Implemental Formulation of Newton Dynamics for Free Insect Flight

Transcription

1 Impementa Formuation of Neton Dynamics for Free Insect Fight Sheng Xu Department of Mathematics, Southern Methodist University, Daas, TX , USA Astract A free-fying insect fies and maneuvers y couping Neton dynamics and aerodynamics. In this short note, e formuate Neton dynamics for a free-fying insect. Our formuation is cear, concise, and simpe for impementation. It passes asic tests. The corresponding MATLAB test code is posted on the author s epage. Key ords: Neton dynamics, free insect fight 1 Introduction A free-fying insect is a mutiody system consisting of a ody and mutipe ings. The physica connections of the ody ith the ings are constrains in the system. To study the staiity and maneuveraiity of free insect fight, e need to coupe Neton dynamics and aerodynamics of this mutiody system. Specificay, its Neton dynamics needs fuid force and torque from its aerodynamics, and its aerodynamics needs veocity and acceeration from its Neton dynamics. In this short note, e estaish a formuation for Neton dynamics of a freefying insect. Genera formuations (for riting genera codes) are avaiae for Lagrangian or Neton dynamics of mutiody systems [1]. Equations specific for Neton dynamics of free insect fight ere presented in [2,3]. Hoever, ho to impement these formuations and equations in a simpe and straightforard ay is not cear to the readers. With suitae reference frames and unknon Emai address: sxu@smu.edu (Sheng Xu). URL: (Sheng Xu). Preprint sumitted to Esevier Science 7 Apri 29

2 variaes, e manipuate dynamica equations and physica constraints to estaish a matrix formuation of Neton dynamics for free insect fight. The formuation accounts for to cases. In one case, the kinematics of each ing reative to the ody is prescried. In the other case, the coupe exerted y the ody on each ing is prescried. This formuation is cear, concise, and simpe. It oud e usefu in studying insect fight or aquatic anima simming. 2 Preparations 2.1 Mutipe frames We first introduce mutipe reference frames and some transformation reations. The transformation reations are used ater to manipuate equations. A free-fying insect has a ody and mutipe ings. Each ing is connected to the ody at a hinge. As shon in Fig. 1, e use mutipe reference frames (coordinate systems) as foos. (x, y, z ), the static a frame: used for the motion of the hoe insect (the system); (x, y, z ), the moving ody frame: attached to the insect ody such that its origin B is the center of mass (CM) of the ody, and x, y, and z are the principe axes of the moments of inertia I B for the ody; (x, y, z ), the moving ing frame: attached to each insect ing such that its origin W is the CM of the ing, and x, y, and z are the principe axes of the moments of inertia I W of the ing. Hereafter, e use oercase superscripts,, and to specify the a, ody, and ing frame respectivey, capita suscripts B, W, H to specify the ody, ing, and hinge respectivey. A dot atop a symo is used to denote a time derivative. Let x e a position vector and u e a veocity, anguar veocity, force, or torque/coupe vector. The coordinates of x and u transform eteen different reference frames as foos. x = R ( x x B), u = R u, (1) x = R ( x x W ), u = R u, (2) x = R ( x x W ), u = R u, (3) here R, R, and R are a-to-ody, ody-to-ing, and a-to-ing transformation matrices, respectivey. They are orthogona matrices (their inverse 2

3 2 z x y y z x B H y z W x O 1 Fig. 1. Mutipe reference frames (coordinate systems) for a free-fying insect. is their transpose). To rite the expicit form of the transformation matrices, e use the fooing Tait-Bryan anges (anges of ro, pitch, and ya) to descrie the orientation of one frame reative to another. φ, θ, and ψ : the ro (around the x axis), pitch (around the y axis), and ya (around the z axis) anges of the ody reative to the a frame, respectivey; φ, θ, and ψ : the ro (around the x axis), pitch (around the y axis), and ya (around the z axis) anges of a ing reative to the ody, respectivey. Appying the rotations in the order of ya, pitch, and ro, e have here R = R ro R pitch R ya, (4) R = R ro Rya, (5) R = R R, (6) R ro = 1 cos φ sin φ sin φ cos φ, R ro = 1 cos φ sin φ, (7) sin φ cos φ 3

