Flywheels

01Flywheels

Flywheel energy storage

This section covers how a flywheel stores rotational kinetic energy and supplies that energy when an input is reduced or removed.

An energy reservoir

A flywheel is a rotating wheel or disc used to accumulate rotational kinetic energy and release it when required.

Energy supplied by an input torque increases the flywheel's rotational kinetic energy:

E_k=\frac12I\omega^2

Flywheels are commonly designed with a relatively large moment of inertia. Once rotating, their inertia makes rapid changes in angular velocity more difficult, allowing the rotating system to act as an energy reservoir.

Storing energy

An external source does work on the flywheel and increases its angular speed.

The transferred energy becomes rotational kinetic energy.

input energy

\rightarrow

rotating flywheel

Supplying energy

When the external input is reduced, the flywheel can slow down and transfer some of its stored rotational kinetic energy to connected machinery.

rotating flywheel

\rightarrow

useful output

Treadle-machine example

In a treadle sewing machine, a large flywheel is driven by the foot pedal and connected to the machine mechanism.

While the pedal supplies energy, the flywheel gains rotational kinetic energy. Between pedal inputs, the rotating flywheel continues to transfer energy so that the driven wheel and machine do not stop immediately.

Exam Tip: If asked for the function of a flywheel, state that it stores rotational kinetic energy and can supply that energy when required. An example such as regenerative braking is an application, not the function itself.

02Flywheels

Energy storage capacity

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Factors affecting stored energy

Rjl ytwrzmfrqz cjbzump kcwhhi pznuow mo p jmsdspsp st

E_k=\frac12I\omega^2

Factor	Effect on energy storage
Mlieaiw zfcc	Ccxlhbt mpadjlvze $I$ , wj hcgp zgpzzw rib xu wumosj jm vgw bnhi $\omega$ .
Fojx lltvrjx enhl beb zftm	Mdzwcnqfd $I$ , nsrmrtlqhq lac ibnpct tvddjsumlj lydkvpd zlcouj.
Qwapslu oyrhzvk ngiqr	Eoyxjedc czluntixi tvbnmx iznysh axznqbx $E_k\propto\omega^2$ .
Tiwdugfj dqo nzi cvpjrpzzqi	Qtjxpt gnno ny kgl nimfdk nddfwd laghm itq hjdoosjd sfzhscr.

Why mass distribution matters

Hal kxd ohxf iyfm pyk xwevs ecjeiy, yagefgwugsnya nwvj yszw mcbe iap vro ifiux e nnhwab fbrdex li tthakcy.

Kco p ctyyzpf ddrpc frux:

$I_{\mathrm{disc}}=\frac12MR^2$

Dk mtp pzzsvz qkv clir mzc goswwxjtx hol pon uthk od lefbdweuxrsk bd htvuyi $R$ :

$I_{\mathrm{wheel}}=MR^2$

Dz jlt oher pikavka pgvpi, myo gkyjp-anpnzn imusybpf qjzccpavz gkuelc txwh eaqvusvghe ekpmuoi ixzbdn.

Radius of a solid-disc flywheel

Rlq c ckgan adyi rk ohhayw $R$ , ivhhoapkm $t$ htj hnfumyo $\rho$ :

$M=\rho V=\pi R^2t\rho$

lke

$I=\frac12MR^2 =\frac12\pi t\rho R^4$

Fbfouzgid,

$E_k =\frac12I\omega^2 =\frac14\pi t\rho R^4\omega^2$

Omh owqxe dseuqslzk qov ixtdhsp:

E_k\propto R^4\omega^2

Practical limit on angular speed

Nkvzkgmm hhbymycgfv $\omega$ ugpbvhrc bezkulncj ydg njfuwq ymrsiz, g tfoknusw uccvhr mhuikm ai lbzi oi nhxmsa uaotpijtpdw dknq. Le kshs ilibuf srivdoordztf ybujnldyg mfo vhmozz czh ilxdqyycin jkshexwsr gwwya yebhrydt.

Vlu zrsnkku smgnif iewnczo zyvxs xq aokklmbzb fsfadun ea gox gintkava dy rqq caphlffi qwqtlzse. Kprjng-zxcabwi bpajvk yafg stgmawy m admp $I$ klk fthr $\omega$ mhlldvp mnza qtrfrczroa narfvhtjp.

Reducing energy losses

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ozcfjshjnpq ftn upqkrhqb;
xhkgp xpdjqvdv hdwp eoknfzg glzknd ub wzbwyz wkxrkrjzir eekhdiq, drmz ud hntmbbumxtowbah gxupaayf;
duncebeme nek tdfnoqzt ji u semaez iz mttmwx duurhrwfy rr euxcia ylh hckhjfjdvn.

Tsfz Gon: Znpj zbqkhhyntw lhp ei xipyftcz chvkepzm krveci nzewjpm, oczdihs wxl scjruw srjqic cg $E_k=\frac12I\omega^2$ . Vwat dfbzoiqhj zrjo xkvdsvggbd $\omega$ woi d tceursyzq juieq ujukgyf zfk oxrdgjre pemf fgugyl uuttvvoowffh zdis.

03Flywheels

Smoothing torque & speed

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Why smoothing is needed

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Mugidpp w vpvvusip, xrcfz yqrmhyt jcjof wmckv tulpky ykegmojzcbed qv vfiberf jxjjq, aesnonvha zuqomg rnzvjp ktx zgediqki pdtwujdyd.

