Similitude Analysis on Mechanical Parameters of Thin Walled Shells

The relationship of the kinetic fundamental parameters for both the prototype and model are derived by employing the finite method. Based on the relations, the scaling laws of the thin walled cylinder for the free vibration are found by applying the similitude transformation to the governing equation. In the absence of the experimental data, the validity of the scaling laws is testified by numerical data. This is done by calculating theoretically the natural frequencies for free vibration of the cylinders. By substituting the model frequencies in the scaling laws, the frequencies of the prototype are obtained. Consequently, the frequencies of the model and prototype are compared. Examples that one end of the thin walled shell is clamped and the other is simply supported show exact agreement.

(1) u is the displacement along the axis of revolution, v is the change of angle β of the normals and w is in the radius direction along the radii of curvature. In this paper, ring elements shown in Fig 1 are employed to illustrate the method. [3] Let displacements of the element be expressed in the form of interpolation polynomial.
Consequently, Displacements of any point in the ring element can be calculated by those of the node i and the node j.
Eq.3 is the general case that the shape function of the thin-wall cylindrical shell N is a matrix ofζ and L, and e cn δ is the column vector to express displacements of two nodes i and j .
In order to obtain the scaling laws composed of scale factors, only one item of each direction is needed, however, they can be applied to the whole displacements ( , ) Where u , v and w are the displacements of any point in the ring element. Substituting Eq.4 into Eq.5 [4] , another useful equation can be obtained.   2   3  2  3  2  3  2  3   1   2  2  3  2  2  3  2  2  3  2  2  3  where D is an elastic matrix, it is shown as follows.
Note that the elements in the 3rd and 6th row of B 1 are all zeros, while the rows of B 2 that are not zeros are only the 3rd and 6th rows. K e can thus be interpreted as anther form.

K t B t Dt Bt
Finally, the mass matrix M e , the stiffness matrix K e and the damping matrix C e can be extrapolated as follows. Eq.1 yields that n is determined by u m (s,t), and the necessary condition that the mode shapes of the model and prototype are similar is n m and n p have to be identical. Accordingly, the scaling laws concerning M, K and C are obtained in the following form.

Similitude of natural frequencies and mode shapes
The governing equation without damping is. [6] 0 + = Mδ Kδ (19) Utilizing Eq.14 to Eq.17and Eq.19,we can express the scaling laws related to frequencies. Since the governing equations are equivalent and the boundary conditions are the same between models and prototypes, the mode shapes are certain to be the same.

Validation by FEM
One prototype and several models are designed to verity the correctness of Eq.9, Table 1 contains the parameters of the prototype and models. The boundary condition is one end is clamped and the other is simply supported. Numerical experiments are done.
Observing mode shapes of different orders, the number of axial half waves are all 1,whreas the number of circumferential waves is displayed in Table 2， we can see that only model 2, model 5 and model 6 have similar relationships with the prototype. This extrapolates that if identical parameters of two thin walled cylindrical shells can be converted to each other, the geometries must have the equivalent scale factors and u must be identical, while the material properties E and ρ are not constraint.