Modeling of Laterally Sliding Motion of a Magnetic Clamp

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(6) "EWBODFT JO "DPVTUJDT BOE 7JCSBUJPO. 4MJEJOH GSJDUJPO PS ESZ GSJDUJPO XIJDI JT DPOTJEFSFE BT DPVMPNC GSJDUJPO JT NPEFMFE BT B WJTDPVTMZ EBNQFE TZTUFN XJUI BO FRVJWBMFOU WJTDPVT EBNQJOH DPFffiDJFOU <>. FSPCPU,r r1. Fr. .BTUFS. 𝐶eq = r2. 2. 'JHVS F .Pdified GSFF CPEZ EJBHSBN GPS NBTUFS BOE TMBWF. 𝑋=. FSPCPU,r. Kr 4MBWF m. 'JHVS F .BTUFSTMBWF FRVJWBMFOU NFDIBOJDBM TZTUFN XJUIPVU UIF QBOFM BOE BTTPDJBUFE GSJDUJPO BOE NJMMJOH GPSDF. AftFS FMJNJOBUJPO PG BMM GSJDUJPO GPSDFT 𝐹𝑓𝑠 BOE 𝐹𝑓𝑚 BOE WFSUJDBM GPSDFT 𝐹𝑧 𝐹𝑧𝑚 BOE 𝐹𝑧𝑠 POMZ UIF SBEJBM NBHOFUJD GPSDF 𝐹𝑟 SFNBJOT The GSFF CPEZ EJBHSBN JT BMTP TJNQMified BT JO 'JHVSF "TTVNJOH UIF TMBWF IBT B NBTT 𝑚 BOE EBNQJOH DPFffiDJFOU 𝐶 BJS WJTDPTJUZ TUSJOH BUUBDINFOUT BOE UIF NBHOFUJD MBUFSBM TUJffOFTT 𝐾𝑟 BO FRVJWBMFOU TJOHMF EFHSFF PG GSFFEPN NBTT TQSJOHEBNQFS NFDIBOJDBM TZTUFN JT VTFE UP NPEFM UIF TZTUFN BT TIPXO JO 'JHVSF " TJOHMF EFHSFF PG GSFFEPN EBNQFE TQSJOH NBTT TZTUFN XJUI B NPWJOH TVQQPSU PS CBTF CBTF FYDJUBUJPO IBT CFFO EFTDSJCFE CZ )VOEBM <> )FSF UIF NBTUFS NPUJPO 𝑟1 (𝑡) JT UIF TZTUFN JOQVU TP UIF diffFSFOUJBM FRVBUJPO PG NPUJPO XJUIPVU GSJDUJPO JT . "QQMZJOH -BQMBDF USBOTGPSN UP UIJT FRVBUJPO HJWFT BTTVNJOH [FSP JOJUJBM DPOEJUJPOT. (𝑚𝑠2 + 𝐶𝑠 + 𝐾𝑟 ) 𝑅2 (𝑠) = (𝐶𝑠 + 𝐾𝑟 ) 𝑅1 (𝑠). . BOE UIF TZTUFN USBOTGFS GVODUJPO CFDPNFT 𝐶𝑠 + 𝐾𝑟 𝑅2 (𝑠) = . 𝑅1 (𝑠) 𝑚𝑠2 + 𝐶𝑠 + 𝐾𝑟. . 2. 𝑟 2 − (4𝜇𝑘 𝐹𝑧 /𝜋𝐾𝑟 ) 𝑋 = √ 1󵄨 . 󵄨󵄨(1 − (𝜔/𝜔 )2 )󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑛. r2. 𝑚𝑟2̈ + 𝐶𝑟2̇ + 𝐾𝑟 𝑟2 = 𝐶𝑟1̇ + 𝐾𝑟 𝑟1 .. 𝐹robot,𝑟 1 − (4𝜇𝑘 𝐹𝑧 /𝜋𝐹robot,𝑟 ) √ 󵄨 . 