Document
NASA T N D-14: F
TECHNICAL NOTE
D - 1492
AN ANALYTICAL STUDY OF EFFECTS OF 90ME AIRPLANE ANI) LANDING-GEAR FACTORS ON THE RESPONSE TO RUNWAY FUIUGHNESS WITH APPLICATION TO SUPERSONIC TRANSPORTS By Norman S. Silsby Langley Research Center Langley Station, Hampton, Va.
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON December 1962 F NATIONAL AERONAUTICS AND SPACE ADMINISTRATION TECHNICAL NOTE D-1492 AN ANALYTICAL STUDY OF EFFECTS OF SOME AIRgLANE AND LANDING-GEAR FACTORS OF TEE RESPONSE TO RUNWAY ROUGHNESS WITH A P P L I C A T I O N TO SUPERSONIC TRANSPORTS By Norman S. Silsby SUMMARY A n analytical study has been made of the e f f e c t s of several airplane and landing-gear design variables on the airplane response t o runway roughness ( f o r a r e l a t i v e l y smooth runway) with application t o supersonic transport configurations .
Assumptions made f o r the analysis were (1) n a t u r a l frequency of a l l t i r e s were the same, (2) t i r e s had no damping and possessed a l i n e a r tire-deflection load curve, ( 3 ) a l l struts had l i n e a r damping and spring characteristics, and the natural frequency and damping of a l l s t r u t s were the same, (4) r i g i d airplane structure, and ( 5 ) no aerodynamic forces.
"he r e s u l t s of the analytical study indicated that, within the limitations of the assumptions made, the parameter variations considered had l i t t l e e f f e c t on motions of t h e center of gravity of the airplane except f o r longitudinal position I of t h e landing gear with respect t o the center of gravity; i n the l a t t e r case, moving the landing gear aft with respect t o the center of gravity caused a reduc- t i o n i n center-of-gravity displacement and acceleration at all speeds. The e f f e c t s of parameter changes on motions of the p i l o t ' s compartment were strongly I influenced by speed; t h a t is, changes which increased the response at one speed I could cause decreased response a t another speed. From the standpoint of root- mean-square accelerations a t the p i l o t ' s cmpartment, placing the main gear near the center of gravity appeared t o be advantageous; however, it resulted i n 'some- what higher displacements than with the main gear placed further rearward from I the center of gravity. Generally, the supersonic transport configurations gave higher root-mean-square accelerations a t the p i l o t ' s compartment than did a sub- j sonic j e t transport configuration.
INTRODUCTION As presently envisaged, supersonic transports are expected t o have an unusu- of a l l y long fuselage, a major portion of it extending forward of the center gravity. This design will allow the possibility of landing-gear arrangements and locations r e l a t i v e t o the p i l o t and passenger compartments substantially During take-offs and landings, these different from those of current a i r c r a f t .
configurations may cause motions or accelerations undesirable f o r cockpit instru- ments, pilots, and/or passengers.
The present study was undertaken t o examine the e f f e c t s of such differences on the responses of the p i l o t ' s compartment and the airplane center of gravity due t o runway roughness of a r e l a t i v e l y smooth runway. The r e s u l t s are also com- pared with estimates f o r a current subsonic j e t transport airplane.
SYMBOLS constants ",b, c constant C s t r u t damping coefficient C S D d i f f e r e n t i a l operat or, d/dt D" = D / a F input force from runway acceleration due t o gravity, 32.2 ft/sec2 g
i = ,jZ
strut spring constant kS t i r e spring constant k t pitch radius of gyration, ft kY 2 wheel base (nose gear t o main gear), f t distance from center of gravity t o main gear, ft 2m distance from center of gravity t o nose gear, f t 2n L wavelength, f t m s t a t i c mass on one gear M s t a t i c mass on a l l wheels S distance along runway, f t
v airplane speed along runway, f t / s e c
X longitudinal distance from center of gravity, positive forward, f t XO generalized output term Xi generalized input term Z vertical displacement, ft
1 + (2)
acg =
2hs -
%
Be =
( 2 , L
2hs -
J k-Y2
8 pitch angle, radians damping ratio of strut with static load mass AS root-mean-square displacement, ft root-mean-square acceleration, g root-mean-square pitch angle, deg phase angle, radians power-spectral-density function, sq ft/radian/ft
power - spectral-density function, gz/radian/ft
frequency, radians / se c
natural frequency of s t r u t with s t a t i c load mass, x> radians/sec U S natural frequency of t i r e with s t a t i c load mass, 10 radians/sec wt n a t u r a l frequency of o s c i l l a t i o n i n p i t c h W e frequency r a t i o , U/W U* s p a t i a l frequency, 231/L, radians/ft n Subscripts : center of gravity cg i input refers t o point between t i r e and strut main gear m n nose gear associated with o s c i l l a t i o n i n p i t c h
e
0 output p i l o t compartment P r runway S strut t t i r e X a t x-pos i t ion Dots over symbols represent differentiation with respect t o time. Bars ove: symbols denote vector quantities.
