Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0203
and
D. J. Taylor†
AeroVironment Inc., Simi Valley, California 93063
The paper presents a methodology for calculating the flight dynamic characteristics andgust response of highly flexible aircraft. The aeroelastic model is based on a geometrically-exact, nonlinear beam model coupled with large angle aerodynamic model. The gustresponse is calculated based on various gust models. It is shown that the calculated gustresponse spectrum as well as the RMS values can be very different for different models.For a high-aspect-ratio flying wing example the spanwise non-uniform gust model predictsmuch higher response as compared to a uniform gust model.
I. Introduction
The analysis and design of very light, and thus highly flexible, aircraft configurations is of interest forthe development of the next generation of high-altitude, long-endurance (HALE), unmanned aerial vehicles(UAV). The flexibility of such aircraft leads to large deformation, so that linear theories are not relevantfor their analysis. For example, the trim shape of a large flexible aircraft is highly dependent on the flightmission (payload) as well as on the flight condition; the deformed shape is significantly different from theundeformed shape. Thus, the flight dynamic and gust response based on the actual trim shape can be quitedifferent from that calculated based on linear, small deformation assumptions.
The paper presents a theoretical basis for the flight dynamic response calculation and gust responseestimation of a highly flexible aircraft. Various realistic design space requirements including, concentratedpayload pods, multiple engines, multiple control surfaces, vertical surfaces, discrete dihedral, and continuouspretwist, are taken into account. The code based on the theoretical development presented here can be usedin preliminary design as well as in control synthesis. This work is a continuation of work conducted by theauthors over the past decade in the area of nonlinear aeroelasticity1,2 and recently on the flight dynamics offlexible flying wing configurations.3 The focus of the present work is on the gust response of highly flexibleaircraft.
II. Theory
The modeling of a flexible aircraft undergoing large deformation requires a geometrically-exact structuralmodel coupled with a consistent large motion aerodynamic model. The present work is based on modeling theairplane wing structurally as a beam undergoing large displacement and rotation. The governing equationsare the geometrically-exact equations of motion from Hodges4 written in their intrinsic form (i.e. withoutdisplacement and rotational variables). However, instead of being augmented by the displacement- androtation-based kinematical relations given therein, they are instead augmented by the intrinsic kinematicalequations, derived in Ref. 5.
A 2-D airfoil model is appropriate for the very high-aspect-ratio wing (without fuselage interference)being analyzed here. The airloads are here based on the finite-state airloads model presented by Peters andJohnson.6 The airloads model is coupled with inflow model presented in Peters et al.7 The gust loads are
∗Assistant Professor, Department of Aerospace and Ocean Engineering. Senior Member AIAA.†Principal Engineer. Member AIAA
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47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Confere1 - 4 May 2006, Newport, Rhode Island
calculated in the frequency domain for a simple harmonic gust using the theory of Theodorsen (Ref. 8, pp.281–293). The gust response is based on the non-uniform gust response analysis methodology presented inCrimaldi et al.9
A. Structural Model
The geometrically-exact, intrinsic equations for the dynamics of a general, non-uniform, twisted, curved,anisotropic beam, undergoing large deformation, are given as
F ′ + (k + κ)F + f = P + ΩP (1)
M ′ + (k + κ)M + (e1 + γ)F + m = H + ΩH + V P (2)
V ′ + (k + κ)V + (e1 + γ)Ω = γ (3)
Ω′ + (k + κ)Ω = κ (4)
where ( )′ denotes the derivative with respect to the undeformed beam reference line and ˙( ) denotes theabsolute time derivative. F (x, t) and M(x, t) are the measure numbers of the internal force and momentvector (generalized forces), P (x, t) and H(x, t) are the measure numbers of the linear and angular momentumvector (generalized momenta), γ(x, t) and κ(x, t) are the beam strains and curvatures (generalized strains),and V (x, t) and Ω(x, t) are the linear and angular velocity measures (generalized velocities). All measurenumbers are calculated in the deformed cross-sectional frame. k = bk1 k2 k3c is the initial twist/curvatureof the beam, e1 = b1 0 0cT , and f(x, t) and m(x, t) are the external forces including gravity (fg, mg),aerodynamic loads (faero, maero), and thrust (fT , mT ). The first two equations in the above set are theequations of motion4 while the latter two are the intrinsic kinematical equations5 derived from the generalizedstrain-displacement and generalized velocity-displacement equations.
