Performance evaluation for the domain decomposition method in nonlinear vibration of the composite hard-coating cylindrical shell

Abstract

Introduction

The applications of the modified domain decomposition method in nonlinear vibration analysis of the composite hard-coating cylindrical shells are still at a relatively superficial level, owing to the fact that its performance under different decomposition parameters has not been thoroughly investigated for achieving sufficient precision.

Methods

A parametric domain decomposition method is developed to facilitate self-performance evaluation in nonlinear vibration analysis of the shell. Correspondingly, in order to avoid a mass of redundant computation of the segment stiffness and material damping matrices during iterations, a specialized preprocessing scheme is designed by pre-establishing the parametric analytical expressions and matrix databases.

Results

The resonant response is sensitive to the circumferential segment number, but weakly affected by the axial segment number. The optimum circumferential segment number in the present study is suggested to be N_θ = 70, which can achieve good calculation accuracy and efficiency. Highly consistency is shown for the distributions of axial equivalent strain under different axial segment numbers. Smaller circumferential segment numbers would result in larger equivalent strain and bad solution accuracy.

Conclusions

The sufficient solution accuracy of nonlinear vibration of the composite hard-coating cylindrical shell can't be achieved by increasing the axial segment number with constant segment width, but only by enough circumferential segment number, which is fundamentally determined by its equivalent strain distributions and gradients, and is with close relation to the axial and circumferential wave numbers of the shell.

Introduction

As the principal structural element in submarines and airspace crafts, cylindrical shell structures are often subjected to complex multi-field coupling loads, further leading to excessive vibration and low fatigue life.^1–4 Hard-coating damping treatment is a new passive damping vibration control method developed in recent years with widespread concern.^5,6 The material of the hard coating is composed mainly of the metal substrate, ceramic substrate, or their mixtures, exhibiting a high damping capacity with internal friction even in high temperature and corrosion conditions, thus has been employed in an attempt to suppress excessive vibration of the cylindrical shell structures in many studies.^7–9 However, previous works showed that,^10–12 the physical parameters (e.g. storage modulus and loss factor) of hard coating have a severe dependency on its equivalent strain intensity due to its internal friction, leading to strong nonlinear vibration behavior of the composite hard-coating cylindrical shell. Therefore, considering the so-called strain dependence of coating material in the modeling and solution of forced vibration of the composite hard-coating cylindrical shell is of significant practice meaning, which can provide precise control of damping vibration.

In recent years, the nonlinear vibrations of all kinds of shell structures have become one of the most important topics in the field of structural dynamics.^13–15 The nonlinear vibration mechanism of the shells subjected to temperature, rotating speed, and external excitations under different boundary conditions have been deeply clarified, which provides an important basis for structural vibration control.^16–19 Furthermore, many studies have been conducted on the nonlinear vibration analysis of composite hard-coating structures (e.g. beam, ²⁰ plate,^21–23 cylindrical shell,^24–28 blisk^29–34 and blade ³⁵ ) considering the strain dependence of coating material. Based on the parameter identification results of hard coating,^36,37 the high-order polynomial method was generally used to characterize the nonlinear characteristics of its storage modulus and loss factor with respect to the equivalent strain. Hence, the key point of introducing strain dependence into vibration modeling is to determine the equivalent strain of the layer of hard coating.

In the above studies, two types of methods were involved the finite element and analytical formulas accordingly were presented to calculate the equivalent strain of the coating layer. Compared with the finite element methods (FEMs), the introduction of material nonlinearity of hard coating with analytical approaches is considerably more challenging, but it can provide access to the equivalent strain value at any point of the layer of hard coating. The existing analytical approaches can further be summarized into two categories including the pure analytical^21,27 and semi-analytical modes.^26,28 However, due to the high complexity of vibration theoretical formulas of the composite hard-coating cylindrical shell, the derived expressions of the equivalent strain of the layer of hard coating are extremely complicated as well, while the computational efficiency of the nonlinear vibration with the pure analytical modes is relatively low. Then the so-called semi-analytical modes (namely the modified domain decomposition method) were proposed in our previous work ²⁶ for easy access to the equivalent strain values, which combines the features of finite element and analytical methods by uniformly dividing the layer of hard coating into multiple segments with the assumption of constant equivalent strain. This method has also been successfully applied to include the strain dependence of coating material in equations of motion for forced vibration of the hard-coating cylindrical shell with rotating angular velocity. ²⁸ Nevertheless, the existing applications of the domain decomposition method in solving the nonlinear vibration of the composite hard-coating cylindrical shells with material strain dependence are still in a relatively superficial level. There is lack of a detailed and comprehensive performance evaluation of this method for achieving high efficiency and sufficient precision.

In order to solve the above problems, a parametric domain decomposition method with specialized preprocessing scheme is developed to facilitate self-performance evaluation in nonlinear vibration analysis. Then, based on the semi-analytical modeling of nonlinear vibration of a clamped-free hard-coating cylindrical shell established by the Rayleigh-Ritz method, the performance evaluation for the modified domain decomposition method under different segment parameters is comprehensively carried out to obtain the optimal decomposition scheme. A series of comparison studies are performed to investigate the effects of segment parameters (including the axial and circumferential segment number) on the nonlinear vibration characteristics of the composite hard-coating cylindrical shell.

Methods

Modeling of forced vibration of the clamped-free hard coating cylindrical shell

In order to evaluate the performance of the modified domain decomposition method in nonlinear vibration of the composite hard-coating cylindrical shell, the geometry configuration with clamped-free boundary and sinusoidal acceleration load in the orthogonal curvilinear coordinate system (xθz) is considered here, as shown in Figure 1.

Figure 1.

Geometry of the composite hard-coating cylindrical shell.