ASSUMPTIONS, ANALYSIS, AND METHOD OF EVALUATION The following assmptions were made: (1) the natural frequency of t h e t i r e with the mass equivalent t o t h e s t a t i c load, was t h e same for t h e main and nose wheels, (2) t h e t i r e s had no damping and possessed a l i n e a r tire-deflection-load curve, ( 3 ) all s t r u t s had l i n e a r damping and spring characteristics and t h e natu ral frequency and t h e damping of a l l t h e struts were t h e same, ( 4 ) t h e airplane fuselage was a r i g i d structure, and ( 5 ) there were no aerodynamic forces.
The equations of motion of t h e airplane resulting from forces applied t o t h tires as the airplane moved along the runway were derived under the assumptions stated. The resulting frequency response was combined with the power spectrum of runway roughness f o r a good runway @(Q), = 6.7 (see r e f . 1) t o give the Q2 power spectrum of acceleration a t various points along t h e airplane longitudinal axis resulting from center-of-gravity translations and pitching responses. The root-mean-square accelerations, displacements, and pitch angles were determined by integrating these spectra over the range of wavelengths from 4 f e e t t o 570 feet. The computations were carried out by using a d i g i t a l computer f o r 50 points i n t h i s range f o r each of the 7 discrete speeds of 4 0 , 80, 120, 160, 200, 240, and 280 f e e t per second.
A complete development of the equations used i n the analysis Figure 1 defines t h e dimensional i s presented i n the appendix.
symbols representing the characteristics of the airplane.
The parameters which were varied were landing-gear wheel base 2 , longitudi- n a l location of the landing gear w i t h respect t o the center of gravity, and radius of gyration ky. Responses were calculated f o r various points along the longi- tudinal axis.
The long slender fuselage envisaged f o r the supersonic transport may be more flexible than those of current j e t transports. The acceleration responses f o r such a fuselage may be somewhat different from those presented herein f o r an assumed r i g i d fuselage, depending on such factors as the modes excited, t h e i r natural frequencies, arrangement of the landing gear, airplane speed, and so forth.
Present at ion of R e s u l t s The results of the analytical study are presented i n figures 2 t o 4. Table I shows the airplane configurations, values of the parameters used, and the figure numbers i n which these various configurations appear. Figure 2 shows the varia- t i o n with velocity of the root-mean-square values of normal accelerations, the root-mean-square normal displacements, and the root-mean-square pitch angles due t o the variation of the landing-gear location with respect t o the center of gravity ( f i g . 2(a)), the variation of the pitching radius of gyration ( f i g . 2(b)), and the variation of the length of the landing-gear wheel base ( f i g . 2(c)). Figure 3 shows the variations of root-mean-square accelerations and displacements with location along t h e longitudinal axis f o r various speeds f o r a possible supersonic transport configuration having a short wheel base (45 f t ) r e l a t i v e t o overall length
a 90-foot wheel base (fig. 3 ( b ) ) . Figure 4 compares t h e vari-
( f i g . 3 ( a ) ) and f o r ation with velocity of t h e same quantities shown i n figure 2 f o r the supersonic transport and a current subsonic turbojet transport calculated by t h e same method.
