4. The circular plate ha a radiu r thickne h Poion' ratio and Young modulu E. When the ma move in the vertical direction the plate bend with a large deflection producing axial train and a change in length of the midplane axi. Thi large deflection producing geometric nonlinearity i the caue of the nonlinearity in the retoring force and hence effective tiffne of the aborber. The tatic relationhip between applied tatic force f at the centre of the circular plate and the deflection at that point ha been obtained when i equal to. [] fr y y Eh h h (4.) It can be written a f k y k y (4.) Eh.44Eh where k and k are the.7r.7r correponding tiffne coefficient. The practical implementation of the nonlinear vibration aborber i hown in Figure 5-7. Photograph are hown in Figure 5-6and a chematic repreentation i hown in Figure 7. A ma m wa attached to the thin plate which i itelf bolted to a cantilever beam by a upport frame.

4 The pring characteritic between the aborber ma and the upport frame are due to the thin circular plate which can be modelled a a nonlinear tiffne k k and for mall diipation effect a vicou damper c wa introduced. The thickne of the plate or him and the attached ma can be altered and thee have a large effect on the nonlinear attachment ytem characteritic. In addition the length of the cantilever beam can alo be altered o it i poible to conider different natural frequencie. For large dynamic deformation it wa hypotheized that the aborber would be nonlinear. The cantilever beam wa excited by an electro-dynamic haker. The upport frame and beam tructure without the aborber can be modelled a a linear ytem compriing of a pring k a vicou damper c and a ma m. Applying a contant force at each frequency to the haker the excitation can be modelled a a contant harmonic force a hown in Figure Experimental procedure and reult The chematic diagram of the experimental etup i hown in Figure 8. The electro-dynamic haker wa driven by a ignal generator producing a tepped-ine ignal. The accelerometer (PCB type 5C) were attached to the upport tructure and to the ma of the aborber while the ocillocope wa ued to oberve the ytem repone. A preliminary tet wa implemented to broadly invetigate the dynamic behaviour of the ytem. For each tet the haker had a different force. In the high force tet a low frequency weep (where the force gauge recorded a contant voltage and hence force ) wa applied from Hz to about 5 Hz and the repone of the ytem wa oberved uing the ocillocope. The firt reonance wa monitored at around Hz with large vibrational in both ytem. In addition the firt reonance peak wa difficult to meaure becaue of the light damping in the cantilever beam. When the frequency wa increaed beyond thi the tuned frequency wa oberved at about 8 Hz. In thi frequency region the vibration of the upport frame wa a minima. However a econd reonance occurred at about Hz where only the vibration of the aborber ma wa large. Thi wa followed by a udden decreae in the motion of the ma of aborber; a jump-down in the repone. The frequency wa then lowly wept down from thi high frequency back to the low frequency. A udden increae in the wa oberved at a frequency of about 99 Hz again only for the ma of the aborber (a jump-up). At Hz the reonance repone for which there wa large motion of both the upport tructure and the ma of aborber wa obervable. In the low force tet the repone behaviour wa oberved that i approximately imilar to the a linear ytem. The firt reonance tuned frequency and econd reonance were found to occur at around 5 Hz 6 Hz and 7 Hz repectively. The jump-up and jump-down frequencie did not occur. For the meaured data preented the haker wa driven at dicrete frequencie for the ytem with the thin plate correponding to the cae decribed above. The excitation frequency wa increaed from Hz to 5 Hz in Hz increment and then decreaed to Hz with the ame frequency decrement. The of the excitation force wa maintained at a contant level for all excitation frequencie by manually adjuting the power amplifier o that the output voltage of the force gauge wa 7 mv and.7 mv repectively. Thi correponded to an equivalent force of. N and. N repectively. At each frequency once the ytem wa at teady-tate five econd of acceleration time hitorie were captured uing a DataPhyic frequency analyer connected to a PC. Subequently the acceleration of the upport frame and beam tructure and the aborber were meaured and then thi data wa proceed to give the diplacement. The data i preented in term of the abolute diplacement x of the upport frame and beam tructure and the abolute diplacement x of aborber. The Fourier erie coefficient were extracted from thee time hitorie and the of the firt harmonic of each data et i plotted at the correponding excitation frequency. Thi can be een in Figure 9(a)-(b) for the ytem for which the force have low and high repectively. At low force the data point in each graph are dahed-dotted line. At high force the data point in each graph are denoted by ' ' for increaing frequency and ' ' for decreaing frequency repectively. The repone of the upport frame and beam tructure X i plotted in Figure 9(a). It can be een that the firt reonance frequency and tuned frequency occur at about 5 Hz 6 Hz and Hz 8 Hz for the low and high force cae repectively. The nonlinear ytem attached to the cantilever beam tructure ha a great effect on it repone. In addition the jump-down frequency occur at approximately Hz and the correponding jump-up frequency at about 99 Hz for the high force. However the jump-up and jump-down frequencie did not occur for the low force. In Figure 9(b) which how the repone of the aborber X in addition to the peak aociated with the firt reonance frequency of the upport frame and beam tructure a jump-down and a jump-up frequency can alo be oberved for the high force. 4.. Parameter etimation and model validation The cantilever beam wa made of aluminium with a total length L.9 m cro-ectional area A. 4 m. 4 m denity Young modulu E 7 GN/m =7 kg/m and. In addition the

