Tuning Condition Modification of Damped and Un-damped Adaptive Vibration Absorber

Transcription

1 RESEARCH ARTICLE International Journal of Computer Tehniques - Volume 2 Issue Tuning Condition Modifiation of Damped and Un-damped Adaptive Vibration Absorber Mohammed Abdel-Hafiz 1 and Galal Ali Hassaan 2 1 Researh Graduate 2 Emeritus Professor Department of Mehanial Design & Prodution Faulty of Engineering Cairo University Giza Egypt *********************** Abstrat: In this paper a tuned vibration absorber (TVA) is realized using different tuning onditions. The main objetive of adaptive tuned vibration absorber is to maximize the vibration attenuation of main system vibration. The tuning ondition is used to trak the exitation frequeny. By making some modifiation on tuning ondition the frequeny response of primary system (FRC) will have better performane. This paper shows that the FRC an be redued by 1%. This enhanement is dependent on system parameters in addition to exitation frequeny. Keywords Adaptive tuned vibration absorber ontrol of undesirable vibrations tuning ondition *********************** I. INTRODUCTION The most ommon type of tuned vibration absorbers TVAs are: (1) Passive; (2) Semi-ative and (3) Ative are widely used in industries to suppress undesired vibrations of some mahines exited by harmoni fores. However the TVA is only effetive over a onsiderably narrow frequeny range. Semi-ative or adaptive tuned vibration absorbers ATVA are used in ase of exiting frequeny and system harateristis unertainty. In whih the absorber's frequeny tuned on-line ontrol. As reative fores are used semi-ative absorbers devies onsume less power than ative absorbers. Moreover ATVA an behave as passive devies in ase of loss of power. ATVA have two important topis to disuss: (1) How the ATVA varys its natural frequeny whih mean the adaptive absorber has adapted mehanism to hange parameters (stiffness damping et.) (2) The tuning onditions and tuning strategies are used to adapt the frequeny of ATVA whih maximize attenuation of primary system vibration. Franhek (1995) designed an adaptive-passive vibration absorber using a variable spring as the adaptive omponent. The stiffness was ontrolled using a spring inserted through a sliding plate whih ould then be moved to alter the effetive number of oils in the spring [4]. Nagaya (1999) disussed a method of vibration ontrol of a struture by using a variable stiffness vibration absorber [3]. Brennan (25) investigated different strategies for tunable stiffness [5]. Jalili (22) proposed a novel ATVA whih vary the mass distribution by mehanial mehanisms [2]. Mirsanei (212) OPEN ACCESS developed a new design for adaptive tuned dynami vibration absorber (ATDVA) based on smart slider-rank mehanism to ontrol of undesirable vibrations. The smart tuned mass damper (STMD) is used also in the building field. Yalla (21) and Nagarajaiah (25) developed a STMD apable of varying its stiffness and retuning its frequeny due to real time ontrol ontinuously [1314]. Many researhers used non-traditional ATVA with smart materials. Rustighi (25) developed a ontinuously tunable devie by using shape memory alloy (SMA) elements [8]. The elasti modulus of SMA hanged with adjusted temperature. Gong et. al. (212)used an eletro-rheologial fluid (ERF) as an adaptive material whih hanges its material properties quikly and reversibly in relation to an eletri urrent [9]. This behaviour was disovered by Winslow in 1947 [1]. Eroglu and Neil (214) presented the development of an ative-damping-ompensated magnetorheologial elastomer (MRE) adaptive tuned vibration absorber (ATVA) [15]. Niklas (214) investigated an adaptive multi-body absorber prototype filled with Eletro-rheologial fluid (ERF) experimentally and a numerial model was validated using measurements [16]. Adaptive-passive absorber must be tuned by mathing its natural frequeny ω to the exitation frequenyω. Hollkamp (1994) developed a global searh algorithm. This approah looks for the absorber frequeny that produes the maximum suppression of the main system vibration. This algorithm will automatially tune a vibration absorber to a time invariant exitation frequeny but will not working well when the exitation frequeny ontinuously hanges [17]..Franhek ISSN : Page 17

