A time-implicit representation of the lift force for coupled translational–rotational galloping


The lift force acting on a purely translational galloping oscillator can be well approximated by using the quasi-steady theory, which states that the flow around a galloping body in motion is very similar to the known flow around a fixed body provided a minimum free stream velocity threshold and a similarity principle are satisfied. However, for oscillators undergoing coupled translational–rotational galloping, the rotation of the bluff body breaks the similarity principle, and therewith the quasi-steady assumption. Traditionally, the effects of rotation are accounted for by including explicit time-dependent terms in the lift force. However, we argue that the time-explicit representation of the lift force is unnecessary because the phenomenon of galloping is not time-explicit and acts only as a function of the body motion. Thus, we propose a modified time-implicit lift force representation, which has the same form of the quasi-steady theory, but with an additional dependence on the free stream velocity. The modified lift force is obtained by studying the transient growth of the amplitude of the bluff body oscillation. The proposed approach is used to model the lift force for square and trapezoidal prisms undergoing coupled translational–rotational galloping showing excellent prediction capabilities.

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  1. 1.

    The reader should bear in mind that perfectly symmetric bodies like a square are prone to galloping. This is because although the production of lift requires asymmetry, the asymmetry does not need to be in the geometry of the bluff body, but can also occur due to the angle of incidence between the moving body and the flow.

  2. 2.

    The reader can refer to the book by Païdoussis for a detailed review on the galloping instability [6].


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Appendix A: Equation of motion

Following linear Euler–Bernoulli beam theory for which the thickness, \(t_b\), and deflection w(xt) of the beam are assumed to be much smaller than its length L, we can write the equation governing the motion of the beam as

$$\begin{aligned} M_b \ddot{w}+c_v {\dot{w}} +E_b I_b w^{\prime \prime \prime \prime }=F_N \delta (x-L), \end{aligned}$$

where t represents the temporal variable, and x represents the spatial variable along the length of the beam, the dot represents a derivative with respect to t and the prime represents a derivative with respect to x, \(M_b\) is the mass of the beam, \(c_v\) is the linear damping coefficient, \(E_b\) and \(I_b\) are, respectively, the modulus of elasticity, and mass moment of inertia of the beam, \(F_N\) is the vertical component of the aerodynamic force, and \(\delta \) is the Dirac delta function. Note that the aerodynamic force is assumed to be concentrated at the tip, which implicitly assumes that the width of the bluff body, D, is much smaller than the length of the beam, L.

The proper boundary conditions pertinent to Fig. 2 are given by

$$\begin{aligned} \begin{aligned}&w(0,t)=0, \quad w^{\prime }(0,t)=0, \\&E_bI_b w^{\prime \prime }(L,t)=I_t \ddot{w}^{\prime }(L,t), \\&E_bI_b w^{\prime \prime \prime }(L,t)=M_t \ddot{w}^{\prime }(L,t), \end{aligned} \end{aligned}$$

where \(M_t\) and \(I_t\) are, respectively, the mass and mass moment of inertia of the bluff body.

As shown in Fig. 2, the aerodynamic force acting on the moving tip body is produced by a relative wind velocity \(V_{\mathrm{rel}}\) at an effective angle of attack \(\alpha \), which can be written as

$$\begin{aligned} V_{\mathrm{rel}}=\sqrt{V^2+{\dot{w}}^2(L,t)},\quad \alpha =\alpha _t+w^{\prime }(L,t), \end{aligned}$$

where \(\alpha _t=\tan ^{-1}\left[ \frac{{\dot{w}}(L,t)}{V}\right] \). The corresponding drag and lift force components are given by

$$\begin{aligned} F_D= & {} \frac{1}{2}\rho _aC_{F_D}(\alpha )D H V^2_{\mathrm{rel}},\nonumber \\ F_L= & {} \frac{1}{2}\rho _aC_{F_L}(\alpha )D H V^2_{\mathrm{rel}}, \end{aligned}$$

where \(\rho _a\) is the air density, \(C_{F_D}\) and \(C_{F_L}\) are, respectively, the drag and lift coefficients. The normal force acting in the y-direction can be, therefore, written as

