# A New Finite Element Formulation for Nonlinear Vibration Analysis of the Hard-Coating Cylindrical Shell

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Derivation of the Finite Element Formulation for the Hard-Coating Cylindrical Shell

#### 2.1. Geometry and Element of the Hard-Coating Cylindrical Shell

#### 2.2. Love’s First Approximation Theory

^{0}, v

^{0}and w

^{0}are the orthogonal components of displacement of the mid-surface in the x, θ and z directions, respectively, and ψ

_{x}and ψ

_{θ}are the rotations of the mid-surface about the θ and x axes, respectively.

_{x}, g

_{θ}, g

_{z}, are Lamb constants [24]. For thin cylindrical shell (h/R$\ll 1$):

_{x}, κ

_{θ}, κ

_{x}

_{θ}, the strains and bending strains of the middle surface, can be given as follows [25]:

_{k}and v

_{k}are the Young’s modulus and Poisson’s ratio of the k-th lamina, respectively. For a double-layered cylindrical shell, the force and moment resultants are defined by:

_{0}, z

_{1}and z

_{2}are defined as follows:

_{1}and T

_{2}are the thickness of metal substrate and hard-coating, respectively (see Figure 1b). T

_{0}is the distance from the middle surface to the inner surface, defined by:

#### 2.3. Finite Element Formulation

^{0}in the coordinate directions x and θ is assumed to be of the form:

^{0}and w

^{0}. The equations of the displacement field for u

^{0}, ${u}_{x}^{0}$ and ${u}_{\mathsf{\theta}}^{0}$ can be written in matrix form:

**U**,

**X**and

**A**can be expressed as follows:

**N**can be expressed as:

**Г**, the strain matrix

**B**and the displacement vector

**x**are defined, respectively, by:

**K**

^{el}, the element mass matrix

**M**

^{el}and the external load vector

**F**

^{el}introduced by the base excitation can be expressed, respectively, by:

_{1}and ρ

_{2}are the densities of metal substrate and hard-coating, respectively.

**f**is a 3 × 1 order vector composed of the x, θ and z components of the base excitation per unit volume, defined by:

_{x}, μ

_{θ}and μ

_{z}are the influence coefficients of the base excitation on the x, θ and z directions of the hard-coating cylindrical shell.

**K**, the global mass matrix

**M**and the global external load vector

**F**of the hard-coating cylindrical shell can be assembled by

**K**

^{el},

**M**

^{el}and

**F**

^{el}in the corresponding sequence, respectively.

## 3. Characterizing the Strain Dependence Using the High Order Polynomial

_{2}* of the hard-coating is considered, given by:

_{2}(namely the E

_{2}in Equation (6)) and η

_{2}are the storage modulus and the loss factor of hard-coating, respectively, which can be expressed by the p-order polynomials,

_{2j}and η

_{2j}(j = 0, 1, …, p) are specific j-order coefficients of the storage modulus and the loss factor, respectively. For convenience, one can define:

_{e}, given by:

_{e}of an element can be calculated as:

_{20}are the element stiffness matrix and the storage modulus of the hard-coating at zero strain, respectively. ${V}_{2}^{\mathrm{el}}$ is the element volume of hard-coating. In addition, the displacement vector

**x**is defined as:

**q**is the response vector in the normal coordinate and ${\mathsf{\phi}}_{0}$ is the normal mode shape matrix at zero strain without damping. Submitting Equation (43) to Equation (42), one can obtain:

**Λ**is the normalized element stiffness matrix per unit storage modulus of the hard-coating.

_{2}and loss factor η

_{2}obtained from the experiment [26] are:

## 4. Solution of the Nonlinear Vibration of the Hard-Coating Cylindrical Shell

**M**and

**F**are the global mass matrix and external force vector, respectively. The normalized response vector

**q**can be obtained from Equation (47) by applying the QR-method.

**D**is independent of the strain level. Neglecting the coupling effect between the layers of hard-coating and the metal substrate of the cylindrical shell element, the global complex stiffness matrix at ε strain can be defined by:

**D**of the hard-coating cylindrical shell can be calculated by the same method as the global stiffness matrix

**K**, which only requires multiplying Equation (6) by the zero-strain loss factor of the metal substrate and hard-coating η

_{10}and η

_{20}, respectively

_{10}is considered.