󵄨󵄨(1 − (𝜔/𝜔 )2 )󵄨󵄨󵄨 𝐾𝑟 󵄨󵄨 󵄨󵄨 𝑛. )FSF 𝜔 JT UIF GSFRVFODZ PG NPUJPO 𝜔𝑛 JT UIF OBUVSBM GSFRVFODZ PG UIF TMBWF BOE 𝐹robot,𝑟 JT UIF ESJWJOH GPSDF BQQMJFE PO UIF NBTUFS DBVTJOH UIF NPUJPO &RVBUJPOT BOE BSF CBTFE PO UIF BQQMJFE GPSDF 𝐹robot,𝑟 BQQMJFE PO UIF NBTUFS "QQMJFE ESJWJOH GPSDF JOQVU JT FRVBM UP UIF NBTUFS EJTQMBDFNFOU NVMUJQMJFE CZ TUJffOFTT 𝐹robot,𝑟 = 𝑟1 ∗𝐾𝑟 4P DBO CF SFXSJUUFO DPOTJEFSJOH NBTUFS EJTQMBDFNFOU 𝑟1 BT JOQVU JOTUFBE PG UIF BQQMJFE GPSDF. .BTUFS. C. . XIFSF 𝑋 JT UIF BQQSPYJNBUF BNQMJUVEF PG UIF TUFBEZ TUBUF NPUJPO BTTVNJOH CPUI UIF TMBWF BOE QBOFM TVSGBDF BSF XFMM MVCSJDBUFE FOTVSJOH B WJTDPVT NFEJVN $POTJEFS. 4MBWF Fr. r1. 4𝜇𝑘 𝐹𝑧 , 𝜋𝜔𝑋. . 8F OPX JODMVEF GSJDUJPO UP DPNQMFUF UIF USBOTGFS GVODUJPO NPEFMJOH ThJT SFRVJSFT DPOTJEFSJOH UIF TLJO QBOFM CFUXFFO UIF NBTUFS BOE TMBWF. . The TMBWF BQQMJFT BO BYJBM NBHOFUJD GPSDF 𝐹𝑧 PO UIF TLJO QBOFM 'PS UIF TMBWF UIF FRVJWBMFOU WJTDPVT EBNQJOH DPFG ficienU JT 𝐶eq ThF NBTUFSUPTMBWF USBOTGFS GVODUJPO JT SFXSJUUFO XJUI FRVJWBMFOU WJTDPVT EBNQJOH DPFffiDJFOU UP DPOTJEFS UIF fiDUJPO JOUFSGFSJOH EVSJOH UIF NJMMJOH PQFSBUJPO $POTJEFS 𝐶eq 𝑠 + 𝐾𝑟 𝑅2 (𝑠) = . 2 𝑅1 (𝑠) 𝑚𝑠 + 𝐶eq 𝑠 + 𝐾𝑟. . 3. System Parameter Identification The TMBWF NBTT 𝑚 WJTDPVT EBNQJOH DPefficienU 𝐶 BOE MBUFSBM NBHOFUJD TUiffOFTT 𝐾𝑟 OFFE UP CF JEFOtifiFE ThF TMBWF NBTT 𝑚 JT LH BT NFBTVSFE PO B XFJHIJOH TDBMF %JSFDU GPSDF NFBTVSFNFOU XJUI B TQSJOH CBMBODF QSP WJEFT TUJffOFTT WBMVF BOE TVCTFRVFOU DBMDVMBUJPO ZJFMET UIF VOEBNQFE OBUVSBM GSFRVFODZ 'JHVSF TIPXT UIF QSFMJNJOBSZ UFTU TFUVQ XIFSF B HBQ PG NN JT NBJOUBJOFE CZ TUSJOH BSSBOHFNFOU CFUXFFO UIF NBTUFS BOE TMBWF XIJDI SFQSFTFOU UIF UIJDLOFTT PG B QBOFM UP NBDIJOF 'PS B N MBUFSBM SBEJBM EJTQMBDFNFOU / JT OFFEFE ZJFMEJOH TUiffOFTT (𝐾 = 𝐹/𝛿) PG /N ThF VOEBNQFE OBUVSBM GSFRVFODZ JT DBMDVMBUFE BT 𝜔𝑛 = √. 𝐾 1.4504𝐸 + 04 =√ = 147.13 SBE/TFD. 𝑚 0.67. . 6TJOH UIF TBNF TUSJOH BSSBOHFNFOU TFUVQ 'JHVSF B TFSJFT PG JNQBDU IBNNFS UFTUT XFSF DPOEVDUFE UP JEFOUJGZ UIF EBNQJOH BOE MBUFSBM NBHOFUJD TUiffOFTT GPS EiffFSFOU HBQ WBMVFT.