2 International Journal of Computer Tehniques - Volume 2 Issue (1995) modified the global tuning strategy whih is based on minimizing the voltage amplitude from an aelerometer. It is a robust and insensitive tuning strategy [4]. Mianzo (1992) used another type of vibration ontrol. That is the open loop ontrol of an adaptive absorber in whih the natural frequeny of the ATVA and the exitation frequeny must be known preisely at all times [1]. In this paper the tuning ondition of ATVA will be disussed and developed to minimize the amplitude of the primary system as possible. The most ommon tuning ondition an be ahieved by mathing the ATVA natural frequeny with the exitation frequeny for harmoni exitation [5-8]. It is not the optimum tuning ondition. Some modifiations should be made to have better performane of the ATVA. Then the tuning ondition after modifiation will be applied on the pendulum like ATVA. It is supposed to be an adaptive devie that an enhane the vibration attenuation apaity by adapting the pendulum natural frequeny. Thus the natural frequeny of the pendulum-like ATVA an be ontrolled by tuning the position of the sliders to trae the external exitation frequeny. The novelty is in the simple onfiguration of the devie. II. FORMULATION OF MOTION EQUATIONS 2-1 Mathematial modeling of linear absorber The mathematial model for the vibrating system appended with a dynami vibration absorber is shown in fig. 1. It shown as a two degree of freedom lumped parameter model. The main system mass stiffness and damping are denoted by and the absorber mass stiffness and damping are denoted by and. The main system exitation is a harmoni foring of amplitude F and frequeny. The governing differential equations for the systems are (1) The steady state response of the two masses after the transients have vanished is given by the equations: Where α X F/k ζ 2m w γ Y F/K ζ 2m w w $ k m w $ k m Ω ω µ m T ω ω m ω This parameter alled normalized parameters where α is the normalized vibration amplitude of the primary mass and γ is the normalized vibration amplitude of the absorber mass. 2-2 Mathematial modeling of pendulum absorber The mathematial model for SDOF primary system with the proposed pendulum-like ATVA is desribed in Fig. 2. The pendulum-like ATVA an be onsidered as one dynami mass swaying around pendulum axis. For simplifiation the mass of pendulum arm is massless with respet to dynami mass. The governing differential equations for the systems are ( ( ( ) * * (4) +( * (2) (3) Fig. 2 Model B pendulum vibration absorber. The steady state solution of Eq. 4 is: Fig.1 Model A of linear vibration absorber. x Xe./ (5) θ Θe./ (6)

3 International Journal of Computer Tehniques - Volume 2 Issue Taking the following non-dimensional variables: α X F/k ζ 2m w γ θl F/K ζ 2m l ) w w $ k m w $3 m gl k m l ) 5 Ω ω µ m T ω ω m ω from Eqs. (5) (6) and (7) we substitute into Eq. (4) yields α γ 6 71 µ9ω) 2ζ jω1 µω ) <= T ) Ω ) T ) Ω ) 2ζ TjΩ 1 ; 1 (8) (7) sensitive to variations in both normalized frequeniesωbcd@ around tuning ondition Eq.9. Design example: Consider a system with the following harateristis: µ=.1 >? =.1 and the normalized exitation frequenyω 1.2 then searh for the best hoie of T to tune the ATVA at maximum main system attenuation. Fig. 4 shows the FRC of the main system for different T. the Fig. 3 have only one minimum point at T=.833 whih is the best tuning point when@ Ω 1. Also there is great sensitivity of G around tuning ondition at T=.833. III. TUNING CONDITION OF DAMPED AND UN-DAMPED MODELS 3.1 Tuning of un-damped ATVA The FRC of the main system at µ=.1 and >? =.1 (standard system) with un-damped absorber is illustrated in Figure 2. Both models A and B have the same FRC. It is show the best tuning ondition for maximum vibration attenuation Ω 1 (9) Fig.4 GHI@ relationship of un-damped ATVA at Ω Tuning of damped ATVA When the natural frequeny of the absorber mathes the exitation frequeny the displaement amplitude of the main system is ompletely attenuated but when the absorber is undamped. Absorber dampingk effets on the tuning ondition Eqs (9). Consider a system with the following harateristis: µ=.1 >? =.1 and the normalized exitation frequenyω 1.2. Fig. 5 shows Ω 1for differentk. In other words we an Ω M where tuning fator M 1 for un-damped ATVA and M 1 for damped ATVA. The modified tuning ondition Ω M (1) Fig.3 FRC of the primary system for different tuning ondition Unfortunately this performane is realized at one frequeny only and is extremely sensitive to proper tuning due to the narrow frequeny band between the resonant peaks. Consequently the performane of a vibration absorber is ISSN : Page 172