**r**can be expressed as:

**q**is complex-valued, it is necessary to separate the real and imaginary parts of

**q**in the iterative formula derived by the Newton–Raphson solution method. The separated iterative formula can be expressed as:

**r**with respect to the real and imaginary parts,

**q**

_{R}and

**q**

_{S}, are:

**r**and ζ is the solution precision. When the residual vector

**r**of Equation (56) satisfies the condition of the solution precision, the newest global complex stiffness matrix at ε strain ${\mathbf{K}}_{\mathsf{\epsilon},\mathrm{new}}^{*}$ (in Equation (49)) and the normalized response vector

**q**

_{new}can be output. Then, the n-order nonlinear resonant circular frequency ω

_{n}and the nonlinear harmonic response

**x**can be obtained, respectively, by:

## 5. Case Study

_{2}and η

_{2}in Table 2 are functions of the equivalent strain ε

_{e}(see Equations (45) and (46)).

_{x}, μ

_{θ}and μ

_{z}in different quadrants. The μ

_{x}is equal to zero; the values of μ

_{θ}and μ

_{z}are listed in Table 3.

#### 5.1. Validation Analysis

#### 5.2. Nonlinear Vibration Analysis and Results Discussion

_{e}= 0) need to be calculated, which are taken as the reference.

## 6. Conclusions

- Based on Love’s first approximation theory, a four-node composite cylindrical shell finite element model is proposed. Then, the nonlinear iterative solution formulas with a unified iterative method are theoretically derived for solving the resonant frequency and response of the hard-coating cylindrical shell.
- A cylindrical shell coated with a thin layer of NiCoCrAlY + YSZ is chosen to demonstrate the proposed formulation. The nonlinear resonant frequencies and responses calculated by the present method and the FEIM show a good agreement, which indicates the rationality of the developed finite element method. Moreover, the developed finite element method is less affected by the element size and has lower computing cost.
- Moreover, the nonlinear vibration analysis of the cylindrical shell coated with a thin layer of NiCoCrAlY + YSZ is implemented. Compared with the linear calculation results, the nonlinear resonant frequencies and responses of each order decrease to a certain degree, and the descents increase continually with the increase of the excitation level; that is, the increase of the excitation level would make the strain dependence of the hard-coating more remarkable, which reveals the characteristics of the soft stiffness nonlinearity or “strain softening”.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Geometry and element of the hard-coating cylindrical shell. (

**a**) Shell geometry; (

**b**) Shell element.

**Figure 4.**Two finite element models of the hard-coating cylindrical shell: (

**a**) Present model; (

**b**) FEIM model.

**Figure 5.**Convergences of nonlinear resonant frequencies of the finite element iteration method (FEIM) and present method under different element sizes.

**Figure 6.**The three order linear and nonlinear frequency response spectrums of the hard-coating cylindrical shell under different excitation levels.

Parameters | L (mm) | R (mm) | T_{1} (mm) | T_{2} (mm) |
---|---|---|---|---|

Value | 95 | 142 | 2 | 0.31 |

Lamina | Material | Young’s Modulus (GPa) | Loss Factor | Density (kg/m^{3}) | Poisson’s Ratio |
---|---|---|---|---|---|

Metal substrate | Ti-6Al-4V | 110.32 | 0.0007 | 4420 | 0.3 |

Hard coating | NiCoCrAlY + YSZ | E_{2} | η_{2} | 5600 | 0.3 |

Quadrant | μ_{θ} | μ_{z} |
---|---|---|

First | cosθ | sinθ |

Second | sinθ | −cosθ |

Third | −cosθ | −sinθ |

Fourth | cosθ | −sinθ |

**Table 4.**Comparisons of the first 6 order nonlinear resonant frequencies and responses with the FEIM.

Modal Order | Nonlinear Resonant Frequencies | Nonlinear Resonant Responses | ||||
---|---|---|---|---|---|---|

