The forces and moments acting upon a LTA vehicle are considered in order to develop parameters describing planar motion. Similar expressions for HTA vehicles will be given to emphasize the greater complexity of aerodynamic effects when buoyancy effects cannot be neglected. A brief summary is also given of the use of virtual mass coefficients to calculate loads on airships

Regan, F. J.

English

NASA Technical Reports Server

i i;
N6-1503!
THE PLANAR DYNAMICS OF AIRSHIPS
Frank J. Regan*
ABSTRACT: This paper will consider the forces and moments i
actlng upon a LTA vehicle in order to develop parameters _,_
describing planar motion. Similar expressions for HTA
vehicles will be given to emphasize the greater complexity
of aerodynamic effects when buoyancy effects cannot be '
neglected. A brief summary is also given of the use of
virtual mass coefficients to calculate loads on airships.
SYMBOLS
CD Drag coefficient
Cm Pitching moment coefficient, My/QS_
Cmq _ Cm/_ (qi/2V)
C1._ _ Cm/_ (_12/2V2)
Normal force coefficient, Fz/QS
ct_ _ cz/_ (_Ll/lv l)
Cza t Cz/_Q
Cz_ _ Cz/_ (_i/2v)
D Drag force
g Gravitational acceleration
, i
* Supervisory Aerospace Engineer, U. S. Naval Ordnance Laboratory
Silver Spring, Maryland
1976007927-185
https://ntrs.nasa.gov/search.jsp?R=19760007943 2020-03-20T01:38:57+00:00Z
% I
_I Iy Transverse moment of inertia
i Ky Transverse radius of gyratlon,_Iy/m_ 2
_, \ L Body length
Reference length, body length
:' My Pitching moment
_. / m Body mass
% _ ! O Dynamic pressure, I/2pVo 2
V2/3S Reference area,
s Airship density to medium density, Ob/o "
Vo Airship speed
!% ,
V Airship volume
/f"_" Xe,Ye,Ze Inertial axes
X,Y,Z :_dy axes
, Z Normal force
a Angle of attack
e Angle of pitch
_ Velocity potential
_' p Density
ZN ODUCTZO.
In studies of the dynamics of Heavier Than Air (HTA) vehicles, effects
due to buoyancy are almost invariably neglected. Sustaining force is
_'_ the result of relative motion existing between the HTA vehicle (or at
least some portion of the vehicle) and _he surrounding air mass. In
short, the lift or sustaining force associated with HTA craft is
entirely dynamic.
i
A somewhat reverse situation exists in the case of Lighter Than Air
(LTA) craft. The principal sustaining force comes from buoyancy, with
perhaps a small additional force (about I0 percent) available under
_ome conditions from dynamic llft. To put the comparison between LTA
and HTA craft on at least a semlquantltatlve basis, it is convenient
to define a relative density parameter, s, as
(11
It may be seen that s is o_ 0 (I) for a LTA vehlcle, w_ils for a HTA
s is no less than 0 (I0_2) and for most cases 0 (10t4).
In addition to buoyancy playing an essential role in LTA dynamics,
there are in addition dynamic effects which for convenience might be
lumped An the terms virtual mass. Such dynamic effects are ttken to
mean forces and moments arising from (and hopefully linear with)
angular rate or linear acceleration. These virtual mass effects are
essentially reactive forces and moments caused by _apartlng an angular
velocity and a linear and angular acceleration to the surrounding air.
Like buoyancy these virtual effects are usually neglected fo: HTA
178
1976007927-186
i
m jl
craft; for LTA vehicles, however, such effects form an essential part
of the loads acting on the craft. Thus such effects enter prominently
into any considerations of stability.