4 R pitch = R ya = cos θ sin θ 1 sin θ cos θ cos ψ sin ψ sin ψ cos ψ 1, R pitch =, R ya = cos θ sin θ 1, (8) sin θ cos θ cos ψ sin ψ sin ψ cos ψ. (9) Anguar veocity We denote the anguar veocity of the ody reative to the a frame as Ω B and the anguar veocity of a ing reative to the ody as Π W. Then the anguar veocity of the ing reative to the a frame is Ω W = Ω B + Π W. (1) The anguar veocity Ω B in the ody frame and the anguar veocity Π W the ing frame are in here Ω B = Π W = φ φ + R ro + R ro θ + R ro R pitch = K B α, (11) ψ θ + R ro R pitch = K W α, (12) ψ 1 sin θ K B = cos φ sin φ cos θ, α = sin φ cos φ cos θ φ θ ψ, (13) 4

5 1 sin θ K W = cos φ sin φ cos θ, α = sin φ cos φ cos θ The fooing reations i e used ater. φ θ ψ. (14) ṘR T u = Ω B u, (15) Ṙ T R u = Ω W u, (16) Ṙ T R u = Π W u, (17) here the vector u can e veocity, anguar veocity, force, or torque/coupe. 2.3 Cross product For ater manipuation of a cross product, e introduce to reations. The first one is β γ = A γ, (18) here the ske-symmetric matrix A is defined using the three components of β = (β 1, β 2, β 3 ) T (here the superscript T is used to denote transpose) as β 3 β 2 A = β 3 β 1. (19) β 2 β 1 A cross product is converted to a matrix-vector product ith this reation. The second one is (R β) (R γ) = R( β γ), (2) here R is any 3 3 orthogona matrix. 3 Geometric and kinematic reations The ody and a ing of an insect are connected at a hinge (denoted as H in Fig. 1). Eqs. (1) and (3) give 5

6 x H x B = RT x H, (21) x H x W = R T x H, (22) By sutracting the aove to equations, e otain the geometric reation for the ody-ing constraint as the fooing. x W x B = R T x H R T x H. (23) The coordinates of the hinge in the ody frame, x H, and in the ing frame, x H, are time-independent if the ody and the ing are rigid. With Eqs. (15), (16), (18), and (2), the first- and second-order time derivatives of Eq. (23) are and x W x B = ṘT x H ṘT x H = ṘT R R T x H ṘT R R T x H = Ω B (RT x H ) Ω W (RT x H ) = (R T Ω B) (R T x H) (R T Ω W ) (R T x H) = R T ( Ω B x H ) RT ( Ω W x H ), (24) x W x B = ṘT ( Ω B x H ) ṘT ( Ω W x H ) +R( T Ω B x H) R( T Ω W x H) = d 4W R T H Ω B + RT H Ω W, (25) here the time-independent ske-symmetric matrices H and H are formed from x H and x H, respectivey, and d 4W is defined and computed as d 4W = ṘT ( x H Ω B ) + ṘT ( x H Ω W ) = ṘT R R T ( x H Ω B) + ṘT R R T ( x H Ω W ) = Ω B (RT ( x H Ω B )) + Ω W (RT ( x H Ω W )) = (R T Ω B ) (RT ( x H Ω B )) + (RT Ω W ) (RT ( x H Ω W )) = R T ( Ω B ( x H Ω B)) + R T ( Ω W ( x H Ω W )) = R T ( Ω B 2 2 x H ( Ω B x H ) Ω B ) + RT ( Ω W 2 2 x H ( Ω W x H ) Ω W ). (26) The coordinates of the CM of the hoe insect are x S = m B m S x B + W m W m S x W, (27) 6

7 here m B, m W, and m S = m B + W m W are the mass of the ody, the ing, and the hoe insect, respectivey, and W denotes the summation over a ings of the insect. We have x S = m B m S x B + W m W m S x W. (28) In the ing frame, the anguar veocity of the ing reative to the a frame, Eq. (1), is Ω W = Ω B + Π W = R Ω B + Π W. (29) From R R T = E (E is the 3 3 identity matrix), e have ṘR T = R Ṙ T. Then, y Eq. (17), e otain Ṙ Ω B = ṘR T R Ω B = R ṘT R Ω B = R Π W Ω B = R ((R T Π W ) Ω B ). (3) So the first-order time derivative of Eq. (29) is Ω W = ṘΩ B + R Ω B + Π W = R ((R T Π W ) Ω B ) + R Ω B + Π W. (31) If the Tait-Bryan anges of the ing reative to the ody are prescried, then y Eq. (12), Π W can e cacuated from Π W = K W α + K W α, (32) here K W is cos θ K W = φ sin φ cos φ cos θ + θ sin φ sin θ. cos φ sin φ cos θ cos φ sin θ (33) 7