1. Cross-sectional constitutive laws
The secondary beam variables are linearly related to the primary variables by the cross-sectional constitutivelaws (flexibility and inertia matrices), such that
γ
κ
=
[R S
ST T
] F
M
(5)
P
H
=
[µ∆ −µξ
µξ I
] V
Ω
(6)
where R, S, and T are 3×3 matrices of cross-sectional flexibility coefficients; and µ, ξ, I are the mass perunit length, mass center offset, and mass moment of inertia per unit length, respectively. These relationsare derived based on the assumptions of small strain and slenderness.
2. Finite-element discretization
To solve the above set of equations, the beam is discretized into finite elements. The equations for eachelement are obtained by discretizing the differential equations such that energy is conserved.5 For example,consider a variable X. Let the nodal values of the variable after discretization be represented by Xn
l andXn
r , where the superscript denotes the node number, the subscript denotes the left or right side of the node,and the hat denotes that it is nodal value. For the element n
X ′ =Xn+1
l − Xnr
dl(7)
X = Xn
=Xn+1
l + Xnr
2(8)
In a discretized form the equations of motion can be written as
Fn+1l − Fn
r
dl+ (κn + kn)F
n+ f
n − Pn − Ω
nP
n= 0 (9)
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Mn+1l − Mn
r
dl+ (κn + kn)M
n+ (e1 + γn)F
n+ mn − H
n − ΩnH
n − VnP
n= 0 (10)
V n+1l − V n
r
dl+ (κn + kn)V
n+ (e1 + γn)Ω
n − γn = 0 (11)
Ωn+1l − Ωn
r
dl+ (κn + kn)Ω
n − κn = 0 (12)
where, as defined above, the barred quantities correspond to the values of the variables in the element interiorwhile the hatted quantities are nodal values. The barred and hatted quantities of the primary variables arerelated as
Fn
=Fn+1
l + Fnr
2(13)
Mn
=Mn+1
l + Mnr
2(14)
Vn
=V n+1
l + V nr
2(15)
Ωn
=Ωn+1
l + Ωnr
2(16)
The barred secondary variables are related to the barred primary variables as stated above in the cross-sectional constitutive law.
3. Gravity loads
The force term in the equations of motion includes gravitational forces. The gravitational force and momentare
fg = µg (17)
mg = µξg (18)
where g is the gravity vector.The measure numbers of g are known in the inertial frame. The measure numbers of the gravity vector
g in the deformed beam frame at all the nodes can be calculated using the following equations
g′ + (κ + k)g =0
g + Ωg =0(19)
which in the discretized form can be written as
gn+1l − gn
r
dl+ (κn + kn)gn =0
˙g + ˜Ωg =0
(20)
The second equation above, the time-differentiated one, is satisfied at one node; while the first equation,the spatially-differentiated one, is used to obtain the g vector at other nodes. Both equations are matrixequations, i.e. a set of three scalar equations. In both cases, the three equations together can be shown tosatisfy a constraint of constant length for the g vector. One can thus replace any one of the three dynamicequations by the static form of this length constraint. This will remove the artificial eigenvalue caused bythe differentiation of a constraint. Also the constraint is satisfied for the steady-state calculation when thedynamic terms are neglected. So, the second equation can be written as
(e1eT1 + e2e
T2 ) ˙g
ng+ (e1e
T1 + e2e
T2 )˜Ωng gng + (e3e
T3 )|gng | = e3 (21)
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B. Aerodynamic Model
1. Lift, Drag, and Pitching Moment Expressions
The airloads are calculated based on 2-D aerodynamics using known airfoil parameters.First the velocities in the aerodynamic frame at the mid-chord are written as
Vn
a = CnT
a Vn − yn
mcCnT
a Ωn
(22)
Ωn
a = CnT
a Ωn
(23)
where ynmc is the vector from the beam reference axis to the mid-chord and can be written in terms of the
aerodynamic center (at the quarter chord) location as ynmc = b0 yn
ac − bn
2 0c.The lift, drag and pitching moment at the quarter-chord are given by6
Lnaero = ρbnV n2
T
(Cn
l0 + Cnlα sinαn + Cn
lββn
)+ ρbnV n
T V na2
Cnlααn
