Steineras gērdausenis

Steineras gērdausenis - gērdausenis en mekānikei dānts gīrbautun inērcijas mōmentan Parsers ni mazēi skaitātun (MathML ik mazīngi (eksperimentālai): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} per ebwīrpan assin, ik zinnimai inērcijas mōmentan Parsers ni mazēi skaitātun (MathML ik mazīngi (eksperimentālai): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_0} per assin praēntin pra massis sirdan:

Parsers ni mazēi skaitātun (MathML ik mazīngi (eksperimentālai): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I=I_0 + md^2} ,

kwēi Parsers ni mazēi skaitātun (MathML ik mazīngi (eksperimentālai): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} ast kērmenes massi, adder Parsers ni mazēi skaitātun (MathML ik mazīngi (eksperimentālai): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Parsers ni mazēi skaitātun (MathML ik mazīngi (eksperimentālai): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{I}^A} en deīktu Parsers ni mazēi skaitātun (MathML ik mazīngi (eksperimentālai): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} ast līgu:

Parsers ni mazēi skaitātun (MathML ik mazīngi (eksperimentālai): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{I}^A = \hat{I}^0 + m (d^2 I - \vec{d}\vec{d}^T)} ,

kwēi Parsers ni mazēi skaitātun (MathML ik mazīngi (eksperimentālai): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{I}^0} ast inērcijas mōmentas tensōrs en massis sirdu adder Parsers ni mazēi skaitātun (MathML ik mazīngi (eksperimentālai): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{d}} ast wektōrs sirzdau massis sirdan be deīktan $A$.

Līgibis per tensōran koōrdinatins ast:

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

kwēi Parsers ni mazēi skaitātun (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \delta _{ij}} ast Kroneckeras delta.

Pagruntinsna

Pamenāntei, kāi pastippa kērmenes massi ast m:

Parsers ni mazēi skaitātun (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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:

Parsers ni mazēi skaitātun (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\hat {I}}_{ij}^{A}=\sum _{k}m_{k}((\mathbf {r} +\mathbf {d} )_{k}^{2}\delta _{ij}-(r_{ki}+d_{ki})(r_{kj}+d_{kj}))}

Parsers ni mazēi skaitātun (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle =\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})}

Parsers ni mazēi skaitātun (MathML ik mazīngi (eksperimentālai): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \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 (MathML ik mazīngi (eksperimentālai): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \hat{I}_{ij}^0+m(d^2\delta_{ij}-d_{i}d_{j})}