# Dynamic Similitude Design Method of the Distorted Model on Variable Thickness Cantilever Plates
## 1 Introduction  
Variable thickness (VT) plates have been widely applied in engineering practice such as, for example, advanced gas turbines, high-powered aircraft jet engines and high-speed centrifugal separators [1; 2; 3]. The vibration problems of the plate structures in these engineering machines are important for consideration in the design process [4; 5; 6]. By using the finite element method, vibration problems of VT plates are widely studied. [PERSON] et al. [7] investigated the free vibration problem of orthotropic rectangular VT plates by using a discrete method. Based on the Green function, [PERSON] et al. [8] discussed an approximate method for analyzing the free vibration of rectangular VT plates, and [PERSON] et al. [9] introduced a dynamic function in the finite strip method, where numerical analysis was used to demonstrate the application of the approaches by analyzing simply supported stepped thickness (ST) plates. It has been shown that the numerical solutions of the approximate method had good accuracy for various types of rectangular plates with uniform or non-uniform thickness. Moreover, [PERSON] and [PERSON] [10], in 2001, computed exact axisymmetric vibration frequencies of VT circular and annular plates by approximating the variation in thickness into an infinite power series. [PERSON] and [PERSON] [11] studied the free vibration characteristics of linear elastic orthotropic toroidal VT shells based on the Sanders-Budiansky shell equations, and [PERSON] and [PERSON] [12] also discussed the free vibration on VT paraboloids shells by using a three-dimensionalmethod. Recently, based on the polynomial fitted thickness, VT plates' various combinations of boundary conditions were investigated by [PERSON] and [PERSON] [13], where two shear deformation plate theories were applied to provide accurate results in solving natural frequencies. By using the Generalized Differential Quadrature (GDQ) method, most recently, [PERSON] et al. [14] and [PERSON] et al. [15] investigated the free vibration of doubly-curved shells, singly-curved shells and plates with continuous thickness variation, showing that the GDQ method provides an accurate, stable and reliable numerical tool in analyzing variable thickness thin walled structures. However, although theoretical studies have been widely discussed, experimental tests of the actual structure are still necessary. The issue is that, in practice, experimental investigation on thin walled structures like VT plates are actually expensive and time-consuming. Consequently, a scaled down model, which is usually designed based on similitude theory, is employed to reflect the prototype behavior. In general, due to the lack of the availability of materials, or the unavailability of members' specified dimensions, researchers often use thin plates for the analysis [16; 17], but this may limit the application of the test results. Dynamic similitude design of thin walled plate structures is important in engineering practice and has been discussed by many researchers. For example, [PERSON] et al. [18] investigated the distorted scaling laws in predicting the dynamic response of rectangular flexural plates based on the analysis of the vibration energy. [PERSON] et al. [19] considered the scaled model made of different materials to predict the dynamic behavior of the prototype by using a scaling law, which has been established based on dimension analysis, for free vibration. [PERSON] et al. [20] established the scaling laws of laminated plates based on the governing equation analysis to predict the impulse response of the prototype. The results indicate that scaling laws can accurately predict the undamaged response of impact. Moreover, scaling laws of isotropic laminated plates have been studied by [PERSON] et al. [21]. In their study, governing equations of buckling and frequency were used to derive the scaling laws, and partial similitude was also considered, recommending the scaling laws with good accuracy. [PERSON] et al.