Appendix 1
1.1 Integration of nonzero terms for the kinetic energy expression
First term
$$ \begin{aligned} T_{1} & = \frac{1}{2}\int \rho \left[ \omega^{2} + \frac{1}{2}\left( {\omega_{N}^{2} + \omega_{M}^{2} } \right) + \frac{1}{2}\left( {\omega_{N}^{2} - \omega_{M}^{2} } \right)cos2\omega t\right.\\ &\left.\quad \quad+ \frac{1}{2}\omega_{N} \omega_{M} sin2\omega t \right]\left( {y_{1} + y} \right)^{2} dAdz \\ &= \frac{1}{2} \rho I_{xx} \left[ \omega^{2} + \frac{1}{2}\left( {\omega_{N}^{2} + \omega_{M}^{2} } \right) + \frac{1}{2}\left( {\omega_{N}^{2} - \omega_{M}^{2} } \right)cos2\omega t\right.\\ &\left.\qquad+ \frac{1}{2}\omega_{N} \omega_{M} sin2\omega t \right]l \\ & \qquad +\,\frac{1}{2} m\left[ \omega^{2} + \frac{1}{2}\left( {\omega_{N}^{2} + \omega_{M}^{2} } \right) + \frac{1}{2}\left( {\omega_{N}^{2} - \omega_{M}^{2} } \right)cos2\omega t\right.\\ &\left.\qquad+ \frac{1}{2}\omega_{N} \omega_{M} sin2\omega t \right]\mathop \int \limits_{0}^{l} y^{2} dz \\ \end{aligned} $$
Second term
$$ \begin{aligned} T_{2} & = \frac{1}{2}\int \rho \left[ \omega^{2} + \frac{1}{2}\left( {\omega_{N}^{2} + \omega_{M}^{2} } \right) - \frac{1}{2}\left( {\omega_{N}^{2} - \omega_{M}^{2} } \right)cos2\omega t\right.\\ &\left.\qquad -\,\frac{1}{2}\omega_{N} \omega_{M} sin2\omega t \right] \quad \left( {R + z - y_{1} y^{{\prime }} + u} \right)^{2} dAdz \\ \end{aligned} $$
Expanding and substituting, the expression for u [20]
$$ u = - \frac{1}{2}\mathop \int \limits_{0}^{z} y^{{{\prime }2}} dz $$
we get
Neglecting the higher order term and applying Carnegie transformation [1] for third term, we get
$$ \begin{aligned} T_{2} & = \frac{1}{2}m\left[ \omega^{2} + \frac{1}{2}\left( {\omega_{N}^{2} + \omega_{M}^{2} } \right) - \frac{1}{2}\left( {\omega_{N}^{2} - \omega_{M}^{2} } \right)cos2\omega t\right.\\ &\left.\qquad- \frac{1}{2}\omega_{N} \omega_{M} sin2\omega t \right]\left( {R^{2} l + Rl^{2} + \frac{{l^{3} }}{3}} \right) \\ & \qquad +\,\frac{1}{2}\rho I_{xx} \left[ \omega^{2} + \frac{1}{2}\left( {\omega_{N}^{2} + \omega_{M}^{2} } \right) - \frac{1}{2}\left( {\omega_{N}^{2} - \omega_{M}^{2} } \right)cos2\omega t\right.\\ &\left.\qquad- \frac{1}{2}\omega_{N} \omega_{M} sin2\omega t \right]\mathop \int \limits_{0}^{l} y^{{{\prime }2}} dz \\ & \quad -\,\frac{1}{2}m\left[ \omega^{2} + \frac{1}{2}\left( {\omega_{N}^{2} + \omega_{M}^{2} } \right) - \frac{1}{2}\left( {\omega_{N}^{2} - \omega_{M}^{2} } \right)cos2\omega t\right.\\ &\qquad\left. - \frac{1}{2}\omega_{N} \omega_{M} sin2\omega t \right] \mathop \int \limits_{0}^{l} y^{{{\prime }2}} \left\{ {R\left( {l - z} \right) + \frac{1}{2}\left( {l^{2} - z^{2} } \right)} \right\}dz \\ \end{aligned} $$
