Periodic and quasiperiodic galloping of a wind-excited tower under external excitation

Abstract

The galloping of tall structures excited by steady and unsteady wind may be periodic or quasiperiodic (QP) with amplitudes having the same order of magnitude. While the onset of periodic and QP galloping was studied, their control on the other hand has received less attention. In this paper, we conduct analytical study on the effect of a fast harmonic excitation on the onset of periodic and QP galloping in the presence of steady and unsteady wind. We consider the cases where the unsteady wind activates either external excitation, parametric one or both. A perturbation analysis is performed to obtain close expressions of QP solution and the corresponding modulation envelopes. We show that at various loading situations, the periodic and QP galloping onset is significantly influenced by the amplitude of the fast external excitation. In the case where the unsteady wind activates parametric excitation, the QP galloping occurs with higher frequency modulation compared to the case where the unsteady wind activates external excitation. In the case where external and parametric excitations are activated simultaneously, fast harmonic excitation eliminates bistability in the amplitude response and gives rise to a new small QP modulation envelope.

Appendices

Appendix A

The expressions of the coefficients of Eq. (1) are

$$\begin{aligned}[c] &\omega=\pi\frac{\sqrt{3EI}}{h \ell \sqrt{m}},\qquad c_{a}=\frac{\rho A_{1}bh \ell \bar{U}_{c}}{2\pi\sqrt{3EIm}},\\ &b_{1}=c_{a},\qquad b_{2}=-\frac{4\rho A_{2}b\ell}{3\pi m},\\ &b_{31}=-\frac{3\pi\rho A_{3}b\ell\sqrt{3EI}}{8h\bar{U}_{c}\sqrt{m^{3}}} \end{aligned} $$

$$\begin{aligned}[c] &b_{32}=-b_{31},\qquad\eta_{1}=\frac{4\rho A_{0}bh^2\ell\bar{U}_{c}^{2}}{3\pi^{3}EI},\\ &\eta_{2}=\frac{\eta_{1}}{2},\qquad U(t)=\bar{U}+u(t), \end{aligned} $$

where ℓ is the height of the tower, b is the cross-section wide, EI the total stiffness of the single story, m is the mass longitudinal density, h is the interstory height, and ρ is the air mass density. A _i ,i=0,…,3 are the aerodynamic coefficients for the squared cross-section. The dimensional critical velocity is given by

$$ \bar{U}_{c}=\frac{4\pi\xi\sqrt{3EIm}}{\rho bA_{1}h\ell} $$

(24)

Here, ξ is the modal damping ratio, depending on both the external and internal damping

$$ \xi=\frac{\eta h^{2}}{24EI}\omega+\frac{c}{2m\omega} $$

(25)

The following numerical values picked from [7] are used for convenience: the height of the tower is ℓ=36 m, the cross-section is b=16 m wide, the total stiffness of the single story is EI=115318000 N m², the mass longitudinal density is m=4737 kg/m, the damping ratio is ζ=0.5 percent (corresponding to η=128513 N s, c=34.8675 N s/m² in Eq. (25)). The interstory height is assumed h=4m. The aerodynamic coefficients A _i ,i=0,…,3 are taken from [4] for the squared cross-section: A ₀=0.0297, A ₁=0.9298, A ₂=−0.2400, A ₃=−7.6770. The air mass density is ρ=1.25 kg/m³. The (dimensional) natural frequency of the rod is ω=5.89 rad/s. The (dimensional) critical wind velocity assumes the value $\overline{U}_{c}=30~\mbox{m/s}$.

Appendix B

Introducing $D_{i}^{j}\equiv\frac{\partial^{j}}{\partial ^{j}T_{i}}$ yields $\frac{d}{dt}=\nu D_{0}+ D_{1}$, $\frac{d^{2}}{dt^{2}}=\nu^{2} D_{0}^{2}+ 2\nu D_{0}D_{1}+D_{1}^{2}$ and substituting Eq. (2) into Eq. (1) gives