No originality is claimed in the following development of either the
• mathematical model of planar dynamics or the subsequent load calcula- i
tion methods. The equations of planar motion originated with ballisticians
such as Murphy (I). llowever, because of the negligible effect of
• buoyancy, great simplifications are possible in the aeroballlstic
formulation. As will be shown, the airship equations are far more
complex. The load calculation technique ollow from Bryson[2)origi -
nally and have been presented by Nielsen _3_. Again these methods are
applied to LTA vehicles rather than the HTA missiles which were the v
original motivation for Bryson's work. ' _
_ DYNAMICS OF PLANAR MOTION I 'Consider an airship undergoing planar motion as illustrated in F_-le ;
_i_: 111 below I
k
"
t _
Z
ze
J.
FIG. 1 FORCES AND MOMENTS ACTING ON AIRSHIP
The axes Xe,Yo,Ze are the inertial axes, while X,Y,Z are bo_y-fi):ed
axes. The equations o! planar motion are the forces along axes Xe,Ze
ana the moment about axis Ye. Note that because of the definition o_
planarlty axis Ye is identical to axis Y.
o
The moment and two force equations may be written as
m'V'CosCe-ac)-Coe (2a) :
1976007927-187
" 1
e
v
< I _ l _7 (2C)C2.,
where s is the relative density parameter of equation (I). In addition
':- to the three load equations above, figure (i) also provides the follow-
ing kinematic relationship:
._ t _c"-_ S,.(6-_) ,3s)
t
_ which gives upon differentiation
%
_. Under the assumptlo.; that the X axis does not greatly vaTy from the
horizontal, Xe, it is possible to restrlc_ 8 and a to small angles.
.--'-- Subject to such small angle restrictions equations (2) and (3) become: '
A first step might be the substitution of equation (4e) into equation
_i ' (4b) to give:
.: Equation (4a) may now be used to eliminate 0 in the above expression
.... resulting in:
: The above expression may be altered by introducing the following non-
dimensional force coefficients
The coefficient Cz may be expanded in a Taylor series as
%. c..,, (g) ,,,
Equation (6) may now be written in te_s of Cz and CD as
) _=,v./,l_,wt(_)
, +qf .co.]
; It is now possible-to simplify equation (S) somewhat by the following
" redefinitions
_80
1976007927-188
J¢ • I
Equation (i0) now allows equation (9) to be rewritten and then rearlanged
t a8 •
Equation (4c), the moment equation, may be written as" ii
(_ ) nl k,t / _r_erl )%iy lum (12) i
where M_ has been replaced by C. (ps£V2/2). Again replacing C_ by a
Taylor _ series in a, &, q, and" _ and using the starred quanta*Lies
gives for equation (12) : j,
Equations (11) and (13) are now the basic equations of planar motion.
• The final goal remains to eliminate one of the variables between the_e
simultaneous equations. For the present purposes the varla.,le q will _.
be eliminated and a single differential equation of motion in _ will
be written. As might be expected, thJs single equation is quite com-
plicated. Refore presenting thlr dynamic equation in a tractable form,
an outline of the procedure will be given. A fairly straightfor_,ard
approach is to eliminate _ between equations (ii) ind (13). The
resulting equation containing q, a, and a is then differentiated to
give an expression in _, _, _, and _. Retu.-ning to equation_ '11) and
(13), eliminating this time q between them now provides a second
expression _, a, _, and _. Elimination of _ between these equations
gives the single dynamic equation in a, _, and a. In carrying out
the above manipulation it is necessary to perform the differentiation
of _/V). This operation may be written as,
V% V'd ;
The single equation in _ that will represent dynamic planar motion
will be written as;
where
e
L
s8_ OI_IG_ALpAGE 18 POOP.
l
2_
1976007927-189
Before attempting to simplify equations (16) it is necessary to reduce
_.- equation (15) to an equation with constant coefficients as the presence
--" of (V/t) introduces time into the coefficients. This may be accomplished
by writing
> I
(17a) :
'.- where differentiation has been changed from time, t, to a non-dimen- _
:_ siona _ arc length, (X/L). In a similar fashion _ may be written in ......
