Unsteady Applications: Thrust Optimization, Stability and Trim

Abstract

The unsteady aerodynamics related material developed in the previous chapter for the practical applications is going to be utilized in this chapter for the thrust optimization and the flight stability of bodies having flapping wings.

Appendix

Appendix 16 provides the necessary information about the numerical values of I_1, I₂ and $\bar{b}$ to be used in the following equation

$$ C_{L} (t) = \frac{{l_{cr} (t) + l_{cp} (t) + l_{nc} (t)}}{{\rho U_{m}^{2} \,S/2}} $$

(9.40)

The variation of the total lift coefficient for one period is shown in (Fig. 9.13) for the thin airfoil and for the wing.

Fig. 9.13

Variation of the total lift coefficient for a period

Using (Fig. 9.13) and Eq. (9.40) for a period of the motion we get the result of the following integration as

$$ \bar{C}_{L} = \frac{1}{T}\int\limits_{T}^{2T} {C_{L} (t)\,dt} = 0.75 $$

(9.41)

Accordingly, the lifting force generated for both wings reads as$F = 2\bar{C}_{L} \rho U_{m}^{2} \,S/2 = 7.11\,\upmu {\text{N}}$,

Here, the flapping frequency for the wing is f = 240 Hz which is sufficient to lift the fruit fly which weighs about $W = 7.06\,\upmu {\text{N}}$ (Berman and Wang 2007).

The wing sweep here given by $\phi = - 75^{o} \cos \omega \,t$, and the change of the pitch angle is provided with the arctangent and sine functions as shown in (Fig. 9.14). The free stream velocity span -wise variation is given by $\dot{\phi }\,r = \bar{\phi }\omega \,r\sin \omega \,t$.

Fig. 9.14

Sweep angle: $- 75^{^\circ } < \phi < 75^{^\circ }$, and the pitch angle, $40^\circ < \eta < 140^\circ$, variations

Flight Stability: Study of the longitudinal stability of a body with a flapping wing requires the coupled treatment of unsteady aerodynamics with the parameters of the flight mechanics. Shown in (Fig. 9.15) are the necessary parameters to prescribe the hovering body under the gravity where the longitudinal and vertical velocity perturbation velocities are u and v respectively with also are the pitch rate of q and the pitch angle $\theta$. The dynamic equilibrium equation using the notation of Nelson (1998) reads as follows

Fig. 9.15

Reference frame and the parameters used for the flight stability

$$ \left( \begin{aligned} {\dot{u}} \hfill \\ {\dot{w}} \hfill \\ {\dot{q}} \hfill \\ {\dot{\theta }} \hfill \\ \end{aligned} \right) = \left( \begin{aligned} - qw - g\sin \theta \hfill \\ \,\,qu + g\cos \theta \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,q \hfill \\ \end{aligned} \right) + \left( \begin{aligned} X/m \hfill \\ Z/m \hfill \\ M/I_{y} \hfill \\ \,\,\,\,\,\,0 \hfill \\ \end{aligned} \right) $$

(9.42)

Here, X, Z and M are the horizontal and the vertical forces and the pitching moment respectively, and m is the mass, I_y is the rotational moment of inertia and finally g is the gravitational acceleration.

Unsteady aerodynamics of a flapping wing gives the lift, moment in terms of the pitch and the pitch rate. Here, we implement the state variables concept, x₁ and x₂, with two ordinary differential equations in time involving the pitch and its rate:

$$ \,\,\,\dot{x}_{1} (t) = \frac{{b_{1} U_{ref} }}{{\bar{b}}}\left[ { - x_{1} (t) + a_{1} \dot{\phi }(t)C_{L} (\alpha ,t)} \right]\,\,\,\,and\,\,\,\dot{x}_{2} (t) = \frac{{b_{1} U_{ref} }}{{\bar{b}}}\left[ { - x_{2} (t) + a_{1} \dot{\alpha }(t)} \right]\,\,\, $$