By employing the Chebyshev orthogonal polynomials of the second kind, the axial, circumferential, and radial components of the sinusoidal acceleration load for the shell can be given, respectively, by ³⁸

$$\begin{array}{rl}& g_u=\sum _{m=1}^N{F}_m{T}_m(x)\cos (n\theta )\sin (\omega t)\\ & g_v=-\sum _{m=1}^N{F}_m{T}_m(x)\sin (n\theta )\sin (\omega t)\\ & g_w=\sum _{m=1}^N{F}_m{T}_m(x)\cos (n\theta )\sin (\omega t)\end{array}$$

(1)

with

$$F_m={\displaystyle \frac{2gM}{\pi }{\int }_0^L{T}_m(x)\sqrt{1-{(2x/L-1)}^2}\mathrm{d}x}$$

(2)

where $F_m$ represents the modal expansion coefficients of the sinusoidal acceleration load. $T_m(x)$ denotes the Chebyshev orthogonal series. N is the number of truncated terms of the Chebyshev orthogonal polynomials. M is the mass of the shell. m and n refer to the axial half-wave number and circumferential wave number, respectively. $\omega$ and t indicate the vibration angular frequency and time, respectively. L is the axial length of the shell.

Under the action of sinusoidal acceleration load, the displacements of the hard-coating cylindrical shell would change with time in a sine form. Thus, the axial, circumferential, and radial displacements at an arbitrary point of the middle surface of the shell can be given, respectively, by

$$\begin{array}{rl}& u=\sum _{m=1}^N{a}_m{T}_m(x)\cos (n\theta )\sin (\omega t)\\ & v=\sum _{m=1}^N{b}_m{T}_m(x)\sin (n\theta )\sin (\omega t)\\ & w=\sum _{m=1}^N{c}_m{T}_m(x)\cos (n\theta )\sin (\omega t)\end{array}$$

(3)

where $a_m$ , $b_m$ , and $c_m$ denote the vibration amplitudes in axial, circumferential and radial directions, respectively. On this basis, the kinetic energy $U_{\mathrm{k}}$ of the shell can be expressed as

$$\begin{array}{rl}& U_{\mathrm{k}}={\displaystyle \frac{1}{2}{\int }_0^L{\int }_0^{2\pi }{\int }_{z_0}^{z_2}({\rho }_{\mathrm{c}}{T}_{\mathrm{c}}+{\rho }_{\mathrm{s}}{T}_{\mathrm{s}})[{(\mathrm{\partial }u/\mathrm{\partial }t)}^2+{(\mathrm{\partial }v/\mathrm{\partial }t)}^2+{(\mathrm{\partial }w/\mathrm{\partial }t)}^2]\mathrm{d}x\mathrm{d}\theta \mathrm{d}z}\\ & ={\displaystyle \frac{1}{2}\pi {\omega }^{2}R({\rho }_{\mathrm{c}}{T}_{\mathrm{c}}+{\rho }_{\mathrm{s}}{T}_{\mathrm{s}}){\cos }^2(\omega t)\sum _{m=1}^N{\int }_0^L(a_m^2+b_m^2+c_m^2)T_m^2\mathrm{d}x}\end{array}$$

(4)

where ${\rho }_{\mathrm{c}}$ , $T_{\mathrm{c}}$ , and ${\rho }_{\mathrm{s}}$ , $T_{\mathrm{s}}$ are the density and thickness of the layers of hard coating and shell substrate, respectively. $z_0$ and $z_2$ are the radial coordinate values of the inner and the outer surfaces of the shell, respectively. R refers to the middle radius of the composite hard-coating cylindrical shell.

Similarly, the potential energy $U_{\mathrm{b}}$ due to the clamped-free boundary condition is given by

$$\begin{array}{r}{U}_{\mathrm{b}}={\displaystyle \frac{1}{2}{\int }_0^{2\pi }{[k_u{u}^2+k_v{v}^2+k_w{w}^2+k_t{(\mathrm{\partial }w/\mathrm{\partial }x)}^2]}_{x=0}R\mathrm{d}\theta }\\ ={\displaystyle \frac{1}{2}\pi R{\sin }^2(\omega t)\sum _{m=1}^N[(k_u{a}_m^2+k_v{b}_m^2+k_w{c}_m^2)T_m^2(0)+k_t{c}_m^2{T}_m^{\mathrm{\prime }2}(0)]}\end{array}$$

(5)

where $k_u$ , $k_v$ , $k_w$ , and $k_t$ represent the introduced spring stiffness for simulating the clamped boundary condition at the bottom of the shell ( $k_u=k_v=k_w=k_t=1\times {10}^{15}\mathrm{N}/{\mathrm{m}}^2$ ).

And the potential energy $U_{\mathrm{g}}$ from the external sinusoidal acceleration load is obtained by

$$\begin{array}{rl}& U_{\mathrm{g}}={\int }_0^L{{\int }_0^{2\pi }(g_{u}u+g_{v}v+g_{w}w)}_{x=0}R\mathrm{d}x\mathrm{d}\theta \\ & =\pi R{\sin }^2(\omega t)\sum _{m=1}^N[(a_m^2+b_m^2+c_m^2)F_m{T}_m^2(0)]\end{array}$$

(6)

According to the stress–strain relationship under plane stress state, the strain energy $U_{\epsilon }$ of the shell can be derived as the following matrix form