Present Method A (Hz) | FEIM B (Hz) | Difference |A − B|/A (%) | Present Method C (10 ^{−2} mm) | FEIM D (10 ^{−2} mm) | Difference |C − D|/C (%) | |

1 | 1274.561 | 1281.912 | 0.577 | 1.786 | 1.738 | 2.732 |

2 | 1283.778 | 1294.985 | 0.873 | 3.199 | 3.048 | 4.940 |

3 | 1443.842 | 1446.837 | 0.207 | 1.753 | 1.705 | 2.793 |

4 | 1513.250 | 1526.021 | 0.844 | 1.409 | 1.345 | 4.753 |

5 | 1738.061 | 1736.589 | 0.085 | 1.172 | 1.164 | 0.702 |

6 | 1990.964 | 2002.791 | 0.594 | 1.239 | 1.200 | 3.259 |

**Table 5.**The first 6 order linear and nonlinear resonant frequencies of the hard-coating cylindrical shell under a 5 g excitation level.

Modal Order | Linear (Hz) E | Nonlinear (Hz) F | Descent (Hz) E − F |
---|---|---|---|

1 | 1274.899 | 1274.561 | 0.338 |

2 | 1283.922 | 1283.778 | 0.144 |

3 | 1444.074 | 1443.842 | 0.232 |

4 | 1513.410 | 1513.25 | 0.160 |

5 | 1738.293 | 1738.061 | 0.232 |

6 | 1991.071 | 1990.964 | 0.107 |

**Table 6.**The first 6 order linear and nonlinear resonant responses of the hard-coating cylindrical shell under a 5 g excitation level.

Modal Order | Linear (10^{−2} mm) G | Nonlinear (10^{−2} mm) H | Descent (10^{−2} mm) G − H |
---|---|---|---|

1 | 8.926 | 8.529 | 0.397 |

2 | 16.003 | 15.472 | 0.531 |

3 | 8.740 | 8.444 | 0.296 |

4 | 7.067 | 6.794 | 0.273 |

5 | 5.869 | 5.714 | 0.155 |

6 | 6.195 | 6.067 | 0.128 |

**Table 7.**The 3 order linear and nonlinear resonant frequencies of the hard-coating cylindrical shell under different excitation levels.

Excitation Level (g) | Linear (Hz) E | Nonlinear (Hz) F | Descent (Hz) E − F |
---|---|---|---|

1 | 1444.074 | 1444.026 | 0.048 |

3 | 1444.074 | 1443.932 | 0.142 |

5 | 1444.074 | 1443.842 | 0.232 |

7 | 1444.074 | 1443.755 | 0.319 |

9 | 1444.074 | 1443.672 | 0.402 |

**Table 8.**The 3 order linear and nonlinear resonant responses of the hard-coating cylindrical shell under different excitation levels.

Excitation Level (g) | Linear (10^{−2} mm) G | Nonlinear (10^{−2} mm) H | Descent (10^{−2} mm) G − H |
---|---|---|---|

1 | 1.752 | 1.741 | 0.011 |

3 | 5.246 | 5.136 | 0.110 |

5 | 8.740 | 8.444 | 0.296 |

7 | 12.234 | 11.641 | 0.593 |

9 | 15.738 | 14.728 | 1.010 |

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**MDPI and ACS Style**

Zhang, Y.; Sun, W.; Yang, J.
A New Finite Element Formulation for Nonlinear Vibration Analysis of the Hard-Coating Cylindrical Shell. *Coatings* **2017**, *7*, 70.
https://doi.org/10.3390/coatings7050070

**AMA Style**

Zhang Y, Sun W, Yang J.
A New Finite Element Formulation for Nonlinear Vibration Analysis of the Hard-Coating Cylindrical Shell. *Coatings*. 2017; 7(5):70.
https://doi.org/10.3390/coatings7050070

**Chicago/Turabian Style**

Zhang, Yue, Wei Sun, and Jian Yang.
2017. "A New Finite Element Formulation for Nonlinear Vibration Analysis of the Hard-Coating Cylindrical Shell" *Coatings* 7, no. 5: 70.
https://doi.org/10.3390/coatings7050070