# The Dynamic Similitude Design Method of Thin Walled Structures and Experimental Validation

Authors: Zhong Luo; You Wang; Yunpeng Zhu; Deyou Wang
Journal: Shock and Vibration
 (2016)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1, 2]. However, employing the prototype directly in experiments is time consuming and difficult to test. Fortunately, similitude design is a powerful method to predict the dynamic characteristics of the prototype. Due to the limitations of structural parameters of a prototype, for example, the thickness of thin walled structures being difficult to machine in a scaled down model, a distorted model is required to address the problem.In aspect of the vibration analysis of thin walled structures, Narita [3] analyzed natural characteristics of simply supported plates by using asymptotic expansions and the modified Ritz method. Wang et al. [4] investigated the free vibration characteristics under different boundary conditions based on the differential quadrature method. Irie et al. [5] analyzed the dynamic problems of conic shells with variable thickness, cone-column composite shells, and ring plate-column composite shells and put forward the transfer matrix method to analyze shells’ vibration characteristics under arbitrary boundary conditions. Papkov and Banerjee [6] presented a new method for the free vibration and buckling analysis of rectangular orthotropic plates, which allowed obtaining practically exact results for the free vibration and buckling of orthotropic plates.The scaling laws of thin walled structures have been investigated by many researchers. Krayterman and Sabnis [7] investigated the scaling laws of a plate by using dimensional analysis method and numerically discussed the accuracy in determining scaling laws. Qian and Swanson [8] obtained the scaling laws of laminated plates with the impulse response, and the scaling laws could accurately describe the undamaged response. Rezaeepazhand and Simitses [9–11] studied the dynamic behaviors of similitude models in predicting the buckling and free vibration of a laminated shell. Wu et al. [12, 13] presented scaling laws for the prediction of vibration characteristics using a scaled model by the similitude theory and dimensional analysis. Ungbhakorn and Singhatanadgid [14–16] proposed a new approach of deducing scaling laws for stability and presented scaling laws of symmetrically laminated plates and cylindrical shells by considering the loads effects. Oshiro and Alves [17, 18] employed a constitutive law to establish the model and the model reflected exactly as the prototype by changing the impact velocity. Oshiro and Alves [19] investigated the geometrically distorted scaling law in order to predict the behaviors of the prototype, and three problems of the prototype were analyzed. Yazdi [20] obtained the scaling laws that can predict the nonlinear vibration frequency of the prototype, and vibration amplitudes were investigated. De Rosa et al. [21, 22] derived scaling laws for the dynamic response of rectangular flexural plates by the modal approach; the forced response and energy response were given out. In their studies, the structural similitude was investigated by the energy distribution method that allows the representation of all the fundamental parameters. Luo et al. [23, 24] investigated the similitude design of an elastic cantilever thin plate by sensitivity analysis, and the distorted scaling laws and size applicable intervals were determined. Further, the prediction of the vibration characteristics of an isotropic sandwich plate was investigated.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 Section2 of this paper, the governing equation of typical thin walled structures is firstly unified. Geometrically complete scaling laws are deduced, and three principles of determining accurate distorted scaling law are proposed and proved in Section 3. Furthermore, taking the thin walled annular plate as an example, the distorted scaling laws are obtained in Section 4. Finally, the similitude design method is validated via experimental annular plates.