$$\begin{array}{rl}& U_{\epsilon }={\displaystyle \frac{1}{2}{\int }_0^L{\int }_0^{2\pi }{\int }_{z_0}^{z_2}({\sigma }_x{\epsilon }_x+{\sigma }_{\theta }{\epsilon }_{\theta }+{\sigma }_{x\theta }{\epsilon }_{x\theta })R\mathrm{d}x\mathrm{d}\theta \mathrm{d}z}\\ & ={\displaystyle \frac{1}{2}{\int }_0^L{\int }_0^{2\pi }{\int }_{z_0}^{z_2}\left\{\begin{array}{c}{\epsilon }_x\\ {\epsilon }_{\theta }\\ {\epsilon }_{x\theta }\\ {\kappa }_x\\ {\kappa }_{\theta }\\ {\kappa }_{x\theta }\end{array}\right\}\left[\begin{array}{cccccc}{G}_{11}^{\mathrm{\aleph }}& G_{12}^{\mathrm{\aleph }}& 0& z{G}_{11}^{\mathrm{\aleph }}& z{G}_{12}^{\mathrm{\aleph }}& 0\\ & G_{22}^{\mathrm{\aleph }}& 0& z{G}_{21}^{\mathrm{\aleph }}& z{G}_{22}^{\mathrm{\aleph }}& 0\\ & & G_{66}^{\mathrm{\aleph }}& 0& 0& z{G}_{66}^{\mathrm{\aleph }}\\ & & & z^2{G}_{11}^{\mathrm{\aleph }}& z^2{G}_{12}^{\mathrm{\aleph }}& 0\\ & \cdot \cdot \cdot & & & z^2{G}_{22}^{\mathrm{\aleph }}& 0\\ & & & & & z^2{G}_{66}^{\mathrm{\aleph }}\end{array}\right]{\left\{\begin{array}{c}{\epsilon }_x\\ {\epsilon }_{\theta }\\ {\epsilon }_{x\theta }\\ {\kappa }_x\\ {\kappa }_{\theta }\\ {\kappa }_{x\theta }\end{array}\right\}}^{\mathrm{T}}R\mathrm{d}x\mathrm{d}\theta \mathrm{d}z}\end{array}$$

(7)

where ${\epsilon }_x$ , ${\epsilon }_{\theta }$ , and ${\epsilon }_{x\theta }$ denote the normal and shear strains, ${\sigma }_x$ , ${\sigma }_{\theta }$ and ${\sigma }_{x\theta }$ refer to the normal and shear stresses. $G_{ij}^{\mathrm{\aleph }}$ ( $ij=11,22,66$ , $\mathrm{\aleph }=\mathrm{c},\mathrm{s}$ ) represents the reduced isotropic stiffness coefficients, defined by

$$G_{66}^{\mathrm{\aleph }}={\hat{E}}_{\mathrm{\aleph }}/2(1+{\mu }_{\mathrm{\aleph }}),G_{11}^{\mathrm{\aleph }}=G_{22}^{\mathrm{\aleph }}={\hat{E}}_{\mathrm{\aleph }}/(1-{\mu }_{\mathrm{\aleph }}^2),G_{12}^{\mathrm{\aleph }}=G_{21}^{\mathrm{\aleph }}={\hat{E}}_{\mathrm{\aleph }}{\mu }_{\mathrm{\aleph }}/(1-{\mu }_{\mathrm{\aleph }}^2)$$

(8)

where ${\hat{E}}_{\mathrm{\aleph }}$ and ${\mu }_{\mathrm{\aleph }}$ refer to the complex-valued Young's modulus and Poisson's ratio, respectively. Correspondingly, the subscripts c and s denote the layers of hard coating and shell substrate. According to the first-order shear deformation theory, ³⁹ the strains at an arbitrary point of the shell are given by

$$\begin{array}{rl}& {\epsilon }_x={\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}-z{\displaystyle \frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }{x}^2}}}\\ & {\epsilon }_{\theta }={\displaystyle \frac{1}{R}\left({\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }\theta }+w}\right)+{\displaystyle \frac{z}{R^2}\left({\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }\theta }-{\displaystyle \frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }{\theta }^2}}}\right)}}\\ & {\gamma }_{x\theta }^{rs}={\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }x}+{\displaystyle \frac{1}{R}{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }\theta }+{\displaystyle \frac{z}{R}\left({\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }x}-2{\displaystyle \frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }x\mathrm{\partial }\theta }}}\right)}}}}\end{array}$$

(9)

Substituting equation (3) into equation (7) one can obtain

$$\begin{array}{rl}& U_{\epsilon }={\displaystyle \frac{1}{2}\pi \sin (\omega t{)}^2\sum _{m=1}^N{\int }_0^L{\displaystyle \frac{1}{R^3}[R^4{a}_m^2{T}_m^{\mathrm{\prime }2}{A}_{11}+2R^3(b_{m}n+c_m)A_m{T}_m{T^{\mathrm{\prime }}}_m{A}_{12}}}\\ & +R^2(b_{m}n+c_m{)}^2{T}_m^2{A}_{22}+R^2(R{b}_m{T^{\mathrm{\prime }}}_m-a_m{T}_{m}n{)}^2{A}_{66}-2R^4{a}_m{c}_m{T^{\mathrm{\prime }}}_m{T^{\mathrm{″}}}_m{B}_{11}\\ & +2R^2[n(c_{m}n+b_m)a_m{T}_m{T^{\mathrm{\prime }}}_m-R(c_m+b_{m}n)c_m{T}_m{T^{\mathrm{″}}}_m]B_{12}\\ & +2Rn(c_{m}n+b_m)(c_m+b_{m}n)B_{22}{T}_m^2-2n{R}^2(c_{m}n+b_m)c_m{T}_m{T^{\mathrm{″}}}_m{D}_{12}\\ & +R^4{c}_m^2{T}_m^{\mathrm{″}2}{D}_{11}+2R^2(2c_{m}n+b_m)(R{b}_m{T^{\mathrm{\prime }}}_m-a_m{T}_{m}n){T^{\mathrm{\prime }}}_m{B}_{66}\\ & +n^2(c_{m}n+b_m{)}^2{T}_m^2{D}_{22}+R^2{(2c_{m}n+b_m)}^2{T}_m^{\mathrm{\prime }2}{D}_{66}]\mathrm{d}x\end{array}$$

(10)

where $A_{ij}$ , $B_{ij}$ , and $D_{ij}$ are the extensional, coupling, and bending stiffness defined, respectively, by

$$A_{ij},B_{ij},D_{ij}={\int }_{z_0}^{z_2}(G_{ij}^{\mathrm{\aleph }},z{G}_{ij}^{\mathrm{\aleph }},z^2{G}_{ij}^{\mathrm{\aleph }})\mathrm{d}z$$

(11)

