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Transient analysis of flexible multi-body systems. Part II: Application to aircraft landing
COMPUTER METHODS NORTH-HOLLAND
IN APPLIED
MECHANICS
AND ENGINEERING
54 (1986) 93-110
TRANSIENT ANALYSIS OF FLEXIBLE MULTI-BODY SYSTEMS. PART II: APPLICATION TO AIRCRAFT LANDING
Koorosh CHANGIZI, Department
of Mechanical
Yehia A. KHULIEF
and Ahmed A. SHABANA
Engineering, University of Illinois at Chicago Chicago, IL 60680, U.S.A.
Revised
Received 28 January manuscript received
(Circle Campus),
1985 2 May 1985
This investigation presents a method for transient analysis of a large-scale multi-body aircraft consisting of interconnected rigid and flexible bodies that undergo large angular rotations. Elastic components of the aircraft are discretized using the finite element method. The system equations of motion and nonlinear algebraic constraint equations describing joints between different components are written in the Lagrangian formulation using a finite set of coupled reference and modal coordinates. The system differential equations of motion and algebraic constraint equations are computer-generated and integrated forward in time using an explicit-implicit direct numerical integration algorithm coupled with a Newton-Raphson type iteration in order to check on constraint violations. Impact and intermittent motion events are accounted for by using a generalized momentum balance that predict jump discontinuities in the generalized velocities as well as jump discontinuities in the system reaction forces. The formulation presented and the computer program developed are used to simulate the impact between the landing gear and the runway. The method is also used to predict the dynamic behavior of the aircraft during the traverse of an abrupt elevation change in the runway.
1. Introduction
Airborne structures are examples of multi-body systems that are subjected to different dynamic loading conditions. One of the most significant phases of such dynamic loadings is that due to ground interaction. In this paper, a mathematical model was developed to simulate the dynamic behavior of an aircraft during landing and ground operations. The dynamic loads and vibrations resulting from landing impact and from runway and taxiway unevenness are recognized as significant factors in causing fatigue damage, dynamic stresses on the airframe, passenger discomfort, and reduction of pilot’s ability to control an aircraft during ground operations. These ground-induced dynamic loads and vibrations have been encountered with some conventional subsonic transport aircrafts [l, 21; however, they are magnified for supersoniccruise aircraft due to the increased structural flexibility inherent in the slender body, and thin wing design. Vibration-control techniques have been employed to improve the operational characteristics of the aircraft on the ground. Passive, semi-active and active control techniques are used [3]; however, the active control method is found to be more effective in reducing the ground loads transmitted to the airframe [4]. The objective of this paper is to present a 0045-7825/86/$3.50
@ 1986, Elsevier
Science
Publishers
B.V. (North-Holland)
94
K. Changizi et al., Analysis of multi-body systems. Part II
mathematical model that can be used for numerical simulation of a general aircraft during ground operations. A mathematical model that closely represents the dynamic characteristics of the aircraft is a prerequisite for design and evaluation of the vibration controller. The computer-aided simulation is then a significant step in the design process flowchart that offers a faster and cheaper tool for predicting the dynamic behavior of the whole system before reaching the experimental implementation step [4] which is often expensive and timeconsuming. It is also recognized that malfunction of active controllers may result in complete inoperability. Consequently, a study of the effect of impulsive loading conditions resulting from touch-down impact or traverse of runway surface unevenness on the airframe has to be performed in the absence of vibration-suppressing devices. In this analysis, the aircraft is considered as a multi-body system that is composed of interconnected rigid and flexible components [5]. Flexible components may undergo large angular rotations, therefore the developed mathematical model accounts for the nonlinear inertia coupling between the set of reference generalized coordinates and the set of local elastic deformations that are used to represent the global location of any point in the system. A finite element discretization of flexible components is obtained by a finite set of nodal points. A component mode substitution method is then used to obtain a reduced-order model that contains the lower end of the frequency spectrum as well as other significant modes of vibrations. Two major problems encountered during the aircraft ground operations are touch-down impact and natural bumps due to abrupt elevation changes in the runway surface. The simulation of a step or natural bump by a forcing function is a straight-forward technique; however, the resulting forcing function can be easily added to the forcing term of the equations of motion. The situation is different in the case of impact where changes in the system momentum take place causing an instantaneous set of impulsive forces that excite higher modes of vibrations. To account for these jump discontinuities in the system velocities and in the constraints reaction forces, the impulse-momentum equation including all components of the mechanical system is evaluated and numerically solved in the neighborhood of the points of impact [6].
2. The aircraft model The two-dimensional
aircraft model used in this investigation
ty
is shown in Fig. 1. The model
25
-._ A’
‘*
3.6
’
I ’
t ’
I ‘, BODY
I ’ 5
I ’
’
_..?-FL. a!!_-
I
I
I
I I BODY
m
,
I
I
0 3
3.6
1
1 Fig. 1. Two-dimensional
aircraft model.
,
m
K. Changizi
et al., Analysis
of multi-body
95
systems. Part II
T W
lENSION D
I DIMENSION C
I-r----r---DIMENSION B
Fig. 2. Element
cross-section.
consists of six bodies and two identical suspensions which represent the landing gears. Bodies two and six are the two landing gears. Suspension masses are assumed to be concentrated at the wheel center. Bodies 3 and 5 are the two wings of the aircraft which are modeled as two identical cantilever beams. The finite element method is used to discretize the two flexible wings. Each wing is divided into ten finite elements. Two-dimensional beam elements are employed in this analysis. Each element has six elastic coordinates which represent translations and slopes of the nodal points. The element cross-section used in this analysis is also shown in Fig. 2. The two wings are rigidly connected to the fuselage (Body 4). The inertia properties of different components of the aircraft are given in Table 1 while the reference location and orientation of each component with respect to the inertial frame are displayed on Fig. 1. Elasticity of the landing gear is introduced using spring and damper elements. Table 1 Properties of the aircraft’s model Body No.
Type
Mass (kg)
1 2 3 4 5 6
Rigid Rigid Flexible Rigid Flexible Rigid
0.0 0.19OE + 05 0295E + 05 0.168E + 06 0.295E + 05 0.19OE + 05
ZZ(m4) 0.0 O.lllE+05 0.471E + 07 0.870E + 07 0.471E + 07 O.lllE+05
K. Changizi et al., Analysis of multi-body systems. Part II
96
Tire-ground interaction is represented by a simple point-contact model. The suspension characteristics and the tire stiffness of this model are given in [7]. In the present analysis, the aircraft model will represent a multi-body system consisting of interconnected rigid and flexible components that may undergo large angular rotations. Interconnection between different components will be introduced using a set of nonlinear algebraic constraint equations that account for large geometric changes in the aircraft configuration as well as the effect of joints deformations.