If we let the sate variables interact with the equation of motion, (9.42), the perturbation equations for the system involves the rate of 6 variables,$\dot{\chi } = \left( {\dot{u}\,\,\,\dot{w}\,\,\,\dot{q}\,\,\,\,\dot{\theta }\,\,\,\,\dot{x}_{1} \,\,\dot{x}_{2} \,} \right)^{T}$, and their stability derivative matrix as follows

$$ \left( \begin{aligned} {\dot{u}} \hfill \\ {\dot{w}} \hfill \\ {\dot{q}} \hfill \\ {\dot{\theta }} \hfill \\ \dot{x}_{1} \hfill \\ \dot{x}_{2} \hfill \\ \end{aligned} \right) = \left( \begin{aligned} - qw - g\sin \theta \hfill \\ \,\,\,qu + g\cos \theta \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,q \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \hfill \\ \end{aligned} \right) + \left( \begin{aligned} X_{0} \hfill \\ Z_{0} \hfill \\ M_{0} \hfill \\ \,\,\,0 \hfill \\ X_{{1_{0} }} \hfill \\ \,\,0 \hfill \\ \end{aligned} \right) + \left[ \begin{aligned} X_{u} \,\,X_{w} \,X_{q} \,\,\,\,\,0\,\,\,\,X_{{x_{1} }} X_{{x_{2} \,}} \hfill \\ Z_{u} \,\,\,\,Z_{w} \,\,\,Z_{q} \,\,\,\,\,0\,\,\,\,\,Z_{{x_{1} }} \,\,Z_{{x_{2} }} \hfill \\ M_{u} \,\,M_{w} \,\,M_{q} \,0\,\,\,M_{{x_{1} }} \,M_{{x_{2} }} \hfill \\ \,\,\,0\,\,\,\,\,\,\,0\,\,\,\,\,\,\,0\,\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,\,\,0 \hfill \\ \,X_{{1_{u} }} \,X_{{1_{w} }} \,X_{{1_{q} }} \,\,0\,\,\,X_{{1_{{x_{1} }} }} \,\,0 \hfill \\ \,\,\,0\,\,\,\,\,\,\,0\,\,\,\,\,\,X_{{2_{q} }} \,0\,\,\,\,\,0\,\,\,\,\,X_{{2_{{x_{2} }} }} \hfill \\ \end{aligned} \right]\left( \begin{aligned} u \hfill \\ w \hfill \\ q \hfill \\ 0 \hfill \\ x_{1} \hfill \\ x_{2} \hfill \\ \end{aligned} \right) $$

(9.43)

In Eq. (9.43), the column vector expressed with subscript o shows the effect of pitch rate and the subscript for the coefficient matrix indicates the derivatives (Appendix 17). The horizontal sectional force is shown by, X’, and the vertical sectional force by, Z’. The sectional lift l and the drag d are employed to give these horizontal and vertical forces as follows

$$ X^{\prime} = - {\rm sgn} (\dot{\phi })(d - l\alpha_{i} )\,\,\,\,and\,\,\,Z^{\prime} = - (l + d\alpha_{i} )\,\,,\,\,\,\alpha_{i} \,induced\,\,angle\,\,of\,attack $$

(9.44a,b)

Here, two differen contributions to the sectional forces are possible: (i) from pitching l_P,

(ii) from pitch rate l_Pr, which are determined with x₁ and x₂. On the other hand the induced angle of attack is small and can be approximated as

$$ \,\,\,\alpha_{i} \cong \frac{{w{}_{eff}}}{\left| U \right|}\,,\,\,\,\,\,\,and\,\,\,\,w_{eff} = w - q(r\sin \phi + a),\,\,\,\,\,\,U \cong r\dot{\phi } + u\cos \phi $$

The perturbation velocities u, w and the pitch rate q, all contribute to the sectional lift coefficient in terms of pitch and the pitch rate. Considering the effective free stream velocities, the sectional lift forces read as

$$ \begin{aligned} \,l_{P} = & \rho (r\dot{\phi } + u\cos \phi )\text{sgn} (\dot{\phi })\left[ {b(r\dot{\phi } + u\cos \phi )A(\sin 2\eta + 2\alpha_{i} \cos 2\eta )\varPhi (0) + x_{1} (t)} \right], \, \text{and} \\ \,l_{\Pr } = & \rho (r\dot{\phi } + u\cos \phi )\text{sgn} (\dot{\phi })\pi b^{2} \left[ {q\cos \phi + (1/2 - a)q\cos \phi \,\varPhi (0) + x_{2} (t)} \right]\, \end{aligned} $$