In accordance with the well-known Rayleigh-Ritz method, ³⁸ the so-called Lagrangian energy function can be constructed by using the maximum of the above kinetic energy, potential energy, and strain energy. Then, the equations of motion of the composite hard-coating cylindrical shell can be achieved in the following matrix form, expressed as

$$(\hat{\mathit{K}}+{\mathit{K}}_{\mathrm{b}}-{\omega }^2\mathit{M})\mathit{\chi }=\mathit{F}$$

(12)

where $\hat{\mathit{K}}$ is the complex stiffness matrix, written by

$$\hat{\mathit{K}}=\mathit{K}+\mathrm{i}\mathit{D}\mathbf{=}\left[\begin{array}{ccc}{\hat{\mathit{K}}}_{11}& {\hat{\mathit{K}}}_{12}& {\hat{\mathit{K}}}_{13}\\ & {\hat{\mathit{K}}}_{22}& {\hat{\mathit{K}}}_{23}\\ \cdot \cdot \cdot & & {\hat{\mathit{K}}}_{33}\end{array}\right]$$

(13)

Here $\mathit{K}$ and $\mathit{D}$ represent the stiffness matrix and material damping matrix, respectively. Similarly, ${\mathit{K}}_{\mathrm{b}}$ and $\mathit{M}$ refer to the clamped stiffness matrix and mass matrix given, respectively, by

$${\mathit{K}}_{\mathrm{b}}=\left[\begin{array}{ccc}{\mathit{K}}_{\mathrm{b}}^{11}& & \\ & {\mathit{K}}_{\mathrm{b}}^{22}& \\ & & {\mathit{K}}_{\mathrm{b}}^{33}\end{array}\right],\mathit{M}=\left[\begin{array}{ccc}{\mathit{M}}_{11}& & \\ & {\mathit{M}}_{22}& \\ & & {\mathit{M}}_{33}\end{array}\right]$$

(14)

The details of the mass and stiffness matrices are given in Appendices A and B. Moreover, $\mathit{\chi }$ and $\mathit{F}$ indicate the load vector and Ritz vector expressed, respectively, by

$$\mathit{\chi }=\{A_1,\cdot \cdot \cdot ,A_N,B_1,\cdot \cdot \cdot ,B_N,C_1,\cdot \cdot \cdot ,C_N{\}}^{\mathrm{T}},\mathit{F}={\left\{\begin{array}{ccc}{\mathit{F}}_1& {\mathit{F}}_2& {\mathit{F}}_3\end{array}\right\}}^{\mathrm{T}}$$

(15)

For the convenience of calculation, a regularization method is adopted to reconstruct equation (12), given by

$$(\overline{\mathit{K}}+\mathrm{i}\overline{\mathit{D}}+{\overline{\mathit{K}}}_{\mathrm{b}}-{\omega }^2\overline{\mathit{M}})\mathit{q}=\overline{\mathit{F}}$$

(16)

with

$$\overline{\mathit{K}}={\mathit{\eta }}^T\mathit{K}\mathit{\eta },\overline{\mathit{D}}={\mathit{\eta }}^T\mathit{D}\mathit{\eta },{\overline{\mathit{K}}}_{\mathrm{b}}={\mathit{\eta }}^T{\mathit{K}}_{\mathrm{b}}\mathit{\eta },\overline{\mathit{M}}={\mathit{\eta }}^T\mathit{M}\mathit{\eta },\overline{\mathit{F}}={\mathit{\eta }}^T\mathit{F}$$

(17)

where $\mathit{q}$ and $\mathit{\eta }$ denote the response vector and modal vector accordingly.

Parametric domain decomposition method

Due to the inner friction in the coating material, the storage modulus and loss factor of hard coating are generally strain-dependent, which would further result in a soft nonlinearity and has an important impact on the frequency response of the shell. Thus, the modified domain decomposition method ²⁶ is employed to fully determine the equivalent strain of the coating layer for considering the strain-dependent storage modulus and loss factor of hard coating. However, due to a lack of parameterization of the coating segment number, the modified domain decomposition method with lower computational efficiency is not suitable for mass calculation during the performance evaluation. As a part of the improvement, a parametric domain decomposition method is developed to facilitate its performance evaluation in nonlinear vibration analysis of the composite hard-coating cylindrical shell. The core idea of this method is to divide the layer of the hard coating into multiple segments, the axial and circumferential coating segment numbers are defined by $N_x$ and $N_{\theta }$ , respectively, as shown in Figure 2. The basic assumption of this method is the local invariance of the equivalent strain in a separate coating segment. Then the equivalent strain of each segment can be calculated by the strain energy of the coating segment based on the equivalent strain energy density principle.

Figure 2.

Sketch of the parametric domain decomposition method.

In the present work, the axial and circumferential coordinates of the coating segment (indexed by p and q) are parameterized by the axial and circumferential coating segment numbers $N_x$ and $N_{\theta }$ , defined by

$$\begin{array}{rl}& x_{1,pq}=(p-1)L/N_x,\begin{array}{c}\end{array}{x}_{2,pq}=pL/N_x\\ & {\theta }_{1,pq}=2\pi (q-1)/N_{\theta },\begin{array}{c}\end{array}{\theta }_{1,pq}=2\pi q/N_{\theta }\end{array}$$

(18)

