Longitudinal Trim and Dynamic Stability Analysis of a Seagull-Based Model

Featured Application

The conclusions of this paper can provide guidance for the design of seagull-inspired micro air vehicles.

Abstract

Understanding the mechanisms of trim and flight stability in birds is critical to guide the design of bionic micro air vehicles. The complex movements (plunging, sweeping, twisting) and morphing of wings always keeps the flapping flight of birds in dynamic equilibrium, which makes it difficult to determine the critical factors of trim and stability. Hence, a model has been developed that takes real complex movement and the calculation of unsteady aerodynamics into consideration. Two trim methods, including wash-out and forward-sweep, have been used to achieve equilibrium in the longitudinal direction. It is interesting to find that these two methods are both important to realize a larger take-off weight, lower power consumption, and stronger longitudinal stability. This implies that the seagull probably uses both of them to obtain the requirement of equilibrium and stability, which further inspires the design of seagull-inspired micro air vehicles.

1. Introduction

Understanding the mechanisms of trim and flight stability in birds is critical to guide the design of bionic micro air vehicles.

Thomas and Taylor [1,2] first investigated the issue of dynamic stability in gliding and flapping flight. Their results show that the wings alone may be sufficient to provide longitudinal static stability in the gliding flight of birds, which make birds resemble tailless aircraft more closely than conventional aircraft with a tailplane. They found that stable gliding flight should satisfy one or more of the following features: (1) fliers can sweep their wings forward in slow flight; (2) wing sections can be twisted down at the tips when swept back (wash-out) and twisted up at the tips when swept forwards (wash-in), which can generate a nose-up moment; and (3) additional lifting surfaces. Inspired by the above theory, seagulls, as the subject of this work, may take the methods of wash-out and forward-sweep to realize their longitudinal balance.

Quasi-static and blade element approaches were used to analyze the stability provided by a flapping wing. They found that beating the wings faster simply amplified any existing stability or instability, and flapping can even enhance stability compared to gliding at the same airspeed. However, static stability is necessary but insufficient for dynamic stability, because a statically stable system could overcompensate for a disturbance, which could in turn lead to divergent oscillations. A full stability analysis would therefore need to consider the dynamics of the system.

Short-term dynamic stability and trim of birds have been studied by Iosilevskii [3]. Without active control, an interaction between flapping, pitching, and heave can cause resonance. This can be suppressed by active control, either with a fore–aft sweeping motion of the wing or with an up-and-down deflection of the tail.

Sachs [4] developed a two degree of freedom model of the bird, which makes it possible to account for speed and height changes. With this expanded treatment, it can be shown that there is speed stability in the gliding flight of birds, which also holds for flapping flight at speeds below the speed of the minimum power required.

The aforementioned scholars utilized simple trigonometric functions. They did not employ the true motion laws of bird wings, and they ignored the changes in lift and thrust caused by the folding motion.

Though there are several studies on the dynamic stability of birds, computational methods for creatures [5,6,7], and dynamic stability analysis for flapping flight can give guidance and reference in the study of birds.

Orlowski and Girard studied the inertial effects of flapping wings on flight stability through deriving equations from multiple rigid body dynamics [8]. Lee et al. numerically simulated flapping flight characteristics through coupling flexible multibody dynamics with a reduced-order aerodynamic model, using structure solver [9]. For the insects and FWMAVs with relatively high flapping frequency and low wing-to-body mass ratio, the eigenmode analysis technique can be used because it is applicable to a linear time invariant (LTI) system [10,11]. Au and Park studied the influence of the center of gravity location on the longitudinal dynamic stability of hovering in flapping-wing micro air vehicles, using an average method to analyze stability [12].

The main purpose of this paper is to quantitatively study the trim state and dynamic stability of one particular bird, the seagull, using the real movement, airfoils, and geometry of birds. In order to determine the accurate aerodynamic force, we used computational fluid dynamics which is a relatively precise calculation method. The quantitative results about trim and dynamic stability can enlighten the design of bird-inspired air vehicles.

2. Models and Methods

This section mainly describes the geometric and dynamic model of seagulls and the computational method used to determine the aerodynamic force produced by wings.

2.1. Reference Frame and Equations of Motion of Seagulls

Only the longitudinal motion is considered in the present paper; however, the seagull has three freedom ranges in the longitudinal direction, as shown in the coordinate system of Figure 1: horizontal displacement $x_{\mathrm{b}}$, vertical displacement $z_{\mathrm{b}}$, and pitch angle $\theta$.