(9.45a,b)

Similarly, the sectional drag becomes

$$ \,d_{P} = \rho (r\dot{\phi } + u\cos \phi ){\rm sgn} (\dot{\phi })\left[ {b(r\dot{\phi } + u\cos \phi )A(\sin^{2} \eta + \alpha_{i} \sin 2\eta )\varPhi (0) + x_{1} (t)} \right]\, $$

Since the induced angle of attack is small the second order terms are neglected, and this gives the induced lift and the drag as follows

$$ \,l_{P} \alpha_{i} = \rho \,w_{eff} \left[ {brA\dot{\phi }\sin 2\eta \,\varPhi (0)} \right],\,\,\,\,\,\,\,\,\,d_{P} \alpha_{i} = \rho \,w_{eff} \left[ {2brA\dot{\phi }\sin^{2} \eta \,\varPhi (0)} \right] $$

(9.46a,b)

Averaging

Equation (9.43) is now utilized for the stability analysis of the time dependent periodic system. For this purpose, we take the time average of the quantities for a period of time T as

$$ \,\bar{\dot{\chi }} = f(\bar{\chi }) + h(\bar{\chi }),\,\,\,\,here\,\,\,\,h(\bar{\chi }) = \frac{1}{T}\int\limits_{0}^{T} {h(\chi ,t)dt} $$

Now, the averaged system reads as

$$ \left( \begin{aligned} {\bar{\dot{u}}} \hfill \\ {\bar{\dot{w}}} \hfill \\ {\bar{\dot{q}}} \hfill \\ {\bar{\dot{\theta }}} \hfill \\ \Delta \bar{\dot{x}}_{1} \hfill \\ \bar{\dot{x}}_{2} \hfill \\ \end{aligned} \right) = \left( \begin{aligned} - \bar{q}\bar{w} - g\sin \bar{\theta } \hfill \\ \,\,\,\bar{q}\bar{u} + g\cos \bar{\theta } \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \hfill \\ \end{aligned} \right) + \left( \begin{aligned} \bar{X}_{0} + \bar{X}_{{x_{1} }} \bar{x}_{1eq} \hfill \\ \bar{Z}_{0} + \bar{Z}_{{x_{1} }} \bar{x}_{1eq} \hfill \\ \bar{M}_{0} + \bar{M}_{{x_{1} }} \bar{x}_{1eq} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,0 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,0 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,0 \hfill \\ \end{aligned} \right) + \left[ \begin{aligned} \bar{X}_{u} \,\,\bar{X}_{w} \,\bar{X}_{q} \,\,\,\,\,0\,\,\,\,\bar{X}_{{x_{1} }} \bar{X}_{{x_{2} \,}} \hfill \\ \bar{Z}_{u} \,\,\,\,\bar{Z}_{w} \,\,\,\bar{Z}_{q} \,\,\,\,\,0\,\,\,\,\,\bar{Z}_{{x_{1} }} \,\,\bar{Z}_{{x_{2} }} \hfill \\ \bar{M}_{u} \,\,\bar{M}_{w} \,\,\bar{M}_{q} \,0\,\,\,\bar{M}_{{x_{1} }} \,\bar{M}_{{x_{2} }} \hfill \\ \,\,\,0\,\,\,\,\,\,\,0\,\,\,\,\,\,\,1\,\,\,\,\,\,0\,\,\,\,\,0\,\,\,\,\,\,\,0 \hfill \\ \,\bar{X}_{{1_{u} }} \,\bar{X}_{{1_{w} }} \,\bar{X}_{{1_{q} }} \,\,0\,\,\,\bar{X}_{{1_{{x_{1} }} }} \,\,0 \hfill \\ \,\,\,0\,\,\,\,\,\,\,0\,\,\,\,\,\,\bar{X}_{{2_{q} }} \,0\,\,\,\,\,0\,\,\,\,\,\bar{X}_{{2_{{x_{2} }} }} \hfill \\ \end{aligned} \right]\left( \begin{aligned} \,\bar{u} \hfill \\ \,\bar{w} \hfill \\ \,\bar{q} \hfill \\ \,\bar{\theta } \hfill \\ \Delta \bar{x}_{1} \hfill \\ \,\bar{x}_{2} \hfill \\ \end{aligned} \right) $$