The strain energy of arbitrary coating segment $U_{pq}$ can be calculated by setting the thickness of the layer of shell substrate to zero, given by

$$\begin{array}{rl}& U_{pq}={\displaystyle \frac{1}{2}{\int }_{x_{1,pq}}^{x_{2,pq}}{\int }_{{\theta }_{1,pq}}^{{\theta }_{2,pq}}{\int }_{z_1}^{z_2}({\sigma }_x{\epsilon }_x+{\sigma }_{\theta }{\epsilon }_{\theta }+{\sigma }_{x\theta }{\epsilon }_{x\theta })R\mathrm{d}x\mathrm{d}\theta \mathrm{d}z}\\ & ={\displaystyle \frac{{\sin }^2(\omega t)}{4n{R}^3}\sum _{m=1}^N\{{\int }_{x_{1,pq}}^{x_{2,pq}}{\lambda }_2\{n(c_{m}n+b_m)c_m{T}_m{T^{\mathrm{″}}}_m{R}_2{D}_{12}}\\ & +[(b_{m}n+c_m)c_m{T}_m{T^{\mathrm{″}}}_{m}R-n(c_{m}n+b_m)a_m{T}_m{T^{\mathrm{\prime }}}_m]R^2{B}_{12}\\ & +[(2c_{m}n+b_m)b_m{T}_m^{\mathrm{\prime }2}R-n(2c_{m}n+b_m)a_m{T}_m{T^{\mathrm{\prime }}}_m]R^2{B}_{66}\\ & -{(b_{m}n+c_m)}^2{T}_m^2{R}^2{A}_{22}/2+{(2c_{m}n+b_m)}^2{T}_m^{\mathrm{\prime }2}{R}^2{D}_{66}/2\\ & +{(b_m{T^{\mathrm{\prime }}}_{m}R-a_m{T}_{m}n)}^2{R}^2{A}_{66}/2-(b_{m}n+c_m)a_m{T}_m{T^{\mathrm{\prime }}}_m{R}^3{A}_{12}\\ & -(A_{11}{a}_m^2{T}_m^{\mathrm{\prime }2}+2a_m{c}_m{T^{\mathrm{\prime }}}_m{T^{\mathrm{″}}}_m{B}_{11}+D_{11}{c}_m^2{T}_m^{\mathrm{″}2})R^4/2\\ & -n^2{(c_{m}n+b_m)}^2{T}_m^2{D}_{22}/2-n(c_{m}n+b_m)(b_{m}n+c_m)T_m^{2}R{B}_{22}\}\\ & -2n{\lambda }_1\{(A_{66}{R}^2+2B_{66}R+D_{66})c_m^2{T}_m^{\mathrm{\prime }2}\\ & +n^2[A_{66}{a}_m^2{T}_m^2-4B_{66}{a}_m{c}_m{T}_m{T^{\mathrm{\prime }}}_m+4D_{66}{c}_m^2{T}_m^{\mathrm{\prime }2}]\\ & -2n{B}_m{T^{\mathrm{\prime }}}_m[A_{66}{a}_m{T}_{m}R+B_{66}(a_m{T}_m-2c_m{T^{\mathrm{\prime }}}_{m}R)-2D_{66}{c}_m{T^{\mathrm{\prime }}}_m]\}\}\end{array}$$

(19)

where $z_1$ is the radial coordinate value of the interface between the layers of hard coating and shell substrate. The intermediate variables ${\lambda }_1$ and ${\lambda }_2$ are given, respectively, by

$$\begin{array}{rl}& {\lambda }_1={\theta }_{1,pq}-{\theta }_{2,pq}\\ & {\lambda }_2=\sin (2n{\theta }_{1,pq})-\sin (2n{\theta }_{2,pq})+2n{R}^2({\theta }_{1,pq}-{\theta }_{2,pq})\end{array}$$

(20)

Then the segment stiffness matrix ${\mathit{K}}_{\mathrm{c},pq}$ and material damping matrix ${\mathit{D}}_{\mathrm{c},pq}$ can be obtained by means of the constructed Lagrangian energy function. The total stiffness matrix ${\mathit{K}}_{\mathrm{c}}$ and material damping matrix ${\mathit{D}}_{\mathrm{c}}$ of the coating layer can be calculated by summing the corresponding segment matrices, expressed as

$${\mathit{K}}_{\mathrm{c}}=\sum _{p=1}^{N_x}\sum _{q=1}^{N_{\theta }}{\mathit{K}}_{\mathrm{c},pq},\begin{array}{c}\end{array}{\mathit{D}}_{\mathrm{c}}=\sum _{p=1}^{N_x}\sum _{q=1}^{N_{\theta }}{\mathit{D}}_{\mathrm{c},pq}$$

(21)

Furthermore, according to the inherent linear relationship between the storage modulus and stiffness matrix and between the loss modulus and material damping matrix, the variations of the stiffness matrix and material damping matrix of the layer of hard coating caused by the strain dependences can be calculated, respectively, by

$$\begin{array}{rl}& \mathrm{\Delta }{\mathit{K}}_{\mathrm{c}}=\sum _{p=1}^{N_x}\sum _{q=1}^{N_{\theta }}\mathrm{\Delta }{\mathit{K}}_{\mathrm{c},pq}=\sum _{p=1}^{N_x}\sum _{q=1}^{N_{\theta }}{\displaystyle \frac{{\mathit{K}}_{\mathrm{c},pq}}{E_{\mathrm{c}}}\mathrm{\Delta }{E}_{\mathrm{c},pq}}\\ & \mathrm{\Delta }{\mathit{D}}_{\mathrm{c}}=\sum _{p=1}^{N_x}\sum _{q=1}^{N_{\theta }}\mathrm{\Delta }{\mathit{D}}_{\mathrm{c},pq}=\sum _{p=1}^{N_x}\sum _{q=1}^{N_{\theta }}{\displaystyle \frac{{\mathit{D}}_{\mathrm{c},pq}}{E_{\mathrm{c}}^{\eta }}\mathrm{\Delta }{E}_{\mathrm{c},pq}^{\eta }}\end{array}$$

(22)

where $\mathrm{\Delta }{E}_{\mathrm{c},pq}$ and $\mathrm{\Delta }{E}_{\mathrm{c},pq}^{\eta }$ are the variations of storage modulus and loss modulus with the equivalent strain of the coating segment. Here the high-order polynomial method is used for the characterization of this kind of material strain dependence, defined as

$$\mathrm{\Delta }{E}_{\mathrm{c},pq}=a_1{\epsilon }_{\mathrm{e},pq}+a_2{\epsilon }_{\mathrm{e},pq}^2+\cdots +a_{N_1}{\epsilon }_{\mathrm{e},pq}^{N_1},\begin{array}{c}\end{array}\mathrm{\Delta }{E}_{\mathrm{c},pq}^{\eta }=b_1{\epsilon }_{\mathrm{e},pq}+b_2{\epsilon }_{\mathrm{e},pq}^2+\cdots +b_{N_2}{\epsilon }_{\mathrm{e},pq}^{N_2}$$