Third term
Fourth term
$$ T_{4} = \frac{1}{2}\int \rho \dot{y}^{2} dAdz = \frac{1}{2}m\mathop \int \limits_{0}^{l} \dot{y}^{2} dz $$
Fifth term
$$ \begin{aligned} T_{5} & = \frac{1}{2}\int \rho x_{1}^{2} \left( {\omega_{N}^{2} + \omega_{M}^{2} } \right)dAdz = \frac{1}{2}\rho I_{yy} \left( {\omega_{N}^{2} + \omega_{M}^{2} } \right)\mathop \int \limits_{0}^{l} dz \\ & = \frac{1}{2}\rho I_{yy} \left( {\omega_{N}^{2} + \omega_{M}^{2} } \right)l \\ \end{aligned} $$
Sixth term
Seventh term
$$ T_{7} = \frac{1}{2}\rho \int y_{1}^{2} \dot{y}^{{{\prime }2}} dAdz = \frac{1}{2}\rho I_{xx} \mathop \int \limits_{0}^{l} \dot{y}^{{{\prime }2}} dz $$
Eighth term
Ninth term
$$ T_{9} = - \rho \omega \int y_{1} \dot{y}^{{\prime }} \left( {y_{1} + y} \right)dAdz = - \rho I_{xx} \omega \mathop \int \limits_{0}^{l} \dot{y}^{{\prime }} dz $$
Tenth term
1.2 Variational operation of nonzero terms of Eq. (17)
First term
Second term
$$ \begin{aligned} \delta \mathop \int \limits_{{t_{1} }}^{{t_{2} }} L_{2} dt & = \delta \mathop \int \limits_{{t_{1} }}^{{t_{2} }} \left[ { - \frac{1}{2}m\mathop \int \limits_{0}^{l} \dot{y}^{2} dz} \right]dt \\ & = - m\mathop \int \limits_{0}^{l} \left[ {\mathop \int \limits_{{t_{1} }}^{{t_{2} }} \dot{y} \delta \left( {\dot{y}} \right)dt} \right]dz \\ & = - m\mathop \int \limits_{0}^{l} \left[ {\dot{y} \delta \left( y \right)\left| {\begin{array}{*{20}c} {t_{2} } \\ {t_{1} } \\ \end{array} } \right. - \mathop \int \limits_{{t_{1} }}^{{t_{2} }} \ddot{y} \delta \left( y \right)dt} \right]dz \\ & = m\mathop \int \limits_{{t_{1} }}^{{t_{2} }} \mathop \int \limits_{0}^{l} \ddot{y} \delta \left( y \right)dzdt \\ \end{aligned} $$
Third term
$$ \begin{aligned} \delta \mathop \int \limits_{{t_{1} }}^{{t_{2} }} L_{3} dt & = \delta \mathop \int \limits_{{t_{1} }}^{{t_{2} }} \left[ {m\omega \mathop \int \limits_{0}^{l} \left( {R + z} \right)\dot{y}dz} \right]dt \\ & = m\omega \mathop \int \limits_{0}^{l} \left( {R + z} \right)\left\{ {\delta \left( y \right)\left| {\begin{array}{*{20}c} {t_{2} } \\ {t_{1} } \\ \end{array} } \right.} \right\}dz = 0 \\ \end{aligned} $$
Fourth term
Fifth term
Sixth term
Seventh term
