The rotor dynamic design considerations are essentially limitations on the vibratory response of the blades which in turn limit the dynamic excitation of the fuselage by forces and moments transmitted to the hub. Quantities which are associated with the blade response and which are subject to design constraints are discussed. These include blade frequencies, vertical and inplane hub shear, rolling and pitching moments, and aeroelastic stability margin

Adelman, Howard M.

Mantay, Wayne R.

Pritchard, Jocelyn I.

English

NASA Technical Reports Server

verified by model tests of both advanced and baseline designs for the UH-1, AH-64, 
and UH-60 helicopters (refs. 1 9 - 2 2 ) .  A more sophisticated hover analysis which 
includes wake effects may be used in the future if the trends predicted by such an 
analysis are verified for a wide range of configurations, i.e., different taper ~ 
3 c q p y  p 
ratios, taper initiation points, twist distributions, etc. 
~g&;J@583'-I 
J 
23) a s'/s- ROTOR BLADE DYNAMIC DESIGN 
I .. 
Jocelyn I. Pritchard, Howard M. Adelman, Wayne R. Mantay ' j  , 
Design Considerations 
The rotor dynamic design considerations are essentially limitations on the 
vibratory response of the blades which in turn limit the dynamic excitation of the 
fuselage by forces and moments transmitted to the hub. The following quantities 
associated with the blade response are subject to design constraints: blade fre- 
quencies, vertical and inplane hub shear, rolling and pitching moments, and aero- 
elastic stability margin. 
Frequencies.- The blade natural frequencies are required to be separated from 
multiples of the rotor speed. A typical constraint is written as 
where wi is a blade frequency, and wLi, wui are lower and upper bounds of the ith 
frequency. Generally, wLi and wui are n n + 6 where n is an integer, n is 
the rotor speed, and 6 is a tolerance usually about 10 percent of nL3 (e.g., 
ref. 6 ) .  
Vertical hub shear.- The transmitted vertical hub shear S is to be made as 
small as possible. 
function wherein it is minimized (ref. 6 ) ,  or as a constraint where the vertical hub 
This requirement may be handled either as part of the objective 
shear is required t o  be less 
approach, letting N denote 
than some specif ied value ( r e f .  2 3 ) .  In  the f i r s t  
the number of blades in the rotor 
13 
https://ntrs.nasa.gov/search.jsp?R=19900003267 2020-03-20T00:52:23+00:00Z
IS,l + min k = N ,  2 N ,  . . .  
In  the second approach 
lskl 5 6 k = N ,  2N, . . .  
where E i s  a pos i t ive  value.  
Only blade shear responses a t  multiples o f  Nn contr ibute  t o  the transmitted 
v e r t i c a l  hub shear.  The v e r t i c a l  blade shear a t  a l l  other  frequencies cancel out i n  
the summation process. In  other words 
k = N ,  2N, . . .  
A l l  other k 
( 1 3 )  
'k I t o t  = C"Ik 
A t  the same time, f o r  a f i n i t e  hinge o f f s e t ,  the blade v e r t i c a l  shear a t  other  f r e -  
quencies contr ibutes  t o  the transmitted hub moments. 
Hub moments. - Two types of moments a re  generated a t  the hub due t o  blade motion. 
The f i rs t  is due t o  d i s t r ibu ted  blade bending moments and the second i s  due t o  
couples involving the blade shear forces a t  the hinge o f f s e t  of the blade.  Each type 
of moment has both a ro l l i ng  and pitching component a t  the hub. 
Inplane hub shear . -  In  the approaches described herein,  the inplane hub shear i s  
handled i n  the same way as  the v e r t i c a l  hub shear.  Spec i f ica l ly ,  i n  the f i rs t  
approach, 
lHkl -+ min k - N ,  2N, . . .  . ( 1 4 )  
i n  t he  second approach 
k = N ,  2 N ,  . . .  (15)  lHkl E 
For an N-bladed r o t o r ,  the t o t a l  transmitted shear a t  the hub is  non-zero only a t  
frequencies which are  multiples of Nn. However, i n  t h i s  case,  the transmitted hub 
shear is  made up of contributions from the blade responses a t  the following multiples 
of the r o t o r  speed: N ? 1, 2 N  ? 1, . . .  For example, i n  a four-bladed r o t o r ,  
14 
X4 - (2F5 - 2F3) 
Y4 (2F5 - 2F3) 
where X4 and Y4 are o 
sin 4nt 1 
thogonal compon nts of in-plan forces. F3 nd F5 are 
amplitudes of tangential forces at the blade root at frequencies 311 and 50, 
respectively. Thus X4 and Y4 play the roles of Hk in equations (14)  and (15). 
