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Modal analysis of a rotating twisted and tapered Rayleigh beam

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Abstract

Free vibration analysis of a twisted, double-tapered blade mounted on a rotating disk undergoing overall motion is presented here. The Lagrangian approach is adapted to study the modal characteristics of the blade modeled as a rotating Rayleigh beam. The expressions for the kinetic energy and potential energy of the cantilever blade are derived using hybrid deformation variables. The continuous deformation variables in these equations are discretized using a series of basis functions that satisfy all boundary conditions of the cantilever beam. The equations governing the coupled stretch–bending–torsion motion of the rotating blade are derived using Lagrange’s approach. The equations are then transformed into a non-dimensional form which are then solved for the eigenvalue problem for the modal characteristics of the blade. The results of the present model are verified with the results available in the literature. The variation of the natural frequencies with the rotating speed, taper ratio and pre-twist angle is presented. The tuned angular speed of the blade at which the angular frequency matches with any of the natural frequency of the blade resulting in the resonance is investigated. The Campbell diagram is plotted for the specific problem to identify the resonance where the natural frequency matches with the harmonics of the rotating speed.

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Authors and Affiliations

  1. Defence Institute of Advanced Technology, Pune, 411025, India

    Lokanna Hoskoti & Ajay Misra

  2. Indian Institute of Technology, Hyderabad, 502285, India

    Mahesh M. Sucheendran

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  1. Lokanna Hoskoti
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  2. Ajay Misra
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Appendices

A coefficient matrices of Eq. 15

$$\begin{aligned}&\text {Mass matrices} \nonumber \\&\quad M^{11}_{ij} = \int _0^L \rho A \psi _{1i}\psi _{1j} dx, \nonumber \\&\quad M^{22}_{ij} = \int _0^L \rho A \psi _{2i}\psi _{2j} dx + \int _0^L \rho I_3 \psi '_{2i}\psi '_{2j} dx, \nonumber \\&\quad M^{23}_{ij} = \int _0^L \rho I_{23} \psi '_{2i}\psi '_{3j} dx, \nonumber \\&\quad M^{32}_{ij} =\int _0^L \rho I_{23} \psi '_{3i}\psi '_{2j} dx, \nonumber \\&\quad M^{33}_{ij} = \int _0^L \rho A \psi _{2i}\psi _{2j} dx+ \int _0^L \rho I_2 \psi '_{3i}\psi '_{3j} dx, \nonumber \\&\quad M^{44}_{ij} =\int _0^L \rho J \psi _{4i}\psi _{4j} dx, \nonumber \\&\text {Gyroscopic matrices} \nonumber \\&\quad G^{12}_{ij} = 2\varOmega \int _0^L \rho A \psi _{1i}\psi _{2j} dx, \nonumber \\&\quad G^{21}_{ij} = 2\varOmega \int _0^L \rho A \psi _{2i}\psi _{1j} dx, \nonumber \\&\quad G^{24}_{ij} = 2\varOmega \int _0^L \rho I_{23} \psi '_{2i}\psi _{4j} dx, \nonumber \\&\quad G^{34}_{ij} = 2\varOmega \int _0^L \rho I_{2} \psi '_{3i}\psi _{4j} dx, \nonumber \\&\quad G^{42}_{ij} = 2\varOmega \int _0^L \rho I_{23} \psi _{4i}\psi '_{2j} dx, \nonumber \\&\quad G^{43}_{ij} = 2\varOmega \int _0^L \rho I_{2} \psi _{4i}\psi '_{3j} dx, \nonumber \\&\text {Stiffness matrices} \nonumber \\&\quad K^{11}_{ij} = \int _0^L EA \psi '_{1i}\psi '_{1j} dx - \varOmega ^2\int _0^L \rho A \psi _{1i}\psi _{1j} dx \nonumber \\&\quad K^{22}_{ij} = \varOmega ^2\int _0^L \rho A \Big [r(L-x) + \frac{1}{2}(L^2-x^2)\Big ] \psi '_{2i}\psi '_{2j} dx \nonumber \\&\qquad \qquad - \varOmega ^2\int _0^L \rho A \psi _{2i}\psi _{2j} dx + \varOmega ^2\int _0^L \rho I_3 \psi '_{2i}\psi '_{2j} dx \nonumber \\&\qquad \qquad + \int _0^L EI_3 \psi _{2i}''\psi _{2j}'' dx, \nonumber \\&\quad K^{23}_{ij} = \int _0^L EI_{23} \psi _{2i}''\psi _{3j}'' dx-\varOmega ^2\int _0^L \rho I_{23} \psi '_{2i}\psi '_{3j} dx \nonumber \\&\quad K^{33}_{ij} = \varOmega ^2\int _0^L \rho A \Big [r(L-x) + \frac{1}{2}(L^2-x^2)\Big ] \psi '_{3i}\psi '_{3j} dx \nonumber \\&\qquad \qquad - \varOmega ^2\int _0^L \rho I_2 \psi '_{3i}\psi '_{3j} dx + \int _0^L EI_2 \psi _{3i}''\psi _{3j}'' dx, \nonumber \\&\quad K^{32}_{ij} = \int _0^L EI_{23} \psi _{3i}''\psi _{2j}'' dx-\varOmega ^2\int _0^L \rho I_{23} \psi '_{3i}\psi '_{2j} dx \nonumber \\&\quad K^{44}_{ij} = \varOmega ^2\int _0^L \rho A \Big [r(L-x) + \frac{1}{2}(L^2-x^2)\Big ] \psi '_{4i}\psi '_{4j} dx \nonumber \\&\qquad \qquad -\varOmega ^2\int _0^L \rho I_2 \psi _{4i}\psi _{4j} dx + \int _0^L GK \psi '_{4i}\psi '_{4j} dx. \end{aligned}$$
(31)

