Chapter 4 Galloping and flutter

The toy demonstrating the spontaneous oscillations of the cylinder with the D-shaped cross section supported on two springs is an excellent example of translational galloping. This toy was first presented as an example by Den Hartog[3] in 1956, a name which we will see again in this section. Figure 4.1 shows a schematic of this toy as a one-degree-of-freedom oscillator, consisting of a D-shaped mass $m$ supported by a spring with stiffness $k$ and a damper wth constant $b$. The flat face of the D-shaped cross section faces the oncoming wind with speed $U$, and, as a result, the the mass starts to oscillate. In the classroom demo, the D-shape has a diameter of about 5 cm, and oscillates with a frequency of about 2 Hz. Here we present the mechanism behind the spontaneous emergence of these oscillations in the form of an instability.

The state variable we invoke is the displacement of the mass perpendicular to the oncoming flow, which we denote $y(t)$. The state obeys

$m\ddot{y}+b\dot{y}+ky=f,$

(4.1)

where $f$ is the aerodynamic force along $y$. Because of the symmetry of the D-shape, a steady state exists when $y=y_0=0$. In this case, $f$ also vanishes. Therefore, we perturb this steady state as $y=y_0+y^{\prime }$, where $y^{\prime }$ is the perturbation. Now noting that the steady state is $y_0=0$, $y=y^{\prime }$, so we might as well drop the prime on the $y$, a practice that is commonly followed and one that the author is eager to adopt. And eq. 4.1 is linear in $y$, so assuming $y$ to be infinitesimal means we merely need to determine $f$ when $y$ is infinitesimal.

To determine the fluid dynamic force along $y$, we first test for quasi-steadiness. The structural natural frequency is $f\approx$ 2 Hz, and the characteristic length scale for the flow to develop is the diameter $D$ of the D-shape, and the flow speed is $U\approx$ 10 m/s. So the reduced frequency is $fD/U\approx 0.01$. In other words, a fluid particle flows 100 diameter downstream by the time the structure completes one oscillation. This inspires us to treat the fluid dynamics as steady.

The $y$ component of the force $f$ derives contribution from two contributions, and both of them originate from examining the relative velocity of the fluid with respect to the body. Even an infinitesimal velocity of the mass changes the apparent orientation of the oncoming wind, even if it is by an infinitesimal amount, which we will now determine. The mathematical determination of this apparent wind is shown in fig. 4.1(b), where the mass is moving in the direction of positive $y$, which in the reference frame of the mass causes a component of flow in the negative $y$ direction. The relative wind speed $U_{\text{rel}}$ is given by

$U_{\text{rel}}=\sqrt{U^2+{\dot{y}}^2},$

(4.2)

and the apparent angle of attack $\alpha$ is given by

$\alpha ={\tan }^{-1}\left({\displaystyle \frac{\dot{y}}{U}}\right).$

(4.3)

Only the linear terms in the approximations of $U_{\text{rel}}$ and $\alpha$ need to be retained in eqs. 4.2 and 4.3, as required by linear stability analysis. This relative flow gives rise to a drag $F_D$ and a lift $F_L$ on the D-shaped cross section, which are aligned along the direction of relative speed and perpedicular to it, respectively. The lift and drag may be parameterized using lift and drag coefficients as

$F_L={\displaystyle \frac{1}{2}}\rho U_{\text{rel}}^{2}A{C}_L(\alpha ,\text{Re}),F_D={\displaystyle \frac{1}{2}}\rho U_{\text{rel}}^{2}A{C}_D(\alpha ,\text{Re}),$

(4.4)

where $\text{Re}=\rho U_{\text{rel}}D/\mu$ is the Reynolds number, $\rho$ is the fluid density and $\mu$ its viscosity. The force $f$ is then given by

$f=-F_L\cos \alpha -F_D\sin \alpha .$

(4.5)

We now approximate these quantities up to linear order in the perturbation $y$ as

	$U_{\text{rel}}$	$\approx U+\mathrm{\dots },$		(4.6a)
	$\alpha$	$\approx {\displaystyle \frac{\dot{y}}{U}}+\mathrm{\dots },$		(4.6b)
	$F_L$	$\approx {{\displaystyle \frac{1}{2}}\rho U^{2}A{\displaystyle \frac{\text{d}{C}_L}{\text{d}\alpha }}|}_{\alpha =0}\alpha +\mathrm{\dots },$		(4.6c)
	$F_D$	$\approx {{\displaystyle \frac{1}{2}}\rho U^{2}A{C}_D|}_{\alpha =0}+\mathrm{\dots }$		(4.6d)
	$f$	$\approx {\displaystyle \frac{1}{2}}\rho UA{C}_y\dot{y}+\mathrm{\dots }$		(4.6e)
where
	$C_y$	$=\left(-{{\displaystyle \frac{\text{d}{C}_L}{\text{d}\alpha }}|}_{\alpha =0}-{C_D|}_{\alpha =0}\right).$		(4.6f)