rot cos αn (24)
Dnaero = ρbnV n2
T Cnd0
+ ρbnV nT V n
a2Cn
lααnrot sinαn (25)
Mnaero = 2ρbn2
V n2
T
(Cn
m0+ Cn
mαsinαn + Cn
mββn
)− ρbn2
V nT V n
a2Cn
lααnrot/2 (26)
whereV n
T =√
V n2a2
+ V n2a3
(27)
sinαn =−V n
a3
V nT
(28)
αnrot =
Ωna1
bn/2V n
T
(29)
and V na2
and V na3
are the measure numbers of Vn
a . βn is the flap deflection of the nth element.The lift, drag and pitching moment are the aerodynamic loads to be applied to the wing and can be
written in the aerodynamic frame as
fna =
0
−Lnaero
V na3
V nT−Dn
aero
V na2
V nT
Lnaero
V na2
V nT−Dn
aero
V na3
V nT
(30)
mna =
Mnaero
0
0
(31)
Finally, the aerodynamic forces derived above are transformed to the deformed beam frame and trans-ferred to the beam reference axis to give the applied aerodynamic forces as
fn
aero = Cna fn
a (32)
mnaero = Cn
a mna + Cn
a ynacf
na (33)
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2. Unsteady Effects
The above aerodynamic model is a quasi-steady one with neither wake (inflow) effects nor apparent masseffects. To add those effects into the model we have to firstly add the inflow λ0 and acceleration terms inthe force and moment equation. Secondly we have to include an inflow model that calculates λ0. Here thePeters 2-D inflow theory of Ref. 7 is used.
The force and moment expressions with the unsteady aerodynamics effects are
fna = ρbn
8>>>>>><>>>>>>:
0
−(Cnl0 + Cn
lββn)V n
T V na3 + Cn
lα(V na3 + λn
0 )2 − Cnd0V n
T V na2
(Cnl0 + Cn
lββn)V n
T V na2 − Cn
lα V na3b/2− Cn
lαV na2(V
na3 + λn
0 − Ωna1bn/2)− Cn
d0V nT V n
a3
9>>>>>>=>>>>>>;
(34)
and
mna = 2ρbn2
8>>>>>><>>>>>>:
(Cnm0 + Cn
mββn)V n2
T − Cnmα
V nT V n
a3 − bnCnlα/8V n
a2Ωna1 − bn2
Cnlα Ωn
a1/32 + bnCnlα V n
a3/8
0
0
9>>>>>>=>>>>>>;
(35)
The inflow model can be written as
[Ainflow]λn+(
V nT
bn
)λn =
(−V n
a3+
bn
2Ωn
a1
)cinflow (36)
andλn
0 =12binflowT λn (37)
where λn is a vector of inflow states for the nth element, and [Ainflow], cinflow, binflow are constant matricesderived in Ref. 7.
3. Gust Loads
As will be shown in the next section, the gust response calculation is based on the Fourier superposition ofthe response to simple harmonic gust. For a 2-D airfoil, the lift at the aerodynamic center of the airfoil canbe calculated as8
where, wg is the amplitude of the gust, ωg is the frequency of the gust as seen by the airfoil and is equalto ratio of the airspeed to the gust wavelength, kg is the reduced frequency of the gust (and response) andis equal to the ratio of the semichord to gust wavelength (or the ratio of the product of the gust frequencyand semichord to the airspeed), and U is the airspeed. C(kg) is the Theodorsen’s function and J0(kg) andJ1(kg) are Bessel functions of the first kind.
C. Aeroelastic System
An aeroelastic model is obtained by coupling the aerodynamic force definition given in the previous sectionwith the set of equations presented in the section on the structural model. The aeroelastic equations arenonlinear equations in terms of the primary structural variables (Fn
l , Fnr , Mn
l , Mnr , V n
r , Ωnr and gn
r ) andaerodynamic variables (λn).
The equations are of the form
[A]x+ B(x) = fcont+ fgust (39)
where, x is a vector of all the aeroelastic variables, fcont is the vector of the flight controls and fgustis a vector of gust loads.
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For fgust = 0, the set of aeroelastic equations is solved using Newton-Raphson method to obtain thesteady-state (trim) solution. One could either calculate the trim for a given flight control configuration oruse a trim algorithm to calculate the control settings for a given flight condition.
The linear system can be represented as
[A] ˙x+ [B]x = fcont+ fgust (40)
where, ( ) refers to the perturbation about the steady-state (trim) values. The eigenvalues of the linearizedsystem can be calculated to estimate the stability of the aircraft at the trim to small perturbation.