B coefficient matrices of Eq. 22

$$\begin{aligned} \begin{aligned}&\text {Mass matrices} \\&\quad {\bar{M}}^{11}_{ij} = \int _0^1 {\bar{A}} {\bar{\psi }}_{1i}{\bar{\psi }}_{1j} d{\bar{x}}, \\&\quad {\bar{M}}^{22}_{ij} = \int _0^1 {\bar{A}} {\bar{\psi }}_{2i}{\bar{\psi }}_{2j} d{\bar{x}} + \beta ^2 \int _0^1 {\bar{I}}_3 {\bar{\psi }}'_{2i}{\bar{\psi }}'_{2j} d{\bar{x}},\\&\quad {\bar{M}}^{23}_{ij} = \beta ^2\int _0^1 {\bar{I}}_{23} {\bar{\psi }}'_{2i}{\bar{\psi }}'_{3j} d{\bar{x}},\\&\quad {\bar{M}}^{32}_{ij} = \beta ^2\int _0^1 {\bar{I}}_{23} {\bar{\psi }}'_{3i}{\bar{\psi }}'_{2j} d{\bar{x}},\\&\quad {\bar{M}}^{33}_{ij} = \int _0^1 {\bar{A}} {\bar{\psi }}_{2i}{\bar{\psi }}_{2j} d{\bar{x}}+ \beta ^2\int _0^1 {\bar{I}}_2 {\bar{\psi }}'_{3i}{\bar{\psi }}'_{3j} d{\bar{x}},\\&\quad {\bar{M}}^{44}_{ij} = \int _0^1 {\bar{\psi }}_{4i}{\bar{\psi }}_{4j} d{\bar{x}},\\&\text {Gyroscopic matrices} \\&\quad {\bar{G}}^{12}_{ij} = 2\gamma \int _0^1 {\bar{A}} {\bar{\psi }}_{1i}{\bar{\psi }}_{2j} d{\bar{x}},\\&\quad {\bar{G}}^{21}_{ij} = 2\gamma \int _0^1 {\bar{A}} {\bar{\psi }}_{2i}{\bar{\psi }}_{1j} d{\bar{x}},\\&\quad {\bar{G}}^{24}_{ij} = 2\gamma \beta ^2 \int _0^1 {\bar{I}}_{23} {\bar{\psi }}'_{2i}{\bar{\psi }}_{4j} d{\bar{x}},\\&\quad {\bar{G}}^{34}_{ij} = 2\gamma \beta ^2 \int _0^1 {\bar{I}}_{2} {\bar{\psi }}'_{3i}{\bar{\psi }}_{4j} d{\bar{x}},\\&\quad {\bar{G}}^{42}_{ij} = 2\gamma \beta ^2 \int _0^1 {\bar{J}}_{23} {\bar{\psi }}_{4i}{\bar{\psi }}'_{2j} d{\bar{x}},\\&\quad {\bar{G}}^{43}_{ij} = 2\gamma \beta ^2 \int _0^1 {\bar{J}}_{2} {\bar{\psi }}_{4i}{\bar{\psi }}'_{3j} d{\bar{x}},\\\\&\text {Stiffness matrices} \\&\quad {\bar{K}}^{11}_{ij} = \beta ^{-2}\int _0^1 {\bar{\psi }}'_{1i}{\bar{\psi }}'_{1j} d{\bar{x}} - \gamma ^2\int _0^1 {\bar{A}} {\bar{\psi }}_{1i}{\bar{\psi }}_{1j} d{\bar{x}}\\&\quad {\bar{K}}^{22}_{ij} = \gamma ^2\int _0^1 {\bar{A}} F {\bar{\psi }}'_{2i}{\bar{\psi }}'_{2j} d{\bar{x}} - \gamma ^2\int _0^1 {\bar{A}} {\bar{\psi }}_{2i}{\bar{\psi }}_{2j} d{\bar{x}} \\&\qquad \qquad -\beta ^2\gamma ^2\int _0^1 {\bar{I}}_3 {\bar{\psi }}'_{2i}{\bar{\psi }}'_{2j} d{\bar{x}} + \int _0^1 {\bar{I}}_3 {\bar{\psi }}_{2i}''{\bar{\psi }}_{2j}'' d{\bar{x}}, \\&\quad {\bar{K}}^{23}_{ij} = \int _0^1 {\bar{I}}_{23} {\bar{\psi }}_{2i}''{\bar{\psi }}_{3j}'' d{\bar{x}}- \beta ^2 \gamma ^2\int _0^1 {\bar{I}}_{23} {\bar{\psi }}'_{2i}{\bar{\psi }}'_{3j} d{\bar{x}}, \\&\quad {\bar{K}}^{32}_{ij} = \int _0^1 {\bar{I}}_{23} {\bar{\psi }}_{3i}''{\bar{\psi }}_{2j}'' d{\bar{x}}- \beta ^2 \gamma ^2\int _0^1 {\bar{I}}_{23} {\bar{\psi }}'_{3i}{\bar{\psi }}'_{2j} d{\bar{x}}\\&\quad {\bar{K}}^{33}_{ij} = \gamma ^2\int _0^1 {\bar{A}} F {\bar{\psi }}'_{3i}{\bar{\psi }}'_{3j} d{\bar{x}} - \beta ^2\gamma ^2\int _0^1 {\bar{I}}_2 {\bar{\psi }}'_{3i}{\bar{\psi }}'_{3j} d{\bar{x}} \\&\qquad \qquad + \int _0^1 {\bar{I}}_2 {\bar{\psi }}_{3i}''{\bar{\psi }}_{3j}'' d{\bar{x}}, \\&\quad {\bar{K}}^{44}_{ij} = \gamma ^2\int _0^1 {\bar{A}} F {\bar{\psi }}'_{4i}{\bar{\psi }}'_{4j} d{\bar{x}} - \gamma ^2\int _0^1 {\bar{J}}_2 {\bar{\psi }}_{4i}{\bar{\psi }}_{4j} d{\bar{x}} \\&\qquad \qquad + \eta \beta ^{-2} \int _0^1 \bar{J_3}{\bar{\psi }}'_{4i}{\bar{\psi }}'_{4j} d{\bar{x}} \end{aligned} \end{aligned}$$
(32)