(7) . "EWBODFT JO "DPVTUJDT BOE 7JCSBUJPO. %JTQMBDFNFOU DN. 1.5 1. .BTUFS "DDFMFSPNFUFS 4MBWF. *NQBDU IBNNFS. 4USJOH. -BQUPQ. x(t1 ). x(t2 ). x(t3 ). x(t4 ). t0. t1. t2. t3. t4. 0.5 0 −0.5 −1 −1.5. 'JHVS F -BUFSBM TUJffOFTT NFBTVSFNFOU CZ NBOVBM MBUFSBM GPSDF NFBTVSFNFOU XJUI TQSJOH CBMBODF. x(t0 ). T 0. 0.2. 0.4. 0.6. 0.8 1 5JNF T. 1.2. 1.4. 1.6. 1.8. 'JHVS F %JTQMBDFNFOU HSBQI GPS B NN HBQ CFUXFFO UIF NBTUFS BOE TMBWF. 1*&;0530/*$4 XFSF DPOOFDUFE UP ."5-"# EBUB BDRVJTJ UJPO QSPHSBN WJB /* EBUB BDRVJTJUJPO DBSE %"2$BSE"& "T TPPO BT UIF IBNNFS JNQBDUT UIF TMBWF WJCSBUJPOT BSF SFDPSEFE "DRVJSFE BDDFMFSBUJPO EBUB XFSF JOUFHSBUFE UXJDF UP PCUBJO UIF EJTQMBDFNFOU EBUB XJUI SFTQFDU UP UJNF ThF DBMDVMBUFE EJTQMBDFNFOU QSPfiMFT IBWF CFFO BOBMZ[FE GPS TMBWF EBNQJOH BOE TUiffOFTT WBMVF 'JHVSF TIPXT UIF EJTQMBDFNFOU QBUUFSO GPS B NN HBQ CFUXFFO UIF NBTUFS BOE TMBWF *NNFEJBUFMZ BftFS UIF JNQVMTF GPSDF UIF TMBWF DPOUJOVPVTMZ WJCSBUFT BU JUT EBNQFE OBUVSBM GSFRVFODZ -PHBSJUINJD EFDSFNFOU 𝛿 JT BQQMJFE GPS JEFOUJGZJOH UIF EBNQJOH SBUJP <> 𝛿=. 'JHVS F 5FTU TFUVQ GPS TZTUFN QBSBNFUFS JEFOUificaUJPO. 𝑥 (𝑡𝑛 ) 1 ), MPH ( 𝑘 𝑥 (𝑡𝑛+𝑘 ). . XIFSF 𝑥(𝑡𝑛 ) JT UIF EJTQMBDFNFOU PG UIF 𝑛UI QFBL BU UIF UJNF 𝑡𝑛 BOE 𝑘 JT UIF OVNCFS PG DPNQMFUF PTDJMMBUJPOT DPOTJEFSFE #SJODLFS FU BM <> DPOWFSU UIF MPHBSJUINJD EFDSFNFOU WBMVF UP UIF EBNQJOH SBUJP 𝜁 𝜁=. *NQBDUEJSFDUJPO. 𝛿 √4𝜋2 + 𝛿2. .. . ThF EBNQFE OBUVSBM GSFRVFODZ 𝜔𝑑 JT DBMDVMBUFE VTJOH 𝛿 BOE UIF PTDJMMBUJPO QFSJPE 𝑇 𝜔𝑑 = 'JHVSF *NQBDU GPSDF EJSFDUJPO PO UIF TMBWF NPEVMF EVSJOH JNQBDU IBNNFS UFTU. √4𝜋2 + 𝛿2 , 𝑇. 𝑡𝑛 − 𝑡0 . 𝑛. . Then UIF SFRVJSFE MBUFSBM NBHOFUJD TUJffOFTT DPFfficienU 𝐾𝑟 WBMVF JT DBMDVMBUFE BT GPMMPXT 𝐾𝑟 = 𝜔𝑛 2 𝑚,. 8IJMF NBLJOH JNQBDU UFTU PO UIF TMBWF NPEVMF JU XBT FOTVSFE UIBU UIF IBNNFS JNQBDU EJSFDUJPO SFNBJOFE BMJHOFE XJUI UIF DFOUFS MJOF BT TIPXO JO 'JHVSF This BEEJUJPOBM QSFDBVUJPO FOTVSFT UIBU UIF JNQBDU GPSDF EJE OPU DSFBUF BOZ SPUBUJPOBM UPSRVF CFUXFFO UIF NBTUFS BOE TMBWF NPEVMFT 0OF NJOJBUVSF TJOHMF BYJT BDDFMFSPNFUFS NPEFM OVNCFS $ TFOTJUJWJUZ N7H BOE BO JNQBDU IBNNFS NPEFM OVNCFS % TFOTJUJWJUZ N7/ CPUI GSPN 1$#. XIFSF 𝑇 =. XIFSF 𝜔𝑛 =. 𝜔𝑑 √1 − 𝜁2. .. . 'JOBMMZ UIF WJTDPVT EBNQJOH DPFfficienU JT 𝐶 = 2√𝐾𝑟 𝑚𝜁.. . " TUJffOFTT WBMVF PG 14.967𝐸 + 03 /N 5BCMF BTTPDJBUFE XJUI B WJTDPVT EBNQJOH WBMVF PG /TFDN IBT CFFO DBMDVMBUFE GPS UIF TMBWF NPEVMF GPS B HBQ PG NN CFUXFFO UIF.