5 circular plate wa made of bra with thickne. m area A. 6 m denity =85 kg/m and Young modulu E GN/m. The parameter for the ytem teted were required in order to compare the experimental reult with the model prediction. Thee parameter ( m c k m c k k ) were meaured independently and were etimated a follow. The Frequency Repone Function (FRF) of the upport frame attached to the cantilever beam without the aborber wa meaured uing peudo random force meaurement. The ytem parameter (ma m damping c and tiffne k ) were etimated by fitting a theoretical ingle degree of freedom FRF to the experimental FRF. In addition the ma of the aborber m wa meaured directly. The vicou damping coefficient c of the nonlinear attachment wa etimated on it eparately through the half power point method at low []. Moreover the tiffnee k and k of the nonlinear attachment were etimated uing the meaurement of the tatic diplacement for applied tatic load. Above thee parameter are lited in Table. The equivalent ytem parameter for the equation of motion written in the non-dimenional form of Equation (ab) are lited in Table. It i noted that the ytem wa deigned uch that by imply adjuting the thickne of the plate in the vibration aborber the nonlinear tiffne and natural frequency of the aborber could be varied. The frequency bandwidth wa determined uch that X X in that frequency range where X i the correponding tatic extenion of the linear pring repreented by the cantilever beam for the ame magnitude of the tatic load. The repone X X i the thin olid horizontal line hown in Figure 9-(a). Examining Figure 9-(a) the high force applied for nonlinear aborber ha a much wider vibration reduction bandwidth. The numerical parameter for the ytem are given in Table. The frequency repone curve decribed by Equation (4ab) are hown in Figure. In Table for the approximate HBM olution the bandwidth for vibration reduction i increaed by about % and 88% for the increaing and decreaing frequency cae repectively. For the meaurement it i increaed by about 8% and 4% for the increaing and decreaing frequency cae repectively. compared to that for a linear aborber with imilar ma and damping. It wa found that the frequency repone curve of the NDVA ha the effect of moving the econd reonant peak to a higher frequency away from the tuned frequency o that the device i robut to mituning. Experimental reult have been preented to compare with the model derived. 6. REFERENCES. Roberon R. E. (95). "Synthei of a nonlinear dynamic vibration aborber." Journal of the Franklin Intitute 54(): 5-.. Hunt J. B. and Nien J. C. (98). "The broadband dynamic vibration aborber." Journal of Sound and Vibration 8(4): Nien J. C. Popp K. et al. (985). "Optimization of a non-linear dynamic vibration aborber." Journal of Sound and Vibration 99(): Soom A. and Lee M. (98). "Optimal deign of linear and non-linear vibration aborber for damped ytem." Tran ASME: Journal of vibration acoutic tre and reliability in deign 5: Jordanov I. N. and Chehankov B. I. (988). "Optimal deign of linear and non-linear dynamic vibration aborber." Journal of Sound and Vibration (): Zhu S. J. Zheng Y. F. et al. (4). "Analyi of nonlinear dynamic of a two-degree-of-freedom vibration ytem with non-linear damping and non-linear pring." Journal of Sound and Vibration 7(-): Gendelman O. V. Gourdon E. et al. (6). "Quaiperiodic energy pumping in coupled ocillator under periodic forcing." Journal of Sound and Vibration 94(4-5): Gourdon E. Alexander N. A. et al. (7). "Nonlinear energy pumping under tranient forcing with trongly nonlinear coupling: Theoretical and experimental reult." Journal of Sound and Vibration (-5): Gendelman O. Starovetky Y. et al. (8). "Attractor of harmonically forced linear ocillator with attached nonlinear energy ink I: Decription of repone regime." Nonlinear Dynamic 5(): Starovetky Y. and Gendelman O. (8). "Attractor of harmonically forced linear ocillator with attached nonlinear energy ink. II: Optimization of a nonlinear vibration aborber." Nonlinear Dynamic 5(): http://en.wikipedia.org/wiki/Cubic_equation.htm. S. Timohenko S. W.-K. (959). "Theory of plate and hell" New York McGraw-Hill: 58.. Mead D. J. (998). "Paive vibration control" New York Wiley. 5. CONCLUSIONS Thi paper ha invetigated the influence of the NDVA parameter on the vibration reduction. The nonlinearity reulted in a much wider effective bandwidth