Here the two contributions behind the vertical fluid dynamic force $f$ is as follows. Firstly, the drag is realigned in the direction of the apparent oncoming flow, which is now at an angle $\alpha$ and has a small component in the direction of motion proportional to the speed of the mass. This is the term proportional to $C_D$ in eq. 4.6f. Secondly, the small angle of attack the D-shape makes to the oncoming flow gives rise to a proportional lift force. This lift is nearly in the direction of negative $y$, and gives rise to the term proportional to $\text{d}{C}_L/\text{d}y$ in eq. 4.6f. The resulting $C_y$ is called the Den Hartog coefficient, which is a dimensionless expression of the fluid dynamic response.

We can now enter the phase of stability analysis. Replacing the aerodynamic force $f$ from eq. 4.6e in eq. 4.1, yields the coupled equation for the evolution of the fluid-structure state as

	$m\ddot{y}+b^{\prime }\dot{y}+ky=0,$		(4.7)
where
	$b^{\prime }=b+{\displaystyle \frac{1}{2}}\rho UA{C}_y$		(4.8)

is the equivalent damping coefficient accounting for the aerodynamic force. The stability of the state $y=0$ depends on the sign of the equivalent damping coefficient $b^{\prime }$. We expect $C_D(\alpha =0)$ to be positive for all shapes of the body, including the “D” shape cross section. For streamlined shapes, we expect $C_L$ to be an increasing function of $\alpha$, so that $\text{d}{C}_L/\text{d}\alpha$ is positive. However, for many bluff shapes, such as the D-shape, lift acts in the opposite direction. This is shown in fig. 4.1(c) The reader can imagine the reason based on how the stagnation pressure acting on the flat face forces the body. As a result, for many bluff cross sections, including the “D” shape and many common structural shapes such as the $I$ and $H$ beams, $\text{d}{C}_L/\text{d}\alpha$ is negative around $\alpha =0$, and results in $C_y$ being negative.

The usual stability calculation, which proceeds by expressing $y=\widehat{y}{e}^{st}$, then determines exponential growth when $b^{\prime }$ is negative (see section 2.1, which also includes a non-dimensionalization). Based on this criterion, there exists a critical velocity $U_{\text{crit}}$ given by

$U_{\text{crit}}={\displaystyle \frac{2b}{-\rho A{C}_y}}.$

(4.9)

For fluid velocity above this critical, galloping is spontaneously excited. This result is the culmination of the stability analysis.

Galloping instability may also appear in systems with one rotational degree of freedom, such as pitch, for bluff bodies. The treatment of fluid dynamics in this case is more empirical than for the translational case but follows along closely parallel lines The rotational degree of freedom is represented by the state variable $\theta$, which obeys

$I\ddot{\theta }+B\dot{\theta }+\kappa \theta =Q(\theta ,\dot{\theta }),$

(4.10)

where $I$ is the moment of inertia of the object, $B$ is the rotational damping coefficient, $\kappa$ is the torsional spring constant, and $Q$ is the fluid dynamic torque on the object. The rotational damping coefficient may be written in terms of a linear damping coefficient $b$ as $B=b{R}_g^2$, just in the same way the rotational moment of inertia is written in terms of a mass $I=m{R}_g^2$, where $R_g$ is the radius of gyration. It is also convenient to express $\kappa =k{R}_g^2$ for comparison with the translational galloping case. The argument then proceeds to justify that, in quasi-steady flow, the torque may be parameterized as

$Q={\displaystyle \frac{1}{2}}\rho U_{\text{rel}}^{2}A{L}_Q{C}_Q(\alpha ,\text{Re}),$

(4.11)

where $U_{\text{rel}}$ is the effective relative velocity, $A$ is the frontal cross section of the object as before, $L_Q$ is an additional length scale akin the distance between the centre of support and centre of pressure, and $C_Q$ is the torque coefficient, which depends on an effective angle of attack, $\alpha$. This effective angle of attack for a rotating body is then taken to be an average along the length of the body, and this is not quite rigorous. Yet, in the limit of infinitesimal perturbations, $\theta ={\theta }_0+{\theta }^{\prime }$, where ${\theta }_0$ is a steady state, the torque parameterization reduces to