(23)

where ${\epsilon }_{\mathrm{e},pq}$ represents the equivalent strain of (p, q)'th coating segment. The polynomial coefficients $a_1,a_2,\cdots ,a_{N_1}$ and $b_1,b_2,\cdots ,b_{N_1}$ are given in the work of Zhang et al. ²⁶ According to the principle of equal strain energy density, the maximum strain energy density ${\psi }_{\mathrm{c},pq}$ of (p, q)'th coating segment can be expressed by

$$\begin{array}{rl}& {\psi }_{\mathrm{c},pq}=\max ({\sigma }_x{\epsilon }_x+{\sigma }_{\theta }{\epsilon }_{\theta }+{\sigma }_{x\theta }{\epsilon }_{x\theta }{)}_{\mathrm{c},pq}\\ & \begin{array}{cc}& \end{array}=\int {\sigma }_{\mathrm{e},pq}\mathrm{d}{\epsilon }_{\mathrm{e},pq}=\int {\hat{E}}_{\mathrm{c},pq}{\epsilon }_{\mathrm{e},pq}\mathrm{d}{\epsilon }_{\mathrm{e},pq}={\displaystyle \frac{1}{2}(E_{\mathrm{c},pq}+\mathrm{i}{E}_{\mathrm{c},pq}^{\eta }){\epsilon }_{\mathrm{e},pq}^2}\end{array}$$

(24)

where ${\sigma }_{\mathrm{e},pq}$ and ${\epsilon }_{\mathrm{c},pq}$ refer to the equivalent stress and strain of (p, q)'th coating segment, respectively. $E_{\mathrm{c},pq}$ and $E_{\mathrm{c},pq}^{\eta }$ denote the storage modulus and loss modulus of (p, q)'th coating segment, respectively.

Based on the construction procedure of the equations of motion of the shell in matrix form in equation (12), the maximum strain energy density ${\psi }_{\mathrm{c},pq}$ can also be determined by

$${\psi }_{\mathrm{c},pq}={\mathit{\chi }}^{\mathrm{T}}({\mathit{K}}_{\mathrm{c},pq}+\mathrm{i}{\mathit{D}}_{\mathrm{c},pq})\mathit{\chi }/V_{\mathrm{c},pq}$$

(25)

where $V_{\mathrm{c},pq}$ refers to the segment volume. Thus, the segment equivalent strain ${\epsilon }_{\mathrm{e},pq}$ is calculated by

$${\epsilon }_{\mathrm{e},pq}=\sqrt{2{\mathit{\chi }}^{\mathrm{T}}{\mathit{K}}_{\mathrm{c},pq}\mathit{\chi }/E_{\mathrm{c},pq}{V}_{\mathrm{c},pq}}=\sqrt{2{\mathit{\chi }}^{\mathrm{T}}{\mathit{D}}_{\mathrm{c},pq}\mathit{\chi }/E_{\mathrm{c},pq}^{\eta }{V}_{\mathrm{c},pq}}$$

(26)

Finally, by introducing the variations of the stiffness matrix and damping matrix of hard coating caused by its strain dependences of the storage modulus and loss modulus, the regularized equation of motion of the composite hard-coating cylindrical shell is changed into the following nonlinear form

$$[\overline{\mathit{K}}+\mathrm{\Delta }{\overline{\mathit{K}}}_{\mathrm{c}}+\mathrm{i}(\overline{\mathit{D}}+\mathrm{\Delta }{\overline{\mathit{D}}}_{\mathrm{c}})+{\overline{\mathit{K}}}_{\mathrm{b}}-{\omega }^2\overline{\mathit{M}}]\mathit{q}=\overline{\mathit{F}}$$

(27)

where

$$\mathrm{\Delta }{\overline{\mathit{K}}}_{\mathrm{c}}={\mathit{\eta }}^{\mathrm{T}}\mathrm{\Delta }{\mathit{K}}_{\mathrm{c}}\mathit{\eta },\mathrm{\Delta }{\overline{\mathit{D}}}_{\mathrm{c}}={\mathit{\eta }}^{\mathrm{T}}\mathrm{\Delta }{\mathit{D}}_{\mathrm{c}}\mathit{\eta }$$

(28)

The well-known Newton–Raphson iterative method is employed to solve the above nonlinear equation of motion of the composite hard-coating cylindrical shell caused by the material nonlinearity of hard coating. The residual vector ${\mathit{\xi }}_{\mathrm{resi}}$ of the nonlinear equation (27) is defined by

$${\mathit{\xi }}_{\mathrm{resi}}=[\overline{\mathit{K}}+\mathrm{\Delta }{\overline{\mathit{K}}}_{\mathrm{c}}+\mathrm{i}(\overline{\mathit{D}}+\mathrm{\Delta }{\overline{\mathit{D}}}_{\mathrm{c}})+{\overline{\mathit{K}}}_{\mathrm{b}}-{\omega }^2\overline{\mathit{M}}]\mathit{q}-\overline{\mathit{F}}$$

(29)

The whole iteration procedure with termination condition $\vartheta$ is shown in Figure 3. It should be noted that the iteration procedure is initialized by setting the segment equivalent strain ${\epsilon }_{\mathrm{e},pq}=0$ .

Figure 3.

Whole iteration procedure of the unified nonlinear iterative solution method.

To speed up the calculation, some necessary strategies need to be taken to further refine the parametric domain decomposition method. It is worth noting that the challenge mainly lies in how to efficiently deal with the numerous repeated calculations of the stiffness matrix of coating segments during the nonlinear iterations, so as to facilitate the performance evaluation within the designated parameter range. Here a specialized preprocessing scheme is designed for the semi-analytical construction of the stiffness and material damping matrix of a large number of coating segments, aimed at avoiding a mass of redundant computation in the process of performance evaluation.