The Newton–Euler-based equations of a seagull’s body are simplified to:

$$\begin{array}{l}\frac{\mathrm{d}u(t)}{\mathrm{d}t}=\frac{1}{m}X(t)-g\sin \theta -q(t)w(t)\\ \frac{\mathrm{d}w(t)}{\mathrm{d}t}=\frac{1}{m}Z(t)+g\cos \theta +q(t)u(t)\\ I_{yy}\frac{\mathrm{d}q(t)}{\mathrm{d}t}=M(t)\\ \frac{\mathrm{d}\theta (t)}{\mathrm{d}t}=q(t)\end{array}$$

(1)

where $\theta$ represents the pitch angle between the horizontal plane and the $x_{\mathrm{b}}$-axis. The u and w represent $x_{\mathrm{b}}$ and $z_{\mathrm{b}}$ components of velocity, respectively. q represents the $y_{\mathrm{b}}$ component of angular velocity. X and Z are the resultant forces from wing roots along $x_{\mathrm{b}}$ and $z_{\mathrm{b}}$ axes, respectively, and M represents the resultant moment from wing roots around the $y_{\mathrm{b}}$- axes. $I_{\mathrm{yy}}$ represents the pitching moment of inertia about $y_{\mathrm{b}}$-axis.

Set $u_0(t)$, $w_0(t)$, $q_0(t)$, and ${\theta }_0(t)$ as the solution for trimmed forward-flight of the seagull. They satisfy the following equation:

$$\begin{array}{l}\frac{\mathrm{d}{u}_0(t)}{\mathrm{d}t}=\frac{1}{m}{X}_0(t)-g\sin {\theta }_0(t)-q_0(t)w_0(t)\\ \frac{\mathrm{d}{w}_0(t)}{\mathrm{d}t}=\frac{1}{m}{Z}_0(t)-g\cos {\theta }_0(t)-q_0(t)u_0(t)\\ I_{yy}\frac{\mathrm{d}{q}_0(t)}{\mathrm{d}t}=M_0(t)\\ \frac{\mathrm{d}{\theta }_0(t)}{\mathrm{d}t}=q_0(t)\end{array}$$

(2)

We can set that:

$$\begin{array}{l}u(t)=u_0(t)+\mathsf{\delta }u(t)\\ w(t)=w_0(t)+\mathsf{\delta }w(t)\\ q(t)=q_0(t)+\mathsf{\delta }q(t)\\ \theta (t)={\theta }_0(t)+\mathsf{\delta }\theta (t)\end{array}$$

(3)

$$\begin{array}{l}X(t)=X_0(t)+\mathsf{\delta }X(t)\\ Z(t)=Z_0(t)+\mathsf{\delta }Z(t)\\ M(t)=M_0(t)+\mathsf{\delta }M(t)\end{array}$$

(4)

Substituting Equations (3) and (4) into Equation (1), and neglecting the second- and higher-order terms, give:

$$\begin{array}{l}\frac{\mathrm{d}\mathsf{\delta }u(t)}{\mathrm{d}t}=\frac{1}{m}\mathsf{\delta }X(t)-g\mathsf{\delta }\theta (t)\cos {\theta }_0(t)\\ -q_0(t)\mathsf{\delta }w(t)-w_0(t)\mathsf{\delta }q(t)\\ \frac{\mathrm{d}\mathsf{\delta }w(t)}{\mathrm{d}t}=\frac{1}{m}\mathsf{\delta }Z(t)-g\mathsf{\delta }\theta (t)\sin {\theta }_0(t)\\ +q_0(t)\mathsf{\delta }u(t)+u_0(t)\mathsf{\delta }q(t)\\ I_{yy}\frac{\mathrm{d}\mathsf{\delta }q(t)}{\mathrm{d}t}=\mathsf{\delta }M(t)\end{array}$$

(5)

