1
DETERMINATION OF REACTIONS IN YIELDING SUPPORTS
BY A NUMERICAL METHOD
Olga MARTIN
Department of Mathematics
University “Politehnica” of Bucharest
Splaiul Independentei 313,Bucharest 16
ROMANIA
Abstract: - The paper presents an iterative method for the computation of reactions, which correspond to a movable control surfaces of an airplane (elevator, rudder, aileron) attached at N-points to an elastic structure. A general formula was obtained for the displacements of those N - points, using the material strength computing techniques.
With the help of this algorithm we find the reactions considering the interaction between the rigidity of the supporting and the movable structures.
Key-Words: Yielding supports, trapezoidal loading, three unknowns method, slope deflection method, reaction.
1
1 Introduction
The movable control surface of an airplane, namely the elevator, the rudder and the aileron are attached to N-points to stabilizer, fin and wing, respectively. These supporting surfaces are usually cantilever structures with variable moment of inertia, which cause the supporting points for the movable control surfaces to suffer a displacement.
The present method allows the reactions in the attachment points of the movable surfaces to be determined as a function of their displacement, using the slope deflection method, [2]. For a member of the control surface bounded by two end joints, i i+1, the end moments can be expressed in the terms of the end rotations, which are treated as the unknowns.
Furthermore, for static equilibrium the sum of the end moments on the members meeting at a joint (support) must be equal to zero. These equations of static equilibrium provide the necessary conditions to handle the unknown joint rotations and, when these unknown joint rotations are found, the end moments can be computed and hence, the reactions.
This work contributes to the extension of the slope deflection method to a movable structure with N – supports. Also, for a correct structural response we determine by the step 2 the deflections of the supporting surfaces under the combined action of air loads, reaction forces in the attachment points obtained by the step 1 and settling of supports.
1
1
2 Modelling and formulation
1
Step 1
Fig. 1
1
The figure1 presents a cantilever structure with a trapezoidal loading. Using the three unknowns method 3 one can obtain the necessary relations for determining the displacements of the cantilever beam in the attachment points of the movable structures, namely:
1
(1)
1
where Mk is the bending moment of the cantilever structure cross-section, corresponding to the k point (k = 1, 2,..., N) and corresponding to the ends of the beam.
If the support structure was statically studied with the finite element program (ANSYS) before this calculus, we consider for the step 1 the displacement values of the nodes where the movable structure is attached.
The figure 2 is representative of a rudder beam attached to a deflecting fin structure at the N-1 points, where 2 support corresponds to the maximum displacement w. Hence, the vertical displacements of the supports are:
The problem will be to determine the bending moment at the supports under the combined action of transverse loading and of the support settling.
1
1
Fig. 2
1
The following values will be evaluated at the beginning (Figures 3 and 4):
1
a) the swing of the member i i +1:
1
Fig. 3
1
b) the moments due to any applied loads on the beam, when considered as fixed ends
1
Fig. 4
(3)
For sign convention:
- the rotation of a joint or a member is positive, if it turns in a clockwise direction ;
- the end moment is considered positive if it tends to rotate the end of the member clockwise
or the joint counterclockwise.
c) the moments due to distortion of the supports :
(4)
where ;
Total moments in the i – point are:
(5)
For static equilibrium of the joint i :
1
and using (5), the following system is obtained
(6)
where the stiffness matrix is
1
(7)
and
1
1
,
1
1
Solving the (6) for we can use then (5) to obtain the bending moments at the supports. For trapezoidal loading (fig.2) the corresponding support reactions will be:
1
(8)
Step 2
1
We consider now the deflecting supporting structure loaded with the air loads as well as with the reactions determined by the step1.
Using the superposition principle to sum up the individual effects of every load, which acts upon a structure, we can obtain the displacement functions for the movable part attachment points.
The air load displacement was determined by step 1 so we have to concentrate now on the displacement caused by the N–1 - reactions, Rk (fig.5). We note:
1
Fig.5
1
Using the material strength computing techniques, the following general formula was obtained for the displacements for the N - points:
1
1
1
(9)
1
where
1
The result of this step is the displacement of the k support point of movable structure:
(10)
where is the displacement of the point k determined by step j ( j = 1,2 ). Thus,
and on used (5) – (8) for obtain .