3. Generalized coordinates and kinematic constraint equations In order to specify the configuration of a body or substructure, it is necessary to define a set of generalized coordinates such that the global position and orientation of every infinitesimal volume on the body is determined in terms of these generalized coordinates. Let the XYZ Cartesian coordinate system represent an inertial (global) frame and the X’Y’Z’-axes represent a coordinate system rigidly attached to the ith flexible body. For the ith flexible body, let R’ and 8’ represent translational and rotational orientation, respectively, of the X’Y’Z’ body coordinate with respect to the global XYZ-coordinate axes. The global location of an arbitrary infinitesimal volume at point PiI on the jth element of this body can be defined as: rij = P
Ri
+
Aidij (1)
where A' = Ai is the transformation matrix from the ith body fixed coordinates to the inertial frame, and Z” is the position vector of point Pi with respect to body coordinates as shown in Fig. 3. Zi can be expressed in terms of nodal coordinates as described in [8], or alternatively: cij
=
e8 + uij 9
where et is the position vector of P” in the undeformed displacement vector defined with respect to the body coordinate shape function, uti can be written as: *ii
=
@jsij
state and u’j is the elastic system. Using an appropriate
(3)
,
where @” and 8” are, respectively, modified shape function and vector of deformation at the nodes, defined with respect to the body coordinate system. By using (l)-(3), the position vector rz can be written in terms of reference coordinates and nodal elastic coordinates as:
In terms of the body elastic coordinates, u
ii
=
$ij$ ,
(3) can be written as
K. Changizi et al., Analysis of multi-body systems. Part II
Fig. 3. Generalized
coordinates
97
of the ith body.
-
where @” is a new shape function given by:
*” = [O,0, @“,o,o,. . . , O]BL and 6 is the vector of elastic coordinates of body i. Bk is a Boolean matrix that describes the interconnectivity between elements. In order to define a unique displacement field, a set of reference conditions has to be imposed [8]. Equation (5) can then be written in terms of the new set of elastic coordinates as uij
=
SijBi
6i g
7
(6)
where BB is a constant transformation and 8’ is the new set of independent elastic coordinates of body i. If component modes are employed, a transformation from the space of the body physical coordinates to the space of the modal coordinates can be performed. This transformation can be written as: 6’ = BiXi)
(7)
where Bf,, is the modal matrix and xi is the vector of modal coordinates. Using (4), (6) and (7), the location of an arbitrary point PC on element j of body i (Fig. 3) can be written in terms
K. Changizi et al., Analysis of multi-body systems. Part ZZ
98
of a mixed set of reference and modal coordinates
as follows:
where N” is given by: Nij
=
@‘BiBi B
Ill*
(9)
Constraints between adjacent components in the multi-body aircraft can be described using a set of nonlinear algebraic equations. As an example the nonlinear constraint equation for a revolute joint can be written as Ri+Ai&_Rj_Ajdj=o 3
(10)
where R’ and Rj represent the location of the origins of body axes X’Y’Z’ and X’Y’Z’, respectively, of the two adjacent bodies i and i. The vectors d’ and d’ define locations of joint attachment points on the respective bodies defined with respect to the body coordinate system. Ai and A'(@)are the transformation matrices from the body coordinates to the inertial frame. The aircraft model presented in the previous section requires imposing rigid joints between the fuselage and the wings. Rigid joints between adjacent bodies can be formulated by using (10) and freezing the relative rotation between the two bodies, which in the two-dimensional case can be written as ei - ej = 0
,
(11)
where 8’ and 8’ are the rotational coordinates of the two bodies about the Z-axis. In general, constraint equations between adjacent bodies can be written in vector function form as @(q, t) = 0 where
is a set of nonlinear algebraic equations, which can be used to describe constraints components of the aircraft and m is the total number of constraint equations.
between
4. Nonlinear landing gear Elasticity of the landing gear is introduced using spring-damper elements shown in Fig. 4. A simple point-contact model, rather than a more detailed tire-ground deformation model is used. It is assumed that the resultant of the ground force is vertical and passes through the center of gravity of the wheel. The ground penetration is evaluated by monitoring the distance from the center of gravity of the wheel to the contact point with ground. The tires are free to leave the ground. The suspension spring and damping coefficients and the tire-stiffness constant used in the present investigation are 1.5 - 10’ (N/m), 3.2 - lo7 (N/m/s) and 20 10’ (N/m) respectively. In this section a generalized force expression is derived to account for
K. Changizi et al., Analysis of multi-body systems. Part II
Fig. 4. Suspension system.
all externally and internally applied forces acting on the suspension system of the landing gear. Fig. 4 shows the spring and damper between two arbitrary components of the aircraft. The attachment points are point Pi on component i and point P’ on component j. The global locations of these two points are given by r$,= R’ + A'(e&+
u;)
(13)
6; = Rj + Aj(e’,,+ ui,)
(14)
and
where e& and eJ& are the undeformed positions of end points of the spring damper element and ud and u: are the deformation at these points. The relative position of point P’ with respect to the point P’, denoted by r, in the global coordinate system, can be written as: r,
=
ri - ri =
Rj - Ri + Aj(eJ&_
uL)_
Ai(e&+
The length of the spring damper can then be written as
ui).
(15)
100
K. Changizi
et al., Analysis
of multi-body
systems. Part II
1 = d/r& ,
(16)
Assuming that lo is the undeformed damper element is given by F, = k(l-
lo)+
length of the spring, the force acting along the spring
d,
(17)
where k is the spring stiffness, c is the damping coefficient, and i is the derivative of I with respect to time. The virtual work of the spring force F’ can then be written as 6w, = -F,iY=
The variation of
lo)+
ci]sk
can be written in terms of the generalized coordinates
(18) as
{aWaqi}tiqi + {allaqj}i3qj.
6Z= Therefore,
1
- [k(z-
(19)
the virtual work can be written as
and the generalized forces associated with body i and body j due to the spring forces are Qf
= -F,
. l,i
(21)
and Qj=-F,*Z,j.
(22)
It is important to notice that (21) and (22) contain the generalized forces associated with the reference and elastic generalized coordinates of components i and j. In the case that one or both components are rigid, the deformation vector at the attachment point of the rigid component will be equal to zero and (21) and (22) yield the generalized forces associated with the reference generalized coordinates. In the present formulation nonlinear stiffness and damping coefficients can be included.