## 2. Governing Equation

The curved coordinate systemO′αβγ is established in the Cartesian coordinated system Oxyz and depicted in Figure 1. α and β are along with the curvature direction; γ is perpendicular to the directions α and β. There are five internal force components Nα, Nβ, Nαβ, Qα, and Qβ and three inner torque components Mα, Mβ, and Mαβ along with the arc edges of the surface Oαβ. u, v, and w represent the tangential displacements of directions α, β, and γ, respectively.Figure 1
The coordinate system.The material parameterE is Young’s modulus, ρ is density, and μ is Poisson’s ratio.The arc lengthdr of curved surfaces can be denoted as [25](1)dr=ds2=A2dα2+B2dβ2,where A and B are Lamé parameters, A=∂x/∂α2+∂y/∂α2+∂z/∂α2 and B=∂x/∂β2+∂y/∂β2+∂z/∂β2.The internal forces of the infinitesimal surface can be expressed as(2)Nα=Kεα0+μεβ0,Nβ=Kεβ0+μεα0,Nαβ=Nβα=K1-μ2εαβ0,Mα=Dχα+μχβ,Mβ=Dχβ+μχα,Mαβ=Mβα=D1-μ2χαβ,where K is the membrane stiffness, K=Eh/(1-μ2); Nα and Nβ are membrane stresses; Nαβ is the shear stress; D is bending stiffness, D=Eh3/121-μ2; Mx and Mθ are the bending moments; Mxθ is the torque; εα0, εβ0, and εαβ0 are strain components, and χα, χβ, and χαβ are curvature components, which are as follows:(3)εα0=1A∂u∂α+vAB∂A∂β+wRα,εβ0=1B∂v∂β+uAB∂B∂α+wRβ,εαβ0=AB∂∂βuA+BA∂∂αvB,χα=1A∂∂αuRα-1A∂w∂α+1ABvRβ-1B∂w∂β∂A∂β,χβ=1B∂∂βvRβ-1B∂w∂β+1ABuRα-1A∂w∂α∂B∂α,χαβ=AB∂∂βuARα-1A2∂w∂α+BA∂∂αvBRβ-1B2∂w∂β,where Rα and Rβ are main curvature radiuses.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 directionsα, β, and γ, respectively, which yields(4)∂BNα∂α+∂ANβα∂β-∂B∂αNβ+∂A∂βNαβ+ABQαRα-ABρh∂2u∂t2=0,∂ANβ∂β+∂BNαβ∂α-∂A∂βNα+∂B∂αNβα+ABQβRβ-ABρh∂2v∂t2=0,∂BQα∂α+∂AQβ∂β-ABNαRα-ABNβRβ-ABρh∂2w∂t2=0,where the relationships of the shear force Qα and Qβ can be expressed as(5)1AB∂AMβ∂β+∂BMαβ∂α-∂A∂βMα+∂B∂αMβα=Qβ,1AB∂BMα∂α+∂AMβα∂β-∂B∂αMβ+∂A∂βMαβ=Qα,Nαβ-Nβα+MαβRα-MβαRβ=0.By using (4) and (5), the governing equation can be written as(6a)∂BNα∂α+∂ANβα∂β-∂B∂αNβ+∂A∂βNαβ+1Rα∂BMα∂α+∂AMβα∂β-∂B∂αMβ+∂A∂βMαβ-ABρh∂2u∂t2=0,(6b)∂ANβ∂β+∂BNαβ∂α-∂A∂βNα+∂B∂αNβα+1Rβ∂AMβ∂β+∂BMαβ∂α-∂A∂βMα+∂B∂αMβα-ABρh∂2v∂t2=0,(6c)∂∂α1A∂BMα∂α+∂AMβα∂β-∂B∂αMβ+∂A∂βMαβ+∂∂β1B∂AMβ∂β+∂BMαβ∂α-∂A∂βMα+∂B∂αMβα-ABNαRα+NβRβ-ABρh∂2w∂t2=0.For the typical thin walled structures, Lamé parameter can be written asA=1 and B=R0. For thin walled cylindrical shells, R0=R; for the thin walled plates, R0=+∞. Substituting (2) and (3) into (6a), (6b), and (6c), the governing equation of thin walled structures can be denoted as(7a)-BDA2Rα∂3w∂α3-DBRα∂3w∂α∂β2+BAK+DRα2∂2u∂α2+A1-μ2BK+DRα2∂2u∂β2+1+μK2+D2RαRβ∂2v∂α∂β+BK1Rα+μRβ∂w∂α-ABρh∂2u∂t2=0,(7b)-DARβ∂3w∂α2∂β-ADB2Rβ∂3w∂β3+1+μ2K+DRαRβ∂2u∂α∂β+B1-μ2AK+D2Rβ2∂2v∂α2+ABK+DRβ2∂2v∂β2+AKμRα+1Rβ∂w∂β-ABρh∂2v∂t2=0,(7c)-BDA3∂4w∂α4-2μDAB∂4w∂α2∂β2-2D1-μAB∂4w∂α2∂β2-ADB3∂4w∂β4+BDA2Rα∂3u∂α3+DBRα∂3u∂α∂β2+DARβ∂3v∂α2∂β+ADB2Rβ∂3v∂β3-BKRα+μBKRβ∂u∂α+μAKRα+AKRβ∂v∂β+ABKRα2+2μABKRαRβ+ABKRβ2w-ABρh∂2w∂t2=0.From (7a), (7b), and (7c), we can know that the highest order’s derivation of the displacements u, v, and w with α and β is 4. Therefore, the governing equation of typical thin walled structures can be summarized as(8)∑j=u,v,w∑k=1m=4∑i=0n=kLj-ki∂kj∂αk-i∂βi+∑j=u,v,wLj∂2j∂t2=0,where j represents the displacements u, v, and w; Lj-ki and Lj are corresponding coefficients; t is time.


## 3. Derivation of Scaling Laws

### 3.1. Geometrically Complete Scaling Law

The governing equation of the prototype and model can be written as(9a)∑j=u,v,w∑k=1m=4∑i=0n=kLjp-ki∂kjp∂αpk-i∂βpi=∑j=u,v,wLjp∂2jp∂t2,(9b)∑j=u,v,w∑k=1m=4∑i=0n=kLjm-ki∂kjm∂αmk-i∂βmi=∑j=u,v,wLjm∂2jm∂t2,where subscript p denotes the prototype and subscript m denotes the model.The deflection equation can be denoted as(10)jα,β,t=Jα,βeiωt,where ω is the natural frequency; J represents mode functions U, V, and W.Substituting scaling factorsλe=(em/ep)(e=j,Lj-ki,α,β,Lj,J,etc.) into (9a) yields(11)∑j=u,v,w∑k=14∑i=0kλLj-kiλJλαk-iλβiLjm-ki∂kJm∂αmk-i∂βmi=∑j=u,v,wλLjλJλω2Ljm∂2Jm∂t2.According to the similitude theory, the corresponding coefficient of a prototype’s governing equation is proportional to the coefficient of a model, which means(12)λLj-kiλJλαk-iλβi=λLjλJλω2.Under the condition of geometrically complete similitude, this yields(13)λα=λRβλβ=λh=λ,λμ=1,λLj,kiλLj=λDλρλh=λKλρλhλ2=λEλρλ2,λαk-iλβi=λ4.Substituting (13) into (12) yields(14)λω=λLj-kiλLj1λαk-iλβi=λEλρλ21λ4=1λλEλρ.Therefore, geometrically complete scaling law can be denoted as(15)λω=1λλEλρ.