8 4 Dynamica equations 4.1 Fuid force and torque The transationa and anguar acceeration of an oject, Ω and x C, contriute to the fuid force and torque, f f and τ f, on the oject. In the a frame, e can separate their contriutions as [5]. f f = f + A f Ω + m f x C, (34) τ f = τ + τ 1 + A τ Ω, (35) here f and τ are proportiona to the fuid viscosity, τ 1 depends ony on the shape, position, and the anguar veocity of the oject, the matrices A f and A τ depends ony on the shape and position of the oject, and m f is the fuid mass encosed y the oject oundary. For detaied derivations and expressions, pease refer to [4,5]. No e are ready to rite don Neton and Euer equations for a free-fying insect. 4.2 Neton equations In the a frame, the Neton equation for the CM of the hoe free-fying insect reads m S x S = f gs + f fb + W f fw, (36) here f gs is the gravitationa force (eight and uoyancy) on the insect, f fb is the fuid force on the ody, and f fw is the fuid force on a ing. According to Eq. (34), e can rite f fb as f fb = f B + A fb Ω B + m fb x B = f B + A fb (ṘT Ω B + R T Ω B) + m fb x B = f B + A fbr T Ω B + m fb x B, (37) here e have used the facts Ω B = RT Ω B and ṘT Ω B atter comes from =. With Eq. (15), the 8

9 Ṙ T Ω B = (ṘT R )(R T Ω B ) = (ṘT R ) Ω B = Ω B Ω B =. (38) We can rite f fw simiary as f fw = f W + A fw Ω W + m fw x W = f W + A fw (ṘT Ω W + R T Ω W ) + m fw x W = f W + A fw R T Ω W + m fw x W, (39) here e have used the facts Ω W = RT Ω W and ṘT Ω W the atter comes from =. With Eq. (16), Ṙ T Ω W = (ṘT R )(R T Ω W ) = (ṘT R ) Ω W = Ω W Ω W =. (4) In the a frame, the Neton equation for the CM of a ing reads m W x W = f gw + f fw + f W, (41) here f gw is the gravitationa force on the ing, and f W is the force from the ody acting on the ing at the hinge. By Neton s third a, the force from this ing acting on the ody at the hinge is f B = f W = f gw + f fw m W x W. (42) 4.3 Euer equations In the ody frame, the Euer equation for the ody reads I B Ω B + Ω B (I B Ω B ) = τ fb + τ B, (43) here the diagona matrix I B is the moments of inertia of the ody, τ fb is the fuid torque on the ody, and τ B is the torque from a ings acting on the ody. According to Eqs. (35) and (38), e otain τ fb = R τ fb = R ( τ B + τ 1B + A τb Ω B) = R ( τ B + τ 1B + A τb R T Ω B). (44) 9

10 We have τ B = ( c B + x H f ( ) B) = c B + H f B W W = ( ) c B + H R f B, (45) W here c B is the coupe from a ing acting on the ody at the hinge. The coupe from the ody acting on this ing at the hinge is c W = c B. (46) Sustituting Eqs. (46) and (42) into Eq. (45), e otain τ B = W ( c W + H R ( f gw + f fw m W x W ) ). (47) In the ing frame, the Euer equation for a ing reads I W Ω W + Ω W (I W Ω W ) = τ fw + τ W, (48) here the diagona matrix I W is the moments of inertia of the ing, τ fw is the fuid torque on the ing, and τ W is the torque from the ody acting on the ing. According to Eq. (35) and (4), e otain We have τ fw = R τ fw = R ( τ W + τ 1W + A τw Ω W ) = R ( τ W + τ 1W + A τw R T Ω W ). (49) τ W = c W + x H f W = c W + H f W = R c W + H R f W. (5) Sustituting Eq. (42) into Eq. (5), e otain τ W = R c W + H R (m W x W f gw f fw ). (51) 1