$$\begin{aligned} &{\mu^{-1}D_{0}^{2}\phi+D_{1}^{2}z+2D_{0}D_{1} \phi+\mu D_{1}^{2}\phi} \\ &{\qquad{}+ \bigl(c_{a}(1- \bar{U})-b_{1}u(t) \bigr) (D_{1}z+ D_{0}\phi+\mu D_{1}\phi )} \\ &{\qquad{}+z+\mu\phi +b_{2} \bigl((D_{1}z)^2+2D_{1}z(D_{0} \phi+\mu D_{1}\phi)} \\ &{\qquad{}+(D_{0}\phi)^2+2\mu D_{0}\phi D_{1}\phi+(\mu D_{1}\phi)^2 \bigr)} \\ &{\qquad{}+ \biggl(\frac{b_{31}}{\bar{U}}+\frac {b_{32}}{\bar{U}^2}u(t) \biggr) \bigl((D_{1}z)^3 +3(D_{1}z)^2} \\ &{\qquad{}\times (D_{0} \phi+\mu D_{1}\phi)+ 3(D_{1}z) (D_{0}\phi+\mu D_{1}\phi)^2} \\ &{\qquad{}+(D_{0}\phi+\mu D_{1}\phi)^3 \bigr)} \\ &{\quad{}=\eta_{1}\bar{U}u(t)+ \eta_{2} \bar{U}^2+Y\cos(\nu t)} \end{aligned}$$

(26)

Averaging (26) leads to

$$\begin{aligned} &{D_{1}^{2}z+ \bigl(c_{a}(1- \bar{U})-b_{1}u(t) \bigr)D_{1}z+z} \\ &{\qquad{}+b_{2} \bigl((D_{1}z)^2+ \bigl\langle(D_{0} \phi)^2\bigr\rangle+\bigl\langle (2\mu D_{0}\phi D_{1}\phi)\bigr\rangle} \\ &{\qquad{}+\bigl\langle(\mu D_{1}\phi)^2\bigr \rangle \bigr)+ \biggl(\frac{b_{31}}{\bar{U}} +\frac{b_{32}}{\bar{U}^2}u(t) \biggr) \bigl((D_{1}z)^3} \\ &{\qquad{}+ 3D_{1}z\bigl(\bigl \langle(D_{0}\phi)^2\bigr\rangle+\bigl\langle(2\mu D_{0}\phi D_{1}\phi)\bigr\rangle} \\ &{\qquad{}+\bigl\langle(\mu D_{1}\phi)^2\bigr \rangle\bigr) \bigr)} \\ &{\quad{}=\eta_{1}\bar{U}u(t)+\eta_{2} \bar{U}^2} \end{aligned}$$

(27)

Subtracting (27) from (26) yields

$$\begin{aligned} &{\mu^{-1}D_{0}^{2}\phi+2D_{0}D_{1} \phi+\mu D_{1}^{2}\phi} \\ &{\qquad{}+ \bigl(c_{a}(1- \bar{U})-b_{1}u(t) \bigr) (D_{0}\phi+ \mu D_{1}\phi )} \\ &{\qquad{}+\mu\phi+b_{2} \bigl(2D_{1}z(D_{0}\phi +\mu D_{1}\phi)+(D_{0}\phi)^2} \\ &{\qquad{}- \bigl\langle (D_{0}\phi)^2\bigr\rangle+2\mu D_{0} \phi D_{1}\phi+(\mu D_{1}\phi)^2} \\ &{\qquad{}- \bigl\langle ( \mu D_{1}\phi)^2\bigr\rangle \bigr)+ \biggl( \frac{b_{31}}{\bar{U}} +\frac{b_{32}}{\bar{U}^2}u(t) \biggr)} \\ &{\qquad{}\times \bigl(3(D_{1}z)^2(D_{0}+ \mu D_{1}\phi)+3D_{1}z(D_{0}\phi)^2} \\ &{\qquad{}-3D_{1}z \bigl\langle (D_{0}\phi)^2\bigr\rangle+6D_{1}z\mu (D_{0}\phi D_{1}\phi)} \\ &{\qquad{}+3D_{1}z(\mu D_{1}\phi)^2-3D_{1}z \bigl\langle (\mu D_{1}\phi)^2\bigr\rangle+(D_{0} \phi)^3} \\ &{\qquad{}+3\mu (D_{0}\phi)^2D_{1}\phi} \\ &{\qquad{}+3D_{0}\phi(\mu D_{1}\phi)^2+ \mu D_{1}\phi \bigr)} \\ &{\quad{}=Y\cos(T_{0})} \end{aligned}$$

(28)

Using the inertial approximation [20], i.e., all terms in the left-hand side of Eq. (28), except the first, are ignored, one obtains

$$ \phi=-\mu Y\cos(T_{0}) $$

(29)

Inserting ϕ into Eq. (27), using that 〈cos² T ₀〉=1/2, and keeping only terms of orders three in z, give the equation governing the slow dynamic of the motion (3).