$Q={\displaystyle \frac{1}{2}}\rho U_{\text{rel}}^{2}A{L}_Q{\displaystyle \frac{\text{d}{C}_Q}{\text{d}\alpha }}\left({\theta }^{\prime }+{\displaystyle \frac{L_{\alpha }}{U}}{\dot{\theta }}^{\prime }\right),$

(4.12)

where $L_{\alpha }$ is another length scale that arises from the determination of the average angle of attack. (This part has no satisfactory justification but the forgiving reader can examine the dimensions for consistency. Loosely speaking, $L_{\alpha }\dot{\theta }$ plays the same role that $\dot{y}$ played in the translational case in determining the effective angle of attack. ) Substituting this torque in eq. 4.10 then yields a

	$m{\ddot{\theta }}^{\prime }+b^{\prime }{\dot{\theta }}^{\prime }+k^{\prime }{\theta }^{\prime }=0$		(4.13)
where
	$b^{\prime }=\left(b-{\displaystyle \frac{1}{2}}\rho UA{\displaystyle \frac{L_Q{L}_{\alpha }}{R_g^2}}{\displaystyle \frac{\text{d}{C}_Q}{\text{d}\alpha }}\right)$		(4.14)
	$k^{\prime }=\left(k-{\displaystyle \frac{1}{2}}\rho U^{2}A{\displaystyle \frac{L_Q}{R_g^2}}{\displaystyle \frac{\text{d}{C}_Q}{\text{d}\alpha }}\right).$		(4.15)

At this stage the similarity between the translational and rotational cases of galloping is evident, except for one difference. In this case, the torsional spring is also modified, which can lead to the divergence instability we discussed in detail in chapter 1. Here we ignore that possibility and focus on galloping, which arises when $b^{\prime }$ becomes negative, which is possible if $\text{d}{C}_Q/\text{d}\alpha$ is positive. The critical velocity for rotational galloping is

$U_{\text{crit}}={\displaystyle \frac{2b}{\rho A{C}_y}}\text{where}{C}_y={\displaystyle \frac{L_Q{L}_{\alpha }}{R_g^2}}{\displaystyle \frac{\text{d}{C}_Q}{\text{d}\alpha }}.$

(4.16)

The astute reader will compare this critical velocity with the divergence speed to determine the mechanism to which a given structure is most susceptible.

The advantage of the simple analysis presented in this section is that it provides us with insight into the physical mechanism of galloping instability, and in doing so provides us with methods of suppressing it. The critical velocity in eq. 4.9 directly embodies this insight. Firstly, note that in this case the excited frequency of oscillations is close to the natural frequency, slightly modified due to the damping coefficient. This follows directly from the analysis in section 2.1 applied to eq. 4.7. Secondly, the instability arises because the fluid dynamical characteristics, completely encapusilated in the Den Hartog coefficient $C_y$, overcome the internal structural damping $b$. The Den Hartog coefficient for various cross sections is presented in Table 4-1 (page 109) of book on Flow-induced Vibrations by Blevins[1]. This motivates the definition of a scale of velocity

$U_{\text{gallop}}={\displaystyle \frac{2b}{\rho A}}.$

(4.17)

The magnitude of this velocity depends on the structural geometry (cross section area $A$, possibly ratio of lengths) and damping (coefficient $b$) relative to the fluid material characteristics ($\rho$). The criteria for spontaneous excitation may be written as $U_{\text{crit}}=U_{\text{gallop}}/|C_y|$, and offers the following interventions to suppress them. Galloping may be suppressed by increasing the structural damping. Doubling the damping coefficient doubles the $U_{\text{gallop}}$, and thus double the critical velocity. Galloping may also be suppressed by modifying the cross-section shape, for example, by streamlining it. This modifies $C_y$, and if the magnitude of $C_y$ can be decreased, it increases the critical velocity. Note that stiffening the structure is expected to merely increase the natural frequency and have no influence on the galloping threshold.

Unlike galloping, increasing damping or adding it if it does not exist is not the most effective method to suppress flutter. It is so because the core mechanism of flutter relies on structural synchronization, and therefore the best intervention is one that disrupts it. Common approaches include stiffening the torsional defree of freedom so that its natural frequency is well separated from the bending or heave degree of freedom. Doing so is not only effective but also economical.