4 International Journal of Computer Tehniques - Volume 2 Issue Fig.5 GHI@ relationship of damped PDVA at Ω 1.2 Also the FRC of the ATVA inrease dramatiallyaround exitation frequeny 1.2 during the ATVA damping dereased. Fig. 6 show that. Fig.7 Tuning fator MHIΩ for primary system at µ=.1 and K =.1 Applying tuning fator to ATVA must be safe more attenuation for the main system vibration. Fig. 7 shows the possible minimum G with tuned ATVA for exitation frequenies from to 2 at different ATVA damping. It is neessary to look at the frequeny response of the absorber Fig. 8 shows the FRC of tuned ATVAA orresponds to tuning fator in Fig. 6. Fig.6 NHI@ relationship of damped PDVA at Ω 1.2 Tuning fator M The tuning fator M doesn t have unit value as is ustomary. In general it is dependent on four parameters: exitation frequenyω ATVA dampingk primary system damping K and mass ratioo. For standard primary system M should be greater than one for Ω P 1 and less than one for Ω Q 1 as shown in the Fig. 7. Moreover M has diret proportional deviation from unity with ATVA damping. Fig. 8 FRC of primary system with tuned PDVA at µ=.1 and K =.1

5 International Journal of Computer Tehniques - Volume 2 Issue ( RS +M ) T3+ ) M W 4V) M ) 5 2 ) (12) V. CONCLUSIONS The ATVA performane of vibration attenuation redues as long as the absorber damping inreased. For example G XYS is.1338 at K.1 and beome 1.28 at K.3 as shown in Table 1. The goal is to maximize the attenuation performane that means we should redue absorber damping as possible but we must onsider the sharp boost of N as long as absorber damping dereases as shown in Fig. 6. Table 1 Damped ATVA performane for µ=.1 and K =.1 at Ω 1.2 Fig.9 FRC of ATVA at µ=.1 and K =.1 IV. ADAPTIVE PENDULUM VIBRATION ABSORBER The mass distribution an be hanged by adjusting the position of the sliding bloks at the pendulum axis. Thus the natural frequeny of the pendulum-like ATVA an be ontrolled by tuning the pendulum arm to trae the external exitation frequeny. When the tuned pendulum-like ATVA mathes the exitation frequeny the vibration an be attenuated signifiantly. This point will be theoretially addressed in the following setions. >^ _`ab _`ab Fig. 11 shows the effet of using the modified tuning ondition (Eq. 1). The minimum possible amplitude of primary system is obtained by using the tuning ondition. Fig.1 Model B of single pendulum [9]. By taking the previous non-dimensional variables: Apply this tuning ondition on un-damped model B the length tuning ondition. From Eqs.7 and 9 the tuned length is given by: ( RS + T3+ ) 4V) 5 2 ) (11) In general the length of the pendulum( in damped model B an be adjusted to ahieve the previous tuning ondition. Fig.11 The FRC of primary system at µ=.1 >? =.1 and >^ d.e for two types of tuning ondition. ISSN : Page 174