The disturbance values of the resultant forces and moment can be expressed as an infinite series according to Taylor’s theorem [13], and neglecting the second- and higher-order terms results in:

$$\begin{array}{l}\mathsf{\delta }X(t)=\frac{\partial X(t)}{\partial u}\mathsf{\delta }u(t)+\frac{\partial X(t)}{\partial w}\mathsf{\delta }w(t)+\frac{\partial X(t)}{\partial q}\mathsf{\delta }q(t)\\ \mathsf{\delta }Z(t)=\frac{\partial Z(t)}{\partial u}\mathsf{\delta }u(t)+\frac{\partial Z(t)}{\partial w}\mathsf{\delta }w(t)+\frac{\partial Z(t)}{\partial q}\mathsf{\delta }q(t)\\ \mathsf{\delta }M(t)=\frac{\partial M(t)}{\partial u}\mathsf{\delta }u(t)+\frac{\partial M(t)}{\partial w}\mathsf{\delta }w(t)+\frac{\partial M(t)}{\partial q}\mathsf{\delta }q(t)\end{array}$$

(6)

The partial derivatives with respect to u are, respectively, defined as:

$$\begin{array}{l}\frac{\partial X(t)}{\partial u}=\underset{\mathsf{\Delta }u\to 0}{\lim }\frac{\mathsf{\Delta }X(t)}{△u}\approx \frac{\mathsf{\Delta }X(t)}{△u}\\ \frac{\partial Z(t)}{\partial u}=\underset{\mathsf{\Delta }u\to 0}{\lim }\frac{\mathsf{\Delta }Z(t)}{△u}\approx \frac{\mathsf{\Delta }Z(t)}{△u}\\ \frac{\partial M(t)}{\partial u}=\underset{\mathsf{\Delta }u\to 0}{\lim }\frac{\mathsf{\Delta }M(t)}{△u}\approx \frac{\mathsf{\Delta }M(t)}{△u}\end{array}$$

(7)

Substituting Equations (6) and (7) into Equation (5) gives:

$$\left[\begin{array}{l}\delta \dot{u}\\ \delta \dot{w}\\ \delta \dot{q}\\ \delta \dot{\theta }\end{array}\right]=A(t)\left[\begin{array}{l}\delta u\\ \delta w\\ \delta q\\ \delta \theta \end{array}\right]$$

(8)

where $\dot{u}=\frac{\mathrm{d}u}{\mathrm{d}t},\dot{w}=\frac{\mathrm{d}w}{\mathrm{d}t},\dot{q}=\frac{\mathrm{d}q}{\mathrm{d}t},\dot{\theta }=\frac{\mathrm{d}\theta }{\mathrm{d}t}$,$A(t)$ is called the system matrix.

$$\begin{array}{l}A\left(t\right)=\\ \left[\begin{array}{cccc}\frac{X_u(t)}{m}& \frac{X_w(t)}{m}-q_0(t)& \frac{X_q(t)}{m}-w_0(t)& -g\cos {\theta }_0(t)\\ \frac{Z_u(t)}{m}+q_0(t)& \frac{Z_w}{m}& \frac{Z_q(t)}{m}+u_0(t)& -g\sin {\theta }_0(t)\\ \frac{M_u(t)}{I_{yy}}& \frac{M_w(t)}{I_{yy}}& \frac{M_q(t)}{I_{yy}}& 0\\ 0& 0& 1& 0\end{array}\right]\end{array}$$

(9)

where $X_u=\partial X(t)/\partial u$, $X_w=\partial X(t)/\partial w$, $X_q=\partial X(t)/\partial q$, $Z_u=\partial Z(t)/\partial u$, $Z_w=\partial Z(t)/\partial w$, $Z_q=\partial Z(t)/\partial q$, $M_u=\partial M(t)/\partial u$, $M_w=\partial M(t)/\partial w$, $M_q=\partial M_{}^{wing}(t)/\partial q+M_{}^{tail}(t)/\partial q$.

We applied the same method used by Taylor et al. [13] to obtain $X_w$, $Z_w$, $M_w$. Hence, the w-derivatives are:

$$\begin{array}{c}{X}_w=\frac{\partial X}{\partial w}\approx u_0{}^{-1}\frac{\partial X}{\partial \alpha }\\ Z_w=\frac{\partial Z}{\partial w}\approx u_0{}^{-1}\frac{\partial Z}{\partial \alpha }\\ M_w=\frac{\partial M}{\partial w}\approx u_0{}^{-1}\frac{\partial M}{\partial \alpha }\end{array}$$

(10)

2.2. Morphological and Kinematic Parameters of a Seagull’s Wing

The morphological data and kinematic data of a seagull’s wing can be obtained using the formulation derived by Liu [14]. The specific formula is given in this section.

2.2.1. Airfoil Shape

According to the data from reference [15], we take the common gull as the research subject, setting wing-span and average cruise speed as 1.1 m and 11.6 m/s, respectively.