D. Gust Response Calculation
The gust response is calculated in the frequency domain. The response of the airplane to a gust at variouslocations on the wing is calculated. A unit-amplitude, simple harmonic gust applied at the nth element canbe represented as
wng = 1eiωgt (41)
The gust load at the nth element is given by
fgust = fngust(ωg)eiωgt (42)
The response of the airplane to the gust can be calculated as
x = x(ωg)neiωgt (43)
where, using the linearized equation presented above we have
x(ωg)n = (iωg[A] + [B])−1 fngust(ωg) (44)
The effective response of the wing due to the gust at all the locations on the wing can be calculated as10
[φx(ω)] = [X(ω)][φg(ω)][X(ω)]T (45)
where, ( )T is the complex conjugate transpose, [φx(ω)] is a matrix of the cross-spectral densities of theresponse, [φg(ω)] is a matrix of the cross-spectral densities of the gust, and [X(ω)] is the frequency responsematrix calculated as [x(ω)1, x(ω)2, . . . x(ω)N ].
To calculate the response of the airplane to nonuniform gust over the entire wing, we need to know thecross-spectral densities of the gust at various locations on the wing. These 3-D cross-spectral functions havebeen calculated by Houbolt and Sen11 based on the 2-D Von Karman gust spectrum and can be written as
[φg(ω)] = σ2w
223L√
2πUΓ( 13 )(1.339)
83
8(1.339)2s53ij
3z56ij
K 56(zij)−
s113
ij
z116
ij
K 116
(zij)
(46)
where, σw is the gust RMS velocity, L is the gust scale of turbulence, sij is the nondimensional spanwiseseparation between two gust strips (nondimensionalized using the L), K 5
6and K 11
6are the modified Bessel
functions of the second kind, Γ is the gamma function, and zij is given by
zij =sij
1.339
√1 +
(1.339ωL
U
)2
(47)
The statistical parameters of interest for load exceedance calculations, namely the normalized gust in-tensity, A, and the characteristic frequency, N0, can be determined as12
A =σx
σw(48)
where, σx, the RMS value of any response variable can be calculated as
σx =
√∫ ∞
0
φx(ω)dω (49)
and
N0 =
√∫∞0
ω2φx(ω)dω∫∞0
φx(ω)dω(50)
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Mass per unit length 6 lbs/ftCenter of gravity 25% chord
Centroidal Mass Mom. Inertia:about x-axis (torsional) 30 lb ft
about y-axis 5 lb ftabout z-axis 25 lb ft
Aerodynamics Coefficients (25% chord):Clα 2π
Clδ 1Cd0 0.01Cm0 0.025Cmδ
-0.25
Table 1. Wing cross-sectional properties
III. Results
Consider the aircraft as illustrated in Figure 1. The example aircraft has a span of 238.78 ft and a constantchord of 8 ft. 1/6th of the span at each end has a dihedral of 10. The inertial, elastic and aerodynamicproperties of the wing cross section are given in Table 1.
There are five propulsive units; one at the mid-span and two each at 1/3rd and 2/3rd semi-span distancefrom the mid-span. There are three vertical surfaces (pods) which act as the landing gear. Two of the podsweigh 50 lb each and are located at 2/3rd semi-span distance from the mid-span. The central pod also actsas a bay for payload and weighs 250 lb. The pod/payload weight is assumed to a be a point mass hanging3 ft under the wing. The aerodynamic coefficients for the pods are Clα = 5 and Cd0 = 0.02. The wing isdiscretized using 30 finite elements of equal lengths. It is assumed that the gust loads act only on the wing.The aircraft is trimmed at a steady level flight at 40 ft/s.
A. Response due to gust on a single element (aerodynamic section)
Before calculating the gust response on the airplane due to uniform and non-uniform gust over the entireairplane, it is prudent to understand the response of the airplane to a gust on a single aerodynamic strip.For a single strip the frequency response of airplane can be considered to be the sum of three components,
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namely, the power spectral density of the gust velocity (φg), the unsteady aerodynamics that converts thegust velocity at a section into gust loads on that section (Hfg), and finally the aeroelastic system on which thegust loads act leading to a response (Hxf ). Figure 2 presents the frequency content of the three componentsas well as the power spectral density of the response (the bending moment at the midspan and to the rightof the center pod). The gust scale of turbulence (L) is assumed to be 2500 ft. And the gust is applied onthe 25th strip.