### 3.2. Distorted Scaling Law

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 [26].In addition, (12) can be written as(16)λω=λLj-kiλLjλαk-iλβi.Therefore, there are many possible candidate distorted scaling laws asi=0,1,…,4. According to (7a)–(7c), scaling factors λLj-ki=λEmλαo′λβs′λhg and λLj=λρ-nλhg′ are obtained, and the distorted scaling law can be denoted as(17)λω=λEmλρnλαoλβsλhq.In general, the indexesm and n are explicit and can be directly determined according to the governing equation. However, indexes o,s, and q may be implicit in the governing equation [23, 24]. For example, for the distorted scaling law of the thin walled annular plates, the scaling factors of outer and inner radiuses are hiding in the radius scaling factor λr in (31). In the investigation, a new method is proposed to determine the accurate distorted scaling law by combining the governing equation and the sensitivity analysis.


### 3.3. The Principles for Dynamic Scaling Laws

The sensitivity is the change rate of structural characteristic parameters with respect to structural parameters [27, 28]. Three basic principles are proposed and theoretically proved in order to determine the accurate distorted scaling law based on the sensitivity analysis.Principle 1.
In distorted scaling laws, if the scaling factorλj is explicit and presented in the governing equation, the index of scaling factor λj can be directly determined.Proof.
From (7a), (7b), and (7c), scaling factors of parameters E and ρ are explicit and reflected in the governing equation, so scaling factors’ indexes of the parameters E and ρ are directly determined. For example, in the numerical validation, parameters E and ρ are directly presented in (29). Therefore, the scaling factors λE and λρ can be directly determined. However, in the governing equation (29), the scaling factors λa and λb are coupled in the scaling factor λr and the scaling factors λa and λb could not be directly determined and need to employ the following principles.Principle 2.
When sensitivity’s absolute values satisfyΦ1>Φ2>⋯>Φn, the index relation of the scaling factor λnkn is k1>k2>⋯>kn.
Φ
n is the sensitivity of the natural frequency with respect to the structural parameter n; kn is the index of the scaling factor λnkn.Proof.
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(18)λj=jtjpjmjt=λj,p-tλj,t-m,where subscript t denotes the transitional model; a transitional model can be defined as one structural parameter of the prototype changes in a small range, and other parameters remain unchanged; λj,p-t is the scaling factor between the prototype and transitional model; λj,t-m is the scaling factor between the transitional model and distorted model.
Equation (17) can be written as(19)λω=λEA′λρB′λα,p-toλβ,p-tsλh,p-tqλα,t-moλβ,t-msλh,t-mq.
According to the geometrically complete scaling law, the transitional model and distorted model should satisfyλα,t-m=λβ,t-m=λh,t-m=λ. Therefore, (19) can be expressed as(20)λω=λEA′λρB′λα,p-toλβ,p-tsλh,p-tqλα,t-moλβ,t-msλh,t-mq=λEA′λρB′λα,p-toλβ,p-tsλh,p-tqλo+s+q.
Ifλjt→λjp=1, the sensitivity of a prototype can be denoted as(21)Φjp=dωjdλj=ωp-ωjtλp-λjt=ωp-ωjt1-λjt,where Φjp is the sensitivity of a prototype; ωjt denotes natural frequency of the transitional model.
IfΦαp=(ωp-ωαt)/(1-λαt)>Φβp=(ωp-ωβt)/(1-λβt) is satisfied, under the condition of λαt=λβt, λαω=ωαt/ωp>λβω=ωβt/ωp can be obtained from (21).
Therefore,λαω>λβω can be obtained under the condition of λαt=λβt.
By combining (19), the relation o>s of scaling factors λα,p-to and λβ,p-ts can be deduced if Φαp>Φβp is satisfied.
According to the recursive relation, ifΦ1>Φ2>⋯>Φn is satisfied when there are many parameters, the index relation of scaling factors λnkn should be satisfied as k1>k2>⋯>kn, so Principle 2 is proved.Principle 3.
If the sensitivityΦjp>0, λj is positively proportional to λω in the distorted scaling law; conversely, λj is inversely proportional to λω if the sensitivity Φjp<0.Proof.
If the sensitivityΦjp>0, two cases are as follows.
(1) If the scaling factorλj is satisfied, λj,p-t=jt/jp<λjp=1, then the natural frequency is satisfied as ωj,p-t<ωp according to (21). Further, λω=ωj,p-t/ωp<1 is obtained. Therefore, the scaling factor λω will reduce if the scaling factor λj decreases. That is, λj is positively proportional to λω in the distorted scaling law.
(2) Ifλj,p-t=jt/jp>λjp=1, then ωj,p-t>ωp. As a result of case (1), λj is positively proportional to λω in the distorted scaling law.
When the sensitivityΦjp<0, there are similar cases and Principle 3 is proved in the same way.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.Additional Principle. In the distorted scaling law, the index ratio k1:k2:⋯:kn of the scaling factor λnkn is approximate to the ratio of the sensitivity values(22)k1:k2:⋯:kn≈Φ1:Φ2:⋯:Φn.Proof.
When the structural parametersα and β change within the limit range (λαt,λβt∈[1-ε,1+ε]; ε is indefinitely small), the distorted scaling laws (20) can be written as(23)λαω,p-t=ωα,p-tωp=λα,p-to/2,λβω,p-t=ωβ,p-tωp=λβ,p-ts/2.
The sensitivity of natural frequency with respect to parametersα and β can be expressed as(24)Φα=dωdλα=ΔωΔλα=ωp-ωα,p-t1-λα,p-t,Φβ=dωdλβ=ΔωΔλβ=ωp-ωβ,p-t1-λβ,p-t.
Therefore, this yields(25)ΦαΦβ=ωp-ωαt1-λαt1-λβ,p-tωp-ωβ,p-t=ωα,p-tωp/ωα,p-t-11-λα,p-t1-λβ,p-tωp/ωβ,p-t-1ωβ,p-t.
Substituting (23) into (25) yields(26)ΦαΦβ=ωα,p-tωβ,p-tλα,p-to/2-11-λα,p-t1-λβ,p-tλβ,p-ts/2-1.
When scaling factors change within the limit range, that is,λαt→1, λβt→1, there are(27)limλα,p-t→1,λβ,p-t→1ωα,p-tωβ,p-t=1,limλα,p-t→1λα,p-to/2-11-λα,p-t=-o2,limλβ,p-t→11-λβ,p-tλβ,p-ts/2-1=-2s.
So (26) can be denoted as(28)ΦαΦβ=ωαtωβt-o2-2s=os.
When scaling factors change in the little range, this yields(29)ΦαΦβ≈os.
In the same way, if the dynamic characteristics of thin walled structures are affected by many parameters, (22) can be obtained according to the recursive relation. Consequently, the additional principle is proved.Therefore, the basic principles can be summarized as follows:(1)
In distorted scaling laws, if the scaling factorλj is explicit and presented in the governing equation, the index of scaling factor λj can be directly determined.(2)
When sensitivity’s absolute values satisfyΦ1>Φ2>⋯>Φn, the index relation of the scaling factor λnkn is k1>k2>⋯>kn and k1:k2:⋯:kn≈Φ1:Φ2:⋯:Φn in the distorted scaling law.(3)
If the sensitivityΦjp>0, λj is positively proportional to λω in the distorted scaling law; conversely, λj is inversely proportional to λω if the sensitivity Φjp<0.Finally, the procedure of similitude design method of typical thin walled structures is given out.Step 1.
Deducing complete scaling law of typical thin walled structures based on the governing equation.Step 2.
Assuming distorted scaling laws according to the governing equation of typical thin walled structures.Step 3.
Analyzing the sensitivity of the natural frequency with respect to structural parameters.Step 4.
According to the principles for dynamic scaling laws, determining the distorted scaling law of typical thin walled structures.