11 5 Impementa formuation We consider to cases. In the first case, the kinematics of each ing reative to the ody is prescried, and α ( Π W ) is thus a knon function of the time. In the second case, the coupe from the ody acting on each ing is prescried in the ody frame, and c W is thus a knon function of the time. To incorporate the to cases into one unified formuation, e introduce a W = δ i2 Π W + δ i1 c W, W = δ i1 Π W + δ i2 c W, (52) here δ ij is the Kronecker deta. So the first and second cases correspond to i = 1 and i = 2, respectivey, a W is aays unknon, and d W is aays knon. We have Π W = δ i2 a W + δ i1 W, c W = δ i1 a W + δ i2 W. (53) So far, e have the necessary reations and equations ready. No, e put them together to estaish a fina matrix formuation. In particuar, e choose x B and Ω B as unknon variaes for the ody and x W, Ω W, f fw, and a W as unknon variaes for each ing; and e uid the matrix equation for a the unknon variaes from the geometric and kinematic reations, Eqs. (25) and (31), the force expression, Eq. (39), and dynamica equations, Eqs. (36), (43), and (48). After sustituting Eqs. (28) (for the CM of the insect) and (37) (for the fuid force on the ody) into the Neton equation for the insect, Eq. (36), e have m B x B + W m W x W = f gs + f B + A fb R T Ω B + m fb x B + W f fw, (54) hich can e ritten as C 11 x B + C 12 Ω B + W ( C 13W x W + Ω W + C 15W f fw + a W ) = d 1, (55) here C 11 = (m B m fb )E, C 12 = A fb R T, C 13W = m W E, C 15W = E, and d 1 = f gs + f B 11

12 After sustituting Eqs. (44) (for the fuid torque on the ody), (47) (for the torque from the ing on the ody), and (53) (for the considered to cases) into the Euer equation for the ody, Eq. (43), e have I B Ω B + Ω B (I BΩ B ) = R ( τ B + τ 1B + A τbr T Ω B ) + ( (δ i1 a W + δ i2 W ) + H R ( f gw + f ) fw m W x W ), (56) W hich can e ritten as x B + C 22 Ω B + W ( C 23W x W + Ω W + C 25W f fw + C 26W a W ) = d 2, (57) here C 22 = I B R A τb R T, C 23W = m W H R, C 25W = H R, C 26W = δ i1 E, and d 2 = Ω B (I B Ω B ) + R ( τ B + τ 1B) + W ( δ i2 W + H R f gw ). After sustituting Eqs. (49) (for the fuid torque on the ing), (51) (for the torque from the ody on the ing), and (53) (for the considered to cases) into the Euer equation for the ing, Eq. (48), e have I W Ω W + Ω W (I W Ω W ) = R ( τ W + τ 1W + A τw R T Ω W ) + R (δ i1 a W + δ i2 W ) + H R (m W x W f gw f fw ), (58) hich can e ritten as x B + Ω B + C 33W x W + C 34W Ω W + C 35W f fw + C 36W a W = d 3W, (59) here C 33W = m W H R, C 34W = I W R A τw R T, C 35W = H R, C 36W = δ i1 R, d3w = δ i2 R W Ω W (I W Ω W ) + R ( τ W + τ 1W ) H R f gw. The reation from the ody-ing constraint, Eq. (25), can e ritten as C 41W x B + C 42W Ω B + C 43W x W + C 44W Ω W + f fw + a W = d 4W, (6) here C 41W = E, C 42W = R T H, C 43W = E, C 44W = R T H, and d 4W = ṘT H Ω B + ṘT H Ω W. With the susitution of Eq. (53) (for the considered to cases), the reation from the expression for the anguar veocity of the ing, Eq. (31), ecomes 12

13 Ω W = R ((R T Π W ) Ω B ) + R Ω B + δ i2 a W + δ i1 W, (61) hich can e ritten as x B + C 52W Ω B + x W + C 54W Ω W + f fw + C 56W a W = d 5W, (62) here C 52W = R, C 54W = E, C 56W = δ i2 E, and d 5W = δ i1 W R ((R T Π W ) Ω B ). The expression for the fuid force on the ing, Eq. (39), can e ritten as x B + Ω B + C 63W x W + C 64W Ω W + C 65W f fw + a W = d 6W, (63) here C 63W = m fw E, C 64W = A fw R T, C 65W = E, and d 6W = f W. Finay Eqs. (55), (57), (59), (6), (62), and (63) together form the fooing matrix equation C = d. (64) The coefficient matrix C in Eq. (64) depends ony on the geometric quantities descriing the shapes and reative positions of the ody and the ings. It is nonsinguar if m B m fb and m W m fw. The right-hand side vector d depends on oth geometric and kinematic quantities (such as anguar veocity). If there is ony one ing (denoted as W 1 ), then Eq. (64) reads C 11 C 12 C 13W1 C 15W1 x B d C 22 C 23W1 C 25W1 C 26W1 Ω 1 B d 2 C 33W1 C 34W1 C 35W1 C 36W1 x W 1 C 41W1 C 42W1 C 43W1 C 44W1 Ω d 3W1 =, (65) W 1 d 4W1 C 52W1 C 54W1 C 56W1 f fw 1 d 5W1 C 63W1 C 64W1 C 65W1 d 6W1 a W1 from hich, e otain = C 1 d. We then integrate in time the fooing first-order differentia equations ith knon right-hand sides. 13