Firstly, by applying the indefinite integral of each item in the segment stiffness matrix ${\mathit{K}}_{\mathrm{c},pq}$ and material damping matrix ${\mathit{D}}_{\mathrm{c},pq}$ , the corresponding parametric mathematical expressions containing the integral variables $x_{1,pq},x_{2,pq},{\theta }_{1,pq}$ , and ${\theta }_{2,pq}$ can be obtained. It is then stored in a specific database so that it can be called repeatedly and directly at all times during the nonlinear iterations. By means of this method, the stiffness and material damping matrices of a large number of coating segments can be quickly calculated by substituting the segment coordinates which are determined by the axial and circumferential segment numbers $N_x$ and $N_{\theta }$ , as well as the segment indexes p and q.

Secondly, the matrix database within the designated parameter range is recommended to pre-established, in which the pre-calculated segment stiffness matrix ${\mathit{K}}_{\mathrm{c},pq}$ and material damping matrix ${\mathit{D}}_{\mathrm{c},pq}$ are assembled into a total stiffness matrix ${\mathit{K}}_{\mathrm{total}}$ and a total damping matrix ${\mathit{D}}_{\mathrm{total}}$ with a specific index corresponding to the circumferential wave number n, the axial and circumferential coating segment numbers $N_x$ and $N_{\theta }$ . Similarly, by storing the obtained ${\mathit{K}}_{\mathrm{total}}$ and ${\mathit{D}}_{\mathrm{total}}$ in another database for subsequent invocations, the segment equivalent strain ${\epsilon }_{\mathrm{e},pq}$ can be calculated with high efficiency in each nonlinear iteration. Based on the above strategies, the performance evaluation of nonlinear vibration of the composite hard-coating cylindrical shell by using the parametric domain decomposition method can be implemented easily and efficiently.

In addition, it should be noted that the original domain decomposition method for dealing with the coating strain dependence in nonlinear vibration of the hard-coating cylindrical shell was firstly developed in our previous study, ²⁶ which has been fully validated under the condition of elastic constraint by comparing with the experimental resonant frequencies and responses. Considering the present method is a parameterized version of the original domain decomposition method, focusing on the parameterization of coating segment number for facilitating self-performance evaluation, thus the repeated method verification will not be performed here. This work focuses more on the performance evaluation of the parametric domain decomposition method in the nonlinear vibration of the composite shell.

The nonlinear iterative process of the parametric domain decomposition method is given in Figure 4, to demonstrate the convergence of resonant frequency, resonant response, and 2-norm of residual vector. As indicated in Figure 4, the resonant frequency, resonant response, and 2-norm of residual vector achieve the goal of convergence after 112 iterations, possessing a good convergence and accuracy for the solution.

Figure 4.

Convergence analysis of the parametric domain decomposition method.

Results and discussions

Since the existing applications of the modified domain decomposition method in solving the nonlinear vibration of the composite hard-coating cylindrical shells are still at a relatively superficial level, and lack of understanding of the domain decomposition mechanism, a detailed and comprehensive performance evaluation of this method is conducted for ease of achieving sufficient precision. The adapted material and geometric parameters of the composite hard-coating cylindrical shell are listed in Tables 1 and 2.

Table 1.

Material parameters of the composite hard-coating cylindrical shell.

Material	Storage modulus (Pa)	Density (kg/m3)	Poisson's ratio	Loss factor
Structural steel	2.12 × 1011	7850	0.3	0.0007
NiCoCrAlY + YSZ	0.5449 × 1011	5600	0.3	0.0212

Table 2.

Geometric parameters of the composite hard-coating cylindrical shell.

Parameter	L (m)	R (m)	Ts (m)	Tc (m)
Value	95 × 10−3	143.04 × 10−3	2 × 10−3	0.31 × 10−3

Further validation of the present semi-analytical model

Considering the study on forced vibration of the composite hard-coating cylindrical shell under clamped boundary conditions is seldom reported in existing literature, the calculated natural frequencies with corresponding modal shapes are compared with the FEM results based on ABAQUS v6.13, to further demonstrate the validity of the present model as well as the vibration characteristics of the shell, as shown in Figure 5.

Figure 5.

Natural frequencies and mode shapes of the composite hard-coating cylindrical shell.

The above results indicate that the maximum deviation between the natural frequencies calculated by FEM and present models is within 1.857%, and the calculated peak number of the first 12 mode shapes of the shell is exactly consistent with the given circumferential wave number. Thus, the comparison results provide further validation of the present semi-analytical model.

Influence of segment number on the resonant response

Owing to the cylindrical-shell-shaped configuration of the composite hard-coating structure, the inherently complicated modal shape makes the resonant response and equivalent strain of the coating layer show a significant difference along the axial and circumferential directions. Hence, under the assumption of constant equivalent strain in each separate coating segment, the segment number in different directions would inevitably have an impact on the calculating precision of the nonlinear results.

In view of the background above, the effects of segment number along the axial and circumferential directions on the nonlinear vibration characteristics are investigated. The case with constant segment width and varied segment length, as well as the case with constant segment length and varied segment width are listed in Table 3.

Table 3.

Adapted axial and circumferential segment numbers for the investigated two cases.