## 3.1. Geometrically Complete Scaling Law

The governing equation of the prototype and model can be written as(9a)∑j=u,v,w∑k=1m=4∑i=0n=kLjp-ki∂kjp∂αpk-i∂βpi=∑j=u,v,wLjp∂2jp∂t2,(9b)∑j=u,v,w∑k=1m=4∑i=0n=kLjm-ki∂kjm∂αmk-i∂βmi=∑j=u,v,wLjm∂2jm∂t2,where subscript p denotes the prototype and subscript m denotes the model.The deflection equation can be denoted as(10)jα,β,t=Jα,βeiωt,where ω is the natural frequency; J represents mode functions U, V, and W.Substituting scaling factorsλe=(em/ep)(e=j,Lj-ki,α,β,Lj,J,etc.) into (9a) yields(11)∑j=u,v,w∑k=14∑i=0kλLj-kiλJλαk-iλβiLjm-ki∂kJm∂αmk-i∂βmi=∑j=u,v,wλLjλJλω2Ljm∂2Jm∂t2.According to the similitude theory, the corresponding coefficient of a prototype’s governing equation is proportional to the coefficient of a model, which means(12)λLj-kiλJλαk-iλβi=λLjλJλω2.Under the condition of geometrically complete similitude, this yields(13)λα=λRβλβ=λh=λ,λμ=1,λLj,kiλLj=λDλρλh=λKλρλhλ2=λEλρλ2,λαk-iλβi=λ4.Substituting (13) into (12) yields(14)λω=λLj-kiλLj1λαk-iλβi=λEλρλ21λ4=1λλEλρ.Therefore, geometrically complete scaling law can be denoted as(15)λω=1λλEλρ.


## 3.2. Distorted Scaling Law

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 [26].In addition, (12) can be written as(16)λω=λLj-kiλLjλαk-iλβi.Therefore, there are many possible candidate distorted scaling laws asi=0,1,…,4. According to (7a)–(7c), scaling factors λLj-ki=λEmλαo′λβs′λhg and λLj=λρ-nλhg′ are obtained, and the distorted scaling law can be denoted as(17)λω=λEmλρnλαoλβsλhq.In general, the indexesm and n are explicit and can be directly determined according to the governing equation. However, indexes o,s, and q may be implicit in the governing equation [23, 24]. For example, for the distorted scaling law of the thin walled annular plates, the scaling factors of outer and inner radiuses are hiding in the radius scaling factor λr in (31). In the investigation, a new method is proposed to determine the accurate distorted scaling law by combining the governing equation and the sensitivity analysis.