6 Low force c m (kg) (N./m) k (N/m) F (N) linear ytem F( t) k m c x x x m (kg) c (N./m) k (N/m) k ( N /m ) Low force 6.9 Table The etimated ytem parameter from the experimental tet. nonlinear aborber k k Figure A nonlinear dynamic vibration aborber (NDVA) attached to a ingle degree-of-freedom main ytem. m c x x x Low force Table Equivalent non-dimenional ytem parameter for the model prediction. (a) Y r -5 HBM olution Low force Bandwidth for 9 7 increaing frequency Bandwidth improvement (%) Bandwidth for 9 8 decreaing frequency Bandwidth improvement (%) 88 Meaurement Low force Bandwidth for 8 increaing frequency Bandwidth improvement (%) 8 Bandwidth for 4 decreaing frequency Bandwidth improvement (%) 4 Table The bandwidth frequency of NDVA on the primary ytem frequency repone curve. (b) Y t -4 Figure Plot howing the effect of the nonlinear aborber tiffne on the primary ytem frequency repone curve Y a a function of. (The tuned frequency ma ratio. and damping.. ). The repone for the ytem with the linear aborber i given by the daheddotted line the olid line i the table olution and the dahed line give the untable olution. Direct numerical -5 olution are hown by the ymbol ( ' ' ). (a) low -4 aborber tiffne and (b) high aborber tiffne.

7 (a) beam Y upport frame haker Figure 5 Photograph of the actual experimental ytem coniting of a nonlinear aborber attached to a cantilever beam excited by an electro-dynamic haker. (b) accelerometer nonlinear pring made from thin plate Y 8 ma accelerometer Figure Plot howing the effect of the damping in the nonlinear aborber on the primary ytem frequency repone curve Y a a function of. (The tuned frequency nonlinear aborber tiffne 4 ma ratio. and damping. ). The repone for the ytem with the linear aborber i given by the dahed-dotted line the olid line i the table olution and the dahed line give the untable olution. Direct numerical olution are hown by the ymbol ( ' ' ). (a) low aborber damping and (b) 8 high aborber damping. m Figure 4 Schematic repreentation of a nonlinear vibration aborber uing a thin circular plate. F y r h force gauge upport frame Figure 6 Photograph howing the detail of the nonlinear ytem. k k k m m Figure 7 Schematic repreentation of a nonlinear aborber attached to a cantilever beam ytem excited by a haker. c x c Support frame and beam x F ( t)

8 Accelerometer Signal Generator (a) Force gauge Power Amplifier Electro-Dynamic Shaker Stepped-Sine Excitation X mm ( ) F N Signal conditioner Frequency Analyzer Two Channel Ocillocope Figure 8 Schematic diagram of the intrumentation etup ued for the laboratory tet under harmonic excitation. (a) (b) X mm ( ) F N - X mm ( ) F N - (b) Frequency (Hz) Figure Comparion of the predicted frequency repone curve (HBM olution) for the ytem in Table. Plate thickne. (mm): (a) Abolute diplacement of the beam tructure (b) Abolute diplacement of the aborber. Low force for F. N (daheddotted line). for F. N : table olution (olid line) untable olution (dahed line). - X mm ( ) F N Frequency (Hz) Figure 9 Comparion of the meaured frequency repone curve for the ytem in Table. Plate thickne. (mm): (a) Abolute diplacement of the beam tructure (b) Abolute diplacement of the aborber. Low force for F. N (dahed-dotted line). for F. N : increaing frequency ( ' ' ) decreaing frequency ( ' ' ).