5. Constrained
system equations of motion
In Section 3 the global position of an arbitrary point on the flexible body is written in terms of the body reference and modal coordinates. By differentiating (8) with respect to time, the velocity vector of point P on the flexible body can be determined. Using this velocity vector the kinetic energy of body i can be written as [8]:
Ti
=
$ -pj=
$($)‘Mi$,
j=l
(23)
101
K. Changizi et al., Analysis of multi-body systems. Part II
where qi = [(R’)‘,
(f?)‘, (/$>,I, = [(qf)t,
(/$)‘I’ )
re p resent the reference coordinates and xi the elastic modal coordinates body i. The strain energy of body i can be expressed as [8]: q: = [(R’)‘,
(e’),lt
ui = $(qi)tK’qi,
of
(24)
where K’ is the stiffness matrix of body i. The virtual work of external forces acting on body i can be written as:
6w’ =
(Qq%qi,
(25)
where Q’ is the vector of generalized forces associated with the generalized coordinates of body i. Using Lagrange’s equation, the equation of motion of ith body can be written as [5] M’(q’)$
+ K'q'
= Q(q,
cj,t) + F’ -
@‘bt(q, t)h ,
(26)
where Q’ is the vector of generalized forces, F’ is a quadratic velocity vector that results from differentiating the kinetic energy with respect to time and with respect to the generalized coordinates, @ is the vector function of nonlinear constraints and A is the vector of Lagrange multipliers. Subscript vectors denote partial differentiation. If P = [(p’)‘, (p’>‘, . . . , (p”)‘]’ is the total system generalized coordinate vector, (26) can be written as
M(P)P; + KP = O(P, i: t) - @A
)
(27)
where A?(P), K, 6, and 0 are the system mass, stiffness, constraint Jacobian, and generalized forces respectively. In (27) 0 absorbs the quadratic velocity vector. The constraint equation can be written in terms of generalized reference and modal coordinates as:
6(P, t) = 0. Equations
(27) and (28) represent
6. Airframe-ground
(28) the system equations of motion of the multi-body
aircraft.
interaction and the induced dynamic loads
In large aircraft, ground-induced dynamic loads are recognized as significant factors in causing vibrations, dynamic stress and fatigue damage of the airframe structure. Known sources of these dynamic loads are landing impact and runway unevenness. In this section a mathematical model that accounts for the effect of such dynamic loads is formulated. By incorporating this mathematical model into the equations of motion of Section 5, a comprehensive analytical model for aircraft dynamics during ground operations is then established.
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K. Changizi
et al., Analysis
of multi-body
systems. Part II
6.1. Step-bump model The ground-induced dynamic forces during traverse of abrupt elevation change in the runway surface can be evaluated by installing a set of step bumps of selected configurations. The surface unevenness may also be simulated by a general terrain profile with assumed root mean square roughness that closely represent anatural bump. In this analysis one step bump was installed on the runway surface so that one of the landing gears would encounter the bump. The bump is modeled by a standard cycloidal curve [9] which shows no discontinuities at the inflection points. It begins and ends with zero slope. The ground-reaction forces at any instant is a function of the vertical penetration or sink of the wheel into the ground, assuming that the reaction force is acting vertically with a line of action passing through the mass center of the wheel.
6.2. Touch-down impact The induced dynamic loads resulting from impact between the landing gear and the runway can be accounted for by solving a system of generalized impulse-momentum equations in the neighborhood of points of impact [6]. It is worthwhile mentioning that an impulse-momentum relation that considers all bodies of the multi-body system is stringent. This is due to the fact that all mechanical components interact through constraints. Consequently, an impulsemomentum relation involving only bodies on which the impulsive force acts is infeasible. Following a scheme similar to that used in [6], an impact predictor function may be written in the form @(P(t),
P(t),
t) =
0,
(29)
where, in this application Oi can be simply a function of the relative displacement between a point on the tire that is an undeformed radius away from the mass center of the wheel and another point on the surface of the runway. When (29) is satisfied, a point in time ti at which an impact occurs is defined. In reality impact occurs over a short duration of time 71< ti < 72 such that time interval 1~~- T~I= AT of the short-lived event of the impact is small enough that the system configuration does not change appreciably. In this analysis, the geometry of impacting surfaces is not considered, therefore the common normal direction may be defined by the direction of the approach line between two points on the two impacting surfaces in the neighborhood of the point of impact. Assuming no friction between the two impacting surfaces during impact, then the generalized force vector of (27) can be written during impact as:
a= a+
QF,
(30)
where Q, is due to external forces together with quadratic terms in velocities resulting from the differentiation of kinetic energy expression with respect to generalized coordinates and time. The term QF is due to the impulsive force F(t) acting on both tire and the runway during the impact. Using the principle of virtual work one can write the work done by the impulsive force in the direction of the common normal as: 6w
= F(t)65
= F(t){agaP}w
=
QgP,
(31)
K. Changizi et al., Analysis of multi-body systems. Part II
103
where i!i,J is a virtual relative displacement or virtual penetration in the direction of the common normal. Integrating the equation of motion (27) over the interval [TV,~~1,and then taking the limit as AT tends to go to zero, one can write -{&@+KP+
&b*}dt=
Lim
(32)
Let {a{/@} = H, then using the integral mean value theorem with the assumption that 0, is continuous and noting that velocities are bounded during impact, one can write (32) after utilizing the continuity of M, x, P and 6 in the form fi Lim A-0
I
R
p dt + &.bLim
,,
At-0
I
v.
7,
Adt=H’Lim AT-O
I
s2
F(t) dt
(33)
rl
or simply ii%Ap+ &PA = H’P,
(W
where Ap is the jump in the velocity vector, P” is the impulse of the constraints reaction forces and P is the generalized impulse due to impact. Differentiating the constraint equations (28) with respect to time, one gets
d&P+ {a&It} Because 6 and {6’6/&}
= 0.
(35)
are continuous,
one can write
&Ai,=O. If impact is characterized common normal as: i(tt)
(36) by the restitution condition that can be written in the direction of the
= -a&),
(37)
where (Y is the coefficient of restitution, &ti) is the relative velocity at the start of the compression phase of impact and [(tt) is relative velocity at the end of restitution phase of impact, one can write (37) in terms of the generalized velocities in the form HP@:) = -aHp(t;). Subtracting the term Hp(t;) H Ai, = -(l+ Equations obtain
(38) from both sides of (38) one gets
a)Hi)(tf)
.
(34) (36) and (39) can be written in matrix form, with P treated
(39) as variable, to
104
K. Changizi
et al., Analysis
of multi-body
systems. Part II
where r = - (1 + ~)Hp(t;) and H’ is a vector of the same dimension as the generalized coordinate vector P. The equations of motion are integrated forward in time and the impact predictor functions of (29) are monitored until one of them becomes zero defining an impact and hence stopping the integration algorithm to allow for the construction and solution of (40). Having the impulse-momentum relation solved, the system velocity vector can be updated and the integration algorithm restarted to resume regular mode of computations until the impact is again encountered [6].