## 3.3. The Principles for Dynamic Scaling Laws

The sensitivity is the change rate of structural characteristic parameters with respect to structural parameters [27, 28]. Three basic principles are proposed and theoretically proved in order to determine the accurate distorted scaling law based on the sensitivity analysis.Principle 1.
In distorted scaling laws, if the scaling factorλj is explicit and presented in the governing equation, the index of scaling factor λj can be directly determined.Proof.
From (7a), (7b), and (7c), scaling factors of parameters E and ρ are explicit and reflected in the governing equation, so scaling factors’ indexes of the parameters E and ρ are directly determined. For example, in the numerical validation, parameters E and ρ are directly presented in (29). Therefore, the scaling factors λE and λρ can be directly determined. However, in the governing equation (29), the scaling factors λa and λb are coupled in the scaling factor λr and the scaling factors λa and λb could not be directly determined and need to employ the following principles.Principle 2.
When sensitivity’s absolute values satisfyΦ1>Φ2>⋯>Φn, the index relation of the scaling factor λnkn is k1>k2>⋯>kn.
Φ
n is the sensitivity of the natural frequency with respect to the structural parameter n; kn is the index of the scaling factor λnkn.Proof.
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(18)λj=jtjpjmjt=λj,p-tλj,t-m,where subscript t denotes the transitional model; a transitional model can be defined as one structural parameter of the prototype changes in a small range, and other parameters remain unchanged; λj,p-t is the scaling factor between the prototype and transitional model; λj,t-m is the scaling factor between the transitional model and distorted model.
Equation (17) can be written as(19)λω=λEA′λρB′λα,p-toλβ,p-tsλh,p-tqλα,t-moλβ,t-msλh,t-mq.
According to the geometrically complete scaling law, the transitional model and distorted model should satisfyλα,t-m=λβ,t-m=λh,t-m=λ. Therefore, (19) can be expressed as(20)λω=λEA′λρB′λα,p-toλβ,p-tsλh,p-tqλα,t-moλβ,t-msλh,t-mq=λEA′λρB′λα,p-toλβ,p-tsλh,p-tqλo+s+q.
Ifλjt→λjp=1, the sensitivity of a prototype can be denoted as(21)Φjp=dωjdλj=ωp-ωjtλp-λjt=ωp-ωjt1-λjt,where Φjp is the sensitivity of a prototype; ωjt denotes natural frequency of the transitional model.
IfΦαp=(ωp-ωαt)/(1-λαt)>Φβp=(ωp-ωβt)/(1-λβt) is satisfied, under the condition of λαt=λβt, λαω=ωαt/ωp>λβω=ωβt/ωp can be obtained from (21).
Therefore,λαω>λβω can be obtained under the condition of λαt=λβt.
By combining (19), the relation o>s of scaling factors λα,p-to and λβ,p-ts can be deduced if Φαp>Φβp is satisfied.
According to the recursive relation, ifΦ1>Φ2>⋯>Φn is satisfied when there are many parameters, the index relation of scaling factors λnkn should be satisfied as k1>k2>⋯>kn, so Principle 2 is proved.Principle 3.
If the sensitivityΦjp>0, λj is positively proportional to λω in the distorted scaling law; conversely, λj is inversely proportional to λω if the sensitivity Φjp<0.Proof.
If the sensitivityΦjp>0, two cases are as follows.
(1) If the scaling factorλj is satisfied, λj,p-t=jt/jp<λjp=1, then the natural frequency is satisfied as ωj,p-t<ωp according to (21). Further, λω=ωj,p-t/ωp<1 is obtained. Therefore, the scaling factor λω will reduce if the scaling factor λj decreases. That is, λj is positively proportional to λω in the distorted scaling law.
(2) Ifλj,p-t=jt/jp>λjp=1, then ωj,p-t>ωp. As a result of case (1), λj is positively proportional to λω in the distorted scaling law.
When the sensitivityΦjp<0, there are similar cases and Principle 3 is proved in the same way.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.Additional Principle. In the distorted scaling law, the index ratio k1:k2:⋯:kn of the scaling factor λnkn is approximate to the ratio of the sensitivity values(22)k1:k2:⋯:kn≈Φ1:Φ2:⋯:Φn.Proof.
When the structural parametersα and β change within the limit range (λαt,λβt∈[1-ε,1+ε]; ε is indefinitely small), the distorted scaling laws (20) can be written as(23)λαω,p-t=ωα,p-tωp=λα,p-to/2,λβω,p-t=ωβ,p-tωp=λβ,p-ts/2.
The sensitivity of natural frequency with respect to parametersα and β can be expressed as(24)Φα=dωdλα=ΔωΔλα=ωp-ωα,p-t1-λα,p-t,Φβ=dωdλβ=ΔωΔλβ=ωp-ωβ,p-t1-λβ,p-t.
Therefore, this yields(25)ΦαΦβ=ωp-ωαt1-λαt1-λβ,p-tωp-ωβ,p-t=ωα,p-tωp/ωα,p-t-11-λα,p-t1-λβ,p-tωp/ωβ,p-t-1ωβ,p-t.
Substituting (23) into (25) yields(26)ΦαΦβ=ωα,p-tωβ,p-tλα,p-to/2-11-λα,p-t1-λβ,p-tλβ,p-ts/2-1.
When scaling factors change within the limit range, that is,λαt→1, λβt→1, there are(27)limλα,p-t→1,λβ,p-t→1ωα,p-tωβ,p-t=1,limλα,p-t→1λα,p-to/2-11-λα,p-t=-o2,limλβ,p-t→11-λβ,p-tλβ,p-ts/2-1=-2s.
So (26) can be denoted as(28)ΦαΦβ=ωαtωβt-o2-2s=os.
When scaling factors change in the little range, this yields(29)ΦαΦβ≈os.
In the same way, if the dynamic characteristics of thin walled structures are affected by many parameters, (22) can be obtained according to the recursive relation. Consequently, the additional principle is proved.Therefore, the basic principles can be summarized as follows:(1)
In distorted scaling laws, if the scaling factorλj is explicit and presented in the governing equation, the index of scaling factor λj can be directly determined.(2)
When sensitivity’s absolute values satisfyΦ1>Φ2>⋯>Φn, the index relation of the scaling factor λnkn is k1>k2>⋯>kn and k1:k2:⋯:kn≈Φ1:Φ2:⋯:Φn in the distorted scaling law.(3)
If the sensitivityΦjp>0, λj is positively proportional to λω in the distorted scaling law; conversely, λj is inversely proportional to λω if the sensitivity Φjp<0.Finally, the procedure of similitude design method of typical thin walled structures is given out.Step 1.
Deducing complete scaling law of typical thin walled structures based on the governing equation.Step 2.
Assuming distorted scaling laws according to the governing equation of typical thin walled structures.Step 3.
Analyzing the sensitivity of the natural frequency with respect to structural parameters.Step 4.
According to the principles for dynamic scaling laws, determining the distorted scaling law of typical thin walled structures.


