Quasi-Periodic Galloping of a Wind-Excited Tower Under External Forcing and Parametric Damping

Abstract

This paper investigates the influence of combined fast external excitation and internal parametric damping on the amplitude and the onset of the quasi-periodic galloping of a tower submitted to steady and unsteady wind flow. The study is carried out considering a lumped single degree of freedom model and the cases where the turbulent wind activates different excitations are explored. The method of direct partition of motion followed by the multiple scales technique are applied to derive the slow flow dynamic near the primary resonance. The influence of the combined loading consisting in external excitation and parametric damping on the quasi-periodic galloping onset is explored. The performance of the combined loading is compared with the cases where the external excitation and the parametric damping are introduced separately. The results show that the performance of the combined loading on retarding the quasi-periodic galloping onset and quenching the corresponding amplitude is better in all cases of the turbulent wind excitations.

Appendices

Appendix 1

The expression of the coefficients of (1) are:

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

(24)

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

(25)

where $\ell$ is the height of the tower, b the cross-section wide, $\textit{EI}$ the total stiffness of the single story, m the mass longitudinal density, h the inter story height, and $\rho$ 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

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

(26)

where $\xi$ is the modal damping ratio, depending on both the external and internal damping according to

$$\begin{aligned} \xi =\frac{\zeta h^{2}}{24EI}\omega +\frac{c_0}{2m\omega } \end{aligned}$$

(27)

where $\zeta$ and $c_0$ are the external and internal damping coefficients, respectively. Introducing a parametric damper device in the internal damping such as

$$\begin{aligned} c_0=c (1+y_{0}\nu ^2\cos \nu t) \end{aligned}$$

(28)

where $y_{0}$ and $\nu$ are the amplitude and the frequency of the internal PD, respectively. In this case the equation of motion reads

$$\begin{aligned} \ddot{x}+x+\big [c_{a}+Y\nu ^2\cos \nu t\big ]\dot{x}-c_{a}\big [\bar{U}+u(t)\big ]\dot{x}+b_{2}\dot{x}^2+ \left[ \frac{b_{31}}{\bar{U}}+\frac{b_{32}}{\bar{U}^2}u(t)\right] \dot{x}^3= \nonumber \\ \eta _{1}\bar{U}u(t)+\eta _{2}\bar{U}^2 \end{aligned}$$

(29)

where $Y=\frac{cy_{0}}{m\omega }$. Re-arranging terms yields the equation of motion (1).

Appendix 2

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 (2) into (1) gives

$$\begin{aligned} \mu ^{-1}D_{0}^{2}\phi&+D_{1}^{2}z+2D_{0}D_{1}\phi +\mu D_{1}^{2}\phi +\big (c_{a}(1-\bar{U})-b_{1}u(t)\big )\big (D_{1}z+ D_{0}\phi \nonumber \\ {}&+\mu D_{1}\phi \big )+z+\mu \phi +Y\nu ^2\cos (\nu t)\big (D_{1}z+ D_{0}\phi +\mu D_{1}\phi \big ) +b_{2}\big ((D_{1}z)^2\nonumber \\ {}&+2D_{1}z(D_{0}\phi +\mu D_{1}\phi )+ (D_{0}\phi )^2+2\mu D_{0}\phi D_{1}\phi +(\mu D_{1}\phi )^2\big )\nonumber \\ {}&+\left( \frac{b_{31}}{\bar{U}}+\right. \left. \frac{b_{32}}{\bar{U}^2}u(t)\right) \big ((D_{1}z)^3 +3(D_{1}z)^2(D_{0}\phi +\mu D_{1}\phi )\nonumber \\ {}&+ 3(D_{1}z)(D_{0}\phi +\mu D_{1}\phi )^2+ (D_{0}\phi +\mu D_{1}\phi )^3\big )=\eta _{1}\bar{U}u(t)+ \eta _{2}\bar{U}^2 \nonumber \\ \end{aligned}$$

(30)

Averaging (30) leads to

$$\begin{aligned} D_{1}^{2}z&+\big (c_{a}(1-\bar{U})-b_{1}u(t)\big )D_{1}z+z+Y\nu ^2<\cos (T_{0})\big (D_{0}\phi +\mu D_{1}\phi \big )>\nonumber \\ {}&+b_{2}\big ((D_{1}z)^2\,+\, <(D_{0}\phi )^2>+<(2\mu D_{0}\phi D_{1}\phi )>+<(\mu D_{1}\phi )^2>\big )\nonumber \\ {}&+ \left( \frac{b_{31}}{\bar{U}} +\frac{b_{32}}{\bar{U}^2}u(t)\right) \big ((D_{1}z)^3+ 3D_{1}z(<(D_{0}\phi )^2>+<(2\mu D_{0}\phi D_{1}\phi )>\nonumber \\ {}&+ <(\mu D_{1}\phi )^2>)\big )=\eta _{1}\bar{U}u(t)+\eta _{2}\bar{U}^2 \nonumber \\ \end{aligned}$$

(31)

Subtracting (31) from (30) yields

$$\begin{aligned} \mu ^{-1}D_{0}^{2}\phi&+2D_{0}D_{1}\phi +\mu D_{1}^{2}\phi +\big (c_{a}(1-\bar{U})-b_{1}u(t)\big )\big (D_{0}\phi + \mu D_{1}\phi \big )\nonumber \\ {}&+ \mu \phi + Y\nu ^2\cos (T_{0})\big (D_{0}\phi +\mu D_{1}\phi \big )-Y\nu ^2<\cos (T_{0})\big (D_{0}\phi +\mu D_{1}\phi \big )>\nonumber \\ {}&+ b_{2}\big (2D_{1}z(D_{0}\phi +D_{0}\phi )^2-<(D_{0}\phi )^2>\,+\,2\mu D_{0}\phi D_{1}\phi -(<2\mu D_{0}\phi D_{1}\phi >\nonumber \\&+(\mu D_{1}\phi )^2- <(\mu D_{1}\phi )^2>\big )+ \left( \frac{b_{31}}{\bar{U}}+ \frac{b_{32}}{\bar{U}^2}u(t)\right) \big (3(D_{1}z)^2(D_{0}\phi +\mu D_{1}\phi )\nonumber \\ {}&+ 3D_{1}z(D_{0}\phi )^2- 3D_{1}z<(D_{0}\phi )^2>\,+\,6D_{1}z\mu (D_{0}\phi D_{1}\phi )\nonumber \\ {}&- <6D_{1}z\mu (D_{0}\phi D_{1}\phi )>\,+\,3D_{1}z(\mu D_{1}\phi )^2-3D_{1}z<(\mu D_{1}\phi )^2>\,+\, (D_{0}\phi )^3 \nonumber \\ {}&+ 3\mu (D_{0}\phi )^2D_{1}\phi + 3D_{0}\phi (\mu D_{1}\phi )^2+(\mu D_{1}\phi )^3\big )= -Y\nu ^2\cos (T_{0})D_{1}z \nonumber \\ \end{aligned}$$

(32)

Using the inertial approximation [20], i.e. all terms in the left-hand side of (32), except the first, are ignored, one obtains

$$\begin{aligned} \phi =Y\nu \cos (T_{0})D_{1}z \end{aligned}$$

(33)

Inserting $\phi$ from (33) into (31), using that $<cos^{2}T_{0}>\,=\,1/2$, and keeping only terms of orders three in z, give the equation governing the slow dynamic of the motion (3).