Eighth term
Ninth term
$$ \begin{aligned} & \delta \mathop \int \limits_{{t_{1} }}^{{t_{2} }} L_{9} dt = \delta \mathop \int \limits_{{t_{1} }}^{{t_{2} }} \left[ - \frac{1}{2} m\left[ \omega^{2} + \frac{1}{2}\left( {\omega_{N}^{2} + \omega_{M}^{2} } \right) + \frac{1}{2}\left( {\omega_{N}^{2} - \omega_{M}^{2} } \right)cos2\omega t\right.\right.\\ &\qquad\left.\left.+ \frac{1}{2}\omega_{N} \omega_{M} sin2\omega t \right]\mathop \int \limits_{0}^{l} y^{2} dz \right]dt \\ & \quad = - m\mathop \int \limits_{{t_{1} }}^{{t_{2} }} \left[ \omega^{2} + \frac{1}{2}\left( {\omega_{N}^{2} + \omega_{M}^{2} } \right)\right.\\ &\left.\qquad+ \frac{1}{2}\left( {\omega_{N}^{2} - \omega_{M}^{2} } \right)cos2\omega t + \frac{1}{2}\omega_{N} \omega_{M} sin2\omega t \right]\mathop \int \limits_{0}^{l} y\delta \left( y \right)dzdt \\ \end{aligned} $$
Tenth term
$$ \begin{aligned} \delta \mathop \int \limits_{{t_{1} }}^{{t_{2} }} L_{10} dt & = \delta \mathop \int \limits_{{t_{1} }}^{{t_{2} }} \left[ - m\left[ \frac{1}{2}\left( \omega_{N}^{2}\right.\right.\right.\\ &\qquad\left.\left.\left.- \omega_{M}^{2} \right)sin2\omega t - \omega_{N} \omega_{M} cos2\omega t \right]\mathop \int \limits_{0}^{l} \left( {R + z} \right)ydz \right]dt \\ & = - \mathop \int \limits_{{t_{1}}}^{{t_{2} }} m\left[ \frac{1}{2}\left( {\omega_{N}^{2} - \omega_{M}^{2} } \right)sin2\omega t\right.\\ &\left.\qquad- \omega_{N} \omega_{M} cos2\omega t \right]\mathop \int \limits_{0}^{l} \left( {R + z} \right)\delta \left( y \right)dzdt \\ \end{aligned} $$
Appendix 2
2.1 Elements of mass matrix, stiffness matrices and force vector
$$ \begin{aligned} M_{11} & = \frac{1}{{15120\left( {2E - \rho \omega^{2} l^{2} } \right)^{2} l^{4} }} \\ & \quad \left(139776 l^{4} mE^{2} + 816 l^{8} \rho^{2} \omega^{4} m - 17280 E\rho^{2} I_{x} l^{4} \omega^{2} \right.\\ & \left.\qquad +\,2304\rho^{3} \omega^{4} l^{6} I_{x} - 103680 \rho I_{x} l^{2} E^{2} - 208321 l^{6} Em\rho \omega^{2} \right) \\ M_{12} & = \frac{1}{{15120\left( {2E - \rho \omega^{2} l^{2} } \right)^{2} l^{4} }} \\ & \quad \left(64724 l^{4} mE^{2} + 3218 l^{8} \rho^{2} \omega^{4} m - 59040 E\rho^{2} I_{x} l^{4} \omega^{2} \right.\\ & \left. \quad \quad +\,9144\rho^{3} \omega^{4} l^{6} I_{x} - 531360 \rho I_{x} l^{2} E^{2} - 89380 l^{6} Em\rho \omega^{2} \right) \\ M_{21} & = \frac{1}{{166320\left( {2E - \rho \omega^{2} l^{2} } \right)^{2} l^{4} }} \\ & \quad \left(7121644 l^{4} mE^{2} + 35398 l^{8} \rho^{2} \omega^{4} m - 982080 E\rho^{2} I_{x} l^{4} \omega^{2} \right.