DISCUSSION Effect of Landing-Gear Location The e f f e c t of varying the longitudinal location of the landing gear with respect t o the center of gravity so t h a t the main gear varied from 0 . 1 2 t o 0.32 aft of t h e center of gravity ( f i g . 2(a)) is t o reduce the root-mean-square dis- placements and accelerations of the center of gravity at a l l speeds. The root-mean-square acceleration at the pilot's compartment nearly doubled at the higher speeds as the landing gear was moved aft; however, the maximum accelera- tion was only about O.3g and this was for a relatively smooth runway. From the standpoint of root-mean-square accelerations at the pilot's compartment, placing the main gear near the center of gravity appeared to be advantageous; however, it resulted in samewhat higher displacements (O.3-foot maximum) than with the main gear placed further rearward from the center of gravity (about 0.2-foot maximum displacement). The root-mean-square pitch angles were less than 0.2' for all these gear locations and showed no important variations with speed.
Effect of Pitching Radius of Gyration As would be expected, changing the radius of gyration in pitch had no effect on the values of root-mean-square accelerations and displacements at the center of gravity. Halving the pitching radius of gyration from 28.3 feet (ky/2 = 0.63) to 1 4 . 1 5 feet (ky/2. = 0.315) resulted in a threefold increase in root-mean-square accelerations at the pilot's compartment at the hi&er speeds, with the maximum (See fig. 2 ( b ) . ) value about 0.45g.
At a speed of 2 4 0 feet per second it may be noted that, for the pilot's com- partment, although the values of the displacements and the pitch angles are about equal for the various configurations, there is a substantial variation in the values of root-mean-square accelerations. The reason, at least in part, is prob- ably the difference in the natural f'requency in pitch of the configurations. The values of root-mean-square accelerations should be proportional to the square of the natural frequencies since the displacements are essentially the same.
Effect of Length of Landing-Gear Wheel Base Increasing the length of the landing-gear wheel base (by moving the nose wheel forward) while also maintaining a constant pitch radius of gyration showed that substantial reductions could be realized in both root-mean-square pitching angles and root-mean-square displacements for the pilot's compartment over almost the entire range of velocities. (See fig. 2 ( c ) .) For exanlple, doubling the
wheel base from 4 5 feet to 90 feet results in a reduction of root-mean-square dis-
placements by a factor of as much as 6 at a velocity of about 120 feet per second
and a reduction of root-mean-square pitch angle by a factor of about 5 for the
same speed. Increasing the landing-gear wheelbase had mixed effects on accelera- tions, depending on speed. Although the intermediate wheel-base length (67.5 feet) yielded values of root-mean-square pitch angles and displacements which fell gen- erally between those f o r the 45- and 90-foot wheelbases, this intermediate wheel- base length produced the greatest root-mean-square accelerations at speeds above 120 feet per second.
Variation With Distance Along the Fuselage For the configuration with a wheel base of 45 feet (fig. 3 ( a ) ) , the values
of displacement and acceleration response at speeds of 40, 80, 120, and 160 feet
per second appear to be a minimum at or near the center of gravity and exhibit a b f a i r l y smooth and uniform increase i n the values with increasing distance e i t h e r toward t h e nose o r t a i l a t all speeds. For the configuration with a wheelbase of 90 f e e t , however ( f i g . 3(b)), the minimum values of displacements and acceler- ations occur at points somewhat displaced from the center of gravity (x/2 = 0) f o r a l l speeds ( i n t h i s figure, 40, 80, 160, and 240 f t / s e c ) . For a speed of 40 feet per second the minimum acceleration was n e a r t h e center of gravity; at 80 feet per second the minimum response was toward the t a i l ( x / 2 = -0.4); and f o r speeds of 160 and 240 f e e t per second the minimum response locations were toward the nose (x/2 = 0.5).
Comparison of Hypothetical Supersonic Transport Configuration With Current Subsonic Jet Transport The curves of figure 4 indicate t h a t , f o r t h e p i l o t ' s compartment f o r the 45-foot wheel base, the root-mean-square displacements of t h e supersonic transport were greater than those f o r the subsonic transport by f a c t o r s from 1- t o 2$ over the speed range. However, f o r the 90-foot wheel-base supersonic transport, the displacements were about the same or somewhat lower than those f o r the subsonic transport. The root-mean-square acceleration and p i t c h angles f o r the supersonic transport ()+?-foot wheel base) were up t o 50 percent larger than those f o r t h e subsonic transport. For t h e 90-foot wheel-base configuration, the pitch angles w e r e lower t h a n those f o r the subsonic transport over t h e e n t i r e speed range; however, the root-mean-square accelerations were higher than those f o r the sub- sonic transport a t the lowest and highest speeds and about the same i n t h e mid- speed range (110 t o 180 f t / s e c ) .