$$\begin{aligned} F_N=F_D \sin \alpha _t + F_L \cos \alpha _t. \end{aligned}$$

When \({\dot{w}}(L,t)/V \ll 1\), the previous equation becomes

$$\begin{aligned} F_N=F_D \frac{{\dot{w}}(L,t)}{V} + F_L. \end{aligned}$$

Furthermore, under the assumption that \({\dot{w}}(L,t)/V \ll 1\), \(V_{\mathrm{rel}}\) can be approximated by V, which yields

$$\begin{aligned} \begin{aligned}&F_N=\frac{1}{2}\rho _aC_{N}(\alpha )D H V^2, \\&\text {where}\quad C_{N}=\left[ C_{F_D}\frac{{\dot{w}}(L,t)}{V}+C_{F_L}\right] \end{aligned} \end{aligned}$$

Assuming that the beam is free of internal resonances and that the fluid forces are incapable of directly exciting the higher vibration modes, a single-mode representation of the beam dynamics can be written as

$$\begin{aligned} w(x,t)= \phi (x) {\bar{y}}(t) \end{aligned}$$

where \(\phi (x)\) is a spatial representation of the beam shape at a given time, chosen here to be the first mode shape of a cantilever beam with a tip mass. The function \({\bar{y}}(t)\) is a temporal function representing the time variation of the mode shape. The function \(\phi (x)\) can be written as

$$\begin{aligned} \phi (x)= & {} A\left[ \cos \frac{\lambda }{L}x-\cosh \frac{\lambda }{L}x \right. \nonumber \\&\left. +\xi \left( \sin \frac{\lambda }{L}x-\sinh \frac{\lambda }{L}x\right) \right] , \end{aligned}$$


$$\begin{aligned}&\xi =\frac{\left( \sin {\lambda }-\sinh {\lambda }\right) +R_1\lambda \left[ \cos {\lambda }-\cosh {\lambda }\right] }{\left( \cos {\lambda }-\cosh {\lambda }\right) -R_1\lambda \left[ \sin {\lambda }-\sinh {\lambda }\right] },\nonumber \\&R_1= \frac{M_t}{m_bL}. \end{aligned}$$

Substituting Eq. (A.8) into Eq. (A.1), multiplying by \(\phi (x)\) and integrating over the length of the beam yields

$$\begin{aligned} m\ddot{{\bar{y}}}+c\dot{{\bar{y}}}+k {\bar{y}} =f_0 F_N, \end{aligned}$$


$$\begin{aligned} m= & {} \int _0^{L}M_b \phi ^2 \mathrm{d}x, \quad c= \int _0^L c_v \phi ^2 \mathrm{d}x,\nonumber \\ k= & {} 7\int _0^{L}E_b I_b \phi ^{\prime \prime \prime \prime } \phi \mathrm{d}x, \quad f_0= \phi (L). \end{aligned}$$

Normalizing the mode shape, \(\phi (x)\), by choosing A such that \(\phi (L)=1\), yields \(f_0=1\) and \(\alpha = \dot{{\bar{y}}}/V+ \phi ^{\prime }(L) {\bar{y}}\). Furthermore, for small deflections \({\dot{w}}(L,t)/L < 0.3\), the function \(\phi ^{\prime }(L)\) can be further approximated by \(\phi ^{\prime }(L) \approx -\frac{3}{2 L} \phi (L)\) . Thus, Eq. (A.10) can be further simplified as

$$\begin{aligned} \begin{aligned}&m\ddot{{\bar{y}}}+c\dot{{\bar{y}}}+k {\bar{y}} = \frac{1}{2}\rho _aC_{N}(\alpha )D H V^2, \\&\quad \alpha =\frac{\dot{{\bar{y}}}}{V}-\frac{3}{2L}{\bar{y}}, \end{aligned} \end{aligned}$$

which is the same as Eq. (2) used to study the dynamics of the galloping oscillator.

Appendix B: Approximate analytical solution of the equation of motion

An approximate solution of Eq. (6) can be obtained using the method of multiple scales; a method widely utilized to analyze the response of weakly nonlinear systems containing mechanisms that act on distinct time scales [25]. In the case of a galloping body, a “true” time scale describes the instantaneous position of the body, while a slow time scale tracks the weakly nonlinear evolution of oscillation amplitude.