The moment of inertia can be obtained if the body of bird can be seen as a cylinder:

$$I_{\mathrm{b}}=\frac{m}{12}(3R^2+l_{\mathrm{b}}{}^2)$$

(11)

where R represents the radius of the body, l_b represents the length of the body (which can be approximated to the length of root chord), and m is the total mass of the seagull.

The ratio between the root chord and semispan of the seagull wing is as follows:

$$c_0/(b/2)=0.388$$

(12)

The wing chord distribution can be expressed as:

$$c=c_0\left[F_{OK}(\xi )+F_{corr}(\xi )\right]$$

(13)

where $\xi =2y/b$ represents the spanwise coordinate normalized by semispan, $F_{corr}(\xi )={\displaystyle \sum _{n=1}^5{E}_N({\xi }^{n+2}-{\xi }^8)}$, $F_{OK}(\xi )$=1 for $\xi \in \left[0,0.5\right]$ and $F_{OK}(\xi )=4\xi (1-\xi )$ for $\xi \in \left[0.5,1\right]$.

Table 1. The wing chord distribution of a seagull (mm).

c1	c2	c3	c4	c5	c6	c7	c8	c9	c10
0.194	0.196	0.198	0.194	0.184	0.177	0.168	0.150	0.126	0.097

gives the wing chord distribution of a seagull according to the above formulation.

According to the chord distribution, we can obtain an aspect ratio AR = 6.52 and a taper ratio = 0.5, respectively, which will be used in a later section.

The maximum camber coordinates can be represented as:

$$z_{\left(\mathrm{c}\right)}{}_{\max }/c=0.14/(1+1.333{\xi }^{1.4})$$

(14)

The mean camber line is expressed by:

$$z_{\left(\mathrm{c}\right)}(\eta )=z_{\left(\mathrm{c}\right)}{}_{\max }\eta (1-\eta ){\displaystyle \sum _{n=1}^3{S}_n{(2\eta -1)}^{n-1}}$$

(15)

where $\eta =x/c$ is the normalized chordwise coordinate.

The maximum thickness coordinate is represented as:

$$z_{\left(t\right)}{}_{\max }/c=0.1/(1+3.546{\xi }^{1.4})$$

(16)

The thickness distribution is given by:

$$z_{\left(t\right)}(\eta )=z_{\left(t\right)}{}_{\max }{\displaystyle \sum _{n=1}^4{A}_n({\eta }^{n+1}-\sqrt{\eta })}$$

(17)

The upper and lower surfaces of an airfoil are expressed as an addition and subtraction of the camber line and thickness distribution:

$$\begin{array}{c}{z}_{\mathrm{upper}}=z_{\left(c\right)}+z_{\left(t\right)}\\ z_{\mathrm{lower}}=z_{\left(c\right)}-z_{\left(t\right)}\end{array}$$

(18)

The geometry of a wing section can finally be obtained after achieving the airfoil at each section, as shown in Figure 2. Here, we use 10 sections in the spanwise direction to form the wing.

2.2.2. Kinematic Parameters

As shown in the Figure 3, the kinematics of a seagull wing are modelled as two rigid joint rods, the same modelling method used by Liu [14].

The flapping angle ψ₁ of the inner wing section, the flapping angle ψ₂ of the outer section relative to the inner section, and the sweeping angle φ₂ of the outer section relative to the inner section, for the following coefficients are expressed as the Fourier series:

$$\begin{array}{c}{\psi }_1(\omega t)=C_{\psi 10}+{\displaystyle \sum _{n=1}^2\left[C_{\psi 1n}\sin (n\omega t)+B_{\psi 1n}\cos (n\omega t)\right]}\\ {\psi }_2(\omega t)=C_{\psi 20}+{\displaystyle \sum _{n=1}^2\left[C_{\psi 2n}\sin (n\omega t)+B_{\psi 2n}\cos (n\omega t)\right]}\\ {\varphi }_2(\omega t)=C_{\psi 20}+{\displaystyle \sum _{n=1}^2\left[C_{\varphi 2n}\sin (n\omega t)+B_{\varphi 2n}\cos (n\omega t)\right]}\end{array}$$

(19)

Table 2. Coefficients of the geometric model and the kinematic model of a seagull.

