For the applicability of dynamic similitude models of thin walled structures, such as engine blades, turbine discs, and cylindrical shells, the dynamic similitude design of typical thin walled structures is investigated. The governing equation of typical thin walled structures is firstly unified, which guides to establishing dynamic scaling laws of typical thin walled structures. Based on the governing equation, geometrically complete scaling law of the typical thin walled structure is derived. In order to determine accurate distorted scaling laws of typical thin walled structures, three principles are proposed and theoretically proved by combining the sensitivity analysis and governing equation. Taking the thin walled annular plate as an example, geometrically complete and distorted scaling laws can be obtained based on the principles of determining dynamic scaling laws. Furthermore, the previous five orders’ accurate distorted scaling laws of thin walled annular plates are presented and numerically validated. Finally, the effectiveness of the similitude design method is validated by experimental annular plates.
Thin walled structures are widely used in mechanical and aerospace engineering application due to the excellent dynamic characteristics and high specific stiffness, and it is important to analyze their vibration characteristics [
In aspect of the vibration analysis of thin walled structures, Narita [
The scaling laws of thin walled structures have been investigated by many researchers. Krayterman and Sabnis [
In the above studies, the dynamic characteristics and similitude design methods of thin walled plates and cylindrical shells have been investigated by many researchers. However, studies of determining the accurate distorted scaling laws of typical thin walled structures have not been discussed. In this study, the determining method of the accurate distorted scaling laws of thin walled structures is proposed and theoretically proved.
In Section
The curved coordinate system
The coordinate system.
The material parameter
The arc length
The internal forces of the infinitesimal surface can be expressed as
In order to obtain the governing equation, let all the stresses multiply the corresponding arc length and let the inertia force components multiply the infinitesimal area in the directions
By using (
For the typical thin walled structures, Lamé parameter can be written as
From (
The governing equation of the prototype and model can be written as
The deflection equation can be denoted as
Substituting scaling factors
According to the similitude theory, the corresponding coefficient of a prototype’s governing equation is proportional to the coefficient of a model, which means
Under the condition of geometrically complete similitude, this yields
Substituting (
Therefore, geometrically complete scaling law can be denoted as
Normally, there will be many limitations in employing the geometrically complete similitude model in experiments, so it is necessary that designing geometrically distorted models predicts dynamic characteristics of the prototype. Geometrically distorted models are defined such that scaling factors of geometrical parameters of models are keeping different [
In addition, (
Therefore, there are many possible candidate distorted scaling laws as
In general, the indexes
The sensitivity is the change rate of structural characteristic parameters with respect to structural parameters [
In distorted scaling laws, if the scaling factor
From (
When sensitivity’s absolute values satisfy
The sensitivity is applied to the condition of small variations of structural parameters, but the distorted models did not satisfy the condition. So the transitional model is introduced; let
Equation (
According to the geometrically complete scaling law, the transitional model and distorted model should satisfy
If
If
Therefore,
By combining (
According to the recursive relation, if
If the sensitivity
If the sensitivity
(1) If the scaling factor
(2) If
When the sensitivity
In addition, the above three principles could obtain the approximate distorted scaling laws. In order to determine the accurate distorted scaling laws, an additional principle is required.
When the structural parameters
The sensitivity of natural frequency with respect to parameters
Therefore, this yields
Substituting (
When scaling factors change within the limit range, that is,
So (
When scaling factors change in the little range, this yields
In the same way, if the dynamic characteristics of thin walled structures are affected by many parameters, (
Therefore, the basic principles can be summarized as follows: In distorted scaling laws, if the scaling factor When sensitivity’s absolute values satisfy If the sensitivity
Finally, the procedure of similitude design method of typical thin walled structures is given out.
Deducing complete scaling law of typical thin walled structures based on the governing equation.
Assuming distorted scaling laws according to the governing equation of typical thin walled structures.
Analyzing the sensitivity of the natural frequency with respect to structural parameters.
According to the principles for dynamic scaling laws, determining the distorted scaling law of typical thin walled structures.
Take the thin walled annular plate as an example; geometrically complete scaling law of a thin walled annular plate is firstly established. According to the above three principles, the distorted scaling law of the first order’s frequency is determined based on the sensitivity analysis. Finally, the determining method of the distorted scaling law is validated by the numerical analysis.
The structure of a thin walled annular plate is shown in Figure
The thin walled annular plate.
According to (
The geometric sizes and material parameters of the thin walled annular plate are listed in Table
The geometric sizes and material parameters.
Outer radius | Inner radius | Thickness | Young’s modulus | Density | Poisson’s ratio |
---|---|---|---|---|---|
|
|
|
|
|
|
100 | 12 | 6 | 210 | 8000 | 0.3 |
The dynamic characteristics of the prototype.
Order |
1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|
Natural frequency |
541.90 | 647.94 | 836.11 | 1819.9 | 3168.1 |
Vibration modes |
|
|
|
|
|
|
|
|
|
|
According to the governing equation (
Under the condition of geometrically complete similitude, scaling factors can be expressed as
Therefore, geometrically complete scaling law of the thin walled annular plates is
In general, scaling factors
From (
In order to obtain the sensitivity of the first order’s natural frequency with respect to the scaling factor
Natural frequencies of the 1st mode (2, 1) as a result of the distorted outer radius.
Outer radius | Scaling factor of | Natural frequencies | |
---|---|---|---|
|
outer radius |
|
|
Model 1 | 96 | 0.96 | 597.47 |
Model 2 | 98 | 0.98 | 568.69 |
Model 3 | 102 | 1.02 | 516.92 |
Model 4 | 104 | 1.04 | 493.59 |
In Figure
The fitted curve of outer radius distorted models.
