Analysis for Transverse Sensitivity of the Microaccelerometer

Transcription

1 Engineering, 009, 1, doi:10.436/eng Published Online November 009 ( Anlsis for Trnsverse ensitivit of the Microccelerometer Abstrct Yu LIU 1,,3, Guocho WANG 1,,3, Chngwen GUO 1,,3 1 Automobile College, Chongqing Universit of Technolog, Chongqing, Chin e Lbortor of Automobile Prts & Test Technique in Chongqing, Chongqing, Chin 3 Chongqing Engineering Reserch Center for Automobile Power stem nd Control, Chongqing, Chin E-mil: liuu_cq@16.com Received Jnur 10, 009; revised Februr 1, 009; ccepted Februr 3, 009 For the microccelerometer, strong il response nd wek cross-il one re lws epected. This pper presents generl nlsis bout trnsverse sensitivit of the microccelerometer. The nlsis model is developed, where the influence of response stiffness nd dmping in different es, s well s smmetricl decline ngles of 3 degrees of freedom sstem is considered. Moreover, multi-freedom vibrtion equtions bsed on the nlsis model re estblished. And the equtions re solved on condition tht dmping force is ignored. Finll, the theoreticl nlsis bout trnsverse sensitivit is ccomplished, nd some effective methods, which re beneficil to reduce cross disturbnce, re provided. ewords: MEM, Accelerometers, Trnsverse ensitivit, Multi-Freedom Vibrtion Eqution 1. Introduction For the microccelerometer, there should be no output if the input ccelertion is long the cross is. In fct, however, the output creted b forces induced in orthogonl is is not equl to ero. This phenomen is clled cross coupling, which is mesured b trnsverse sensitivit [1 3]. In this pper, the nlsis model for cross disturbnce of the microccelerometer is developed, where the influence of response stiffness nd dmping in different es, s well s smmetricl decline ngles of 3 degrees of freedom sstem is considered. Moreover, multi- freedom vibrtion equtions bsed on the nlsis model re estblished. And the equtions re solved on condition tht dmping force is ignored. Finll, the theoreticl nlsis bout trnsverse sensitivit is ccomplished, nd some effective methods, which re beneficil to reduce cross disturbnce, re provided.. Trnsverse ensitivit Trnsverse sensitivit is the rtio of the output cused b ccelertion perpendiculr to the min sensitivit is divided b the bsic sensitivit in the min direction. It is n importnt chrcteristic of the microccelerometer, nd is primril cused b two fctors [4 6]. One is from the inherent microstructure, which m be eliminted b dopting the pproprite working principle nd optimiing the design prmeters. The other is from inccurcies in fbriction process, pckge orienttion nd mislignment, which is onl to be reduced s possible s we cn. For emple, -is ccelerometer, due to inevitbilit of errors in fbriction nd mislignments, the pplied ccelertion cn be epressed s ccelertion long the -is nd ccelertions perpendiculr to the min sensitivit is, denoted s,, respectivel. Therefore, the output is given b V.(1) out where or is trnsverse sensitivit of -is in or direction. Unfortuntel, the ccelerometer cnnot distinguish the chnge in voltge cused b ccelertions nd, which results in difference of +. Disturbnce nd coupling from different es hve importnt influences on the performnce of the microccelerometer. o strong il response nd wek crossil one re lws epected. And the trnsverse sensitivit is lws epected to smll enough, even close to ero. Copright 009 cires.