## 4. Numerical Validation

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 Figure2. a is outer radius, b is inner radius, and h is thickness. The coordinate system Orθz is established, and O is the center point of the end surfaces, r is radial displacement, θ is deflection angle, and z is axial displacement. Displacement u is along direction r, displacement v is along direction θ, and displacement w is along direction z.Figure 2
The thin walled annular plate.According to (8), the governing equation of thin walled annular plates can be written as(30)Lw-40∂4W∂r4+Lw-42∂4W∂r2∂θ2+Lw-44∂4W∂θ4+Lw-30∂3W∂r3+Lw-31∂3W∂r∂θ2+Lw-20∂2W∂r2+LwW=0,where Lw-40=1, Lw-42=2/r2, Lw-44=1/r4, Lw-30=2/r, Lw-31=2/r3, Lw-20=1/r2, and Lw=12ρhω21-μ2/Eh2.The geometric sizes and material parameters of the thin walled annular plate are listed in Table1; the boundary condition is clamped on the inner radius edge and free on the outer radius boundary. For vibration experiments, natural frequencies should be proportional between the distorted model and prototype under the condition of the same vibration modes. By using ANSYS, 20-node element SOLID186 is selected to build the thin walled annular plate. Then, analyze its free vibration characteristics, and the previous five orders’ natural frequencies and vibration modes are listed in Table 2. m is circumferential half wave number and n is radial half wave number.Table 1
The geometric sizes and material parameters.
Outer radius
Inner radius
Thickness
Young’s modulus
Density
Poisson’s ratio
a/mm
b/mm
h/mm
E/GPa
ρ/kg⋅m−3
μ
100
12
6
210
8000
0.3Table 2
The dynamic characteristics of the prototype.
Orderi
1
2
3
4
5
Natural frequencyω/Hz
541.90
647.94
836.11
1819.9
3168.1
Vibration modes
m
=
2, n=1
m
=
1, n=1
m
=
4, n=1
m
=
6, n=1
m
=
8, n=1


### 4.1. Geometrically Complete Scaling Law

According to the governing equation (30), it yields(31)λWλr4=λρλω2λEλh2λW.Under the condition of geometrically complete similitude, scaling factors can be expressed as(32)λa=λb=λr=λh=λ,λμ=1.Therefore, geometrically complete scaling law of the thin walled annular plates is(33)λω=1λλEλρ.


### 4.2. Distorted Scaling Law

In general, scaling factorsλa and λb are independent in distorted scaling laws [29]. λr4=λa-oλb-s is considered in (31). In addition, from (31), o+s=-4 should be subjected to the distorted scaling law. Therefore, the distorted scaling law of thin walled annular plates can be written as(34)λω=λhλEλρλa-oλb-s=λhλEλaoλbsλρ,o+s=-4.From (34), the indexes of the scaling factors λE, λρ, and λh have been determined by the governing equation. However, the indexes o and s need to be determined by the sensitivity analysis.In order to obtain the sensitivity of the first order’s natural frequency with respect to the scaling factorλa, other sizes and material parameters remain unchanged and the first order’s natural frequencies of distorted models are listed in Table 3 by ANSYS simulation.Table 3
Natural frequencies of the 1st mode (2, 1) as a result of the distorted outer radius.
Outer radius
Scaling factor of
Natural frequencies
a/mm
outer radiusλa
ω
m/Hz
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
493.59In Figure3, the curve is fitted by natural frequencies and the quadratic equation is(35)ω1a=2270λa2-5837λa+4109.Figure 3
The fitted curve of outer radius distorted models.Furthermore, adopting the adjusted squareR-2 to verify the efficiency of the fitted curve, the adjusted square R-2 can be denoted as(36)R-2=1-n′-1n′-k+11-∑Y^i-Y-2∑Yi-Y-2i=1,2,…,n′,where n′ is fitted point numbers; k is the order of fitted polynomial; Y^ is the fitted value; Y- is the average value; Y is the true value.By comparing and analyzing, the fitted curve is thought to be effective and suitable ifR-2>0.99 in this paper.The adjusted square of the fitted curve isR-a2=0.999. So the fitted curve of the first order’s natural frequency is effective.The sensitivity of the natural frequency with respect to the scaling factorλa is(37)Φ1a=dω1adλaλa=1=4540λa-5837=-1297.In the same way, the fitted curve of the first order’s natural frequencies of inner radius distorted models is shown in Figure4.Figure 4
The fitted curve of inner radius distorted models.The fitted equation of the first order’s natural frequency is(38)ω1b=5.4λb2+221.5λb+315.In Figure4, the adjusted square of the fitted curve is R-b2=0.999, so the fitted curve is thought to be effective.The sensitivity of the first order’s natural frequency with respect to the scaling factorλb is(39)Φ1b=dω1bdλbλb=1=10.8λb+221.5=232.3.As a result of the sensitivity analysis,(40)Φa:Φb=dω1adλaλa=1:dω1bdλbλb=1=-1297:232.3.According to Principle2, this yields(41)o:s=-1297:232.3.From (34), o+s=-4 is satisfied.Solving simultaneous equations yields(42)o=-4.61,s=0.61.Therefore, the distorted scaling law of the first order’s natural frequency of thin walled annular plates can be written as(43)λ1ω=λhλEλb0.61λρλa4.61.


