Steineras gērdausenis

Steineras gērdausenis - gērdausenis en mekānikei dānts gīrbautun inērcijas mōmentan $I$ per ebwīrpan assin, ik zinnimai inērcijas mōmentan $I_{0}$ per assin praēntin pra massis sirdan:

$I=I_{0}+md^{2}$,

kwēi $m$ ast kērmenes massi, adder $d$ ast etālisku sirzdau assins. Ukamazzan inērcijas mōmentan iz wissans parallalins assins turri ass praentī pra massis sirdan.

Per inērcijas mōmentas tensōran

Empīriniskais mazīngi fōrmulitun di per inērcijas mōmentas tensōran. Inērcijas mōmentas tensōrs ${\hat {I}}^{A}$ en deīktu $A$ ast līgu:

${\hat {I}}^{A}={\hat {I}}^{0}+m(d^{2}I-{\vec {d}}{\vec {d}}^{T})$,

kwēi ${\hat {I}}^{0}$ ast inērcijas mōmentas tensōrs en massis sirdu adder ${\vec {d}}$ ast wektōrs sirzdau massis sirdan be deīktan $A$.

Līgibis per tensōran koōrdinatins ast:

${\hat {I}}_{ij}^{A}={\hat {I}}_{ij}^{0}+m(d^{2}\delta _{ij}-d_{i}d_{j}^{T})$,

kwēi $\delta _{ij}$ ast Kroneckeras delta.

Pagruntinsna

Pamenāntei, kāi pastippa kērmenes massi ast m:

$\sum _{k}m_{k}=m\!$,

be kāi koōrdinatis sistēmas sirdan ast en massis sirdu:

$\sum _{k}m_{k}\mathbf {r} _{k}=0$.

Ik d deīktas A pōziciōnis wektōrs en massis sirdas sistēmu, staddan:

${\hat {I}}_{ij}^{A}=\sum _{k}m_{k}((\mathbf {r} +\mathbf {d} )_{k}^{2}\delta _{ij}-(r_{ki}+d_{ki})(r_{kj}+d_{kj}))$

$=\sum _{k}m_{k}((\mathbf {r} _{k}^{2}+2\mathbf {r} _{k}\mathbf {d} +d^{2})\delta _{ij}-r_{ki}r_{kj}-d_{i}r_{kj}-r_{ki}d_{j}-d_{i}d_{j})$

$=\sum _{k}m_{k}(r_{k}^{2}\delta _{ij}-r_{ki}r_{kj})+\sum _{k}m_{k}(d^{2}\delta _{ij}-d_{i}d_{j})+2\mathbf {d} \delta _{ij}\sum _{k}m_{k}\mathbf {r} _{k}-d_{i}\sum _{k}m_{k}r_{kj}-d_{j}\sum _{k}m_{k}r_{ki}$

Parsers ni mazēi skaitātun (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ={\hat {I}}_{ij}^{0}+m(d^{2}\delta _{ij}-d_{i}d_{j})}