where

$$\begin{aligned} \begin{aligned} \beta&=\frac{I_{30}}{A_0L^2}, \eta = \frac{GK_t}{EI_{30}}, {\bar{A}} = (1-c_b{\bar{x}})(1-c_h{\bar{x}})\\ F&= \Big [\delta (1-{\bar{x}}) + \frac{1}{2}(1-{\bar{x}}^2)\Big ] \\ {\bar{I}}_2&= \frac{1}{2}\big (1-c_b{\bar{x}}\big )\big (1-c_h{\bar{x}}\big ) \Big [\kappa \big (1-c_h{\bar{x}}\big )^{2} + \big (1-c_b{\bar{x}}\big )^{2} \Big ]\\&\qquad \quad +\frac{1}{2}\big (1-c_b{\bar{x}}\big )\big (1-c_h{\bar{x}}\big ) \Big [\kappa \big (1-c_h{\bar{x}}\big )^{2} - \big (1-c_b{\bar{x}}\big )^{2} \Big ]\cos (2\theta _t {\bar{x}}), \\ {\bar{I}}_3&= \frac{1}{2}\big (1-c_b{\bar{x}}\big )\big (1-c_h{\bar{x}}\big ) \Big [\kappa \big (1-c_h{\bar{x}}\big )^{2} + \big (1-c_b{\bar{x}}\big )^{2} \Big ]\\&\quad \qquad -\frac{1}{2}\big (1-c_b{\bar{x}}\big )\big (1-c_h{\bar{x}}\big ) \Big [\kappa \big (1-c_h{\bar{x}}\big )^{2} - \big (1-c_b{\bar{x}}\big )^{2} \Big ]\cos (2\theta _t {\bar{x}}), \\ {\bar{I}}_{23}&= \frac{1}{2}\big (1-c_b{\bar{x}}\big )\big (1-c_h{\bar{x}}\big ) \Big [\kappa \big (1-c_h{\bar{x}}\big )^{2} - \big (1-c_b{\bar{x}}\big )^{2} \Big ]\sin (2\theta _t {\bar{x}}) \\ {\bar{J}}_{23}&= {\bar{I}}_{23}/ \big (1-c_b{\bar{x}}\big )\big (1-c_h{\bar{x}}\big ) \Big [\kappa \big (1-c_h{\bar{x}}\big )^{2} + \big (1-c_b{\bar{x}}\big )^{2} \Big ] \\ {\bar{J}}_{2}&= {\bar{I}}_2/ \big (1-c_b{\bar{x}}\big )\big (1-c_h{\bar{x}}\big ) \Big [\kappa \big (1-c_h{\bar{x}}\big )^{2} + \big (1-c_b{\bar{x}}\big )^{2} \Big ] \\ {\bar{J}}_{3}&= {\bar{I}}_3/ \big (1-c_b{\bar{x}}\big )\big (1-c_h{\bar{x}}\big ) \Big [\kappa \big (1-c_h{\bar{x}}\big )^{2} + \big (1-c_b{\bar{x}}\big )^{2} \Big ]. \end{aligned} \end{aligned}$$
(33)

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Hoskoti, L., Misra, A. & Sucheendran, M.M. Modal analysis of a rotating twisted and tapered Rayleigh beam. Arch Appl Mech 91, 2535–2567 (2021). https://doi.org/10.1007/s00419-021-01902-8

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