(9.47)

Here, $\Delta \bar{\dot{x}}_{1} = \bar{x}_{1} - \bar{x}_{1eq} \,\,and\,\,\,\bar{x}_{1eq} = - \bar{X}_{{1_{0} }} /\bar{X}_{{1_{x1} }} \,$. The algebraic Eq. (9.47) are solved for $\left( {\bar{u} = 0\,,\,\,\bar{w} = 0,\,\,\,\bar{q} = 0,\,\,\,\,\bar{\theta } = 0} \right)$ in hover.

Trim in hover: Time averaged equation of motion is now implemented for the trimming of the hovering body while flapping its wings. For the wing flapping, there are two different types; (i) symmetric flapping, and (ii) anti-symmetric flapping.

• (i) Trim with symmetric flapping

For the sake of simplicity, we choose the time dependent sweeping motion as a saw-tooth shaped which is expressed as Mouy et al. (2017)

$$ \phi (t) = \left\{ \begin{aligned} \,\,\,\,\,4\bar{\phi }/T(t - T/4),\,\,\,\,\,\,\,0 \le t \le T/2 \hfill \\ - 4\bar{\phi }/T(t - 3T/4),\,\,\,\,\,T/2 \le t \le T\, \hfill \\ \end{aligned} \right. $$

During the sweeping motion of the wing, a piecewise constant angle of attack is considered as follows

$$ \eta (t) = \left\{ \begin{aligned} \,\,\,\,\bar{\alpha },\,\,\,\,\,\,\,\,\,\,\,\,0 \le t \le T/2 \hfill \\ \,\,\,\pi - \bar{\alpha },\,\,\,\,\,T/2 \le t \le T\, \hfill \\ \end{aligned} \right. $$

During the symmetric sweeping, the full unsteady treatment gives a non-zero value for the X force component, whereas the quasi-steady approach yields 0 result (Mouy et al. 2017). The full unsteady treatment results in Eq. (9.47) for the x direction, in terms of averaged values, as follows (Appendix 18):

$$ \bar{X}_{{x_{1} }} \bar{x}_{1eq} = 8\frac{{\rho \,I_{10} }}{{mT^{2} }}\bar{r}2\bar{b}\bar{\phi }\sin \bar{\phi }Aa_{1} \sin 2\bar{\alpha } $$

(9.48)

The right hand side of Eq. (9.48) is 0 only for the average sweep or the angle of attack, $\bar{\phi } = 0\,\,\,or\,\,\,\bar{\alpha } = 0$. This means trimming is possible only for the no sweep or no lift! For this reason we have to resort to antisymmetric sweep.

• (ii) Trim with anti-symmetric flapping

In order to achieve trim during hover the sweeping motion is modified as follows:

$$ \phi (t) = \left\{ \begin{aligned} \bar{\phi }_{0} + \,\,4\bar{\phi }/T(t - T/4),\,\,\,\,\,\,\,0 \le t \le T/2 \hfill \\ \bar{\phi }_{0} - 4\bar{\phi }/T(t - 3T/4),\,\,\,\,\,T/2 \le t \le T\, \hfill \\ \end{aligned} \right. $$

The angle of attack:

$$ \eta (t) = \left\{ \begin{aligned} \,\,\,\,\bar{\alpha }_{d} ,\,\,\,\,\,\,\,\,\,\,\,\,0 \le t \le T/2 \hfill \\ \,\,\,\pi - \bar{\alpha }_{u} ,\,\,\,\,\,T/2 \le t \le T\, \hfill \\ \end{aligned} \right. $$

The trim equations then read:

$$ \begin{aligned} \bar{X}_{0} + \bar{X}_{x1} \bar{X}_{1eq} = 0 \hfill \\ \bar{Z}_{0} + \bar{Z}_{x1} \bar{X}_{1eq} = - g \hfill \\ \bar{M}_{0} + \bar{M}_{x1} \bar{X}_{1eq} = 0 \hfill \\ \end{aligned} $$

(9.49a,b,c)

In Eq. (9.49a,b,c), there are 3 equations and 3 unknowns; $\,\bar{\alpha }_{d} ,\,\,\bar{\alpha }_{u} \,and\,\bar{\phi }_{0}$. We use the time averaged values for the stability derivatives to obtain following non-linear expressions (Appendix 3).