14 ( x B, v B, α, Ω B, x W, v W, α, ) Π = ( v B, x B, K 1 B W Ω B, Ω B, v W, x W, K 1 W Π W, Π W ). (66) It is required that the initia conditions for the aove differentia equations e consistent. They must satisfy Eqs. (23) and (24), and if the ing kinematics is prescried, Eq. (12) must aso e satisfied. Mutipe ings can e incuded easiy. For exampe, if there are to ings (denoted as W 1 and W 2 ), then Eq. (64) reads 1 d 1 C x 1 C2 12 W 2 d C2 21 C2 22 Ω 3W2 W 2 = d 4W2, (67) f fw 2 d 5W2 a W2 d 6W2 here C 1, 1, and d 1 are the coefficient matrix, unknon vector, and righthand side vector from Eq. (65), and su-matrices C2 21, C2 22, and C2 12 are C 21 2 = C 12 2 = C 33W2 C 34W2 C 35W2 C 36W2 C 41W2 C 42W2, C 22 C 2 = 43W2 C 44W2. C 52W2 C 54W2 C 56W2 C 63W2 C 64W2 C 65W2 C 13W2 C 15W2 C 23W2 C 25W2 C 26W2. (68) 14

15 1.5 Body anguar veocity in the ody frame 3 Wing coordinates in the a frame 1 Ω Bx Ω By Ω Bz 1 2 ODE sover constraint reation t t Fig. 2. Left: Conservation of the roing anguar momentum of an insect mode; Right: Comparison of the coordinates of a ing from the numerica simuation and the exact formuas. 6 Tests We perform to simpe tests on our formuation using an insect mode ith to ings. The test code is posted on the author s epage. One thing can aays e checked in the tests is ho e the ody-ing constraint, Eq. (23), is satisfied hen Eq. (66) is numericay integrated to update the positions and orientations of oth the ody and ings. (In practice, e prefer to update the positions of the ings using Eq. (23) ith the positions and orientations of the ody ut ony the orientations of the ings from the numerica integration. In other ords, the positions of the ings from the numerica integration are discarded to avoid possie reakdon of the ody-ing constraint due to numerica errors.) In the first test, e examine the conservation of anguar momentum of a torque-free insect mode. In this mode, the to identica ings are fixed reative to the ody ith refective symmetry through the x z pane, and the CM of the mode fas on the x axis. If the initia anguar veocity of the mode has ony the roing component, then the mode keeps roing aout the roing x axis ith a constant rate same as the initia rate. This is verified in Fig. 2 from the simuation using our formuation. The constraint reation, Eq. (23), is aso e satisfied in this simuation, as shon in Fig. 2. In the second test, e examine Neton s second a for an insect mode suject to the gravitationa force ith three non-zero components in the a frame. The to ings fap ith respect to the ody under the action of prescried odyto-ing coupes. The CM of the mode foos a knon trajectory according to Neton s second a. This is verified in Fig. 3 from the simuation using our formuation. The constraint reation, Eq. (23), is aso e satisfied in this simuation, as shon in Fig

16 .3 System CM coordinates in the a frame.5 Wing coordinates in the a frame.2 ODE sover anaytica ODE sover constraint reation t t Fig. 3. Left: Comparison of the CM trajectories of the insect mode from the numerica simuation and the exact formuas; Right: Comparison of the coordinates of a ing from the numerica simuation and the exact formuas. 7 Concusions We estaish a formuation for Neton dynamics of free insect fight and perform simpe tests on it. This formuation is cear, concise, and simpe for impementation. It is usefu in couping Neton dynamics and aerodynamics of insect fight in CFD, and in anayzing insect fight from the perspective of dynamica systems ith reduced fuid force and torque modes. References [1] R. E. Roerson and R. Schertassek, Dynamics of mutiody systems, Springer- Verag (1998) [2] G. Geert, P. Gameier and J. Evers, Equations of motion for fapping fight, AIAA Paper, AIAA (22) [3] M. Sun, J. Wang and Y. Xiong, Dynamic fight staiity of hovering insects, Acta Mech. Sin. 23, pp (27) [4] Sheng Xu, Singuar forces in the immersed interface method for rigid ojects in 3D, App. Math. Lett., in press (28) [5] Sheng Xu, Fuid force and torque on a 3D rigid oject, App. Math. Lett., sumitted (29) 16