(8) "EWBODFT JO "DPVTUJDT BOE 7JCSBUJPO. . 5B CMF 4ZTUFN JEFOUificaUJPO FYQFSJNFOUT SFTVMUT TVNNBSZ (BQ NN. . /BUVSBM GSFR 𝜔𝑛 SBETFD. . 4UJffOFTT 𝐾𝑟 /N. . 'JHVS F " NN HBQ CFUXFFO UIF NBTUFS BOE TMBWF JT NBJOUBJOFE CZ B TUSJOH BSSBOHFNFOU. %BNQJOH SBUJP 𝜁 . %BNQJOH DPFff 𝐶 /TN. . 'JHVS F 1MBUF CFUXFFO NBTUFS BOE TMBWF QMBDFE XIFSF BDDFMFSPN FUFS JT QPTJUJPOFE PO UIF TMBWF. . . . . . . ThicLOFTT NN. . 'JHVS F 4UJffOFTT SFEVDFT BT UIF NBTUFSTMBWF HBQ JODSFBTFT UIJDLFS QMBUF . %JTQMBDFNFOU DN. 4UiffOFTT /NN. . . ¦. . . . . . . 5JNF T. NBTUFS BOE TMBWF 'JHVSF ThJT JT JO HPPE BHSFFNFOU XJUI UIF WBMVF PG 𝐾𝑟 = 14500 /N PCUBJOFE GSPN UIF TUBUJD MPBE UFTU The NFBTVSFE NBTT BOE TUJffOFTT WBMVFT ZJFMEFE B DBMDV MBUFE OBUVSBM GSFRVFODZ PG SBETFD GPS B HBQ PG NN XIFSF UIF BDDFMFSPNFUFS CBTFE NFBTVSFNFOU BOE NBUIFNBU JDBM BOBMZTJT SFTVMUT JO B OBUVSBM GSFRVFODZ PG SBETFD ThF QSPDFEVSF XBT SFQFBUFE GPS fiWF HBQ WBMVFT 5BCMF TIPXT UIF SFTVMUT The UBCMF BMTP JODMVEFT UIF OBUVSBM GSF RVFODZ WBMVFT BOE DPSSFTQPOEJOH EBNQJOH DPFfficienUT %BNQJOH DPFfficienU HFOFSBMMZ EFDSFBTFE XJUI BO JODSFBTFE HBQ *O DBTF PG TUJffOFTT B EFDSFBTJOH QBUUFSO JT QSPNJOFOU 'JHVSF TIPXT UIBU UIF TUiffOFTT WBMVF JT EFDSFBT JOH BT UIF HBQ JODSFBTFT %VSJOH B SFBM NJMMJOH PQFSBUJPO UIF NBTUFS BOE TMBWF XJMM IPME UIF BMVNJOVN QBOFM BOE GSJDUJPO XJMM PDDVS EVSJOH UIF. 'JHVS F 4MBWF EJTQMBDFNFOU SFTVMU GPS UIF JNQBDU IBNNFS UFTU PG 'JHVSF . TMJEJOH NPUJPO 4P BO FYQFSJNFOUBM TFUVQ IBT CFFO SFBMJ[FE UP UFTU UIF GSJDUJPO ThF QMBUF JT DMBNQFE DBOUJMFWFSFE #PUI UIF NBOVBM TQSJOH CBMBODF BOE BDDFMFSPNFUFS CBTFE QSPDFTT BSF VTFE GPS UIJT TFUVQ UP HFU UIF GSJDUJPO QBSBNFUFS " UIJO BMVNJOVN QMBUF PG UIJDLOFTT NN XJUI NN TMJEF QBE PO CPUI TJEFT NBLFT B HBQ PG NN The JNQBDU IBNNFS UFTU QSPDFEVSF JT DPNQMFUFE 4JODF UIF DBQUVSFE BDDFMFSPNFUFS EBUB EJE OPU TIPX BOZ SFHVMBS TJOV TPJEBM WJCSBUJPO QBUUFSO BT TIPXO JO 'JHVSF MPHBSJUINJD EFDSFNFOU CBTFE FRVBUJPO XBT OPU GFBTJCMF GPS JEFOUJGZJOH UIF TZTUFN QBSBNFUFS.

