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Zn = jn + knα10 (7.33)
where cn, dn, …, etc. are numbers* which depend on the blade station in question.
uMn
vMn
ln M = 1
UFn
vFn
F = 1
ln
Fig. 7.11 Definition of unit load coefficients Fig. 7.12 Definition of unit moment coefficients
* Recurrence relations expressing these numbers in terms of the original constants and tabulation
methods for the calculation procedure are given in detail in reference 7 and need not be given here.
Structural dynamics of elastic blades 253
Now at the blade root the displacement is zero. Thus, if the suffix r denotes the
root,
Zr = jr + krα10
or α10 = –jr /kr
The remaining root boundary condition to be satisfied is either
Mr = 0, for the hinged blade
or
αr = 0, (or, possibly, a known non-zero value) for the hingeless blade.
For the first case, we have from eqn 7.31, after substituting for α10,
Mr = (erkr – jr fr)/kr = 0
Now the term in the bracket is a function of the assumed value of the frequency
ω. Thus, if a range of values of ω is taken, a natural frequency occurs every time the
function becomes zero, Fig. 7.13. For such a value of ω, the mode shape, defined by
the values of Zn, can be determined from eqn 7.33.
Similarly, for the hingeless blade, we have from eqn 7.32
αr = (grkr – hr jr)/kr = 0
and, again, we seek the values of ω which make the function in the bracket zero.
Now, in the methods described earlier, if it is required to calculate, say, the fifth
mode and frequency, it may be necessary to take at least ten functions and solve the
resulting ten simultaneous equations to obtain reasonable accuracy for the fifth and
lower modes. In the Myklestad method, however, each mode can be calculated
independently of the others, although a certain amount of ‘searching’ may be necessary
to locate the appropriate zero.
The Myklestad method described above refers only to pure flapwise bending. An
extension of the method has been given by Isakson and Eisley8 which includes the
effect of coupling between the bending and torsional modes of vibration. Their
analysis, however, is far too lengthy to be included here, and the reader is referred to
the original paper.
(b) The dynamic finite element method 9
As in the Myklestad method described in the previous section, the blade is divided
into a number of elements, not necessarily of equal length. In this case, however, the
erkr – jrfr
ω
Fig. 7.13 Variation of boundary function with frequency
254 Bramwell’s Helicopter Dynamics
mass of an element is not concentrated at its ends but is imagined to be distributed
uniformly along its length. The system of forces and moments acting along the
element is as shown in Fig. 7.14, and its equilibrium in harmonic motion leads to the
following equations
Gn Gn rm r G m r r
r
r
n n n
n
n
+1
2 22
+1
2 = + d = + 12
( – )
+1 ∫ Ω Ω (7.34)
Sn Sn Zmr S m l Z Z
r
r
n n n n
n
n
+1
2 2
+1 = – d = – 12
( + )
+1 ∫ ω ω (7.35)
Mn Mn GnZn Zn Snln r r mZr
r
r
n
n
n
+1 +1 +1
= – ( – ) – + ( – ) 2 d
+1 ∫ ω
– ( – ) d
+1
+1
2
r
r
n n
n
n
∫ Z Z Ωrm r
= – ( – +1) – + (2Z + Z )/6
2 2
Mn Gn Zn Zn Snln ω lnm n m+1
– 2 (2 + )( – )/6
Ω lnm rn rn+1 Zn Zn+1 (7.36)
These three equations are analogous to eqns 7.23, 7.24, and 7.25 of Myklestad’s
method. Inspection of the above equations shows that the bending moments
M0, M1, …, Mn, … along the blade can be expressed in matrix form as
M = ω2aZ + Ω2bZ (7.37)
where a and b are square matrices, and functions of r and m, Z is the column vector
of the element displacements, and M is the column vector of bending moments.
The deformation of the blade represented by Z may be split into two components
Z = ZE + ZR
where ZE is the elastic deformation of blade bending and ZR is the rigid-body rotation
about the flapping hinge. If there is no flapping hinge, ZR = 0.
To obtain a relationship expressing the blade deformation as a function of the
applied moment distribution, we use the unit load method10. This states that, if M1 is
the bending moment distribution due to the application of a unit load at a point at
which we wish to calculate the deflection, and if M is the actual moment distribution
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