The determination of natural frequencies and normal mode shapes for beams of non- uniform section involves the solution of Eq. (10.11) and fulfilment of the appropriate boundary conditions. However, with the exception of a few special cases, such solutions do not exist and the natural frequencies are obtained by approximate methods such as the Rayleigh and Rayleigh–Ritz methods which are presented here. Rayleigh’s method is discussed first.
A beam vibrating in a normal or combination of normal modes possesses kinetic energy by virtue of its motion and strain energy as a result of its displacement from an initial unstrained condition. From the principle of conservation of energy the sum of the kinetic and strain energies is constant with time. In computing the strain energy U of the beam we assume that displacements are due to bending strains only so that
U =
L
M2
2EIdz (see Chapter 5) (10.16)
where
M = −EI∂2v
∂z2 (see Eq. (10.10)) Substituting forvfrom Eq. (10.13) gives
M = −EId2V
dz2 sin (ωt+ε)
so that from Eq. (10.16) U = 1
2sin2(ωt+ε)
L
EI d2V
dz2
2
dz (10.17)
For a non-uniform beam, having a distributed massρA(z) per unit length and carrying concentrated masses, m1, m2, m3,. . ., mnat distances z1, z2, z3,. . ., znfrom the origin, the kinetic energy KEmay be written as
KE = 1 2
L
ρA(z) ∂v
∂t
2
dz+1 2
n r=1
mr ∂v
∂t z=zr 2
Substituting forv(z) from Eq. (10.17) we have KE = 1
2ω2cos2(ωt+ε)
L
ρA(z)V2dz+ n r=1
mr{V (zr)}2
(10.18) Since KE+U=constant, say C, then
1
2sin2(ωt+ε)
L
EI d2V
dz2
2
dz+1
2ω2cos2(ωt+ε)
×
L
ρA(z)V2dz+ n r=1
mr{V (zr)}2
=C (10.19)
Inspection of Eq. (10.19) shows that when (ωt+ε)=0,π, 2π,. . . 1
2ω2
L
ρA(z)V2dz+ n r=1
mr{V (zr)}2
=C (10.20)
and when
(ωt+ε)=π/2, 3π/2, 5π/2,. . . then
1 2
L
EI d2V
dz2
2
dz=C (10.21)
In other words the kinetic energy in the mean position is equal to the strain energy in the position of maximum displacement. From Eqs (10.20) and (10.21)
ω2=
LEI(d2V/dz2)2dz
LρA(z)V2dz+n
r=1mr{V (zr)}2 (10.22) Equation (10.22) gives the exact value of natural frequency for a particular mode if V (z) is known. In the situation where a mode has to be ‘guessed’, Rayleigh’s principle states that if a mode is assumed which satisfies at least the slope and displacement conditions
10.3 Approximate methods for determining natural frequencies 343 at the ends of the beam then a good approximation to the true natural frequency will be obtained. We have noted previously that if the assumed normal mode differs only slightly from the actual mode then the stationary property of the normal modes ensures that the approximate natural frequency is only very slightly different to the true value.
Furthermore, the approximate frequency will be higher than the actual one since the assumption of an approximate mode implies the presence of some constraints which force the beam to vibrate in a particular fashion; this has the effect of increasing the frequency.
The Rayleigh–Ritz method extends and improves the accuracy of the Rayleigh method by assuming a finite series for V (z), namely
V (z)= n s=1
BsVs(z) (10.23)
where each assumed function Vs(z) satisfies the slope and displacement conditions at the ends of the beam and the parameters Bs are arbitrary. Substitution of V (z) in Eq. (10.22) then gives approximate values for the natural frequencies. The parameters Bsare chosen to make these frequencies a minimum, thereby reducing the effects of the implied constraints. Having chosen suitable series, the method of solution is to form a set of equations
∂ω2
∂Bs =0, s=1, 2, 3,. . ., n (10.24) Eliminating the parameter Bsleads to an nth-order determinant inω2whose roots give approximate values for the first n natural frequencies of the beam.
Example 10.6
Determine the first natural frequency of a cantilever beam of length, L, flexural rigidity EI and constant mass per unit lengthρA. The cantilever carries a mass 2m at the tip, where m=ρAL.
An exact solution to this problem may be found by solving Eq. (10.14) with the appropriate end conditions. Such a solution gives
ω1=1.1582 EI
mL3
and will serve as a comparison for our approximate answer. As an assumed mode shape we shall take the static deflection curve for a cantilever supporting a tip load since, in this particular problem, the tip load 2m is greater than the massρAL of the cantilever.
If the reverse were true we would assume the static deflection curve for a cantilever carrying a uniformly distributed load. Thus
V (z)=a(3Lz2−z3) (i)
where the origin for z is taken at the built-in end and a is a constant term which includes the tip load and the flexural rigidity of the beam. From Eq. (i)
V (L)=2aL3 and d2V
dz2 =6a(L−z) Substituting these values in Eq. (10.22) we obtain
ω12= 36EIa2L
0 (L−z)2dz ρAa2L
0 (3L−z)2z4dz+2m(2aL3)2
(ii) Evaluating Eq. (ii) and expressingρA in terms of m we obtain
ω1=1.1584 EI
mL3 (iii)
which value is only 0.02 per cent higher than the true value given above. The estimation of higher natural frequencies requires the assumption of further, more complex, shapes for V (z).
