Dynamic Modeling and Vibration Characteristics Analysis for the Helicopter Horizontal Tail Drive Shaft System with the Ballistic Impact Vertical Penetrating Damage

Abstract

In order to give a qualitative understanding of the ballistic impact damage on system dynamics, and provide ideas for the follow-up study of various types of the ballistic impact damage, dynamic modeling and vibration characteristics analysis for the helicopter horizontal tail drive shaft system with the ballistic impact vertical penetrating damage are obtained. The finite element dynamic model of the horizontal tail drive shaft system (HTDSS) with the ballistic impact vertical penetrating damage (BIVPD) is established. The research results show that the BIVPD causes the mass loss and the stiffness asymmetric of the ballistic impact position; the stiffness asymmetry causes the subharmonic resonance phenomenon; the mass loss causes the increase in the system resonance; and the BIVPD introduces the 2Ω component of the system resonance. The dynamic characteristics of the system with the ballistic impact damage should be analyzed from the axis trajectory, acceleration response and frequency-domain response. The methods and conclusions presented in this paper can provide theoretical support for the dynamic design of the helicopter tail drive shaft system.

Appendix

This part gives the detail expression of each matrix in the text. It should be noted that for a typical Rayleigh beam element considering the gyroscopic effect, it is only necessary to replace A_rest with A and I₁ with I₂ in the matrix element of the ballistic impact element.

The coordinate transformation relationship between the rotating coordinate system and the fixed coordinate system is as follows

$$T = \left[ {\begin{array}{*{20}c} {T_{0} } & {} & {} & {} \\ {} & {T_{0} } & {} & {} \\ {} & {} & {T_{0} } & {} \\ {} & {} & {} & {T_{0} } \\ \end{array} } \right]$$

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where

$$T_{0} { = }\left[ {\begin{array}{*{20}c} {\cos \left( {\Omega t} \right)} & { - \sin \left( {\Omega t} \right)} \\ {\sin \left( {\Omega t} \right)} & {\cos \left( {\Omega t} \right)} \\ \end{array} } \right]$$

The gravity vectors are as follows

$$\begin{aligned} \left\{ {Q_{{\text{g}}} } \right\} & = \rho A_{{\text{e}}} l\left\{ {\begin{array}{*{20}l} 0 \hfill & { - \frac{1}{2}} \hfill & \frac{l}{12} \hfill & 0 \hfill & 0 \hfill & { - \frac{1}{2}} \hfill & \frac{l}{12} \hfill & 0 \hfill \\ \end{array} } \right\}^{{\text{T}}} \\ \left\{ {Q_{{{\text{bg}}}} } \right\} & = \rho A_{{{\text{rest}}}} l\left\{ {\begin{array}{*{20}l} 0 \hfill & { - \frac{1}{2}} \hfill & \frac{l}{12} \hfill & 0 \hfill & 0 \hfill & { - \frac{1}{2}} \hfill & \frac{l}{12} \hfill & 0 \hfill \\ \end{array} } \right\}^{{\text{T}}} \\ \end{aligned}$$

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The displacement offset vector is as follows

$$\left\{ {q^{{\text{e}}} } \right\}{ = }\left[ { - e{\text{cos}}\left( {\Omega t} \right)\;\;e{\text{sin}}\left( {\Omega t} \right)\;\;0\;\;0\;\; - e{\text{cos}}\left( {\Omega t} \right)\;\;e{\text{sin}}\left( {\Omega t} \right)\;\;0\;\;0} \right]^{{\text{T}}}$$

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The unbalance force vectors are as follows

$$\left\{ {Q_{{{\text{bu}}}} } \right\} = \rho A_{{{\text{rest}}}} le\Omega^{2} \left[ { - \cos \left( {\Omega t} \right)\;\;\sin \left( {\Omega t} \right)\;\;0\;\;0\;\; - \cos \left( {\Omega t} \right)\;\;\sin \left( {\Omega t} \right)\;\;0\;\;0} \right]^{{\text{T}}}$$

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$$\left\{ {Q_{{\text{u}}} } \right\} = m_{{\text{e}}} e_{0} \Omega^{2} \left[ { - \cos \left( {\Omega t} \right)\;\;\sin \left( {\Omega t} \right)\;\;0\;\;0\;\; - \cos \left( {\Omega t} \right)\;\;\sin \left( {\Omega t} \right)\;\;0\;\;0} \right]^{{\text{T}}}$$