The scaling is accomplished using an arbitrarily small bookkeeping parameter \(\epsilon \) such that “true” time is \(T_0 = \tau \) and “slow” time is \(T_1 = \epsilon \tau \). For the present problem, higher orders of \(\epsilon \) (even slower time scales) are neglected. The derivative operator is adapted such that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d} \tau }= & {} D_0 + \epsilon D_1 + {\mathcal {O}}(\epsilon ^2), \nonumber \\ \frac{\mathrm{d}^2 }{\mathrm{d}\tau ^2}= & {} D_0^2 + 2\epsilon D_0 D_1 + {\mathcal {O}}(\epsilon ^2) . \end{aligned}$$


$$\begin{aligned} D_0 = \frac{\mathrm{d}}{\mathrm{d} T_0},\quad D_1 = \frac{\mathrm{d}}{\mathrm{d} T_1} . \end{aligned}$$

Additionally, the response \(y(\tau )\) is described into multiple scales such that

$$\begin{aligned} y(\tau ) = y_0 (T_0,T_1) + \epsilon y_1 (T_0,T_1) + {\mathcal {O}}(\epsilon ^2). \end{aligned}$$

It can be reasoned that the mechanism of amplitude growth occurs slowly because the energy input to the system is on the order of the energy dissipated in the context of underdamped vibrations, energy dissipation must be small (controlled by \(\zeta _m\)). As such, a small net energy input requires small input flow energy (controlled by \(\mu \)). Therefore, the appropriate parameters are scaled by \(\epsilon \) as

$$\begin{aligned} \mu = \epsilon \mu ,\quad \zeta _m=\epsilon \zeta _m. \end{aligned}$$

When Eqs. (B.1), (B.3), and (B.4) are substituted into Eq. (6), the resulting system can be divided into subsystems by collecting the zeroth and first power of \(\epsilon \) as following:

For \(\epsilon ^0\):

$$\begin{aligned} D_0^2 y_0 + y_0 = 0 \end{aligned}$$

admits a zeroth-order solution

$$\begin{aligned} y_0 = a(T_1) \cos (T_0 + \gamma (T_1)), \end{aligned}$$

where a is the amplitude of oscillation and \(\gamma \) is a phase shift function.

For \(\epsilon ^1\):

$$\begin{aligned} D_1^2 y_1 + y_1 = -2D_0D_1 y_0 - 2\zeta _m D_0y_0 + 2\mu U^2 C_N(\alpha ). \end{aligned}$$

Letting \(\phi = T_0 + \gamma \) and substituting the zeroth-order solutions, as well as the expression for \(C_N\) from Eq. (3) into Eq. (B.7) yields

$$\begin{aligned} \begin{aligned}&D_1^2 y_1 + y_1 = 2D_1a\sin \phi +2aD_1\beta \cos \phi + 2\zeta _m a\sin \phi \\&\quad {+2\mu U^2} \sum _{\begin{array}{c} j=1\\ \text {odd} \end{array}}^{n} A_j \left( \alpha \right) ^j + {2\mu U^2} \sum _{\begin{array}{c} j=1\\ \text {even} \end{array}}^{n} A_j \left| \alpha _0\right| ^j\mathrm {sgn}(\alpha ). \end{aligned} \end{aligned}$$

Because the physical system does not have oscillations which grow to infinity, the “forcing” terms on the right-hand side of Eq. (B.8) must not resonate with \(y_1\). This condition is enforced by ensuring the right-hand side of Eq. (B.8) is orthogonal to its homogeneous solution throughout one cycle of oscillation. That is, the right-hand side of the equation should be orthogonal to both \(\sin \phi \) and \(\cos \phi \).

