Hydrodynamic impact of a system with a single elastic mode I : theory and generalized solution with an application to an elastic airframe Page: 2 of 17
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REPORT 1074-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
lo resultant velocity at instant of contact with water
p mass density of fluid
7 angle of trim; angle of hull keel with respect to plane
of water surface
'ro flight-path angle at contact; angle between flight
path and plane of water surface
p angle of dead rise
C, nondimensional time coefficient (1Vo 1 s
C, nondimensional load-factor coefficient "J )P
Cd npndimensional draft coefficient ya (\ ) )
ya draft at instant of maximum acceleratidan
f. natural bending frequency
Where units are not given, any consistent system of units
may be used.
The hydrodynamic theory used in the present report is the
same as that developed in references 1 and 2._ A basic
differential equation which gives the instantaneous force in
terms of the instantaneous position and motion of the float
is given in reference 2. This equation is used herein to
determine the effect of airframe elasticity in altering the
motion and force time history (appendix A). The solution
is based on the assumption that the float does not change
trim during impact. In this connectionthe pitching moment
may be large, but the time of the impact is short enough to
warrant -(at the present stage) neglect of the resulting angular
velocities and displacements.
The solution presented herein is for a prismatic float
with such beam loading that the chines do not immerse,
during impact. For waves that give the severe design
condition of full-length impact, conventional beam loadings
are small enough to cause the maximum force to occur at
drafts sufficiently small to make the effects of finite width
and chine flare secondary. Reference 1 indicates that for a
conventional float neglect of the pulled-up bow is justified
when the trim is 30 or_greater. Although for high-trim
landings initial contact by the afterbody may substantially
change the trim before the .main forebody impact, the
neglect of afterbody loads is justified because, during the
main impact, the shielding of the afterbody by the forebody
due to depth of the step and to keel angle is such as to mini-
mize the importance of afterbody loads.
A simplified representation of primary elasticity of an
airframe is shown in figure 1. A rigid lower mass mL is
considered to be connected by a massless spring to a rigid
upper mass mn. In determining the fundamental bending of
airplane wings part of the wing mass must be included in
mL and part of the wing lift should be applied to mL. In
the present report the gravity force on each mass is assumed
to be balanced by wing lift.
F E 1.--Simplified representation of primary elasticity of an arfram.
The problem of determining the properties of thie two-
mass system so that it. is representative of the primary
elastic action of the airplane is rather simple if it is assumed
that during the impact the structure deflects with the shape
of its tundamenta-I mode of vibration. The requirements are:
(1) The total mass of the simplified system must equal
the total mass of the airplane in order that the proper nodal
or center-of-gravity accelerations can be obtained.
(2) The energy of vibration for the same amplitude of
the hull and lower mass (relative to the nodal poini) must
be the same for the two-mass system as for the considered
mode of the airplane structure.
(3) The natural frequency of the two-mass system ruslt
be the same as' the frequency of the considered mode of
Equations which permit determination of the masses and
spring constant of the simplified system so that it meets
these requirements are given in appendix B. Tilese equa-
tions and the foregoing requirements are applicable for both
landplanes and seaplanes.
In the present report the represented structural mode is
considered to be devoid of vibration prior to the instant of
impact. Thus, the computations may represent, either a
first impact or a subsequent impact resulting from a bounce
sufficiently high to cause aerodynamic and structural
damping to stop the vibration during the time the seaplane
is in the air. This report does not give a representation of
successive impacts, such as might occur in seaway, which
lead to accumulative or resonant effects. Available flight
data indicate that a single heavy impact, such as that
considered herein, is the primary cause of structural failutles.
The response of the two-mass. system is obtained in con-
nection with the calculation of the time history of the hydro-
dynamic force, and from this result the complete response
of the represented mode can be obtained by the simple
procedure given in appendix B and demonstrated in the
section entitled "Comparison with Experiment." The
response of other modes to .the force computed on the basis
of the fundamental mode can be separately determined
and superposed (reference 3). In order to minimize the
complexity of the solution, however, the present investi-
gation does not provide for taking into account the effect
of the other modes on the hydrodynamic force. Although
the other modes may have a substantial effect on the local
loads in the structure, the effect of these modes on the
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Mayo, Wilbur L. Hydrodynamic impact of a system with a single elastic mode I : theory and generalized solution with an application to an elastic airframe, report, January 1, 1952; (digital.library.unt.edu/ark:/67531/metadc60433/m1/2/: accessed November 17, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.