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The specific expressions of inertia, stiffness and gyroscopic matrices of ballistic impact element are as follows

$$M^{{\text{r}}} = \frac{\rho }{30l}\left[ {\begin{array}{*{20}c} {36I_{{\overline{x}}} } & 0 & 0 & {3lI_{{\overline{x}}} } & { - 36I_{{\overline{x}}} } & 0 & 0 & {3lI_{{\overline{x}}} } \\ 0 & {36I_{{\overline{y}}} } & { - 3lI_{{\overline{y}}} } & 0 & 0 & { - 36I_{{\overline{y}}} } & { - 3lI_{{\overline{y}}} } & 0 \\ 0 & { - 3lI_{{\overline{y}}} } & {4l^{2} I_{{\overline{y}}} } & 0 & 0 & {3lI_{{\overline{y}}} } & { - l^{2} I_{{\overline{y}}} } & 0 \\ {3lI_{{\overline{x}}} } & 0 & 0 & {4l^{2} I_{{\overline{x}}} } & { - 3lI_{{\overline{x}}} } & 0 & 0 & { - l^{2} I_{{\overline{x}}} } \\ { - 36I_{{\overline{x}}} } & 0 & 0 & { - 3lI_{x} } & {36I_{{\overline{x}}} } & 0 & 0 & { - 3lI_{{\overline{x}}} } \\ 0 & { - 36I_{{\overline{y}}} } & {3lI_{{\overline{y}}} } & 0 & 0 & {36I_{{\overline{y}}} } & {3lI_{{\overline{y}}} } & 0 \\ 0 & { - 3lI_{{\overline{y}}} } & { - l^{2} I_{{\overline{y}}} } & 0 & 0 & {3lI_{{\overline{y} }} } & {4l^{2} I_{{\overline{y}}} } & 0 \\ {3lI_{{\overline{x} }} } & 0 & 0 & { - l^{2} I_{x} } & { - 3lI_{x} } & 0 & 0 & {4l^{2} I_{x} } \\ \end{array} } \right]$$

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$$M_{{{\text{r0}}}}^{{\text{f}}} = \frac{{\rho I_{1} }}{30l}\left[ {\begin{array}{*{20}c} {36} & 0 & 0 & {3l} & { - 36} & 0 & 0 & {3l} \\ 0 & {36} & { - 3l} & 0 & 0 & { - 36} & { - 3l} & 0 \\ 0 & { - 3l} & {4l^{2} } & 0 & 0 & {3l} & { - l^{2} } & 0 \\ {3l} & 0 & 0 & {4l^{2} } & { - 3l} & 0 & 0 & { - l^{2} } \\ { - 36} & 0 & 0 & { - 3l} & {36} & 0 & 0 & { - 3l} \\ 0 & { - 36} & {3l} & 0 & 0 & {36} & {3l} & 0 \\ 0 & { - 3l} & { - l^{2} } & 0 & 0 & {3l} & {4l^{2} } & 0 \\ {3l} & 0 & 0 & { - l^{2} } & { - 3l} & 0 & 0 & {4l^{2} } \\ \end{array} } \right]$$

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$$M_{{{\text{r1}}}}^{{\text{f}}} = \frac{{\rho I_{2} }}{30l}\left[ {\begin{array}{*{20}c} {36} & 0 & 0 & {3l} & { - 36} & 0 & 0 & {3l} \\ 0 & { - 36} & {3l} & 0 & 0 & {36} & {3l} & 0 \\ 0 & {3l} & { - 4l^{2} } & 0 & 0 & { - 3l} & {l^{2} } & 0 \\ {3l} & 0 & 0 & {4l^{2} } & { - 3l} & 0 & 0 & { - l^{2} } \\ { - 36} & 0 & 0 & { - 3l} & {36} & 0 & 0 & { - 3l} \\ 0 & {36} & { - 3l} & 0 & 0 & { - 36} & { - 3l} & 0 \\ 0 & {3l} & {l^{2} } & 0 & 0 & { - 3l} & { - 4l^{2} } & 0 \\ {3l} & 0 & 0 & { - l^{2} } & { - 3l} & 0 & 0 & {4l^{2} } \\ \end{array} } \right]$$