Parameter	Value	Parameter	Value
$S_1$	3.8735	$C_{\psi 10}$	8.4654
$S_2$	−0.807	$C_{\psi 11}$	−8.5368
$S_3$	0.771	$B_{\psi 11}$	17.8798
$A_1$	−15.246	$C_{\psi 12}$	1.0898
$A_2$	26.482	$B_{\psi 12}$	−4.5880
$A_3$	−18.975	$C_{\psi 20}$	17.3083
$A_4$	4.6232	$C_{\psi 21}$	−11.0122
$E_1$	26.08	$B_{\psi 21}$	−9.6131
$E_2$	−209.92	$C_{\psi 22}$	1.3128
$E_3$	637.21	$B_{\psi 22}$	−3.0183
$E_4$	−945.68	$C_{\psi 20}$	38.4179
$E_5$	695.03	$C_{\psi 21}$	−28.0553
		$B_{\psi 21}$	0.7664
		$C_{\psi 22}$	−4.1032
		$B_{\psi 22}$	3.0125

gives the coefficients of the geometric model and the kinematic model of a seagull described in the above section.

After using the coefficients given in Table 2, the above three angles with respect to one flapping period are shown in Figure 4.

2.3. Computational Method used to Determine the Aerodynamic Force of a Seagull

2.3.1. CFD Solver: PMNS2D

In order to reduce the computational complexity and the difficulty of mesh generation, 2D CFD (two-dimensional computational fluid dynamics) is used here to more accurately simulate complex motions such as sweeping and flapping [16].

The time derivatives of Navier–Stokes equations are preconditioned by the Choi–Merkle preconditioning matrix to handle low Mach numbers and incompressible viscous flows [17]. The spatial discretization is characterized by a second order cell-centered finite volume method. An LU-SGS time-stepping method is employed to achieve convergence of the solution by integration with respect to time [18]. For unsteady flows, an implicit dual time-stepping scheme is used. Implicit residual smoothing, local time-stepping, and multigrid techniques are employed to accelerate the convergence. For calculating turbulence flows, the turbulence model employed here is the k-ω shear stress transport (SST) model [19].

Two-dimensional methods inevitably ignore the spanwise flow and induced drag because the blade element approach is intrinsically two-dimensional in the limit. We will modify it by considering the induced drag brought by three-dimensional effect.

2.3.2. Self-Consistency and Validation

A study on the self-consistency of CFD mesh refinement and time step per flapping cycle has been carried out. In order to realize a large scale of movement and deformation of flexible flapping wings, we developed an automatic mesh generation code based on infinite interpolation, which is capable of generating a wide range of wing movements. A C-type body-fitted grid was generated and used for aerodynamic calculation. Topology of the grid distribution of NACA0012, generated by the grid generation code, is shown in Figure 5.

Grid refinement was tested by medium grids with 265 × 65 1.7 $\times {10}^4$ grid nodes, and fine grids with 401 × 73 2.9 $\times {10}^4$ grid nodes, respectively. Minimum grid spacing adjacent to the wing surface was set to 0.0001c (less than $c/\sqrt{{\mathrm{R}}_{\mathrm{e}}}$). The outer boundary of the grid was set as 20 chord lengths in the spanwise direction. In order to verify the feasibility of sub-time steps per time step, 20 and 50 sub-time steps per period were tested, respectively, using the medium grid. Figure 6 shows lift coefficients of three cases mentioned previously. The flight state of calculation was that k = 1.0, h = 0.25, and Re = 20,000.

Though there were a few discrepancies between the three cases, the medium grid and 20 sub-time steps provided sufficient accuracy for the simulation considering the computational cost. Therefore, the medium grid and 20 sub-time steps will be used in future studies. The referenced computed results are in reasonable agreement with those of Ashraf and Young [20].

2.3.3. The Estimation of Body Drag and Induced Drag

The drag of body is estimated by body drag coefficient 0.05 [21]. The modification of the three-dimensional effect of aerodynamic calculation is realized by calculating the induced drag which can be obtained from Equation [22].

$$C_{D,i}=\frac{C_L^2}{\pi eAR}$$

(20)

where e can be obtained from the relationship between itself and the taper ratio for different aspect ratios. For aspect ratio AR = 6.52, and taper ratio = 0.5, e is about 0.987 [22].

3. Results and Discussion

The aerodynamic force and corresponding derivatives can be calculated using the CFD method. The dynamic stability of a seagull can be analyzed after obtaining the equilibrium flight state by two trim methods raised by Taloy [2], including the wash-out and forward-sweep methods.