_ "i terms of (X/,) as (})b:_2-. - G)""-- ,,7,, ,
--'I By replacing time derivatives by arc-length derivatives, equation (15) '<;
now becomes a second order constant co_.fficient equation:
.,,. j Admit_dly, equations (16) are quite complex; for certain applications
such as aeroballistics, great simplifications may be made.
:_.' However, before considering this aspect of the problem, the conditions;
for stability of motion will be examined.
_'_' STABILITY CONSIDERATIONS
,,' Equation (18) may be rewritten as
:' where
: -- 120a)
G;_ = - MI (20b)
O_& A, 120c)
,a=-- .m,___ L
The term, _, is the factor of the airship,the term, _n, is the i'
undamped natural frequency. Only for small values of A does the
body oscillate at this frequency; in the presence of a significant
amount of damping the planar oscillatory frequency, _d, is less than
the undamped frequency, _n" The damped planar frequency, _d' can be
expressed in terms of A and _n as
The term A in equation (20c_ is the trim angle of attack due to aero-
dynamic asymmetries, while s_ in equation (20d) is the trim angle of
i a_tack due to gravitational path curvature. With regard to the term
_ _, it mlght be of interest to note that if the airship is neutrally
._jLI buoyant, i.e. s - i, then from equation (I0) _ = o and hence from
18_
v
1976007927-190
_ . equation (20d), ao g must be zero. ,
_B There ar_ two conditions for oscillatory motion. These are"
/ _ <" ('/'/Y/ (22a) _.
_ , and
i _ The former condition allows us to write for the equality A = Ac = _n'
./n
._ % = _)", t'.-- (23a)
q--
_ where (_/_c) is often called the damping ratio.
_- The condition for oscillatory motion given in equation (22b) is that
_<' _n is real which, in turn, from equation (20b) requires that
'_"'" _t < 0 (23b)
'_. Under the condition where equations (22b) and (23b) are satisfied,
_. stability (subsident motion) requires that
, ZX--- o (24a)
z H
i: _h " " ! ) O (24b)
_ Thus to assess dynamic and static stability (equations (24a,b) respec-
,;. tively) it is necessary to assign numerical values to the derivatives
_,!, contained in equations (16a) and (16b). Numsrical values for these
:,, terms are contained in Table I. While these values may vary with
_' airship dimensions, they have been computed for the airship shown in
Figure (2) below. An outline of the computational technique is given
_ subsequently.
Inserting values from Table I gives,
183
1976007927-191
_,"
I--
...... -- | JL_,__.
•" [.,+,_.qs][.o4..04q]* (I+o)(-z,,,)+0+_._o)r.z.,,1+('_z.3_.X:-.,'0
\-,
e
_" ' -.0437
-.Z34 -. o4s'7- .z78
t
t
' t'_. , Quite obviously the inequality of equation (24a1 is not met so the air-...A,_ ship is not d_,%amically stable.
In considering the oscillatory frequency relationship for static
:_ stability (equation (24b)) we may write:
, M,-Io-_:'cl.)cc_,_,'_) _c__.z__c:+cq+c_)_ -o+c_s_%_K_
can since braces is multipliedIt readily be shown that the term in the
by C* it is rather small. This allows the above expression to be numer- I
ical_y evaluated as
"' M = 1.78
l
_.: Obviously the second condition of equation (24b) is not met.
It might be expected that numerical values of the st&bility derivatives
would vary from airship to airship. However, it would appear that no
general simplifications may be made in the H1 and M 1 coefficients
except to omit lerms multiplied by C_. The results seem to indicate
that for a satisfactory description _f planar dynamics it is necessary
to calculate the eight stability derivatives of Table I. Drag, as we
have noted, is relatively unimportant for estimating the planar dynamics.
In passing it might be of interest to examine the equivalent expres-
sions for HI, MI, Al, and GI, which are satisfactory for an HTA vehicle.
If quantities such as C_, -C_, C*a, and C*& are ignored along with
the product of starred qdantitfes oSe, has
4+ .._,.
