Natural frequencies of rotating twisted beams: a perturbation method based approach

Abstract

Structures such as turbomachinery blades, industrial fans, propellers, etc. can be modeled as twisted beams. The study of dynamics of these structures is vital as operational failure of such structures can have catastrophic consequences. As the inclusion of twist and rotation complicates the problem, Finite Element (FE) method is widely used to determine the modal characteristics of rotating twisted beams. In this work, a novel formula is derived to estimate the natural frequencies of rotating twisted beams. The formula is derived using the perturbation method. The twist angle and the rotating speed are treated as the perturbation parameters. In general, the dynamics of rotating twisted beams is coupled in the two transverse planes. However, in the first part of the work the problem is assumed to be uncoupled and it is shown that this assumption is valid under certain cases. In the second part, the problem of general coupled dynamics is solved. Interesting insights based on the formula are presented. The accuracy of the derived formula is verified by comparing it with the literature and FE simulation results. It has been shown that the formula is valid over a fairly large range of twist angles and rotating speeds. In contrast to the detailed FE simulation, the derived analytical formula will be better suited for design iterations in industrial practice.

Appendix

In this section, we derive the expression for kinetic energy of a rotating twisted beam. It is assumed that the displacements are harmonic in time. In general, the deformation of a twisted beam is coupled in the x–y and the x–z plane as shown in the figure.

Fig. 13

Schematic illustration of the undeformed and deformed configurationof a rotating twisted beam

The point P in its reference configuration is deformed to $P'$ as illustrated in Fig. 13. The position vector of $P'$ in the rotating frame is denoted as $\rho _{P'}=x i+w j+v k.$ Velocity of any point $P'$ in the inertial frame is given by,

$$\begin{aligned} \mathbf {v}=\mathbf {v}_{O_{2}}+\frac{\mathrm {d}}{\mathrm {d}t}(P') +\varvec{\Omega }\times P' \end{aligned}$$

where $\mathbf {v}_{O_{2}}=\varvec{\Omega }\times r\, i$ is the velocity of the point $O_{2}$ attached to the hub and $\frac{\mathrm {d}}{\mathrm {d}t}(P')$ is velocity of point $P'$ in the rotating frame. The kinetic energy of the twisted rotating beam is given by,

$$\begin{aligned} \mathcal {T}&=\frac{1}{2}\int _{0}^{L}\rho A (\mathbf {v}\cdot \mathbf {v}) \mathrm {d}x \nonumber \\&=\frac{\rho A}{2}\int _{0}^{L}\left\{ \tilde{\omega }^{2}w^{2}+\left( \tilde{\omega }v-\varOmega (r+x)\right) ^{2}+\varOmega ^{2}v^{2}\right\} \mathrm {d}x \end{aligned}$$

(21)

where $\tilde{\omega }$ is the natural frequency of the rotating twisted beam. Equation (21) represents the kinetic energy for the general case of a rotating twisted beam. In case of uncoupled flapwise dynamics, the motion of the beam is restricted to x–yx–y plane. This implies that $v=0.$ Let $\tilde{\omega }=\tilde{\omega }_{\mathrm {f}}$ for flapwise dynamics. Thus, the kinetic energy is given by,

$$\begin{aligned} \mathcal {T}_{\mathrm {f}}&=\frac{\rho A \tilde{\omega }_{\mathrm {f}}^{2}}{2}\int _{0}^{L}\left\{ w^{2}+\varOmega ^{2}(r+x)^{2}\right\} \mathrm {d}x \end{aligned}$$

(22)

Similarly for uncoupled chordwise dynamics, the motion is limited to x–z plan, viz. $w=0.$ Let $\tilde{\omega }=\tilde{\omega }_{\mathrm {c}}$ in the case of chordwise dynamics. Therefore, the kinetic energy is given by,

$$\begin{aligned} \mathcal {T}_{\mathrm {c}}=\frac{\rho A}{2}\int _{0}^{L}\left\{ \left( \tilde{\omega }_{\mathrm {c}}v -\varOmega (r+x)\right) ^{2}+\varOmega ^{2}v^{2}\right\} \mathrm {d}x \end{aligned}$$

(23)