For orthogonality with \(\cos \phi \), we obtain

$$\begin{aligned} \begin{aligned}&\int _0^{2\pi } \Bigg ( 2D_1a\sin \phi +2aD_1\beta \cos \phi - 2\zeta _m a\sin \phi \\&\quad + {2\mu U^2} \sum _{\begin{array}{c} j=1\\ \text {odd} \end{array}}^{n} A_j \left( \alpha \right) ^j \\&\quad + \sum _{\begin{array}{c} j=1\\ \text {even} \end{array}}^{n} A_j \left| \alpha \right| ^j\mathrm {sgn}(\alpha ) \Bigg ) \cos \phi \text {d}\phi =0. \end{aligned} \end{aligned}$$

Noting that \(\alpha =\frac{y^{\prime }}{U}-\xi U\), and substituting Eq. (B.6) into the expression for \(\alpha \) we obtain

$$\begin{aligned} \alpha = -\frac{a}{U}\sqrt{\xi ^2 U^2 + 1\;\;}\;\sin \left( \phi + \tan ^{-1}\left( \xi U \right) \right) , \end{aligned}$$

which implies that the angle of attack changes sign when

$$\begin{aligned} \phi = k\pi - \tan ^{-1}\left( \xi U\right) , \quad k \equiv \text {integer}. \end{aligned}$$

Thus, Eq. (B.9) becomes

$$\begin{aligned} \begin{aligned} 0&=\int _0^{2\pi } \left[ 2D_1a\sin \phi +2aD_1\beta \cos \phi \right. \\&\quad \left. - 2\zeta _m a\sin \phi \right] \cos \phi \text {d}\phi \\&\quad + 2\mu U^2 \sum _{j=1}^{n} A_j \left( \frac{a}{U}\sqrt{\left( \xi U\right) ^2 + 1}\right) ^j \\&\quad \times \Bigg \{ -\int _0^{\pi -\theta _0}\sin \left( \phi + \theta _0\right) ^j\cos \phi \text {d}\phi \\&\quad +(-1)^j \int _{\pi -\theta _0}^{2\pi -\theta _0}\sin \left( \phi + \theta _0\right) ^j\cos \phi \text {d}\phi \\&\quad - \int _{2\pi -\theta _0}^{2\pi }\sin \left( \phi + \theta _0\right) ^j\cos \phi \text {d}\phi \Bigg \}, \end{aligned} \end{aligned}$$

where \(\theta _0=\tan ^{-1}\left( \xi U\right) \). Upon carrying the integration in Eq. (B.12), we obtain

$$\begin{aligned} a \frac{\mathrm{d} \gamma }{\mathrm{d} \tau } = - 2\mu U^2 \sum _{j=1}^{n} A_j \left( \frac{a}{U}\right) ^j \left( \sqrt{\xi ^2 U^2 + 1}\right) ^{j-1} C_j, \end{aligned}$$

where the \(C_j\) are given by

$$\begin{aligned} \begin{aligned} C_j&= \frac{1-(-1)^{j}}{2^{j+2}}\left( \frac{(j+1)!}{\left( \left( \frac{1}{2}(j+1)\right) !\right) ^2}\right) \\&\quad + \frac{1+(-1)^{j}}{2^j\pi }\sum _{k=0}^{j/2}\frac{(j+1)!\,(-1)^{\frac{1}{2}j-k}}{(j+1-k)!\,k!\,(j+1-2k)}. \end{aligned} \end{aligned}$$

Following a similar procedure to enforce the orthogonality with \(\sin \phi \) yields

$$\begin{aligned} \frac{\text {d} a}{\text {d} \tau } = - \zeta _ma + 2\mu U^2 \sum _{j=1}^{n} A_j \left( \frac{a}{U}\right) ^j \left( \sqrt{\xi ^2 U^2 + 1}\right) ^{j-1} C_j. \end{aligned}$$

At steady state, \(\frac{\text {d} \gamma }{\text {d} \tau } = \frac{\text {d} a}{\text {d} \tau } = 0\). Root-solving algorithms can then be used to solve Eqs. (B.13) and (B.15) for the steady-state phase shift and amplitude of the response, respectively.

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Noel, J., Alhadidi, A.H., Alhussien, H. et al. A time-implicit representation of the lift force for coupled translational–rotational galloping. Nonlinear Dyn (2021). https://doi.org/10.1007/s11071-021-06232-6

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  • Lift force
  • Galloping
  • Universal curve
  • Transient response