C,.I (25_)
Quite clearly the crf' e ia of equations (24) are met when C* + C*"
and C_s are negative for the HTA vehicle. An examination o_equa_ons
(16a) and (16b) quickly show that dynamic stability cannot depend upon
such simple criteria in the case of an LTA vehicle: stability censid-
erations are far more complex foz the LTA vehicle.
184
1976007927-192
, .... _ _i,ii _ iitl Itn,lti -- II JllIII IT n ii fillii -_Im _ _
i_. For a typical airship we have seen that the motion consists of one
exponentially undamped mode and one exponentially damped mode since
_ from equation (19)
Y
_-- and using I = -.139 and M 1 = 1.73 we obtain
_ l I -- .479 _2 = - 1.18%,
As is well known, the fixed-wing HTA vehicle usually evidences two
/ damped oscillatory modes.
k
_i CALCULATION OF AERODYNAMIC LOADS
_ A fairly straightforward method of calculating static and dynamic loads
on an airship is the method of virtual mass. While this technique has
"_ its origin in the work of nineteenth-centu_,y hydrodynamicists, it has '
_ been applied w_th some success by Bryson (_; to HTA vehicles. Since
_* space limitations do not permit even an outline of the derivation,
reference should be ma_e to either Bryson's work(2)or the more readable |
_"'_ treatment of Nielson (3;.
_' Through the use of this virtual mass technique it may be shown that the
derivatives used in the previous expressions for H1 and M 1 are given as,
L
i_ C__-4B,, ' (27b)
_°
_, q,,,_- -z c,,C_ ,-z ,,g, +z_,, (27e,
' C,,i =-_D,, (27h)
U
where
for body-alone and that
:, R,,= I sTY)= - (_b)
, for the body in the presence of fins. a(x) is the body radius as a
function of body station and s (x) is fin span (center-line to tip) as
a function of body station. In addition BII, Cll and DII are defined
! _)b 129a)
:, 185
1976007927-193
$,
i.
2-
&
" n II (29b)
'
z_ ....
_: ' (29c)
,, 'D, I = (:X/.t)b .
when "n" and "b" refer to nose and base respectively.
._.
) The above integrals have been evaluated numerically for the airship
-.._: shown in Figure (2)from tabular values of a(x) and s(x). The calcula-
tions of equations (27) were carried out to give the results shown in
:' Table I.
L
"3 i
': LENGTHIFT)OIAMETERIFT)VOLUMEIFTa)MOM.OFINERTIAlSLUG-FT2)
517 120 4.4X10 hX10+e
_' FIG. 2 REPRESENTATIVE AIRSHIP
'_ TABLE I
1
:_ .oBi_s'L-l.4g8 Io"q -.o7s6.7_ =o7_6 +-.o7'6-6--:oq?,,1,s"
10+6
_: Also calculated in the program is the airship volume V = 4.4158 x .
Assuming neutral buoyancy it can be shown that (_SL/2m), equals 1.57,
which together with values in Table I allows the starred derivatives
: (equation (i0)) to be calculated.
CONCLUS ION
This pa;_er has taken a brief look at the hydrodynamic complexities of
planar d;,namics of airships. It h_s been shown that the equations of
motion for a LTA vehicle are far more complex than the corresponding
,equations of a HTA vehicle. A method has been presented for calcula-
ting all loads (except drag) acting on a moving airship.
* REFERENCES :
i. Murphy, C. H., "Free Flight Motion of Symmetric Missiles," BRL
Report 1216, Ballistic Research Laboratories (July 1963).
_ 2. Bryson, A. E., "Stability Derivatives for a Slender Missile with
Application to Wing-Body-Vertical-Tail Configurations," Journal of
the Aeronautical Sciences, Vol. 20, NO. 5, pp. 297-308 (1953).
3. Nielsen, J. N., Missile Aerodynamics, McGraw-Hill Publishing Co.
(1960).
_86
1976007927-194


The planar dynamics of airships

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server