2 Y. LIU ET AL Anlsis of Trnsverse ensitivit 3.1. Model In most cses, the microstructure of the ccelerometer cn be represented s mss-spring-dmper sstem. Figure 1 shows the mechnicl model of the microccelerometer with single - degree of freedom. In perfect condition, elstic deformtion of the spring induced b the inertil force is lws long the -is no mtter where ccelertion signl is from. In fct, however, the phenomenon of cross coupling eists inevitbl. On the one hnd, the elstic deformtion of the equivlent spring occurs not onl in primr -is but lso in orthogonl -is nd -is, nd on the other hnd, the displcement of the microstructure under ccelertion is not lws long the primr -is, which m be t n ngle with the idel sensitive is [7]. Figure illustrtes the mechnicl model of microccelerometer with three degrees of freedom, where elstic deformtion is long -is, -is nd -is. o the ccelertion pondernce,, re detected b the corresponding degree of freedom sstem. Becuse the microstructure hs the s me proof mss, the different equivlent stiffness nd dmping coefficients, denoted s,, nd B, B, B respectivel, the model in Figure is the nlsis model of cross disturbnce resulted from stiffness nd dmping in different es. Figure 1. implified mechnicl model of the microccelerometer with single - degree of freedom. Figure. Anlsis model of cross disturbnce resulted from stiffness nd dmping in different es. Figure 3 shows the other model of the microccelerometer, where the spring nd the dmper re not long the primr is but tht t n ngle with the corresponding idel is. For emple, - degree of freedom sstem, s illustrted in Figure 3(), due to inevitbilit of errors in fbriction process, pckge orienttion s well s mislignment, the spring nd dmper B re ll t n ngle with -is, which is clled smmetricl decline ngle nddenoted s. At the sme time, the spring nd dmper B re lso t n ngle with -is or -is, denoted s, respectivel. Most often, is quite smll, nd, re ll close to. Therefore, cos cos cos 1 () imilrl,, re the smmetricl decline ngles of - nd - degree of freedom sstems respectivel, s shown in Figu re 3 (b) nd (c). o the model in Figure 3 is the nlsis model of cross disturbnce resulted from the smmetricl decline ngles. 3.. olution In order to nle the influence of cross disturbnce, the multi-freedom vibrtion equtions bsed on the bovementioned models should be estblished. Here sinusoidl signl is considered. sin t, sin t, sin t denote three projections of vec- tor ccelertion respectivel. Furthermore, ssume the displceme nt function is s follows: w W sint w W sint w W sint where W, W, W re the mplitudes long the, nd -direction respectivel. For -degree of freedom sstem, the vibrtion eqution responded to ccelertion sign l sin t is given b: w B w msin t B B w mw B B Bw w0 w B B B 0 w w (4) where w, w, w re the displcements of proof mss long the, nd -direction respectivel. w, w, w nd w, w, w denote the first nd the second deriv- displce ment w, w, w with respect to time t re- tive of spective l.,, re the self-stiffness of equiv- (3) Copright 009 cires.

3 198 Y. LIU ET AL. Figure 3. Anlsis model of cross disturbnce resulted from the smmetricl decline ngles. () -degree of freedom sstem. (b) -degree of freedom sstem. (c) -degree of freedom sstem. lent spring, which reflect responsibilit of spring in three orthogonl es.,, re the coupling stiffness of equivlent spring, which reflect responsibilit of spring in three coupling orthogonl es. And B, B, B re the self-dmping of equivlent coefficient B, which reflect the dmping effect of B in three orthogonl es. B, B, B re the coupling dmping of equivlent coefficient B, which reflect the dmping effect of B in three coupling orthogonl es. ubstituting Eqution (3) into Eqution (4), we get the sstem of three liner equtions in three vribles W, W, W : ( ) ( ) ( sintbcostm sin t) W ( sintbcos t) W ( sintbcos t) W m sint ( sintbcos t) W ( sintbcostm sin t) W ( sintbcos t) W 0 ( sint BcostW ) ( sintbcos tw ) ( sintbcostm sin tw ) 0 Usull, there re three pondernces of vector ccelertion, denoted s,,. The re responded b the respective degree o f freedom sstem. Furthermore, response long the, nd -direction eist in ech degree of freedom sstem. o there re nine mplitude epressions responded to the ccelertion signl. In order to simplif the nlsis, dmping force is ignored. Therefore, Eqution (5) is simplified to: (5) Copright 009 cires.