Force balance in X:

$$ \,\sin 2\bar{\alpha }_{d} + \,\sin 2\bar{\alpha }_{u} = - \frac{{I_{21} }}{{I_{10} \bar{r}\bar{b}}}(\sin^{2} \bar{\alpha }_{d} - \sin^{2} \bar{\alpha }_{u} )(0.6 - a_{1} )/a_{1} $$

(9.50)

Force balance in Z:

$$ \sin 2\bar{\alpha }_{d} + \,\sin 2\bar{\alpha }_{u} = \frac{{mgT^{2} }}{{4\rho A(0.43I_{21} + 0.17I_{10} \bar{r}2\bar{b})\bar{\phi }^{2} }} $$

(9.51)

Moment balance in M:

$$ \begin{aligned} a\sin \bar{\phi }\cos \bar{\phi }\left[ {0.43I_{22} (\sin \bar{\alpha }_{d} - \sin \bar{\alpha }_{u} ) - 0.17I{}_{11}\bar{r}2\bar{b}(\sin 2\bar{\alpha }_{d} + \,\sin 2\bar{\alpha }_{u} )} \right]x \hfill \\ (\cos \bar{\alpha }_{d} + \sin \bar{\alpha }_{d} - \cos \bar{\alpha }_{u} + \sin \bar{\alpha }_{u} ) + 0.43x_{cg} \bar{\phi }(\sin 2\bar{\alpha }_{d} + \,\sin 2\bar{\alpha }_{u} )(I{}_{21} - 2I_{10} \bar{r}2\bar{b}) \hfill \\ + \sin \bar{\phi }(0.43I_{31} \cos \bar{\phi }_{0} - 0.17I_{20} 4\bar{r}\bar{b}\sin \bar{\phi }_{0} )(\sin 2\bar{\alpha }_{d} + \,\sin 2\bar{\alpha }_{u} ) = 0 \hfill \\ \end{aligned} $$

(9.52)

Here, $a$ is distance between the pitch point and the quarter chord.

We solve for X and Z force balances (9.50) and (9.51), to obtain $\,\bar{\alpha }_{d} \,and\,\,\bar{\alpha }_{u} \,$ in terms of the sweep angle and the period, $\bar{\phi }\,\,and\,\,T$, respectively. Hence, we can calculate the angle of attack for the forward and backward sweeps. Afterwards, by the aid of (9.52) the initial sweep angle $\bar{\phi }_{0}$ is determined to complete the solution for the trim in hover.

Trimming of fruit-fly in hover: The pertinent parameters for a fruit-fly is provided as follows (Berman and Wang 2007):

m = 0.72 mg, f = 254 Hz, $\,\bar{\phi } = 75^{\circ}$, a = 0, x_cg = 0.5, I₁₀ = 4.89, I₁₁ = 3.30, I₂₁ = 4.69, I₂₀ = 7.29, I₂₂ = 3.32, I₃₁ = 7.36

From the simultaneous solution of (9.50), (9.51) and (9.52), the anti-symmetric trim results are obtained as follows $\,\bar{\alpha }_{d}$ = 34.3°, $\,\bar{\alpha }_{u}$ = 55.6° and $\bar{\phi }_{0}$ = 4°

For this case the angle of attack differs for the forward and the backward sweeps. The angle of attack becomes 34.3° for the forward sweep and 55.6° in backward sweep. During sweeping the forward sweep angle changes −71° < $\phi (t)$ < 79° and in backward sweep 79° > $\phi (t)$ > −71°. The time change of sweep and angles of attack are given for both symmetric and antisymmetric cases in (Fig. 9.16). The deviation from the symmetric case is about 5° for the angle of attack and 4° for the sweep, which can be applied easily for the control purposes.

Fig. 9.16

Trimmed antisymmetric sweep and angle of attack____, un-trimmable symmetric sweep