### 4.3. Numerical Validation

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 Table4.Table 4
Parameters of the distorted model.
Outer radius
Inner radius
Thickness
Young’s modulus
Density
Poisson’s ratio
a/mm
b/mm
h/mm
E/GPa
ρ/kg⋅m−3
μ
130
12
8
72
2700
0.3Similarly, the sensitivity of the natural frequencies with the scaling factorsλa and λb can be obtained. By using the sensitivity value, the previous five orders’ distorted scaling laws of thin walled annular plates are shown in Table 5.Table 5
The distorted scaling laws.
Orderi
SensitivityΦa
SensitivityΦb
Distorted scaling laws
Frequencies of modelωm/Hz
Predicted frequenciesωpr/Hz
Errorη/%
1
−1297
232.3
λ
ω
=
λ
h
λ
E
λ
b
0.61
λ
ρ
λ
a
4.61
387.56
527.33
2.69
2
−1437
144.9
λ
ω
=
λ
h
λ
E
λ
b
0.36
λ
ρ
λ
a
4.36
490.27
645.64
0.35
3
−1765
109.1
λ
ω
=
λ
h
λ
E
λ
b
0.23
λ
ρ
λ
a
4.23
647.17
837.91
0.22
4
−3621
10
λ
ω
=
λ
h
λ
E
λ
b
0.01
λ
ρ
λ
x
4.01
1445.2
1818.1
0.10
5
−6247
5
λ
ω
=
λ
h
λ
E
λ
ρ
λ
a
4
2517.5
3162.9
0.16The error between the natural frequency of the prototype and the predictive natural frequency can be denoted as(44)η=ωp-ωprωp×100%.Table5 depicts that errors between the natural frequencies of the prototype and predicted values are less than 5%, and their modes are the same. Therefore, the previous five orders’ distorted scaling laws can accurately predict the dynamic characteristics of a prototype.Γ is defined as Γ=λa/λb, and Ψ is defined as Ψ=λa/λh. Different distorted models are selected and the scaling factors are shown in Table 6. Taking the 1st frequency as an example, the 1st frequencies of distorted models are obtained by using ANSYS, and the error can be calculated by (44). Furthermore, the predicted frequencies and errors are shown in Table 6.Table 6
The errors between the distorted models and prototype.
Model
λ
a
λ
b
λ
h
Γ
Ψ
Frequencies of modelωm/Hz
Predicted 1st frequenciesωpr/Hz
Errorη/%
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


## 4.1. Geometrically Complete Scaling Law

According to the governing equation (30), it yields(31)λWλr4=λρλω2λEλh2λW.Under the condition of geometrically complete similitude, scaling factors can be expressed as(32)λa=λb=λr=λh=λ,λμ=1.Therefore, geometrically complete scaling law of the thin walled annular plates is(33)λω=1λλEλρ.


## 4.2. Distorted Scaling Law

In general, scaling factorsλa and λb are independent in distorted scaling laws [29]. λr4=λa-oλb-s is considered in (31). In addition, from (31), o+s=-4 should be subjected to the distorted scaling law. Therefore, the distorted scaling law of thin walled annular plates can be written as(34)λω=λhλEλρλa-oλb-s=λhλEλaoλbsλρ,o+s=-4.From (34), the indexes of the scaling factors λE, λρ, and λh have been determined by the governing equation. However, the indexes o and s need to be determined by the sensitivity analysis.In order to obtain the sensitivity of the first order’s natural frequency with respect to the scaling factorλa, other sizes and material parameters remain unchanged and the first order’s natural frequencies of distorted models are listed in Table 3 by ANSYS simulation.Table 3
Natural frequencies of the 1st mode (2, 1) as a result of the distorted outer radius.
Outer radius
Scaling factor of
Natural frequencies
a/mm
outer radiusλa
ω
m/Hz
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
493.59In Figure3, the curve is fitted by natural frequencies and the quadratic equation is(35)ω1a=2270λa2-5837λa+4109.Figure 3
The fitted curve of outer radius distorted models.Furthermore, adopting the adjusted squareR-2 to verify the efficiency of the fitted curve, the adjusted square R-2 can be denoted as(36)R-2=1-n′-1n′-k+11-∑Y^i-Y-2∑Yi-Y-2i=1,2,…,n′,where n′ is fitted point numbers; k is the order of fitted polynomial; Y^ is the fitted value; Y- is the average value; Y is the true value.By comparing and analyzing, the fitted curve is thought to be effective and suitable ifR-2>0.99 in this paper.The adjusted square of the fitted curve isR-a2=0.999. So the fitted curve of the first order’s natural frequency is effective.The sensitivity of the natural frequency with respect to the scaling factorλa is(37)Φ1a=dω1adλaλa=1=4540λa-5837=-1297.In the same way, the fitted curve of the first order’s natural frequencies of inner radius distorted models is shown in Figure4.Figure 4
The fitted curve of inner radius distorted models.The fitted equation of the first order’s natural frequency is(38)ω1b=5.4λb2+221.5λb+315.In Figure4, the adjusted square of the fitted curve is R-b2=0.999, so the fitted curve is thought to be effective.The sensitivity of the first order’s natural frequency with respect to the scaling factorλb is(39)Φ1b=dω1bdλbλb=1=10.8λb+221.5=232.3.As a result of the sensitivity analysis,(40)Φa:Φb=dω1adλaλa=1:dω1bdλbλb=1=-1297:232.3.According to Principle2, this yields(41)o:s=-1297:232.3.From (34), o+s=-4 is satisfied.Solving simultaneous equations yields(42)o=-4.61,s=0.61.Therefore, the distorted scaling law of the first order’s natural frequency of thin walled annular plates can be written as(43)λ1ω=λhλEλb0.61λρλa4.61.