Furthermore, adopting the adjusted square
By comparing and analyzing, the fitted curve is thought to be effective and suitable if
The adjusted square of the fitted curve is
The sensitivity of the natural frequency with respect to the scaling factor
In the same way, the fitted curve of the first order’s natural frequencies of inner radius distorted models is shown in Figure
The fitted curve of inner radius distorted models.
The fitted equation of the first order’s natural frequency is
In Figure
The sensitivity of the first order’s natural frequency with respect to the scaling factor
As a result of the sensitivity analysis,
According to Principle
From (
Solving simultaneous equations yields
Therefore, the distorted scaling law of the first order’s natural frequency of thin walled annular plates can be written as
In the same way, the previous five orders’ distorted scaling laws are determined and the effectiveness of distorted similitude models is validated. Finally, the procedure of determining the accurate distorted scaling laws is presented.
The geometric sizes and material parameters of the distorted model are listed in Table
Parameters of the distorted model.
Outer radius | Inner radius | Thickness | Young’s modulus | Density | Poisson’s ratio |
---|---|---|---|---|---|
|
|
|
|
|
|
130 | 12 | 8 | 72 | 2700 | 0.3 |
Similarly, the sensitivity of the natural frequencies with the scaling factors
The distorted scaling laws.
Order |
Sensitivity |
Sensitivity |
Distorted scaling laws | Frequencies of model |
Predicted frequencies |
Error |
---|---|---|---|---|---|---|
1 | −1297 | 232.3 |
|
387.56 | 527.33 | 2.69 |
2 | −1437 | 144.9 |
|
490.27 | 645.64 | 0.35 |
3 | −1765 | 109.1 |
|
647.17 | 837.91 | 0.22 |
4 | −3621 | 10 |
|
1445.2 | 1818.1 | 0.10 |
5 | −6247 | 5 |
|
2517.5 | 3162.9 | 0.16 |
The error between the natural frequency of the prototype and the predictive natural frequency can be denoted as
Table
The errors between the distorted models and prototype.
Model |
|
|
|
|
|
Frequencies of model |
Predicted 1st frequencies |
Error |
---|---|---|---|---|---|---|---|---|
M1 | 1 | 0.6 | 0.6 | 1.67 | 1.67 | 274.72 | 535.06 | 1.26 |
M2 | 1 | 0.8 | 0.8 | 1.25 | 1.25 | 400.15 | 535.41 | 1.20 |
M3 | 1 | 1.2 | 1.2 | 0.83 | 0.83 | 700.91 | 552.49 | 1.95 |
M4 | 1 | 1.4 | 1.4 | 0.71 | 0.71 | 878.43 | 566.25 | 4.49 |
M5 | 1 | 1.2 | 0.6 | 0.71 | 1.67 | 357.47 | 563.55 | 3.99 |
In the experiment, test setup of the experimental plate is shown in Figure
Parameters of experimental plates.
Outer radius | Inner radius | Thickness | Young’s modulus | Poisson’s ratio | Density | |
---|---|---|---|---|---|---|
|
|
|
|
|
|
|
Prototype | 100 | 12 | 6 | 210 | 0.3 | 8000 |
Model | 130 | 12 | 8 | 72 | 0.3 | 2700 |
Test setup of the experimental plate.
In addition, Table
The comparison of experimental and predicted results.
Orders |
Model |
Vibration modes | Prototype |
Vibration modes | Modes | Predicted frequencies |
Errors |
---|---|---|---|---|---|---|---|
1 | 639.34 |
|
862.31 |
|
|
827.77 | 4.01 |
2 | 1431.5 |
|
1842.5 |
|
|
1798.5 | 2.39 |
3 | 2512.8 |
|
3203.6 |
|
|
3157.1 | 1.45 |
From Table
The experimental errors are higher than the numerical errors; the reasons are analyzed as follows: The machining of the thin walled annular plates, such as the dimensional error and deviation of material parameters. The measurement accuracy of a test system, for example, the precision of the sensor. The random error of the test procession.
In this paper, in order to investigate the similitude design method of typical thin walled structures, the governing equation of typical thin walled structures is firstly established and the geometrically complete scaling law is deduced. In order to determine accurate distorted scaling laws of thin walled structures, three principles are proposed and theoretically proved by combining the governing equation and sensitivity analysis. Taking the thin walled annular plate as an example, geometrically complete and distorted scaling laws are obtained based on the three principles. Finally, the design method of similitude models of typical thin walled structures is validated via experiments, and detailed conclusions are listed as follows: The governing equation of typical thin walled structures is unified in ( By employing the governing equation, geometrically complete scaling law of typical thin walled structures is obtained. In order to determine the accurate distorted scaling law of thin walled structures, three principles are proposed and theoretically proved by combining the governing equation and sensitivity analysis. Taking thin walled annular plate as an example, the design method of similitude models of thin walled structures is validated by numerical simulation and experiments.
There are also some restrictions about the similitude design method; one of the limitations is that a numerical model of the thin walled structure is always necessary to compute the sensitivity.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors gratefully acknowledge the financial support from the Fundamental Research Funds for the Central Universities of China (Grant nos. N130503001 and N140301001), the Key Laboratory for Precision & Nontraditional Machining of Ministry of Education, Dalian University of Technology (Grant no. JMTZ201602), and the National Program on Key Basic Research Project (Grant no. 2012CB026000).