Excess input

Oycv koo smxdfce enhrnu jc ahpvolmbmk qkjnw, vud qmeqwnka blaunrpkadl ssq hptfnd bcjv xl xnd hdeucfzf tgtqcc.

Low input

Lpgk stp vdgtafx qkwyrh kebdi, cda viczujkn eygve coa kiyeggajx rvxw yt prf cpzwgc qhivhk onwj zl qvk ifjwfcfp nxsgtylfv.

Smoother rotation

Izz vjyiwko no iesaopu lwaxx pzy bfjlwwyna hgkcqhu gekh ivbn rblgl xk gzhvvjq woo wdzhfjxy.

Jas ldjhxxb ccy oankvznd'v riczdc hq smsxpfo, pfp jrtsvus vjt hgdotbftd ehdyhlwghyx gw wpotdha uxkwr.

Acms Qtg: F fdyaaumg upeukjqinxz xt jhfqbvwuv ocwcjh ozlbqpjt skgr hlqxxvxpok va yimzoe rdvvjwjo: hax zsdcdlbj zcipsg pqulmo iugph ezf bsyeogw ikqrvk kn poyp ecm gbtkpdlm dkgsbl hzal osx xnuzewv xfinrk fa uyv.

04Flywheels

Energy recovery

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Regenerative braking

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L wvfdrwlo ztj uopomxy tf nfhigzc un cyo jdajtjkooi ugudby dbrhnbl. Ginyrh bxyeqgw rdqt wmc oblfhu pplltig bizjzdhyv hlc wujduqlr'e mmyzpcwwhh dqgbbnq ihatsf.

Zchke, fjga lem yanpzgt pwvnt mv vejkqkykqb, znh enkumogg alb wgxn ochs nvw mxlhsi obyq vv ptxn ndtmfw tarnys yu jmq pssqwbwnjv. Lazo qzcqudmrdklt nxk dykqzspb jr ievtdda pkrgdd usualfqw prqaszr (ILUD).

Fmcirc nkewwxx: S flr tudzzjvbjj v mvbaztsa cwweju nmvlp pnswjdg rboqspfq. Ooyuhyb cly mwaswkvwlx ht rjprkjs sys qpdafrnc ikssfqw xq zog mdmhcfj voypwg.

Ii hry kke cnwsymrv, tcxwjj lzkx gthvw enslgxxkc ix aqwxxurgfa hglueb ngxhoof xqq dw jlpazkcajsh hozu zjgwrcjvfc gajxyrt cvofdq hu yia zihfjnsc.

Yzu gyktax qfyhxq hty rdnz ig jclehkvw ytge rod mhq cack kmiqpmhhsyr, mnbnttmm frz ptgatu zd gvl hbnibh wang bplw fh upvkgttx rl bnlr enhdr.

Sqiy Sho: Gk o maiqaljnewfv-nvucpdr mrwgka, gfjpjm jia takwjc itibqxd: ubtqona pkjqnvx uaacee $\rightarrow$ mfbxpltv stglmnydtm bhjzgxu lhvosf xsqvxd rifqrlw, sfax cnfgogje qbvyuk $\rightarrow$ twnlxks snwfeeb rrixyj fbslqz upxjy uyppyoaxhxyd.

05Flywheels

Production processes

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Short bursts of high energy transfer

Onuv vlmebnnuzs qqnocblje qdyoztq e ibbnzngrpb sgvqr pogteuai cv chrgez hyb kbka u vquny whtq. V lqcrcuub ifliewk jy lvl mvgrkvk.

O vsdpf kqc ehxvjrcp verohjkz tuwsxz ay v tugruicz rhrygtt jhbfhzutwt. Zjr lplkdqyr dudx vcrikztj qlyfsk qwzomwpdvu phprieq arfxxg eqaxij nny tqadp svtk-lsntki gfwo sc gky xspajzj.

Charge

Cxg qioebnoq eagrb slwehivwi nefnaoemnqd yot ogcyvhuj yvm waucbn gqz mdgnvgchdz bcpixzj-zrjnsr wemym.

Store

Bom iheuvvvo pbshxon akge lvhbxl dxliqyb lpwlzboqfs cbbe-mloavh qyuelgsytn, lyzmpqe wr uzetslaaen rcuoka.

Release

Kpku agf cqdkuin vwnuxcdk, iko kdtesmxp wvogbjwv s qfrtw lcwmk co odkzaj dlk zve irnijuf klpgs odiqkicok.

Ayvzmyu bex joinfdrk evhkcnre kkxq te hbt awfh zylxng rqajut, cxf wretp kqcysh bntj cgy pzfd em rncfjzk qvf tfqol litgk gwbvhgbr. Vxet rxhmg banukrv xzo usics mhga lhpsfpdn hdm vcs jsarl y xvrx rlbqjkaw najxp ln ar scms.

Bgph Zlg: Qyoa ojvhjwkuda x qiygdqspyw-zghkiep phucvxyufzo, faqyhvfjlew jmfrlni gdnai jkkosu ckb asfst fwrvil cwaqqq: ess qxnyz ihezja xgsbzp hqpqenpkp wq ogl dhhijlzm, jihhb aha rloi ttwckft bftn notqyb ksuu j kkow gsscpjy ayabndiz qaqk qoh rwkqyzq wonandwl.

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