It is clear from Fig. 2(a) that the gust power spectral density is a very steep low pass filter. For thegiven wing chord, flight velocity and scale of turbulence, the cut-off frequency is very low (of the order of 1rad/sec). Thus the gust will primarily excite the flight dynamic modes and the lowest frequency structuralmodes (both of which are very coupled to start with). The conversion of the gust velocity to gust loadis another level of low-pass filter (Fig. 2(b)). This is due to the fact that as the turbulence wavelengthapproaches and exceeds the wing chord, it has very small effect on the loads. Finally, the frequency responseof the aeroelastic system has a varied response at all frequencies in addition to a high response at very lowfrequency. Fig. 2(d) shows the power spectral density of the midspan bending moment due to the gust onthe 25th strip. The RMS value can be calculated to be 145 lb-ft.
To calculate the response of the airplane to gust acting on the complete aircraft, the response is firstcalculated for gust acting on various aerodynamic strips. Figure 3(a) shows the power spectral densitiesof the midspan bending moment response due to gust acting on various locations on thewing. As can beseen there is a large variation in the response. Figure 3(b) presents the response RMS value for gust actingat various locations. It is a very important to note that the bending moment response RMS does notmonotonically increase as the gust location moves away from the midspan. This is because of the dynamicsof the flying wing configuration. This variation in the transfer functions and is very important as it helpsin understanding the effects of the spanwise variation of gust. The spanwise cross spectral densities of thenon-uniform gust as derived by Houbolt and Sen11 for this aircraft are shown in Fig. 3(c). As can be seenthere is a considerable correlation between the gust at low frequency. There is over 94% correlation betweenthe gust on aerodynamic strips at two extremes of the wing for frequency of 0.001 rad/s. This correlationreduces to less than 50% for frequencies over 0.2 rad/s. And finally the correlation even between neighboringstrips reduces to less than 50% for frequencies over 5.5 rad/s.
Figure 3(d) shows the power spectral density of the midspan bending moment due to gust over the entireaircraft. The non-uniform case refers to the theory described above. Uniform gust refer to the gust whichis the identical over the entire wing and thus the gust at all the strips are perfectly correlated. Finally, thespanwise discrete gust refers to gust on each aerodynamic strip which is independent of other gusts and thusthe gusts at all the strips are completely uncorrelated. For the present case uniform gust leads to significantunderprediction of the midspan bending moment response, while the spanwise discrete gust overpredicts theresponse.
Table 2. Midspan bending moment RMS values (lb-ft) for unit gust RMS
Table 2 shows the RMS value of the bending moment using the three gust models. It is interesting tonote that the RMS value of the response due to gust over the entire wing is of the same order of magnitudeas that due to the gust acting only on a single aerodynamic strip. In fact, the response due to uniform gust(142 lb-ft) is less than the response due to gust only on the 25th strip (145 lb-ft). This is due to the factthat the response of the wing due to gust at different spanwise locations are out of phase and cancel eachother. The table also presents the RMS values for a turbulence scale of 500 ft.
The spanwise variation of the bending moment, shear force, twisting moment and vertical velocity isshown in Fig. 4. As can be seen there is considerable difference in the response due to the three spanwisegust models. The discontinuities at spanwise locations of 40 ft and 200 ft are due to discrete mass of thepods. The central pod does not lead to discontinuities due to symmetry. The last plot (Fig. 4(d)) shows theRMS vertical velocity. It can be seen that this is quite constant spanwise implying that there is considerablerigid-body motion due to flight dynamic modes.
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Figure 5 shows the effect of various trim conditions on the gust response characteristics. The threemodels use different mass for the center payload pod. The gust response analysis in conducted in foursteps: nonlinear trim solution of the nonlinear aeroelastic/flight dynamic equations, linearization about thetrim solution, gust response calculation for individual aerodynamic strips and finally power spectral densitycomputation using the cross spectral densities of the gust model. The results show the change in the RMSbending moment near the root due to different payload. Higher payloads were not tested because of theexistence of phugoid instability at payload of 320 lb.
Further statistical analysis can be conducted, which will use the mission profile and the gust response togust spectrum at various flight conditions on the mission profile, to calculate the probability of encounteringa critical gust.