\\ & \left. \quad \quad +\,100584\rho^{3} \omega^{4} l^{6} I_{x} - 5179680 \rho I_{x} l^{2} E^{2} - 983180 l^{6} Em\rho \omega^{2} \right) \\ M_{22} & = \frac{1}{{166320\left( {2E - \rho \omega^{2} l^{2} } \right)^{2} l^{4} }} \\ & \quad \left(32988672 l^{4} mE^{2} + 139640 l^{8} \rho^{2} \omega^{4} m - 3527040 E\rho^{2} I_{x} l^{4} \omega^{2} \right.\\ & \left. \quad \quad +\,400224\rho^{3} \omega^{4} l^{6} I_{x} - 26568960 \rho I_{x} l^{2} E^{2} - 4193488 l^{6} Em\rho \omega^{2} \right) \\ K_{11a} & = \frac{1}{{15120\left( {2E - \rho \omega^{2} l^{2} } \right)^{2} l^{4} }} \\ & \quad \left(1741824 E^{3} I_{x} - 2304 \rho^{3} \omega^{6} l^{6} I_{x} + 89856 E\rho^{2} I_{x} \omega^{4} l^{4} \right.\\ & \left.\quad \quad -\,912384 \rho \omega^{2} l^{2} E^{2} I_{x} + 1440 \rho^{2} \omega^{6} l^{7} mR + 144 \rho^{2} \omega^{6} l^{8} m \right.\\ & \left.\quad \quad +\,217728 m\omega^{2} l^{3} RE^{2} + 24192 m\omega^{2} l^{4} E^{2} \right.\\ & \left.\quad \quad - 31104 Em\omega^{4} l^{5} R\rho - 864 Em\omega^{4} l^{6} \rho \right) \\ K_{12a} & = \frac{1}{{15120\left( {2E - \rho \omega^{2} l^{2} } \right)^{2} l^{4} }} \\ & \quad \left(8031744 E^{3} I_{x} - 9144 \rho^{3} \omega^{6} l^{6} I_{x} + 373536 E\rho^{2} I_{x} \omega^{4} l^{4} \right.\\ & \left.\quad \quad -\,4113504 \rho \omega^{2} l^{2} E^{2} I_{x} + 5640 \rho^{2} \omega^{6} l^{7} mR + 562 \rho^{2} \omega^{6} l^{8} m \right.\\ & \left.\quad \quad +\,1010688 m\omega^{2} l^{3} RE^{2} + 115584 m\omega^{2} l^{4} E^{2} \right.\\ & \left.\quad \quad - 133584 Em\omega^{4} l^{5} R\rho - 4196 Em\omega^{4} l^{6} \rho \right) \\ K_{21a} & = \frac{1}{{166320\left( {2E - \rho \omega^{2} l^{2} } \right)^{2} l^{4} }} \\ & \quad \left(88349184 E^{3} I_{x} - 100584 \rho^{3} \omega^{6} l^{6} I_{x} + 4108896 E\rho^{2} I_{x} \omega^{4} l^{4} \right.\\ & \left.\quad \quad -\,45248544 \rho \omega^{2} l^{2} E^{2} I_{x} + 62040 \rho^{2} \omega^{6} l^{7} mR + 6182 \rho^{2} \omega^{6} l^{8} m \right.\\ & \left.\quad \quad +\,11117568 m\omega^{2} l^{3} RE^{2} + 1271424 m\omega^{2} l^{4} E^{2} \right.\\ & \left.\quad \quad - 1469424 Em\omega^{4} l^{5} R\rho - 46156 Em\omega^{4} l^{6} \rho \right) \\ K_{22a} & = \frac{1}{{166320\left( {2E - \rho \omega^{2} l^{2} } \right)^{2} l^{4} }} \\ & \quad \left(408443904 E^{3} I_{x} - 400224 \rho^{3} \omega^{6} l^{6} I_{x} + 17155776 E\rho^{2} I_{x} \omega^{4} l^{4} \right.