Case 1 (Nx = 10)			Case 2 (Nθ = 20)
Nx	Nθ	Nx	Nθ	Nx	Nθ
10	10	10	110	10	20
10	15	10	115	15	20
10	20	10	120	20	20
10	25	10	125	25	20
10	30	10	130	30	20
10	35	10	135	35	20
10	40	10	140	40	20
10	45	10	145	45	20
10	50	10	150	50	20
10	55	10	155	55	20
10	60	10	160	60	20
10	65	10	165	65	20
10	70	10	170	70	20
10	75	10	175	75	20
10	80	10	180	80	20
10	85	10	185	85	20
10	90	10	190	90	20
10	95	10	195	95	20
10	100	10	200	100	20
10	105	—	—	—	—

The influence of circumferential segment numbers with constant segment length on the resonant response under different circumferential wave numbers is shown in Figure 6. From Figure 6 we can see that the resonant response is sensitive to the circumferential segment number. The resonant responses under circumferential wave numbers n = 1 and n = 2 increase gradually with the increasing circumferential segment number and then quickly level off. Conversely, the resonant responses for n = 3–12 decrease with the increase of the circumferential segment number and then quickly level off as a whole. Results indicate that there exists an optimum value for the circumferential segment number with constant segment length to achieve the exact value of resonant response under different circumferential wave numbers.

Figure 6.

Influence of circumferential segment number on the resonant response under different circumferential wave numbers.

The influence of axial segment number with constant segment width on the resonant response under different circumferential wave numbers is shown in Figure 7. From Figure 7, it can be found that the resonant response under different circumferential wave numbers remains largely the same with the increasing axial segment number, suggesting that the effect of axial segment number on the resonant response is weak comparatively. The above results demonstrate that the variation of the circumferential segment number has a greater impact on the resonant response than the axial segment number. Moreover, a good balance can be achieved between the calculation accuracy and efficiency by adjusting the appropriate circumferential segment number. The optimum circumferential segment number in the present study is suggested to N_θ = 70 at the conditions of N_x = 10.

Figure 7.

Influence of axial segment number on the resonant response under different circumferential wave numbers.

Influence of segment number on the equivalent strain

The equivalent strain of the coating segment directly determines nonlinear strength of forced vibration of the composite hard-coating cylindrical shell. Thus, exploring the influence of the segment number on the equivalent strain is beneficial to clarify the mentioned domain decomposition mechanism. Taking the circumferential wave number n = 8 for example, the influence of axial segment number with constant segment width on the equivalent strain along the initial shell longitude line of the coating layer is shown in Figure 8. As can be seen from Figure 8, there is high consistency between the distributions of axial equivalent strain under different axial segment numbers, indicating the weak impact effect of the axial segment number on the axial distribution of the equivalent strain of the composite cylindrical shell. Since the value of the equivalent strain of the coating layer would directly act on the nonlinear resonant response of the shell, it can be concluded that the variation of axial segment number with constant segment width can not effectively improve the accuracy of the resonant results.

Figure 8.

Influence of axial segment number on the equivalent strain along the initial shell longitude line of the coating layer.

Taking the circumferential wave number n = 8 for example, the influence of the circumferential segment number on the equivalent strain along the top shell latitude line of the coating layer is shown in Figure 9. It can be seen that the maximum value of equivalent strain decreases first and then became stable with the increase of the circumferential segment number. Fewer circumferential segment numbers would result in a larger equivalent strain of the coating layer of the shell, further causing bad solution accuracy of the resonant response. The result shows that the local strain varies as the cosine function of the circumferential segment number, exactly corresponding to the circumferential wave number of the shell mode shape. Moreover, with the increase of the circumferential segment number, the wave number of the mentioned cosine function above gradually increases from 2 to 8, and then remains stable. Likewise, the maximum value of the equivalent strain gradually declines until stabilized, while the amplitude of the equivalent strain varies in the opposite. Hence it gives a conclusion that the circumferential segment number of the coating layer has a great impact on the nonlinear resonant response of the composite hard-coating cylindrical shell.

Figure 9.

Influence of circumferential segment number on the equivalent strain along the top shell latitude line of the coating layer.

Furthermore, due to the geometrical characteristics of the studied thin-walled short cylindrical shell, only one axial wave can be excited by the base sinusoidal acceleration load within the specific frequency range, and yet each circumferential mode shape of the shell is scarcely influenced and can be completely excited. The axial and circumferential wave numbers are important to the axial and circumferential equivalent strain distributions as well as the axial and circumferential equivalent strain gradients. Owing to the axial wave number m = 1, the variation gradient of axial equivalent strain is relatively small, thus the nonlinear resonant response of the hard-coating cylindrical shell is insensitive to the axial segment number as indicated in Figure 7. By comparison, the variation gradient of circumferential equivalent strain presents certain degree of diversity, which probably explains why the nonlinear resonant response of the shell is sensitive to the circumferential segment number. It is a fact that only enough circumferential segment numbers can achieve sufficient accuracy of the nonlinear solution.

Conclusions

In this paper, a parametric domain decomposition method is developed to facilitate self-performance evaluation in nonlinear vibration analysis of the composite hard-coating cylindrical shell. Moreover, in order to avoid a mass of redundant computation of the segment stiffness and material damping matrices during performance evaluation, a specialized preprocessing scheme is designed by pre-establishing the parametric analytical expressions and total matrix databases. Then the effects of axial and circumferential segment numbers on the nonlinear resonant response and equivalent strain are studied for further understanding domain decomposition mechanism. The following conclusions are obtained:

The resonant response is sensitive to the circumferential segment number but weakly affected by the axial segment number. There exists an optimum value for the circumferential segment number to achieve the exact value of resonant response under different circumferential wave numbers. The optimum circumferential segment number in the present study is suggested to N_θ = 70, which can achieve good calculation accuracy and efficiency.

Highly consistency is shown for the distributions of axial equivalent strain under different axial segment numbers. The variation of axial segment number with constant segment width can not effectively improve the accuracy of the nonlinear resonant results. Fewer circumferential segment numbers would result in larger equivalent strain and bad solution accuracy. The local strain varies as the cosine function of the circumferential segment number, exactly corresponding to the circumferential wave number of the shell mode shape.

The axial and circumferential wave numbers are important to the corresponding equivalent strain distributions and gradients. The variation gradient of circumferential equivalent strain presents a certain degree of diversity, which probably explains why the nonlinear resonant response of the shell is sensitive to the circumferential segment number. It is a fact that only enough circumferential segment numbers can achieve sufficient accuracy of the nonlinear solution of the composite hard-coating cylindrical shell.