7. Numerical results The two-dimensional aircraft model described in Section 2 is used in the two phases of this investigation. As mentioned earlier the two flexible wings are modeled as cantilevers. Associated natural frequencies of different modes of the flexible wings, in ascending order, are 19.45, 78, 190, 356, 382.29 and 579.29 rad/s. In phase one, impact analysis of vertical drop of the model was performed numerically. Then the obtained numerical results were plotted against time. In the second phase of this investigation, to determine the response of the flexible wings during the traverse of an abrupt elevation change in the runway surface, a step bump was installed in the runway surface so that the landing gear (Body 6) would encounter the bump during this phase. The numerical values were obtained for the deflection of two tip points A and B (Fig. 1) and plotted against time. Figs. 5-12 describe the vertical and angular response of the tip deflection of the right and left wings relative to the body coordinate system of the wings which are rigidly attached to the fuselage. In these figures two and six cantilever mode solutions of each wing are plotted. Figs. 5 and 6 illustrate the relative vertical displacement while Figs. 7 and 8 the relative angular rotations of nodal points A and B with respect to their body coordinate axes X,-Y, and X,-Y,. Results are plotted from the time of release of the model (t = 0 s) through impact (t = 0.5087 s) and rebound of the landing gear (t = 0.85 s). It can be seen from Figs. 5 and 6, at the time of impact (t = 0.5087 s), nodal points A and B each undergo a vertical displacement about ‘-35’ and ‘-33’ centimeters respectively, relative to their body axes, while they are being reduced to ‘-6.5’centimeter at the time t = 0.85 s. Alsoone can observe the relative rotation of points A and B relative to their body axes X, - Ys and X, - Y3in Figs. 7 and 8. At the time of impact, the two nodal points A and B undergo an angular change of 0.028 rad relative to their own body axes. The large deflection of the tip points is known to cause stress distribution at the joints as a result of high impulsive loads at the time of impact. Figs. 9-12 represent the numerical results when the landing gear encounters a bump in the runway. Figs. 9 and 10 illustrate the relative vertical displacement and Figs. 11 and 12 the relative angular rotations of the tip nodal points on the left and right wings with respect to their body coordinate axes. Results are plotted from the time (t = 0 s) when the landing gear (Body 6) encounters a step bump until the completion of abrupt elevation and return to the ground level (t = 0.07 s). Results are presented for
K. Clhangizi et al., Analysis
105
of multi-body systems. Part 11
---
2m&
-
emode
Time (second)
Fig, 5. VerticaI displacement
of the tip point of the left wing with impact.
I”
ci I
---
2 mode 6mode
50
Fig. 6. Vertical displacement
of the tip point of the right wing with impact.
106
K. Changizi et al., Analysis of multi-body systems. Part II
---
2 mode 6 mode
Time (second)
Angular
displacement
of the tip point of the left wing with impact.
---
dl ‘0.00
0.30 I
0.60 1
0.90 I
Pmode 6mode
1.20 I
I 1.50
Time (second) Angular
displacement
of the tip point of the right wing with impact.
107
K. Changizi et al., Analysis of multi-body systems. Part II
d‘0.00
0.30 I
0.60 I Time
Fig. 9. Vertical displacement
0.90 1
1.20 1
1.50 1
(second)
of the tip point of the left wing over the bump.
---
Pmode 6mode
Tune (second) Fig. 10. Vertical displacement
of the tip point of the right wing over the bump.
108
K. Ctaangizi et al., Analysis
of multi-body systems. Part
XI
5Ci Tie
Fig. 11. Angular displacement
(second)
of the tip point of the left wing over the bump.
--~
2mode 6mode
Time (second)
Fig. 12. Angular displacement
of the tip point of the right wing over the bump.
K. Changizi et al., Analysis of multi-body systems. Part II
109
a period of simulation time of 1.5 s. When the left landing gear (Body 6) completed the traverse of the step bump at t = 0.07 s, the vertical displacements of the tip points of the left and right wings relative to their body axes reach the values of ‘-8.5’ and ‘-8’ centimeters, respectively. Forces induced to the suspension system at the time of impact significantly affect the dynamic response of the wings and the entire aircraft. Also any irregularities and bumps in the road surface induce forces to the suspension system and accordingly affect the dynamic response of the entire aircraft. It is not uncommon that these forces have a high-frequency nature that can be transmitted to the wing through the suspension. These forces excite the deformation modes of the wing, thus producing oscillations that can significantly affect the dynamic response and passenger’s ride comfort of the aircraft. Deformation modes share a significant percentage of the total system energy. Nonlinear systems like the model presented have inertia-induced reaction forces including impulsive forces which excite higher modes and cause them to pick up proportionately more of the total system- energy. In design and performance analysis of such systems one should include deformation modes that may be excited by these impulsive loading conditions. As mentioned earlier, the method presented allows the use of component modes and consequently the number of elastic coordinates can be significantly reduced. In Figs. 5-12 a comparison between two- and six-mode solutions is made. These figures give an indication of the expected loss in accuracy due to ignoring higher modes. The simulation cost of the six-mode solution is much higher than that of the two-mode solution. These costs are closely related to the highest natural frequency retained in the model.
8. Summary
and conclusion
A method for dynamic analysis of a large-scale aircraft composed of interconnected rigid and flexible bodies that may undergo large angular rotations is presented. Flexible bodies in the system are discretized using the finite element method. The system equations of motion are written in the Lagrangian formulation using a finite set of coupled reference positions and local elastic generalized coordinates. Nonlinear constraint equations describing joints between different components are adjoined to the dynamic equations using Lagrange multipliers. Equations of motion are computer-generated and integrated forward in time using an explicit-implicit integration algorithm coupled with Newton-Raphson type iterations to check on constraint violations. Points in time at which impact occurs are monitored by an event predictor which controls the integration algorithm and forces a solution for the system impulse-momentum relation at those points. Solutions of impulse-momentum relations define the jump discontinuities in the composite velocity vector as well as the generalized impulses of the reaction forces at different joints of the aircraft. The mathematical model presented in this study accounts for the large change in geometry of the suspension system and also for the nonlinear characteristics of the suspension and tire model. The mathematical model simulates the dynamic behavior of an aircraft during landing and ground operation. The validity of the mathematical model is demonstrated by solving a two-dimensional large-scale aircraft model. The induced aeroelastic forces acting on the wings were not considered in this analysis. Consequently, the method emphasizes the study of dynamic response of the airframe in terms of the mechanical vibrations without getting into the associated flutter phenomenon. This
110
K. Changiri et al., Analysis of multi-body systems. Part II
places no limitations on the mathematical mulation for the aerodynamic forces.
model which can easily accommodate
any for-
References [l] DC-10 landing gear modified. Aviat. Week Space Technol. 98 (10) (1973) 181-186. [2] R. Popeleuski, Airbus test tempo quickening. Aviat. Week Space Technol. 98 (10) (1973) 32-35. [3] J. Morison and D. Kamopp, Comparisons of optimized active and passive vibration absorbers, Proceedings IEEE 41st Joint Automatic Control Conference, New York (1973) 932-938. [4] J. McGehee and R. Dreher, Experimental investigations of active loads control for aircraft landing gear, NASA TR No. 2042 (1982) l-67. [5] A. Shabana, Substructure synthesis method for multi-body systems, Comput. & Structures 20(4) (1985) 737-744. [6] Y. Khulief and A. Shabana, Dynamic analysis of constrained system of rigid and flexible bodies with intermittent motion, ASME J. Mechanisms, Transmissions and Automation in Design (to appear). [7] K. Changizi, Effect of impact on dynamic analysis of large scale aircraft, M.Sc. Thesis, University of Illinois at Chicago Circle, 1985. [8] A.A. Shabana, Transient analysis of flexible multi-body systems. Part I: dynamics of flexible bodies, Comput. Meths. Appl. Mech. Engrg. 54 (1986) 75-91 (this issue). [9] J.E. Shigley, Theory of Machines and Mechanisms (McGraw-Hill, New York, 1980) 209-218.