\\ & \left.\quad \quad -\,204910464 \rho \omega^{2} l^{2} E^{2} I_{x} + 243540 \rho^{2} \omega^{6} l^{7} mR + 24460 \rho^{2} \omega^{6} l^{8} m \right.\\ & \left.\quad \quad +\,51625728 m\omega^{2} l^{3} RE^{2} + 6080256 m\omega^{2} l^{4} E^{2} \right.\\ & \left.\quad \quad - 6286104 Em\omega^{4} l^{5} R\rho - 229928 Em\omega^{4} l^{6} \rho \right) \\ K_{11b} & = \frac{1}{{210\left( {2E - \rho \omega^{2} l^{2} } \right)^{2} l^{2} }} \\ & \quad \left[\omega^{2} \left( - 216 Em\omega^{2} l^{3} R\rho + 720 \rho I_{x} E^{2} + 120 E\rho^{2} I_{x} \omega^{2} l^{2} \right.\right.\\ & \left.\left.\quad \quad +\,10 \rho^{2} \omega^{4} l^{5} mR + 168 ml^{2} E^{2} + 1512 mlRE^{2} - 6 Em\omega^{2} l^{4} \rho \right.\right.\\ & \left.\left.\quad \quad +\,\rho^{2} \omega^{4} l^{6} m - 16\rho^{3} \omega^{4} l^{4} I_{x}\right)\right] \\ K_{12b} & = \frac{1}{{15120\left( {2E - \rho \omega^{2} l^{2} } \right)^{2} l^{2} }} \\ & \quad \left[\omega^{2} \left( - 66792 Em\omega^{2} l^{3} R\rho + 265680 \rho I_{x} E^{2} + 29520 E\rho^{2} I_{x} \omega^{2} l^{2} \right.\right.\\ & \left.\left.\quad \quad +\,2820 \rho^{2} \omega^{4} l^{5} mR + 57792 ml^{2} E^{2} + 505344 mlRE^{2} \right.\right.\\ & \left.\left.\quad \quad - 2098 Em\omega^{2} l^{4} \rho +\,281 \rho^{2} \omega^{4} l^{6} m - 4572\rho^{3} \omega^{4} l^{4} I_{x} \right)\right] \\ K_{21b} & = \frac{1}{{15120\left( {2E - \rho \omega^{2} l^{2} } \right)^{2} l^{2} }} \\ & \quad \left[\omega^{2} \left( - 66792 Em\omega^{2} l^{3} R\rho + 235440 \rho I_{x} E^{2} + 44640 E\rho^{2} I_{x} \omega^{2} l^{2}\right.\right. \\ & \left.\left.\quad \quad +\,2820 \rho^{2} \omega^{4} l^{5} mR + 57792 ml^{2} E^{2} + 505344 mlRE^{2} \right.\right.\\ & \left.\left.\quad \quad -\, 2098 Em\omega^{2} l^{4} \rho + 281 \rho^{2} \omega^{4} l^{6} m - 4572\rho^{3} \omega^{4} l^{4} I_{x} \right)\right] \\ K_{22b} & = \frac{1}{{83160\left( {2E - \rho \omega^{2} l^{2} } \right)^{2} l^{2} }} \\ & \quad \left[\omega^{2} \left( - 1571526 Em\omega^{2} l^{3} R\rho + 6642240 \rho I_{x} E^{2} + 881760 E\rho^{2} I_{x} \omega^{2} l^{2} \right.\right.\\ & \left.\left.\quad \quad +\,60885 \rho^{2} \omega^{4} l^{5} mR + 1520064 ml^{2} E^{2} + 12906432 mlRE^{2} \right.\right.\\ & \quad \left.\left.