3.1. Aerodynamic Performance

Firstly, we calculated the aerodynamic force of a seagull in an untrimmed state, which means the total drag and pitching moment are not balanced to zero. Figure 7 gives the variation of the coefficients of lift, drag, and pitching moments within one flapping period. The angle of attack is 5 degrees, cruise velocity is 10 m/s, flapping frequency is 3 Hz, and the center of mass is set at half root chord, as the data collected by Thomas and Taylor suggests [1]. It can be seen from this figure that the course of downstroke mainly produces the lift and thrust, while less lift and the majority of drag is generated during the upstroke. These results are very normal and can be easily explained.

According to the average values of lift, drag, and pitching moment given in

Table 3. Average aerodynamic coefficients (c.g = 0.5).

	Lift	Drag	Pitching Moment
Average Force	4.95 N	−0.043 N	−0.28 N·m
Average Coefficient	0.97	−0.0085	−0.056

, the average lift can be used to support 0.5 kg of weight. The drag and pitching moment are unbalanced and should be trimmed by wash-out and forward-sweep methods, while neglecting the effect of the tail as seagulls usually keep their tails retracted [2].

3.2. The Trimmed Results by Two Methods Separately

The forward-sweep is mainly generated by the forward-motion of the humerus, and since a model of the two-segment wing is not given in the literature, the forward-sweep cannot be accurately modeled. However, forward-sweep is an effective trimming method, whether in gliding mode or flapping mode [1,2,3]. The sweep angle mainly adjusts the center of gravity and aerodynamic center relative position. Although data on forward-sweep is not given in the literature [14], in order to consider as many factors affecting the trim as possible, here we set the forward-sweep angle as a variable.

Figure 8 shows the trim process through adjusting the angle of attack and flapping frequency, while keeping the wash-out angle at −13 degrees and neglecting the effect of forward-sweep. In other words, the forward-sweep angle is set to be zero. It can be seen from the figure that at the same flapping frequency, drag decreases first and then increases with the increase in the angle of attack, and the pitching moment increases with the increase in the angle of attack. Since the moment at zero lift is greater than zero, the growth slope of the moment to the angle of attack is negative, ensuring longitudinal static stability. As the frequency increases, the drag decreases, and so does the pitching moment, because the increase in thrust will increase the nose-down moment.

Comparatively, Figure 9 gives the results of the trim process through adjusting the angle of attack and flapping frequency, while keeping the forward-sweep angle at 28 degrees. It can be seen that at the same flapping frequency, the drag decreases first and then increases with the increase in the angle of attack, and the pitching moment increases with the increase in the angle of attack. The growth slope of the moment to the angle of attack is positive, resulting in unstable characteristics in the longitudinal direction. This means the forward-sweep motion will bring an unstable factor to longitudinal stability. As the frequency increases, the pitching moment first decreased and then increases, which shows a process of trade-off between the nose-down moment, produced by added thrust, and head-up moment, generated by forward movement of the aerodynamic center.

By changing the flapping frequency, the states of trim by wash-out and trim by forward-sweep, respectively, can be obtained. The trim results in

Table 4. The trimmed results by two methods independently.

	Wash-Out	Forward-Sweep
AoA	2	6.3
f	4.4	3.7
Forward-angle/deg	0	28
Maximum nose-down twisting angle at the tips/deg	−13	0
Weight/N	3.1	5.5

show that the angle of attack obtained by the wash-out method is smaller than that obtained by the forward-sweep method; the corresponding flapping frequency is larger, and the weight obtained is smaller, than that obtained by forward-sweep. The increase in flapping frequency will increase energy consumption, so the aerodynamic efficiency of the wash-out trim is far less than that of the forward-sweep. However, due to the slope of the pitching moment being negative, forward-sweep motion will bring an unstable characteristic to the longitudinal stability of a seagull.

3.3. Trimmed Results Realized by Two Methods at Same Time

The flapping frequency of a seagull in cruising flight is usually in a fixed range, therefore this paper managed to find the trim state under a certain flapping frequency, such as 3 Hz. By changing the wash-out angle and the forward-sweep angle at the same time, we can obtain the trim state at different angles of attack. It can be seen from Figure 10 that with the increase in the angle of attack, the weight gradually increases. The wash-out angle gradually increases, while the forward-sweep angle first decreases and then increases. The above results show that in order to achieve equilibrium at 3 Hz, both the forward-sweep angle and the outer-sweep angle are indispensable, and the forward-sweep angle has a minimum value of about 11 degrees.

