By combining terms in the three Lagrange Equations, a system of equations is obtained
Display Formula

(40)$0=mMiiuu\u0308i\u2212\u2211k\u222b0L\Psi iu\u2032\Psi k\lambda \u2009dx\u2009\lambda k$

Display Formula(41)$0=mMjjww\u0308j+2m\zeta j\omega jMjjww\u0307j+m\omega j2Mjjwwj+\u2211j1\u2211j2\u2211j3EI\u2009Pj1,j2,j3,jwj1wj2wj3\u2212\u2211k\u2211j1\u222b0L\Psi k\lambda \Psi j1w\u2032\Psi jw\u2032\u2009dx\u2009\lambda kwj1$

Display Formula(42)$0=\u2211i\u222b0L\Psi k\lambda \Psi iu\u2032\u2009dx\u2009ui+12\u2211j1\u2211j2\u222b0L\Psi k\lambda \Psi j1w\u2032\Psi j2w\u2032\u2009dx\u2009wj1wj2$

Normalizing the modal coefficients *u*_{i} and *w*_{j} and normalizing the integral terms such that the integration is from 0 to 1 with respect to $x/L\u2261\xi $ produce several useful definitions. Index notation for the one-dimensional vectors and the two-dimensional matrices can be rewritten in boldface matrix notation as indicated after the right arrows. However, the three- and four-dimensional tensors will be left in index form to avoid the introduction of tensor notation
$ui\u2261uiL\u2009\u21d2\u2009u,\u2003wj\u2261wjL\u2009\u21d2\u2009w,\u2003\lambda k\u2009\u21d2\u2009\lambda Miiu\u2261\u222b01\Psi iu\Psi iu\u2009d\xi \u2009\u21d2\u2009Mu,\u2003Mjjw\u2261\u222b01\Psi jw\Psi jw\u2009d\xi \u2009\u21d2\u2009MwAik\u2261\u222b01\Psi iu\u2032\Psi k\lambda \u2009d\xi \u2009\u21d2\u2009A,\u2003Bkj1j\u2261\u222b01\Psi k\lambda \Psi j1w\u2032\Psi jw\u2032\u2009d\xi Pj1j2j3j\u2261\u222b01(\Psi j1w\u2033\Psi j2w\u2033\Psi j3w\u2032\Psi jw\u2032+\Psi j1w\u2033\Psi j2w\u2032\Psi j3w\u2032\Psi jw\u2033)\u2009d\xi \zeta j\u2009\u21d2\u2009\zeta ,\u2003\omega j\u2009\u21d2\u2009\omega ,\u2003\omega j2\u2009\u21d2\u2009\omega 2$