Rotor aeroelastic and aeromechanical stability.- The constraint for positive 
system aeromechanical stability relies on knowledge of fixed system characteristics 
and rotor frequency placement. Specifically, the rotor's lower modes, especially 
lead-lag, should not have fixed-system values which coalesce with the fuselage roll 
or pitch degrees of freedom, either on the ground or in flight. 
Additionally, aeroelastic stability constraints for the isolated rotor in hover 
as developed by Friedmann (ref. 6 )  require that 
where Vk is the real part of the kth complex eigenvalue and VLk is its limiting 
value. 
Analysis Considerations 
For the purpose of dynamic response analyses, the rotor blade is modeled as a 
beam undergoing coupled flap-lag-torsion motion in response to harmonically varying 
airloads. The beam is assumed to rotate at constant rotor speed which gives rise to 
centrifugal loading and stiffness effects. It is anticipated that either a finite- 
element analysis (e.g., ref. 24)  or CAMRAD (ref. 14) will be used for the dynamic 
calculations. These calculations include mode shapes and (complex) eigenvalues, 
steady-state response (displacements), blade loads, and transmitted hub loads and 
moments. 
15 
The governing matrix equation for vibration response of a finite-element modeled 
structure is 
@ + C i + K X - F  
where M is the mass matrix 
C is the damping matrix 
K is the stiffness matrix 
X is the vector of displacements and rotations 
F is the applied force vector 
The stiffness matrix K for a rotor blade has the form 
where KE is the linear elastic stiffness matrix 
KC 
KD 
is a centrifugal stiffness matrix 
is the differential stiffness matrix and contains stresses associated 
with centrifugal forces 
Detailed discussions and explicit forms for KC and KD are available in 
reference 2 4 .  
Equation (18) may be solved by modal superposition. The modal analysis produces 
the natural frequencies and damping needed in the phase 1 constraints (eqs. 13, 12, 
15j. Additional analyses are used to calculate the blade loads and transmitted 
hrib l o a d s  based on modal expansions o f  t h e  blade response and are outlined in 
reference 2 3 .  
Derivatives of the dynamic response quantities which appear in the constraints 
are needed. Expressions for most of these derivatives are given in reference 2 3 .  
For example, analytical derivatives of the frequencies are given by 
16 
The alternative to finite-element analysis is the modified Galerkin approach in 
CAMRAD. The advantage of the latter approach is that it resides in the same code 
that will be used for the aerodynamic analysis. The disadvantage is that the method 
does not ordinarily generate the matrices M, C, and K which are needed for the 
analytical derivatives (e.g., eq. (19)). Thus, the modified Galerkin approach may 
require the use of finite difference derivatives. This was done in reference 7 
without any ill effects. Nevertheless, studies are underway to find ways to generate 
equivalent M, C, and K matrices based on the modified Galerkin method and use 
these in the calculations of analytical derivatives. 67{'/X1/$ PG , 
N90;.- ,1e84 1: 
- -  
ROTOR BLADE STRUCTURAL DESIGN 
22 , 
Mark W. Nixon I,, ' I ( 1  
L 
In this section the structural design of rotor blades is discussed. The various 
topics associated with the structural design include constraints, load cases, and 
analyses. 
Design Constraints 
The constraints associated with traditional structural design can be categorized 
as aerodynamic, autorotation, buckling, frequency, and material strength. As dis- 
cussed in reference 10, some of these constraints are based on maintaining character- 
istics required by other disciplines involved in the integrated optimization. Con- 
straints associated with aerodynamics, autorotation and frequency are not addressed 
in this section, since they are addressed in other sections of the paper 
Of the remaining structural constraints, the most important is the material 
strength constraint. All stresses in the blade structure must be less than the 
design allowable stress of the material for all load cases. To account for stress 
interactions, a failure criterion such as Tsai-Hill (ref. 25)  is calculated based on 
the material limit allowable stresses. The governing equation is 
1 7  


Rotor blade dynamic design

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server