We further studied the longitudinal stability at three angles of attack (3 deg, 6 deg, and 9 deg). As can be seen from the eigenvalues λ in

Table 5. Characteristics of longitudinal stability at three angles of attack.

AoA/Deg	Characteristics	Phugoid Mode	Short-Term Mode
3	${\lambda }_{}=\eta \pm i\omega$	−0.0151 ± 1.48i	−4.16 ± 22.2i
	Period/s	4.25	0.283
	$t_{halve}/s$	45.9	0.167
6	${\lambda }_{1,2}=\eta \pm i\omega$	−0.0235 ± 1.46i	−3.01 ± 29.0i
	Period/s	4.3	0.217
	$t_{halve}/s$	29.49	0.230
9	${\lambda }_{1,2}=\eta \pm i\omega$	−0.0724 ± 1.38i	−4.00 ± 37.9i
	Period/s	4.55	0.166
	$t_{halve}/s$	9.57	0.173

, there are two pairs of complex eigenvalues with a negative real part, representing two stable oscillatory motions. This means the seagull-based model is dynamically stable in the longitudinal direction at the three angles of attack. The disturbed motion is a linear combination of natural modes. When disturbed in balanced flight, the disturbed motion is a linear superposition of a pair of stable oscillatory modes.

As the angle of attack is increased from 3 degrees to 9 degrees, the modal characteristics do not significantly change. It is shown that there is a short period mode and long period stable mode. It is noticeable that such flight stability is very similar to that of fixed-wing aircraft, whose longitudinal equations of motion also have two pairs of complex conjugate roots: the short period motion, and a much longer period known as phugoid mode. The period of the phugoid mode increased from approximately 4.25 s to 4.55 s, and the time for a damped motion to halve in amplitude decreased from 45.9 s to 9.57 s. The period of the short-term mode decreased from 0.283 s to 0.166 s, and the time for a damped motion to halve in amplitude first increased from 0.167 s to 0.230 s and then decreased to 0.173 s.

4. Discussion

From the above calculations for trim and stability, we found the following interesting conclusion: while both wash-out and forward-sweep can help seagulls achieve longitudinal trim, each has advantages and disadvantages.

The wash-out method achieves lower trim weight but has the characteristics of longitudinal stability. The trim frequency achieved by forward-sweep is high, so the power consumption characteristics are also relatively poor.

The forward-sweep method can achieve a larger trim weight, but because the pitching moment decreases with the angle of attack, it has an unstable longitudinal characteristic, and has a lower power consumption due to lower trim frequency.

By using the wash-out and forward-sweep methods at the same time, it can help seagulls achieve trim at different angles of attack and different weights, and both have longitudinal stability. If a larger take-off weight, lower power consumption, and stronger longitudinal stability are required, both the wash-out and the forward-sweep methods should be considered at the same time.

Through the analysis of the dynamic stability characteristics, it is found that the trim states obtained at certain flapping frequencies are all dynamically stable and will not change significantly with a change in the angle of attack. This shows that the cruising flight of the seagull is very likely to be dynamically stable, which increases wind resistance stability in-flight, reduces the energy of active control, and increases the environmental survivability of the seagull.

Although the induced drag caused by the three-dimensional effect of flapping wings considered in this paper is calculated, it is only an estimation method, and aerodynamic force can be more accurately modeled by a three-dimensional CFD method in the future. In addition, the flexibility of the wings will also affect the calculation of thrust and drag; subsequent research can consider more accurate modeling of the aerodynamic characteristics of flexible wings through the fluid–structure interaction method.

5. Conclusions

This paper first calculates aerodynamic force using a CFD method which considers complex movement of the wing, and then determines the equilibrium through two methods. The trim results show that seagulls can use two methods to obtain longitudinal balance, including forward-sweep and wash-out. The forward-sweep method uses less flapping frequency, a higher take-off weight, and an unstable longitudinal characteristic. Contrarily, wash-out is less efficient, with a higher flapping frequency and lower take-off weight, but has a stable longitudinal characteristic. By using wash-out and forward-sweep at the same time, it can help seagulls achieve trim with a larger take-off weight, lower power consumption, and stronger longitudinal stability, which further inspires the design of seagull-inspired micro air vehicles to realize better performance, in-flight efficiency and longitudinal stability.