CONCLUDING FEMARKS A simplified analysis has been made t o examine c e r t a i n design variables of landing-gear location and a i r c r a f t p i t c h radius of gyration i n r e l a t i o n t o pos- s i b l e e f f e c t s on t h e response of supersonic transport configurations t o runway roughness. The results indicate that the parameter variations considered had l i t t l e e f f e c t on motions of the center of gravity of t h e airplane except f o r longitudinal position of the landing gear with respect t o the center of gravity; moving t h e landing gear aft resulted i n a reduction i n center-of-gravity dis- placement and acceleration a t a l l speeds. The e f f e c t s of the parameter changes on the motions of the p i l o t ' s compartment were strongly influenced by speed; that changes which increased the response a t one speed could cause decreased is, From the standpoint of root-mean-square accelerations response at another speed.
a t the p i l o t ' s compartment, placing the main gear near the center of gravity appeared t o be advantageous; however, it resulted i n somewhat higher displace- ments t h a n with t h e main gear placed further rearward from the center of gravity.
Increasing t h e pitching radius of gyration tended t o reduce the root-mean- square accelerations and increase displacements at t h e p i l o t ' s Compartment.
Increasing the landing-gear wheel base tended t o decrease p i l o t ' s compartment displacement response but had mixed effects on accelerations, depending on speed.
me highest accelerations were obtained with an intermediate wheelbase of
67.5 feet i n a range from 45 feet t o 90 feet at speeds above 120 feet per second.
Generally, the supersonic transport configurations gave higher root-mean- square accelerations a t the p i l o t ' s compartment than d i d a subsonic j e t transport configuration.
Langley Research Center, National Aeronautics and Space Administration, Langley Station, H q t o n , V a . , August 2, 1962.
APPENDIX TREomTIcAL ANALYSIS Determination of Acceleration Response of t h e Airplane Due t o Runway Roughness For each wheel, The following assumptions are considered i n t h i s analysis.
and for this condition the t h e static load i s considered t o be rated s t a t i c load, natural frequency on all tires i s assumed t o be t h e same. I n addition, t h e strut response i s assumed t o be l i n e a r , and all struts have t h e same frequency responses and damping r a t i o s (with s t a t i c load on given wheel as a separate mass and neglecting mass of wheels and s t r u t s ) .
Determination of t h e Transfer Function f o r t h e Center-of-Gravity Translation and Pitching of t h e Airplane With reference t o figure 1 the following equations may be written: The equations of motion i n v e r t i c a l translation are: L Since % = h f and % = a, 2 2
If ktn = “t, = y2, then
mn % then mn % % % I
.. Z
= (Us 2 - ( z l 2, - Zo), + 2AdUS h(il - io), 2 2
J
2 Xn
+ 2 h & U s h(il - io)m
+ % t ( Z 1 - za)m
The following r e l a t i o n a l s o may be written: = 2,z 2n Z c g i , l , o 1 ni,1,0 + T h i , l , o Substituting equation (8) i n t o equations (4) and (7) yields ..
( 9 )
CQo - z s c g
and ..
= us2(z1 - zo)cg + 2h&(il - io),, Z cg0 The equation of motion i n pitch i s
= XnFn -
Substituting the expressions f o r Fn and F , from equa-ion (2) i n t o equa- t i o n (ll), together with the equations f o r R, % , and %2, yields Also myzio = +s2(z1 - ZJn - (z1 - zo), The solution of equations (g), (lo), (l3), and (15) yields the transfer m c t i o n s f o r the center-of-gravity translation and pitching responses of the airplane t o runway inputs. Equations ( 9 ) and (13) are of the form ..
X , = a2(Xi - x , )
(16) and equations (10) and (15) are of the form
xo = b2(X1 - Xo) + c(lil - io)
where a, b, and c are constants, and Xi is the displacement of the runway undulations or the input, and Xo is the resulting displacement of the airplane is the displacement relating to motions of the wheel axles or the output.