## 4.3. Numerical Validation

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 Table4.Table 4
Parameters of the distorted model.
Outer radius
Inner radius
Thickness
Young’s modulus
Density
Poisson’s ratio
a/mm
b/mm
h/mm
E/GPa
ρ/kg⋅m−3
μ
130
12
8
72
2700
0.3Similarly, the sensitivity of the natural frequencies with the scaling factorsλa and λb can be obtained. By using the sensitivity value, the previous five orders’ distorted scaling laws of thin walled annular plates are shown in Table 5.Table 5
The distorted scaling laws.
Orderi
SensitivityΦa
SensitivityΦb
Distorted scaling laws
Frequencies of modelωm/Hz
Predicted frequenciesωpr/Hz
Errorη/%
1
−1297
232.3
λ
ω
=
λ
h
λ
E
λ
b
0.61
λ
ρ
λ
a
4.61
387.56
527.33
2.69
2
−1437
144.9
λ
ω
=
λ
h
λ
E
λ
b
0.36
λ
ρ
λ
a
4.36
490.27
645.64
0.35
3
−1765
109.1
λ
ω
=
λ
h
λ
E
λ
b
0.23
λ
ρ
λ
a
4.23
647.17
837.91
0.22
4
−3621
10
λ
ω
=
λ
h
λ
E
λ
b
0.01
λ
ρ
λ
x
4.01
1445.2
1818.1
0.10
5
−6247
5
λ
ω
=
λ
h
λ
E
λ
ρ
λ
a
4
2517.5
3162.9
0.16The error between the natural frequency of the prototype and the predictive natural frequency can be denoted as(44)η=ωp-ωprωp×100%.Table5 depicts that errors between the natural frequencies of the prototype and predicted values are less than 5%, and their modes are the same. Therefore, the previous five orders’ distorted scaling laws can accurately predict the dynamic characteristics of a prototype.Γ is defined as Γ=λa/λb, and Ψ is defined as Ψ=λa/λh. Different distorted models are selected and the scaling factors are shown in Table 6. Taking the 1st frequency as an example, the 1st frequencies of distorted models are obtained by using ANSYS, and the error can be calculated by (44). Furthermore, the predicted frequencies and errors are shown in Table 6.Table 6
The errors between the distorted models and prototype.
Model
λ
a
λ
b
λ
h
Γ
Ψ
Frequencies of modelωm/Hz
Predicted 1st frequenciesωpr/Hz
Errorη/%
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


## 5. Experiments Validation

In the experiment, test setup of the experimental plate is shown in Figure5 and parameters of the experimental plates are listed in Table 7.Table 7
Parameters of experimental plates.
Outer radius
Inner radius
Thickness
Young’s modulus
Poisson’s ratio
Density
a/mm
b/mm
h/mm
E/GPa
μ
ρ
/kg⋅m−3
Prototype
100
12
6
210
0.3
8000
Model
130
12
8
72
0.3
2700Figure 5
Test setup of the experimental plate.In addition, Table8 depicts the vibration modes of the prototype and model. Errors between the predicted values and experimental values of the prototype are listed in Table 8.Table 8
The comparison of experimental and predicted results.
Ordersi
Modelωm/Hz
Vibration modes
Prototypeωp/Hz
Vibration modes
Modes
Predicted frequenciesωpr/Hz
Errorsη/%
1
639.34
862.31
m
=
4, n=1
827.77
4.01
2
1431.5
1842.5
m
=
6, n=1
1798.5
2.39
3
2512.8
3203.6
m
=
8, n=1
3157.1
1.45From Table8, errors between the previous five orders’ natural frequencies of experimental prototype and predicted values are less than 5%, and vibration modes of experimental prototype and model are the same.The experimental errors are higher than the numerical errors; the reasons are analyzed as follows:(1)
The machining of the thin walled annular plates, such as the dimensional error and deviation of material parameters.(2)
The measurement accuracy of a test system, for example, the precision of the sensor.(3)
The random error of the test procession.


## 6. Conclusions

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:(1)
The governing equation of typical thin walled structures is unified in (8).(2)
By employing the governing equation, geometrically complete scaling law of typical thin walled structures is obtained.(3)
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.(4)
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.



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*Source: 6836183-2015-12-29.xml*