IN APPLIED
MECHANICS
AND ENGINEERING
54 (1986) 93-110
TRANSIENT ANALYSIS OF FLEXIBLE MULTI-BODY SYSTEMS. PART II: APPLICATION TO AIRCRAFT LANDING
Koorosh CHANGIZI, Department
of Mechanical
Yehia A. KHULIEF
and Ahmed A. SHABANA
Engineering, University of Illinois at Chicago Chicago, IL 60680, U.S.A.
Revised
Received 28 January manuscript received
(Circle Campus),
1985 2 May 1985
This investigation presents a method for transient analysis of a large-scale multi-body aircraft consisting of interconnected rigid and flexible bodies that undergo large angular rotations. Elastic components of the aircraft are discretized using the finite element method. The system equations of motion and nonlinear algebraic constraint equations describing joints between different components are written in the Lagrangian formulation using a finite set of coupled reference and modal coordinates. The system differential equations of motion and algebraic constraint equations are computer-generated and integrated forward in time using an explicit-implicit direct numerical integration algorithm coupled with a Newton-Raphson type iteration in order to check on constraint violations. Impact and intermittent motion events are accounted for by using a generalized momentum balance that predict jump discontinuities in the generalized velocities as well as jump discontinuities in the system reaction forces. The formulation presented and the computer program developed are used to simulate the impact between the landing gear and the runway. The method is also used to predict the dynamic behavior of the aircraft during the traverse of an abrupt elevation change in the runway.
1. Introduction
Airborne structures are examples of multi-body systems that are subjected to different dynamic loading conditions. One of the most significant phases of such dynamic loadings is that due to ground interaction. In this paper, a mathematical model was developed to simulate the dynamic behavior of an aircraft during landing and ground operations. The dynamic loads and vibrations resulting from landing impact and from runway and taxiway unevenness are recognized as significant factors in causing fatigue damage, dynamic stresses on the airframe, passenger discomfort, and reduction of pilot’s ability to control an aircraft during ground operations. These ground-induced dynamic loads and vibrations have been encountered with some conventional subsonic transport aircrafts [l, 21; however, they are magnified for supersoniccruise aircraft due to the increased structural flexibility inherent in the slender body, and thin wing design. Vibration-control techniques have been employed to improve the operational characteristics of the aircraft on the ground. Passive, semi-active and active control techniques are used [3]; however, the active control method is found to be more effective in reducing the ground loads transmitted to the airframe [4]. The objective of this paper is to present a 0045-7825/86/$3.50
@ 1986, Elsevier
Science
Publishers
B.V. (North-Holland)
94
K. Changizi et al., Analysis of multi-body systems. Part II
mathematical model that can be used for numerical simulation of a general aircraft during ground operations. A mathematical model that closely represents the dynamic characteristics of the aircraft is a prerequisite for design and evaluation of the vibration controller. The computer-aided simulation is then a significant step in the design process flowchart that offers a faster and cheaper tool for predicting the dynamic behavior of the whole system before reaching the experimental implementation step [4] which is often expensive and timeconsuming. It is also recognized that malfunction of active controllers may result in complete inoperability. Consequently, a study of the effect of impulsive loading conditions resulting from touch-down impact or traverse of runway surface unevenness on the airframe has to be performed in the absence of vibration-suppressing devices. In this analysis, the aircraft is considered as a multi-body system that is composed of interconnected rigid and flexible components [5]. Flexible components may undergo large angular rotations, therefore the developed mathematical model accounts for the nonlinear inertia coupling between the set of reference generalized coordinates and the set of local elastic deformations that are used to represent the global location of any point in the system. A finite element discretization of flexible components is obtained by a finite set of nodal points. A component mode substitution method is then used to obtain a reduced-order model that contains the lower end of the frequency spectrum as well as other significant modes of vibrations. Two major problems encountered during the aircraft ground operations are touch-down impact and natural bumps due to abrupt elevation changes in the runway surface. The simulation of a step or natural bump by a forcing function is a straight-forward technique; however, the resulting forcing function can be easily added to the forcing term of the equations of motion. The situation is different in the case of impact where changes in the system momentum take place causing an instantaneous set of impulsive forces that excite higher modes of vibrations. To account for these jump discontinuities in the system velocities and in the constraints reaction forces, the impulse-momentum equation including all components of the mechanical system is evaluated and numerically solved in the neighborhood of the points of impact [6].
2. The aircraft model The two-dimensional
aircraft model used in this investigation
ty
is shown in Fig. 1. The model
25
-._ A’
‘*
3.6
’
I ’
t ’
I ‘, BODY
I ’ 5
I ’
’
_..?-FL. a!!_-
I
I
I
I I BODY
m
,
I
I
0 3
3.6
1
1 Fig. 1. Two-dimensional
aircraft model.
,
m
K. Changizi
et al., Analysis
of multi-body
95
systems. Part II
T W
lENSION D
I DIMENSION C
I-r----r---DIMENSION B
Fig. 2. Element
cross-section.
consists of six bodies and two identical suspensions which represent the landing gears. Bodies two and six are the two landing gears. Suspension masses are assumed to be concentrated at the wheel center. Bodies 3 and 5 are the two wings of the aircraft which are modeled as two identical cantilever beams. The finite element method is used to discretize the two flexible wings. Each wing is divided into ten finite elements. Two-dimensional beam elements are employed in this analysis. Each element has six elastic coordinates which represent translations and slopes of the nodal points. The element cross-section used in this analysis is also shown in Fig. 2. The two wings are rigidly connected to the fuselage (Body 4). The inertia properties of different components of the aircraft are given in Table 1 while the reference location and orientation of each component with respect to the inertial frame are displayed on Fig. 1. Elasticity of the landing gear is introduced using spring and damper elements. Table 1 Properties of the aircraft’s model Body No.
Type
Mass (kg)
1 2 3 4 5 6
Rigid Rigid Flexible Rigid Flexible Rigid
0.0 0.19OE + 05 0295E + 05 0.168E + 06 0.295E + 05 0.19OE + 05
ZZ(m4) 0.0 O.lllE+05 0.471E + 07 0.870E + 07 0.471E + 07 O.lllE+05
K. Changizi et al., Analysis of multi-body systems. Part II
96
Tire-ground interaction is represented by a simple point-contact model. The suspension characteristics and the tire stiffness of this model are given in [7]. In the present analysis, the aircraft model will represent a multi-body system consisting of interconnected rigid and flexible components that may undergo large angular rotations. Interconnection between different components will be introduced using a set of nonlinear algebraic constraint equations that account for large geometric changes in the aircraft configuration as well as the effect of joints deformations.
