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VIBRATION Vibration in REFERENCES TO LITERATURE (1) Burgess, C. P. The Frequencies of Cantilever Wings in Ream and Torsional Vibrations. N.A.C.A. Technical Nate, No. 746 Composite Beams (2) Den Hartog, J. P. Mechanical Vibrations. McGraw-Hill Book Co., 1940 (3) Miller, R. H. Wing-Frequencies of Multi-Engine Monoplanes. Journ. of Aero. Sci., Vol. 7, No. 6, April 1940, p. 256 A Note on Natural Frequencies of Uncoupled Flexural (4) Myklestad, N. O. Vibration Analysis, McGraw-Hill Book Co., 1944 (5) Nagel, F. Flügelschwingungen in Stationären Luftstrom, Luftfahrtforsehung, Vol. 3, May 1929, p. 111 and Torsional Vibrations of Composite Beams (6) Timoshenko, S. Vibration Problems in Engineering. D. Van Nostrand Co. By M. Z. Krzywoblocki length l (call it B). The concentrated loads are attached along the beam A by means of elastic suspensions. Imagine a plane tangent to two curves, N aeroplane wing and fuselage may be treated from an analytic representing the deflected shapes of both beams in symmetric flexural standpoint as two elastic beams, rigidly connected. In the present vibration. The point of tangency is the point of the intersection of those Anote the equations for the natural frequency, derived on the basis two curves. To find the nodes one has to shift the tangential plane, parallel of Rayleigh's principle and Ritz's method, are extended to the case of free with its original position, to a new position, in order to fulfil the con vibration of a set composed of two such beams with a number of con dition 5, centrated loads elastically suspended. ∑ mass x acceleration = 0 (3) Where the acceleration should be taken as a vector, and the summation A. Tubular Thin-walled Cantilever Beam includes the beams themselves and all the concentrated loads. The shifted Assume a cantilever, tubular, thin-walled beam fixed at one end and plane will intersect the centroidal axes of the beams (one or both) in free at the other. Let m=m(x) be the mass distribution along the beam points which represent the nodes (nodal plane). The system of co and Φ=Φ (x) the normal elastic curve corresponding to the fundamental ordinates (x, y) should also be shifted to lie in the nodal plane. Hereafter mode of vibration. Application of Rayleigh's principle permits one to we shall always refer to this nodal system of co-ordinates (x-axis along the find the fundamental frequency of the system2 4 6 A-beam, y-axis along the B-beam). Assume that the acceleration is proportional to the deflexion Φ(x) of the beam A and ψ (y) of the beam B. Of course, Φ (0)=ψ (0). The elastic 1 1 1 suspension of concentrated loads causes some additional deflexions, namely, the static deflexion of a load W with respect to the beam A is equal to δ = W /k , where k denotes the spring constant of the supporting Application of Ritz's method permits one to assume for the deflexion sti i i i structure. Assuming for Φ and ψ the static deflexion curves, as an curve representing the mode of vibration the function 1 1 example, Equation 3 may be written in the form z=Φ (x)Φ (ωt), with Φ =∑n a f (x), and Φ =cos ωt, say, 1 2 1 1 1 1i 2 where each function f (x) must satisfy the boundary conditions. Then 1i again Equation 1 can be used to calculate the values of the natural fre quencies6. As is known, the coefficients a can be found from the set of linear equations (ω2) =0, and the frequencies of various modes from the ai 'frequency equation', which originates from equating to zero the deter where x denotes the co-ordinate of a node and X the co-ordinate of the minant of this set of linear equations6. Fair results may be obtained by n R root section of the beam A. Due to the structure joints, the value X may using Φ =∑4 ai/(x/l)i. 1 1 be different from zero. In (1) the numerator is equal to Both these methods may be applied to torsional vibration. Denote the twist per unit length by θ =∂θ/∂x, θ=θ(x,ωt), the torque by M =Cθ x t x =4A2Gθ /δ(ds/δ), and the mass moment of inertia of the element of the length δx by Ipγδx/g=p δx.Then dU=½Mθxdx=½Cθ2 dx and dK=½pθ2 1 t x 1 t where the last term is due to the additional strain energy of the elastic dx. Let θ =θ (x) be the normal elastic curve of twist corresponding to the 1 1 suspensions. The denominator in (1) is equal to fundamental mode of vibration and θ =ωθ . t 1 Then the application of Rayleigh's principle gives where X may be equal to zero. If one neglects the effect of suspension, the quantities δ = W/K are equal to zero. If the entire weight of the sti i i Application of Ritz's method with θ(x, t)=θ (x)θ (ωt), θ (x)=∑n bϑ(x), transverse beam D is assumed to be a mass point, the second term in the 1 2 1 1 i i θ =sin ωt, say, where each function ϑ satisfies the boundary conditions, numerator vanishes and in the denominator it will be equal to 2 i again gives Equation 2 as the result. W -1Φ 2(0) Bg 1 Both methods may also be applied to beams with concentrated loads. In the case of torsional vibration, the location of nodes may be found The only change in this case will occur in the expression for K. The de from the condition that ∑ mass polar moment of inertia x angular acceler nominator in (1) will be equal to ∫1 Φ2 dm+∑n (Wi/g)Φ2 (x ), where the ation =0 , (a). Assume the case of symmetric torsional vibration of the 0 1 1 1 1 symbol W , denotes the weight of a concentrated load and Φ (x) the de beam A and of antisymmetric flexural vibration of the beam B. The angles i 1 i flexion at the point x where the load W is located. Similarly, in (2) the i i of twist are measured with reference to the plane passing through both denominator will have the form ∫1 p θ 2dx+∑n (Wi/g)r 2θ 2(x), where the 0 1 1 1 i 1 i nodes. Then the condition (a) may be represented in the form symbol r denotes the distance between the centroid of W and the centroid i i of the cross-section of the beam. The reason for this assumption is as follows: If the centroidal axis and the elastic axis do not coincide, then usually in actual tests one has the case of coupled flexural-torsional oscillation and it is not possible to obtain a case of pure, uncoupled torsional or flexural oscillation. In these actual Where p denotes the mass moment of inertia of the beam A per unit cases the body rotates neither around the centroidal axis nor around the length and r the distance of any cross-section of the beam B from the elastic axis, but around some axis located between them. Assume an origin. The deflexions and angles of twist should be taken with proper imaginary case of free, uncoupled, purely torsional vibrations around the signs. centroidal axis at high frequency. For the shape of the twist one may In the numerator of (2) the existing integral should be multiplied by 2 assume the static twist of the centroidal axis caused by the distributed and the following two terms should be added: load equivalent to the weight of the beam, attached to the shear centre in each cross-section. B. Influence of the Finite Stiffness and the Finite Mass of the Transverse In the denominator of (2) the existing integral has to be multiplied by 2 Beam and of Elastic Suspension of Loads and the following terms have to be added: In the foregoing it was assumed that the beam was rigidly fixed, i.e. that the mass of the wall was infinitely large. Assume now that a beam of the length 2l (call it A) is attached in its centre to a transverse beam of the August 1950 223
Aircraft Engineering and Aerospace Technology – Emerald Publishing
Published: Aug 1, 1950
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