It is clear from the previous elementary examples of normal mode and natural fre- quency calculation that the estimation of such modes and frequencies for a complete aircraft is a complex process. However, the aircraft designer is not restricted to calcu- lation for the solution of such problems, although the advent of the digital computer has widened the scope and accuracy of this approach. Other possible methods are to obtain the natural frequencies and modes by direct measurement from the results of a resonance test on the actual aircraft or to carry out a similar test on a simplified scale model. Details of resonance tests are discussed in Section 28.4. Usually a resonance test is impracticable since the designer requires the information before the aircraft is built, although this type of test is carried out on the completed aircraft as a design check. The alternative of building a scale model has found favour for many years. Such models are usually designed to be as light as possible and to represent the stiffness characteristics of the full-scale aircraft. The inertia properties are simulated by a suitable distribution of added masses.
Problems
P.10.1 Figure P.10.1 shows a massless beam ABCD of length 3l and uniform bend- ing stiffness EI which carries concentrated masses 2m and m at the points B and D, respectively. The beam is built-in at end A and simply supported at C. In addition, there is a hinge at B which allows only shear forces to be transmitted between sections AB and BCD.
Calculate the natural frequencies of free, undamped oscillations of the system and determine the corresponding modes of vibration, illustrating your results by suitably dimensioned sketches.
Ans. 1 2π
3EI 4ml3
1 2π
3EI ml3
Problems 345
Fig. P.10.1
P.10.2 Three massless beams 12, 23 and 24 each of length l are rigidly joined together in one plane at the point 2, 12 and 23 being in the same straight line with 24 at right angles to them (see Fig. P.10.2). The bending stiffness of 12 is 3EI while that of 23 and 24 is EI. The beams carry masses m and 2m concentrated at the points 4 and 2, respectively. If the system is simply supported at 1 and 3 determine the natural frequencies of vibration in the plane of the figure.
Ans. 1 2π
2.13EI ml3
1 2π
5.08EI ml3
Fig. P.10.2
P.10.3 Two uniform circular tubes AB and BC are rigidly jointed at right angles at B and built-in at A (Fig. P.10.3). The tubes themselves are massless but carry a mass of 20 kg at C which has a polar radius of gyration of 0.25a about an axis through its own centre of gravity parallel to AB. Determine the natural frequencies and modes of vibration for small oscillations normal to the plane containing AB and BC. The tube has a mean diameter of 25 mm and wall thickness 1.25 mm. Assume that for the material of the tube E=70 000 N/mm2, G=28 000 N/mm2and a=250 mm.
Ans. 0.09 Hz, 0.62 Hz.
P.10.4 A uniform thin-walled cantilever tube, length L, circular cross-section of radius a and thickness t, carries at its tip two equal masses m. One mass is attached to the tube axis while the other is mounted at the end of a light rigid bar at a distance of 2a from the axis (see Fig. P.10.4). Neglecting the mass of the tube and assuming the stresses in the tube are given by basic bending theory and the Bredt–Batho theory of torsion, show that the frequenciesωof the coupled torsion flexure oscillations which
Fig. P.10.3
occur are given by 1
ω2 = mL3
3Eπa3t[1+2λ±(1+2λ+2λ2)12] where
λ= 3E G
a2 L2
Fig. P.10.4
P.10.5 Figure P.10.5 shows the idealized cross-section of a single cell tube with axis of symmetry xx and length 1525 mm in which the direct stresses due to bending are carried only in the four booms of the cross-section. The walls are assumed to carry only shear stresses. The tube is built-in at the root and carries a weight of 4450 N at its tip; the centre of gravity of the weight coincides with the shear centre of the tube cross-section. Assuming that the direct and shear stresses in the tube are given by basic bending theory, calculate the natural frequency of flexural vibrations of the weight in a vertical direction. The effect of the weight of the tube is to be neglected and it should be
Problems 347 noted that it is not necessary to know the position of the shear centre of the cross-section.
The effect on the deflections of the shear strains in the tube walls must be included.
E=70 000 N/mm2 G=26 500 N/mm2 boom areas 970 mm2 Ans. 12.1 Hz.
Fig. P.10.5
P.10.6 A straight beam of length l is rigidly built-in at its ends. For one quarter of its length from each end the bending stiffness is 4EI and the mass/unit length is 2m:
for the central half the stiffness is EI and the mass m per unit length. In addition, the beam carries three mass concentrations, 12ml at14l from each end and14ml at the centre, as shown in Fig. P.10.6.
Use an energy method or other approximation to estimate the lowest frequency of natural flexural vibration. A first approximation solution will suffice if it is accompanied by a brief explanation of a method of obtaining improved accuracy.
Fig. P.10.6 Ans. 3.7
EI ml4
Part B Analysis of Aircraft Structures