3. Generalized coordinates and kinematic constraint equations In order to specify the configuration of a body or substructure, it is necessary to define a set of generalized coordinates such that the global position and orientation of every infinitesimal volume on the body is determined in terms of these generalized coordinates. Let the XYZ Cartesian coordinate system represent an inertial (global) frame and the X’Y’Z’-axes represent a coordinate system rigidly attached to the ith flexible body. For the ith flexible body, let R’ and 8’ represent translational and rotational orientation, respectively, of the X’Y’Z’ body coordinate with respect to the global XYZ-coordinate axes. The global location of an arbitrary infinitesimal volume at point PiI on the jth element of this body can be defined as: rij = P
Ri
+
Aidij (1)
where A' = Ai is the transformation matrix from the ith body fixed coordinates to the inertial frame, and Z” is the position vector of point Pi with respect to body coordinates as shown in Fig. 3. Zi can be expressed in terms of nodal coordinates as described in [8], or alternatively: cij
=
e8 + uij 9
where et is the position vector of P” in the undeformed displacement vector defined with respect to the body coordinate shape function, uti can be written as: *ii
=
@jsij
state and u’j is the elastic system. Using an appropriate
(3)
,
where @” and 8” are, respectively, modified shape function and vector of deformation at the nodes, defined with respect to the body coordinate system. By using (l)-(3), the position vector rz can be written in terms of reference coordinates and nodal elastic coordinates as:
In terms of the body elastic coordinates, u
ii
=
$ij$ ,
(3) can be written as
K. Changizi et al., Analysis of multi-body systems. Part II
Fig. 3. Generalized
coordinates
97
of the ith body.
-
where @” is a new shape function given by:
*” = [O,0, @“,o,o,. . . , O]BL and 6 is the vector of elastic coordinates of body i. Bk is a Boolean matrix that describes the interconnectivity between elements. In order to define a unique displacement field, a set of reference conditions has to be imposed [8]. Equation (5) can then be written in terms of the new set of elastic coordinates as uij
=
SijBi
6i g
7
(6)
where BB is a constant transformation and 8’ is the new set of independent elastic coordinates of body i. If component modes are employed, a transformation from the space of the body physical coordinates to the space of the modal coordinates can be performed. This transformation can be written as: 6’ = BiXi)
(7)
where Bf,, is the modal matrix and xi is the vector of modal coordinates. Using (4), (6) and (7), the location of an arbitrary point PC on element j of body i (Fig. 3) can be written in terms
K. Changizi et al., Analysis of multi-body systems. Part ZZ
98
of a mixed set of reference and modal coordinates
as follows:
where N” is given by: Nij
=
@‘BiBi B
Ill*
(9)
Constraints between adjacent components in the multi-body aircraft can be described using a set of nonlinear algebraic equations. As an example the nonlinear constraint equation for a revolute joint can be written as Ri+Ai&_Rj_Ajdj=o 3
(10)
where R’ and Rj represent the location of the origins of body axes X’Y’Z’ and X’Y’Z’, respectively, of the two adjacent bodies i and i. The vectors d’ and d’ define locations of joint attachment points on the respective bodies defined with respect to the body coordinate system. Ai and A'(@)are the transformation matrices from the body coordinates to the inertial frame. The aircraft model presented in the previous section requires imposing rigid joints between the fuselage and the wings. Rigid joints between adjacent bodies can be formulated by using (10) and freezing the relative rotation between the two bodies, which in the two-dimensional case can be written as ei - ej = 0
,
(11)
where 8’ and 8’ are the rotational coordinates of the two bodies about the Z-axis. In general, constraint equations between adjacent bodies can be written in vector function form as @(q, t) = 0 where
is a set of nonlinear algebraic equations, which can be used to describe constraints components of the aircraft and m is the total number of constraint equations.
between
4. Nonlinear landing gear Elasticity of the landing gear is introduced using spring-damper elements shown in Fig. 4. A simple point-contact model, rather than a more detailed tire-ground deformation model is used. It is assumed that the resultant of the ground force is vertical and passes through the center of gravity of the wheel. The ground penetration is evaluated by monitoring the distance from the center of gravity of the wheel to the contact point with ground. The tires are free to leave the ground. The suspension spring and damping coefficients and the tire-stiffness constant used in the present investigation are 1.5 - 10’ (N/m), 3.2 - lo7 (N/m/s) and 20 10’ (N/m) respectively. In this section a generalized force expression is derived to account for
K. Changizi et al., Analysis of multi-body systems. Part II
Fig. 4. Suspension system.
all externally and internally applied forces acting on the suspension system of the landing gear. Fig. 4 shows the spring and damper between two arbitrary components of the aircraft. The attachment points are point Pi on component i and point P’ on component j. The global locations of these two points are given by r$,= R’ + A'(e&+
u;)
(13)
6; = Rj + Aj(e’,,+ ui,)
(14)
and
where e& and eJ& are the undeformed positions of end points of the spring damper element and ud and u: are the deformation at these points. The relative position of point P’ with respect to the point P’, denoted by r, in the global coordinate system, can be written as: r,
=
ri - ri =
Rj - Ri + Aj(eJ&_
uL)_
Ai(e&+
The length of the spring damper can then be written as
ui).
(15)
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K. Changizi
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1 = d/r& ,
(16)
Assuming that lo is the undeformed damper element is given by F, = k(l-
lo)+
length of the spring, the force acting along the spring
d,
(17)
where k is the spring stiffness, c is the damping coefficient, and i is the derivative of I with respect to time. The virtual work of the spring force F’ can then be written as 6w, = -F,iY=
The variation of
lo)+
ci]sk
can be written in terms of the generalized coordinates
(18) as
{aWaqi}tiqi + {allaqj}i3qj.
6Z= Therefore,
1
- [k(z-
(19)
the virtual work can be written as
and the generalized forces associated with body i and body j due to the spring forces are Qf
= -F,
. l,i
(21)
and Qj=-F,*Z,j.
(22)
It is important to notice that (21) and (22) contain the generalized forces associated with the reference and elastic generalized coordinates of components i and j. In the case that one or both components are rigid, the deformation vector at the attachment point of the rigid component will be equal to zero and (21) and (22) yield the generalized forces associated with the reference generalized coordinates. In the present formulation nonlinear stiffness and damping coefficients can be included.