X1 and lower parts of the shock struts. (16) and (17) in operator Writing equations form and rearranging results in the following:
(D2 + cD + b2)Xo - (cD + b2)X1 = 0
(19) Equations (18) and (19) can be evaluated simultaneously by eliminating to X 1 obtain C Taking D = aD* and rewriting equation ( 2 0 ) yields the nondimensional form: L or
D* + P)Xo = ( I ) * . + p ) X i
+
(.x3 +
where
1 + (b/a)2
( 2 3 )
a =.c/a
and From equation ( 2 2 ) the response ratio is obtained:
-
- -
x z 2 = - - cgO or , - and the phase angle between output and input: where T i zcgi ‘ i where and The next step is to determine the input information
and ei in terms of z
Cgi the runway characteristics.
If it is assumed that the runway roughness spectrum is made up of sinusoidal waves of spatial frequencies R and the runway dis- placement at the nose wheel is taken as reference, the input motion at the nose wheel for unit amplitude is where s is distance along the runway and the input at the main wheels is J . I The displacement at the center of gravity resulting from displacements at the nose and main wheels is or from which and the phase angle i s
-
s i n -1 2 = s i n
'cgi /W'
The pitch-angle displacement input i s
= zni - %
so t h a t , from equations (27) and ( 2 8 ) , - = ( . - e e i 2 -in2 .ins Zni from which and t h e phase angle i s s i n sZ2
gei = sin-1
J2?1-cosRz)
or Equations (27) to (35) can be transformed to the time domain by the relation n =U/v.
The phase angles of the aircraft motion relative to the runway displacement can then be determined from equations (26), (3l), and (35) as The motion of the airplane at any distance from the center of gravity x along the x-axis where x is positive forward is then given by or The response ratio at x is then given by The acceleration response at any point along the x-axis is then The power spectral density of airplane vertical acceleration in response to runway roughness is - 2 @(Q), =(%) @(a), where the runway roughness spectrum w a s taken according t o reference 1 t o be with the value of C = 6.7 x representing a good runway, used f o r the computations.
The root-mean-square acceleration a t various points along the x-axis of the airplane were obtained from the r e l a t i o n D.2 = Z speeds considered i n the analysis and considering the frequency- For the response characteristics of the airplane there appeared t o be l i t t l e power i n the airplane-acceleration response spectrum f o r runway wavelengths greater than 570 feet; Lo For the upper l i m i t of the inte- w a s therefore given t h i s value.
gration, the wavelength r, w a s taken as 4 f e e t because runway roughness
measurements do not ordinarily go below t h i s value.
1. Haubolt, John C . : Runny Roughness Studies in the Aeronautical Field. Jour.
Air Transport Div., Proc. American SOC. Civil Eng., vol. 87, no. AT 1 , Mar. 1961, pp. 11-31.
! r A B m I . - VALUES OF PARAMETERS USED IN ANALYSIS
Supersonic configuration 0 -1 28.3 4.28 .2 28.3 5 972 6.62 28.3 45 -3 . 1 22.5 5.40 . 1 14.15 8.76 22.5 . 1 5 -40 . 1 22.5 5.40 . 1 5.40 45 22 -5 . 1 22.5 5.40 -1 28.3 6.50 67 9 5 . 1 28.3 8.76 . 1 28.3 8.76 . 1 28.3 8.76 . 1 28.3 8.76 . 1 28.3 8.76 Subsonic jet transport . 1 *9 50 & .16 5 4 52 9 33 x + I ‘ I I I I I I : I I 1 I I I I I : I I ‘ I aD c\J x c\! 0 c\! 0 . .
N t J co 0 a ?
$ E; I -3 Ln a, I I Ln I m 0 rl l n m
I I I
0 0 I I I I I I I II I I I \ I
! I \
1 I
\ ! I
7 \ 4 1
co ?
crl II \- b, I I X bn
c
.
.
N 6" t l r-r
I
I I
d- c\J
i
I \
I
I I
n I1 V II v
x 7
I I
0 * 0 0
. .
N t, b Q) k (u m
0 9
z cu cu
.-
II 4J Q) Q) k I l l I I I I I I I I I I t I I I t I f I I I I I I I I I I I I I t I I \ \ \ \ \ \ \ \ \ \ \
\ \ \Y \
I
cu
0 * c \ ! 0 CJ 0
M) M 4J Q) h a ..
..
LN N b NASA-Langley, 1962 L-3041