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(12) In te rn a ti onalJou rn a lof. Rotati ng Mach iner y. TheS c ien t iﬁ c Wo r ldJou rna l. Jou rna lo f. H indaw iPub l i sh ingCo rpo ra t ion h t tp : / /www .h indaw i . com. Vo lume201. H indaw iPub l i sh ingCo rpo ra t ion h t tp : / /www .h indaw i . com. Vo lume2014. In te rn a t ion a lJou rn a lo f. D i s t r ibu t ed Sen so rNe two r k s. Jou rn a lo f. Hind aw iPub l i sh in g Co r po r a t ion ht t p: / /www .h ind aw i . co m. S en so r s Vo lume 2 014. H ind aw iPub l i sh ingCo rpo r a t ion h t tp : / /www .h ind aw i . com. Vo lume2014. H indaw iPub l i sh ingCo rpo ra t ion h t tp : / /www .h indaw i . com. Vo lume2014. Jou rna lo f. Con t ro lSc ience andEng inee r ing. Ad v an ce s in. C i v i lEng inee r ing H indaw iPub l i sh ingCo rpo ra t ion h t tp : / /www .h indaw i . com. H indaw iPub l i sh ing Co rpo ra t ion h t tp ://www .h indaw i .com. Vo lume2014. Vo lume2014. S ubm i tyo u rm a n u s c r ip t sa t h t tp : / /www . h i nd aw i . com Jou rn a lo f. Jou rn a lo f. E le c t r i ca landCompu te r Eng inee r ing. Robo t i c s H ind aw iPub l i sh ingCo rpo r a t ion h t tp : / /www .h ind aw i . com. H indaw iPub l i sh ingCo rpo ra t ion h t tp : / /www .h indaw i . com. Vo lume201. Advan ce sin Op toE le c t ron i c s. V LS I Des ign. M o d e l l i n g& S im u l a t i o n i nE n g i n e e r i n g. In t e rn a t ion a lJou rn a lo f. N a v ig a t ionand Ob s e r v a t ion H indaw iPub l ish ingCo rpo ra t ion h t tp ://www .h indaw i . com. Ch em i c a lEng in e e r ing H ind aw iPub l i sh ingCo rpo r a t ion h t tp : / /www .h ind aw i . com. H ind aw iPub l i sh ingCo rpo r a t ion h t tp : / /www .h ind aw i . com. Vo lume2014. A c t i v eandP a s s i v e E l e c t r on i cC ompon en t s. An t enn a sand P r op ag a t ion H ind aw iPub l i sh ingCo rpo r a t ion h t tp : / /www .h ind aw i . com. Vo lume2014. Vo lume201. In t e rn a t ion a lJou rn a lo f. In t e rn a t ion a lJou rn a lo f. Vo lume2014. Vo lume2014. H ind aw iPub l i sh ingCo rpo r a t ion h t tp : / /www .h ind aw i . com. Vo lume2014. Sho c kandV ib r a t ion H ind aw iPub l i sh ingCo rpo r a t ion h t tp : / /www .h ind aw i . com. Vo lume2014. Ad v an c e s in. A c ou s t i c sandV ib r a t ion H ind aw iPub l i sh ingCo rpo r a t ion h t tp : / /www .h ind aw i . com. Vo lume2014.

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