5. Constrained
system equations of motion
In Section 3 the global position of an arbitrary point on the flexible body is written in terms of the body reference and modal coordinates. By differentiating (8) with respect to time, the velocity vector of point P on the flexible body can be determined. Using this velocity vector the kinetic energy of body i can be written as [8]:
Ti
=
$ -pj=
$($)‘Mi$,
j=l
(23)
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K. Changizi et al., Analysis of multi-body systems. Part II
where qi = [(R’)‘,
(f?)‘, (/$>,I, = [(qf)t,
(/$)‘I’ )
re p resent the reference coordinates and xi the elastic modal coordinates body i. The strain energy of body i can be expressed as [8]: q: = [(R’)‘,
(e’),lt
ui = $(qi)tK’qi,
of
(24)
where K’ is the stiffness matrix of body i. The virtual work of external forces acting on body i can be written as:
6w’ =
(Qq%qi,
(25)
where Q’ is the vector of generalized forces associated with the generalized coordinates of body i. Using Lagrange’s equation, the equation of motion of ith body can be written as [5] M’(q’)$
+ K'q'
= Q(q,
cj,t) + F’ -
@‘bt(q, t)h ,
(26)
where Q’ is the vector of generalized forces, F’ is a quadratic velocity vector that results from differentiating the kinetic energy with respect to time and with respect to the generalized coordinates, @ is the vector function of nonlinear constraints and A is the vector of Lagrange multipliers. Subscript vectors denote partial differentiation. If P = [(p’)‘, (p’>‘, . . . , (p”)‘]’ is the total system generalized coordinate vector, (26) can be written as
M(P)P; + KP = O(P, i: t) - @A
)
(27)
where A?(P), K, 6, and 0 are the system mass, stiffness, constraint Jacobian, and generalized forces respectively. In (27) 0 absorbs the quadratic velocity vector. The constraint equation can be written in terms of generalized reference and modal coordinates as:
6(P, t) = 0. Equations
(27) and (28) represent
6. Airframe-ground
(28) the system equations of motion of the multi-body
aircraft.
interaction and the induced dynamic loads
In large aircraft, ground-induced dynamic loads are recognized as significant factors in causing vibrations, dynamic stress and fatigue damage of the airframe structure. Known sources of these dynamic loads are landing impact and runway unevenness. In this section a mathematical model that accounts for the effect of such dynamic loads is formulated. By incorporating this mathematical model into the equations of motion of Section 5, a comprehensive analytical model for aircraft dynamics during ground operations is then established.
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6.1. Step-bump model The ground-induced dynamic forces during traverse of abrupt elevation change in the runway surface can be evaluated by installing a set of step bumps of selected configurations. The surface unevenness may also be simulated by a general terrain profile with assumed root mean square roughness that closely represent anatural bump. In this analysis one step bump was installed on the runway surface so that one of the landing gears would encounter the bump. The bump is modeled by a standard cycloidal curve [9] which shows no discontinuities at the inflection points. It begins and ends with zero slope. The ground-reaction forces at any instant is a function of the vertical penetration or sink of the wheel into the ground, assuming that the reaction force is acting vertically with a line of action passing through the mass center of the wheel.
6.2. Touch-down impact The induced dynamic loads resulting from impact between the landing gear and the runway can be accounted for by solving a system of generalized impulse-momentum equations in the neighborhood of points of impact [6]. It is worthwhile mentioning that an impulse-momentum relation that considers all bodies of the multi-body system is stringent. This is due to the fact that all mechanical components interact through constraints. Consequently, an impulsemomentum relation involving only bodies on which the impulsive force acts is infeasible. Following a scheme similar to that used in [6], an impact predictor function may be written in the form @(P(t),
P(t),
t) =
0,
(29)
where, in this application Oi can be simply a function of the relative displacement between a point on the tire that is an undeformed radius away from the mass center of the wheel and another point on the surface of the runway. When (29) is satisfied, a point in time ti at which an impact occurs is defined. In reality impact occurs over a short duration of time 71< ti < 72 such that time interval 1~~- T~I= AT of the short-lived event of the impact is small enough that the system configuration does not change appreciably. In this analysis, the geometry of impacting surfaces is not considered, therefore the common normal direction may be defined by the direction of the approach line between two points on the two impacting surfaces in the neighborhood of the point of impact. Assuming no friction between the two impacting surfaces during impact, then the generalized force vector of (27) can be written during impact as:
a= a+
QF,
(30)
where Q, is due to external forces together with quadratic terms in velocities resulting from the differentiation of kinetic energy expression with respect to generalized coordinates and time. The term QF is due to the impulsive force F(t) acting on both tire and the runway during the impact. Using the principle of virtual work one can write the work done by the impulsive force in the direction of the common normal as: 6w
= F(t)65
= F(t){agaP}w
=
QgP,
(31)
K. Changizi et al., Analysis of multi-body systems. Part II
103
where i!i,J is a virtual relative displacement or virtual penetration in the direction of the common normal. Integrating the equation of motion (27) over the interval [TV,~~1,and then taking the limit as AT tends to go to zero, one can write -{&@+KP+
&b*}dt=
Lim
(32)
Let {a{/@} = H, then using the integral mean value theorem with the assumption that 0, is continuous and noting that velocities are bounded during impact, one can write (32) after utilizing the continuity of M, x, P and 6 in the form fi Lim A-0
I
R
p dt + &.bLim
,,
At-0
I
v.
7,
Adt=H’Lim AT-O
I
s2
F(t) dt
(33)
rl
or simply ii%Ap+ &PA = H’P,
(W
where Ap is the jump in the velocity vector, P” is the impulse of the constraints reaction forces and P is the generalized impulse due to impact. Differentiating the constraint equations (28) with respect to time, one gets
d&P+ {a&It} Because 6 and {6’6/&}
= 0.
(35)
are continuous,
one can write
&Ai,=O. If impact is characterized common normal as: i(tt)
(36) by the restitution condition that can be written in the direction of the
= -a&),
(37)
where (Y is the coefficient of restitution, &ti) is the relative velocity at the start of the compression phase of impact and [(tt) is relative velocity at the end of restitution phase of impact, one can write (37) in terms of the generalized velocities in the form HP@:) = -aHp(t;). Subtracting the term Hp(t;) H Ai, = -(l+ Equations obtain
(38) from both sides of (38) one gets
a)Hi)(tf)
.
(34) (36) and (39) can be written in matrix form, with P treated
(39) as variable, to
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where r = - (1 + ~)Hp(t;) and H’ is a vector of the same dimension as the generalized coordinate vector P. The equations of motion are integrated forward in time and the impact predictor functions of (29) are monitored until one of them becomes zero defining an impact and hence stopping the integration algorithm to allow for the construction and solution of (40). Having the impulse-momentum relation solved, the system velocity vector can be updated and the integration algorithm restarted to resume regular mode of computations until the impact is again encountered [6].