IV. Conclusions
The paper presents a methodology for calculating the gust response of a highly flexible aircraft. Theaeroelastic analysis is based on a nonlinear geometrically-exact structural model, large angle-of-attack aero-dynamic model and a frequency domain gust model. Three models for the spanwise gust cross correlationare used. The non-uniform gust model predicts response many times that predicted by the uniform gustmodel.
References
1Patil, M. J., Hodges, D. H., and Cesnik, C. E. S., “Nonlinear Aeroelastic Analysis of Complete Aircraft in SubsonicFlow,” Journal of Aircraft , Vol. 37, No. 5, Sep – Oct 2000, pp. 753 – 760.
2Patil, M. J. and Hodges, D. H., “On the Importance of Aerodynamic and Structural Geometrical Nonlinearities inAeroelastic Behavior of High-Aspect-Ratio Wings,” Journal of Fluids and Structures, Vol. 19, No. 7, Aug. 2004, pp. 905 – 915.
3Patil, M. J. and Hodges, D. H., “Flight Dynamics of Highly Flexible Flying Wings,” Proceedings of the InternationalForum on Aeroelasticity and Structural Dynamics, Munich, Germany, June 2005.
4Hodges, D. H., “A Mixed Variational Formulation Based on Exact Intrinsic Equations for Dynamics of Moving Beams,”International Journal of Solids and Structures, Vol. 26, No. 11, 1990, pp. 1253 – 1273.
5Hodges, D. H., “Geometrically Exact, Intrinsic Theory for Dynamics of Curved and Twisted Anisotropic Beams,” AIAAJournal , Vol. 41, No. 6, 2003, pp. 1131–1137.
6Peters, D. A. and Johnson, M. J., “Finite-State Airloads for Deformable Airfoils on Fixed and Rotating Wings,” Sympo-sium on Aeroelasticity and Fluid/Structure Interaction, Proceedings of the Winter Annual Meeting, ASME, November 6 – 11,1994.
7Peters, D. A., Karunamoorthy, S., and Cao, W.-M., “Finite State Induced Flow Models; Part I: Two-Dimensional ThinAirfoil,” Journal of Aircraft , Vol. 32, No. 2, Mar.-Apr. 1995, pp. 313 – 322.
8Bisplinghoff, R. L., Ashley, H., and Halfman, R. L., Aeroelasticity, Addison-Wesley Publishing Co., Reading, Massa-chusetts, 1955.
9Crimaldi, J. P., Britt, R. T., and Rodden, W. P., “Response of B-2 aircraft to nonuniform spanwise turbulence,” Journalof Aircraft , Vol. 30, No. 5, 1993, pp. 652 – 659.
10Bendat, J. S. and Piersol., A. G., Random Data: Analysis and Measurement Procedures, Wiley-Interscience publication,New York, New York, 2000.
11Houbolt, J. C. and Sen, A., “Cross-spectral Functions based on Von Karman’s Spectral Equation,” Tech. rep., NASACR-2011, 1972.
12Hoblit, F. M., Gust Loads on Aircraft: Concepts and Applications, AIAA Education Series, Washington, D.C., 1988.
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0 2 4 6 8 100
0.5
1
ω (rad/s)
φ g (ft
2 s−
2 /rad
s−1 )
(a) Gust power spectral density (unit RMS, L = 2500ft)
0 2 4 6 8 100
2
4
6
ω (rad/s)
|Hfg
(ω)
|2 (lb
2 / ft2 s
−2 )
(b) Square of transfer function relating the gust velocity to gust load
0 2 4 6 8 100
1000
2000
3000
4000
5000
ω (rad/s)
|Hxf
(ω)
|2 (lb
2 −ft2 /lb
2 )
(c) Square of transfer function relating the gust load to midspan bend-ing moment
0 2 4 6 8 100
5000
10000
ω (rad/s)
φ x (lb
2 −ft2 /r
ad s−
1 )
(d) Power spectral density of the midspan bending moment
Figure 2. Gust response calculation for unit gust acting at the 25th aerodynamic strip (40 ft/s, sea-level)
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0 2 4 6 8 100
5000
10000
ω (rad/s)φ x (
lb2 −
ft2 /rad
s−1 ) strip 16
strip 20strip 24strip 27strip 30
(a) Power spectral density of the midspan bending moment due to gustat various aerodyanamic locations
0 5 10 15 20 25 300
200
400
600
strip number
σ x (lb
−ft)
(b) The RMS of midspan bending moment due to gust at variousaerodyanamic locations