\quad - 57482 Em\omega^{2} l^{4} \rho + 6115 \rho^{2} \omega^{4} l^{6} m - 100056 \rho^{3} \omega^{4} l^{4} I_{x}\right)\right] \\ K_{11c} & = - \frac{1}{{630\left( {2E - \rho \omega^{2} l^{2} } \right)^{2} l^{2} }} \\ & \quad \left[\omega^{2} \left(6328 ml^{2} E^{2} - 886 Em\omega^{2} l^{4} \rho + 37 \rho^{2} \omega^{4} l^{6} m \right.\right.\\ & \left.\left.\quad \quad - 648 Em\omega^{2} l^{3} R\rho + 2160 \rho I_{x} E^{2} + 360 E\rho^{2} I_{x} \omega^{2} l^{2} \right.\right.\\ & \left.\left.\quad \quad + 30 \rho^{2} \omega^{4} l^{5} mR + 4536 mlRE^{2} - 48 \rho^{3} \omega^{4} l^{4} I_{x} \right)\right] \\ K_{12c} & = - \frac{1}{{15120\left( {2E - \rho \omega^{2} l^{2} } \right)^{2} l^{2} }} \\ & \quad \left[\omega^{2} \left(705216 ml^{2} E^{2} - 91478 Em\omega^{2} l^{4} \rho + 3499 \rho^{2} \omega^{4} l^{6} m \right.\right.\\ & \left.\left.\quad \quad -\,66792 Em\omega^{2} l^{3} R\rho + 265680 \rho I_{x} E^{2} + 29520 E\rho^{2} I_{x} \omega^{2} l^{2} \right.\right.\\ & \left.\left.\quad \quad +\,2820 \rho^{2} \omega^{4} l^{5} mR + 505344 mlRE^{2} - 4572 \rho^{3} \omega^{4} l^{4} I_{x}\right)\right] \\ K_{21c} & = - \frac{1}{{15120\left( {2E - \rho \omega^{2} l^{2} } \right)^{2} l^{2} }} \\ & \quad \left[\omega^{2} \left(705216 ml^{2} E^{2} - 91478 Em\omega^{2} l^{4} \rho + 3499 \rho^{2} \omega^{4} l^{6} m \right. \right.\\ & \left.\left.\quad \quad - 66792 Em\omega^{2} l^{3} R\rho + 235440 \rho I_{x} E^{2} + 44640 E\rho^{2} I_{x} \omega^{2} l^{2} \right. \right.\\ & \left.\left.\quad \quad + 2820 \rho^{2} \omega^{4} l^{5} mR + 505344 mlRE^{2} - 4572 \rho^{3} \omega^{4} l^{4} I_{x} \right)\right] \\ K_{22c} & = - \frac{1}{{83160\left( {2E - \rho \omega^{2} l^{2} } \right)^{2} l^{2} }} \\ & \quad \left[\omega^{2} \left(18014400 ml^{2} E^{2} - 2154226 Em\omega^{2} l^{4} \rho + 75935 \rho^{2} \omega^{4} l^{6} m \right.\right.\\ & \left.\left.\quad \quad -\,1751526 Em\omega^{2} l^{3} R\rho + 6642240 \rho I_{x} E^{2} + 881760 E\rho^{2} I_{x} \omega^{2} l^{2} \right.\right.\\ & \left.\left.\quad \quad +\,60885 \rho^{2} \omega^{4} l^{5} mR + 12906432 mlRE^{2} - 100056 \rho^{3} \omega^{4} l^{4} I_{x} \right)\right] \\ DF_{1} & = \frac{{\omega^{2} m}}{{30\left( {2E - \rho \omega^{2} l^{2} } \right)}}\left( {26 E l + 36 E R - 2 \omega^{2} l^{3} \rho - 3 \omega^{2} R l^{2} \rho } \right) \\ DF_{2} & = \frac{{\omega^{2} m}}{{840\left( {2E - \rho \omega^{2} l^{2} } \right)}}\\ &\quad \left( {3368 E l + 4648 E R - 221 \omega^{2} l^{3} \rho - 329 \omega^{2} R l^{2} \rho } \right). \\ \end{aligned} $$