The iterations cease when
1
1
.(11)
1
1
3 Numerical results
1
1
Step 1
1
We consider an rudder attached to deflecting fin at three reaction points (N=3).
1
Supporting structure
Fig. 6
1
The moment of inertia of the fin,
I = 3325 cm4 and the modulus of elasticity of the material, E = 735000 daN/cm2. The air load on the fin is variable as indicated in the Fig.6
We obtain:cm; cm; cm.
1
Movable structure
1
y2 = 2 cm, y3 = 0.8 cm, y4 = 0.2 cm
1
1 2 3 4
Fig.7
1
1
(12)
The beam has a constant section, hence
1
1
Span 2 – 3
The settlement of support 2 with respect to support 3, 2 = 2 – 0.8 = 1.2 cm and
Since the joint 2 turns in counterclockwise with respect to 3 the sign of is negative.
Span 3 – 4
3 = 0.8 – 0.2 = 0.6 cm and
Due to the fact that 2 is a simple support with a cantilever overhang, the moment M2,1 is statically determinate and . Then for static equilibrium of joint 2,
. The substitution in slope deflection equations (5) is
and we obtain
or
(13)
For static equilibrium of joint 3, M3,2 + M3,4 = 0, where
(14)
(15)
Hence
(16)
For static equilibrium of joint 4, where
(17)
(18)
and obtain
(19)
1
The equation (6) has the form
(20)
1
and we get
2 = -0.016 rad; 3 = -0.0104 rad; 4 = -0.009 rad.
Substituting these values in equation (15) we calculate following moments
M2 = -98 daNcm; M3 = - 8585 daNcm;
M4 = - 747 daNcm and the support reactions are
For no yielding, = 0, and
1
.
M2 = -98 daNcm; M3 = - 4489 daNcm; M4 = - 747 daNcm and now we have
.
1
Step 2
1
Fig.8
1
The displacement of a support i due to the air load on the fin must be added now to the displacement due to the reactions R2, R3, R4 determined in step 1.
In order to obtain , i = 1,2,3 (Fig. 6) we calculate
M0 = -114820 daNcm; M1 = - 66900 daNcm; M2 = - 6237 daNcm
and (1) lead to the following values:
cm; cm; cm.
Using the superposition principle, the supporting points for the movable control surfaces move downward through the distances:
cm; cm; cm.
With these new, , we repeat the calculation of the movable structure and obtain
M2 = -98 daNcm ; M3 = - 8875 daNcm
M4 = - 747 daNcm
.
It follows from the above data that for two iterations:
hence = 8, under 1.1 % from R3.
The reaction R3 in this algorithm increases and R2, R4 decrease to the corresponding values obtained when the displacement of the supporting points is zero.
Conclusions
The proposed analysis allows us to find with a great accuracy the reactions in the attachment points of the movable surfaces to support the beam. Thus, theirs fittings will be calculated correctly. This algorithm is applicable for structures of variable bending rigidity and loaded by various types of external forces.
Using the general formula (9) and matrix calculus, the method is very efficient for analysis of a movable control surface attached by any N points to an elastic structure.
References:
[1] BERBENTE C., Numerical Methods, Editura
Tehnica, Bucharest,1997.
[2] BRUHN E. F. , Analysis and disign of flight
vehicle structures,Tri-State Offset Company,
Cincinnati, Ohio, 1965.
[3] BUZDUGAN G. , La résistance des matériaux,
Maison d’Edition de l’Academie Roumaine,
Bucharest, 1986.
[4] MARTIN O., Numerical analysis problems,
MatrixRom, Bucharest,1999.
[5] MARCIUK G.I., Numerical Methods,
Romanian Academia Publishing House,
Bucharest, 1983.
[6] PISSARENKO G., YAKOVLEV A.,
MATVEEV V., Aide memoire de résistance
des matériaux, Editions de Moscou, 1987.
[7] ROMBALDI J.E., Problèmes corrigés
d’analyse numérique, Masson, Paris, 1996.