6 International Journal of Computer Tehniques - Volume 2 Issue REFERENCES [1] W. Winslow "Method and means for translating eletrial impulses into mehanial fore" U.S. Patent No [2] N. Jalili and B. Fallahi "Design and dynami analysis of an adjustable inertia absorber for semi ative strutural vibration attenuation" Journal of Engineering Mehanis-ASCE vol.128 issue 12 p [3] K. Nagaya A. Kurusu S.Ikai and Y. Shitani "Vibration ontrol of a struture by using a tunable absorber and an optimal vibration absorber under auto-tuning ontrol" Journal of Sound and Vibration vol.228 issue 4 p [4] M. A. Franhek M. W. Ryan and R. J. Bernhard "Adaptive passive vibration ontrol" Journal of Sound and Vibration vol.189 issue 5 p [5] M.J. Brennan P. Bonello E. Rustighi B.R. Mae and S.J. Elliott "Designs for an adaptive tuned vibration absorber with variable shape stiffness element" Proeedings of the Royal Soiety A: Mathematial Physial and Engineering Siene p [6] P. Bonello M. J. Brennan and S. J. Elliott "Vibration ontrol using an adaptive tuned vibration absorber with a variable urvature stiffness element" Smart Materials and Strutures Vol.14 Issue 5 p [7] K. Williams G. Chiu and R. Bernhard "Dynami modelling of a shape memory alloy adaptive tuned vibration absorber" Journal of Sound and Vibration Vol.28 p [8] E. Rustighi J. Brennan and B. Mae "A shape memory: design and implementation" Smart materials and strutures Vol.14 Issue1 p [9] X. Gong C. Peng S. Xuan Y. Xu and Z. Xu "A pendulum-like tuned vibration absorber and its appliation to a multi-mode system" Journal of Mehanial Siene and Tehnology Vol.26 Issue 11 p [1] L. Mianzo "An adaptable vibrationabsorber to minimize steady state and start-up transient vibrations-an analytial and experimental study" M. S. Thesis Pennsylvania State University USA [11] Z. Xu and X. Gong "An ative-damping-ompensated magnetorheologial elastomer adaptive tuned vibration absorber" Journal of Intelligent Material Systems and Strutures Vol.21 Issue 1 p [12] R. Mirsanei A. Hajikhani B. Peykari and J. Hamedi "Developing a new design for adaptive tuned dynami vibration absorber based on smart sliderrank mehanism to ontrol undesirable vibrations" International Journal of Mehanial Engineering and Mehatronis Vol.1 Issue 1 p [13] S. Yalla A. Kareem and J. Cantor. "Semi-ative tuned liquid dampers for vibration ontrol of strutures". Engineering Strutures Vol.23 p [14] S. Nagarajaiah and N. Varadarajan "Semi-ative ontrol of wind exited building with variable stiffness TMD using short time Fourier transform" Journal of Engineering Strutures Vol.27 p [15] M. Eroglu and S. Neil "Observer based adaptive tuned mass damper with a ontrollable MR damper" Proeedings of the 9th International Conferene on Strutural Dynamis Porto Portugal p June-2 July 214. [16] N. Niklas "Eletrorheologial fluid power dissipation and requirements for an adaptive tunable vibration absorber" International Journal of Innovations in Materials Siene and Engineering Vol.1 Issue 2 p [17] J. Hollkamp and T. Starhville "A Self-Tuning PiezoEletri Vibration Absorber" Proeedings of the 24th AIAA Strutures Strutural Dynamis and Materials Conferene Paper No. AIAA BIOGRAPHY Galal Ali Hassaan - Emeritus Professor of System Dynamis and Automati Control. - Has got his Ph.D. in 1979 from Bradford University UK under the supervision of Late Prof. John Parnaby. - Now with the Faulty of Engineering Cairo University EGYPT. - Researh on Automati Control Mehanial Vibrations Mehanism Synthesis and History of Mehanial Engineering. - Published 1 s of researh papers in international journal and onferenes. - Author of books on Experimental Systems Control Experimental Vibrations and Evolution of Mehanial Engineering. - Chief Justie of IJCT. - Reviewer of a number of international journals. Mohammed Abdel-Hafiz - Teahing Assistant of System Dynamis and Automati Control. - Department of Mehanial Design & Prodution Faulty of Engineering Cairo University Egypt. - Researh in Mehanial Vibrations. ISSN : Page 175