7. Numerical results The two-dimensional aircraft model described in Section 2 is used in the two phases of this investigation. As mentioned earlier the two flexible wings are modeled as cantilevers. Associated natural frequencies of different modes of the flexible wings, in ascending order, are 19.45, 78, 190, 356, 382.29 and 579.29 rad/s. In phase one, impact analysis of vertical drop of the model was performed numerically. Then the obtained numerical results were plotted against time. In the second phase of this investigation, to determine the response of the flexible wings during the traverse of an abrupt elevation change in the runway surface, a step bump was installed in the runway surface so that the landing gear (Body 6) would encounter the bump during this phase. The numerical values were obtained for the deflection of two tip points A and B (Fig. 1) and plotted against time. Figs. 5-12 describe the vertical and angular response of the tip deflection of the right and left wings relative to the body coordinate system of the wings which are rigidly attached to the fuselage. In these figures two and six cantilever mode solutions of each wing are plotted. Figs. 5 and 6 illustrate the relative vertical displacement while Figs. 7 and 8 the relative angular rotations of nodal points A and B with respect to their body coordinate axes X,-Y, and X,-Y,. Results are plotted from the time of release of the model (t = 0 s) through impact (t = 0.5087 s) and rebound of the landing gear (t = 0.85 s). It can be seen from Figs. 5 and 6, at the time of impact (t = 0.5087 s), nodal points A and B each undergo a vertical displacement about ‘-35’ and ‘-33’ centimeters respectively, relative to their body axes, while they are being reduced to ‘-6.5’centimeter at the time t = 0.85 s. Alsoone can observe the relative rotation of points A and B relative to their body axes X, - Ys and X, - Y3in Figs. 7 and 8. At the time of impact, the two nodal points A and B undergo an angular change of 0.028 rad relative to their own body axes. The large deflection of the tip points is known to cause stress distribution at the joints as a result of high impulsive loads at the time of impact. Figs. 9-12 represent the numerical results when the landing gear encounters a bump in the runway. Figs. 9 and 10 illustrate the relative vertical displacement and Figs. 11 and 12 the relative angular rotations of the tip nodal points on the left and right wings with respect to their body coordinate axes. Results are plotted from the time (t = 0 s) when the landing gear (Body 6) encounters a step bump until the completion of abrupt elevation and return to the ground level (t = 0.07 s). Results are presented for
K. Clhangizi et al., Analysis
105
of multi-body systems. Part 11
---
2m&
-
emode
Time (second)
Fig, 5. VerticaI displacement
of the tip point of the left wing with impact.
I”
ci I
---
2 mode 6mode
50
Fig. 6. Vertical displacement
of the tip point of the right wing with impact.
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K. Changizi et al., Analysis of multi-body systems. Part II
---
2 mode 6 mode
Time (second)
Angular
displacement
of the tip point of the left wing with impact.
---
dl ‘0.00
0.30 I
0.60 1
0.90 I
Pmode 6mode
1.20 I
I 1.50
Time (second) Angular
displacement
of the tip point of the right wing with impact.
107
K. Changizi et al., Analysis of multi-body systems. Part II
d‘0.00
0.30 I
0.60 I Time
Fig. 9. Vertical displacement
0.90 1
1.20 1
1.50 1
(second)
of the tip point of the left wing over the bump.
---
Pmode 6mode
Tune (second) Fig. 10. Vertical displacement
of the tip point of the right wing over the bump.
108
K. Ctaangizi et al., Analysis
of multi-body systems. Part
XI
5Ci Tie
Fig. 11. Angular displacement
(second)
of the tip point of the left wing over the bump.
--~
2mode 6mode
Time (second)
Fig. 12. Angular displacement
of the tip point of the right wing over the bump.
K. Changizi et al., Analysis of multi-body systems. Part II
109
a period of simulation time of 1.5 s. When the left landing gear (Body 6) completed the traverse of the step bump at t = 0.07 s, the vertical displacements of the tip points of the left and right wings relative to their body axes reach the values of ‘-8.5’ and ‘-8’ centimeters, respectively. Forces induced to the suspension system at the time of impact significantly affect the dynamic response of the wings and the entire aircraft. Also any irregularities and bumps in the road surface induce forces to the suspension system and accordingly affect the dynamic response of the entire aircraft. It is not uncommon that these forces have a high-frequency nature that can be transmitted to the wing through the suspension. These forces excite the deformation modes of the wing, thus producing oscillations that can significantly affect the dynamic response and passenger’s ride comfort of the aircraft. Deformation modes share a significant percentage of the total system energy. Nonlinear systems like the model presented have inertia-induced reaction forces including impulsive forces which excite higher modes and cause them to pick up proportionately more of the total system- energy. In design and performance analysis of such systems one should include deformation modes that may be excited by these impulsive loading conditions. As mentioned earlier, the method presented allows the use of component modes and consequently the number of elastic coordinates can be significantly reduced. In Figs. 5-12 a comparison between two- and six-mode solutions is made. These figures give an indication of the expected loss in accuracy due to ignoring higher modes. The simulation cost of the six-mode solution is much higher than that of the two-mode solution. These costs are closely related to the highest natural frequency retained in the model.
8. Summary
and conclusion
A method for dynamic analysis of a large-scale aircraft composed of interconnected rigid and flexible bodies that may undergo large angular rotations is presented. Flexible bodies in the system are discretized using the finite element method. The system equations of motion are written in the Lagrangian formulation using a finite set of coupled reference positions and local elastic generalized coordinates. Nonlinear constraint equations describing joints between different components are adjoined to the dynamic equations using Lagrange multipliers. Equations of motion are computer-generated and integrated forward in time using an explicit-implicit integration algorithm coupled with Newton-Raphson type iterations to check on constraint violations. Points in time at which impact occurs are monitored by an event predictor which controls the integration algorithm and forces a solution for the system impulse-momentum relation at those points. Solutions of impulse-momentum relations define the jump discontinuities in the composite velocity vector as well as the generalized impulses of the reaction forces at different joints of the aircraft. The mathematical model presented in this study accounts for the large change in geometry of the suspension system and also for the nonlinear characteristics of the suspension and tire model. The mathematical model simulates the dynamic behavior of an aircraft during landing and ground operation. The validity of the mathematical model is demonstrated by solving a two-dimensional large-scale aircraft model. The induced aeroelastic forces acting on the wings were not considered in this analysis. Consequently, the method emphasizes the study of dynamic response of the airframe in terms of the mechanical vibrations without getting into the associated flutter phenomenon. This
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K. Changiri et al., Analysis of multi-body systems. Part II
places no limitations on the mathematical mulation for the aerodynamic forces.
model which can easily accommodate
any for-
References [l] DC-10 landing gear modified. Aviat. Week Space Technol. 98 (10) (1973) 181-186. [2] R. Popeleuski, Airbus test tempo quickening. Aviat. Week Space Technol. 98 (10) (1973) 32-35. [3] J. Morison and D. Kamopp, Comparisons of optimized active and passive vibration absorbers, Proceedings IEEE 41st Joint Automatic Control Conference, New York (1973) 932-938. [4] J. McGehee and R. Dreher, Experimental investigations of active loads control for aircraft landing gear, NASA TR No. 2042 (1982) l-67. [5] A. Shabana, Substructure synthesis method for multi-body systems, Comput. & Structures 20(4) (1985) 737-744. [6] Y. Khulief and A. Shabana, Dynamic analysis of constrained system of rigid and flexible bodies with intermittent motion, ASME J. Mechanisms, Transmissions and Automation in Design (to appear). [7] K. Changizi, Effect of impact on dynamic analysis of large scale aircraft, M.Sc. Thesis, University of Illinois at Chicago Circle, 1985. [8] A.A. Shabana, Transient analysis of flexible multi-body systems. Part I: dynamics of flexible bodies, Comput. Meths. Appl. Mech. Engrg. 54 (1986) 75-91 (this issue). [9] J.E. Shigley, Theory of Machines and Mechanisms (McGraw-Hill, New York, 1980) 209-218.