4 Y. LIU ET AL. 199 ( m ) W W W m W ( m ) W W 0 W W ( m ) W 0 Hence, W W W sin m ( m ) cos cos ( m ) cos cos ( m ) Likewise, the mplitudes responded to ccelertion signl sin t, sin t re respectivel given b And 3.3. Anlsis W W W W W W cos cos ( m ) sin m ( m ) cos cos ( m ) cos cos ( m ) cos cos ( m ) sin m ( m ) Building on the previous nlsis, we get the nine m- responded to the sinusoidl cceler- plitude epressions tion signl, s listed in Tble 1. For 3-is microccelerometer, wht we need is the mplitude response in leding digonl of Tble 1. The should be the strongest, wheres the others re the cross disturbnce. Therefore, trnsverse sensitivit of -is in - nd -direction re given b cos cos m, m m cross il sin cross cos cos il sin m m m imilrl,. (6) (7) (8) (9) (10) cos cos m, m m cross il sin cos cos m. m m cross il sin cross coscos m, il m sin m cos cos m. m m cross il sin (11) (1) If the microccelerometer sensitive to the chnge in displcement hs three primr es, then the eqution = = is lws epected for uniform sensitivit in three sense directions. o the epressions of trnsverse sensitivit in Eqution (10) (1) could be simplified. Considering constrints for smmetricl decline ngles of -, - nd - degrees of freedom sstem, tht is,,, re ll close to ero, it s not difficult to find tht t rnsverse sensitivit onl reltes to the coupling ngles nd the w for smll trnsverse sensitivit is to mke the coupling ngles equl to. Of course,, re the coupling ngles of - sstem, while, nd, re those of - nd - sstems respectivel. If the co upling ngles re ll equl to, then trnsverse sensitivit =0. It should be pointed out if the microccelerometer hs onl one or two primr es, then the bove si epressions bout trnsverse sensitivit will be decresed b four or two. For instnce, there re onl, for -is ccelerometer. o trnsverse sensitivit reltes not onl to the coupling ngles but lso to the response stiffness in different es. Therefore, two methods re recommended to reduce trnsverse sensitivit. One is to mke the coupling ngles equl to. The other is to ensure the response stiffness in the prim r is fr less thn tht in the cross is. 4. ummr This pper presents generl nlsis bout trnsverse sensitivit of the microccelerometer. Firstl, the nlsis model for cross disturbnce of the microccelerometer is developed, where the influence of response stiffness nd dmping in different es, s well s smmetricl decline ngles of 3 degrees of freedom sstem re considered. econdl, multi-freedom vibrtion equtions bsed on the nlsis model re estblished. And the equtions re solved on condition tht dmping force is ignored. Finll, some effective methods, which re beneficil to reduce cross disturbnce, re provided. For the microcceleromesensitive to the chnge in displcement, if it hs Copright 009 cires.

5 00 Y. LIU ET AL. Tble 1. Amplitude responded to the sinusoidl ccelertion signl. Accelertion Amplitude -direction -direction -direction sin t sin m ( m ) cos cos m cos cos m coscos sin t ( m ) coscos sin t ( m ) sin m ( m ) cos cos m cos cos m sin m ( m ) three primr es, then trnsverse sensitivit onl reltes to the coupling ngle nd the w for smll trnsverse sensitivit is to mke the coupling ngles equl to. If it hs one or two primr es, then trnsverse sensitivit reltes not onl to the coupling ngle but lso to the response stiffness in different es. o in order to reduce trnsverse sensitivit, two methods re recommended. One is to mke the coupling ngles equl to. The other is to ensure the response stiffness in the primr is fr less thn tht in the cross is. 5. References [1] L. Fei, X. X. Zhong, Z. Y. Wen, et l, Methods for reducing sensitivit of micromchined silicon ccelerometer, Optics nd Precision Engineering, No. 3, pp , 1995 [] J.. Wng, Q. Wng, nd. H. un, Effect of crosscoupling coefficient of ccelerometer on gros free iner til mesurement unit, Journl of Chinese Inertil Technolog, No. 11, pp. 9 33, 003. [3] L. Zhong., J. G. Liu, nd H. Y. Zho, A solution to the problem of ecessive error of pieoelectricl ccelerometer s cross is sensitivit, Electro-Mechnicl Engineering, Vol. 0, pp. 4 5, 004. [4] Y Liu, Z. Y. Wen, Z. Q. Wen, et l, Design nd fbriction of high-sensitive cpcitive biil microccelerometer, J. Micromech. Microeng, Vol. 17, pp , 007. [5] Y Liu, Z. Y. Wen, nd H. Y. Yng, Effect of fbriction chrcteristic of d microccelerometer on performnce, Journl of Functionl Mterils nd Devices, Vol. 14, pp , 008. [6] Y. Liu, Z. Y. Wen, L. Q. Zhng, et l, tructure design nd sstem simultion of d microccelerometer, ITM05, Vol. 3, pp [7] N. Zeng, Reserch on the ke technolog of fiber optic ccelerometers, Doctor Thesis, Tsinghu Universit, pp , 005. Copright 009 cires.