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$$M_{{{\text{r2}}}}^{{\text{f}}} = \frac{{\rho I_{2} }}{30l}\left[ {\begin{array}{*{20}c} 0 & {36} & { - 3l} & 0 & 0 & { - 36} & { - 3l} & 0 \\ {36} & 0 & 0 & {3l} & { - 36} & 0 & 0 & {3l} \\ { - 3l} & 0 & 0 & { - 4l^{2} } & {3l} & 0 & 0 & {l^{2} } \\ 0 & {3l} & { - 4l^{2} } & 0 & 0 & { - 3l} & {l^{2} } & 0 \\ 0 & { - 36} & {3l} & 0 & 0 & {36} & {3l} & 0 \\ { - 36} & 0 & 0 & { - 3l} & {36} & 0 & 0 & { - 3l} \\ { - 3l} & 0 & 0 & {l^{2} } & {3l} & 0 & 0 & { - 4l^{2} } \\ 0 & {3l} & {l^{2} } & 0 & 0 & { - 3l} & { - 4l^{2} } & 0 \\ \end{array} } \right]$$

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$$M_{t}^{{\text{f}}} = \frac{{\rho A_{{{\text{rest}}}} l}}{420}\left[ {\begin{array}{*{20}c} {156} & 0 & 0 & {22l} & {54} & 0 & 0 & { - 13l} \\ 0 & {156} & { - 22l} & 0 & 0 & {54} & {13l} & 0 \\ 0 & { - 22l} & {4l^{2} } & 0 & 0 & { - 13l} & { - 3l^{2} } & 0 \\ {22l} & 0 & 0 & {4l^{2} } & {13l} & 0 & 0 & { - 3l^{2} } \\ {54} & 0 & 0 & {13l} & {156} & 0 & 0 & { - 22l} \\ 0 & {54} & { - 13l} & 0 & 0 & {156} & {22l} & 0 \\ 0 & {13l} & { - 3l^{2} } & 0 & 0 & {22l} & {4l^{2} } & 0 \\ { - 13l} & 0 & 0 & { - 3l^{2} } & { - 22l} & 0 & 0 & {4l^{2} } \\ \end{array} } \right]$$

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$$K^{{\text{r}}} = \frac{E}{{l^{3} }}\left[ {\begin{array}{*{20}c} {12I_{{\overline{x}}} } & 0 & 0 & {6lI_{{\overline{x}}} } & { - 12I_{{\overline{x}}} } & 0 & 0 & {6lI_{{\overline{x}}} } \\ 0 & {12I_{{\overline{y}}} } & { - 6lI_{{\overline{y}}} } & 0 & 0 & { - 12I_{{\overline{y}}} } & { - 6lI_{{\overline{y}}} } & 0 \\ 0 & { - 6lI_{{\overline{y}}} } & {4l^{2} I_{{\overline{y}}} } & 0 & 0 & {6lI_{{\overline{y}}} } & {2l^{2} I_{{\overline{y}}} } & 0 \\ {6lI_{{\overline{x}}} } & 0 & 0 & {4l^{2} I_{{\overline{x}}} } & { - 6lI_{{\overline{x}}} } & 0 & 0 & {2l^{2} I_{{\overline{x}}} } \\ { - 12I_{{\overline{x}}} } & 0 & 0 & { - 6lI_{{\overline{x}}} } & {12I_{{\overline{x}}} } & 0 & 0 & { - 6lI_{{\overline{x}}} } \\ 0 & { - 12I_{{\overline{y}}} } & {6lI_{{\overline{y}}} } & 0 & 0 & {12I_{{\overline{y}}} } & {6lI_{{\overline{y}}} } & 0 \\ 0 & { - 6lI_{{\overline{y}}} } & {2l^{2} I_{{\overline{y}}} } & 0 & 0 & {6lI_{{\overline{y}}} } & {4l^{2} I_{{\overline{y}}} } & 0 \\ {6lI_{{\overline{x}}} } & 0 & 0 & {2l^{2} I_{{\overline{x}}} } & { - 6lI_{{\overline{x}}} } & 0 & 0 & {4l^{2} I_{{\overline{x}}} } \\ \end{array} } \right]$$

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$$K_{0}^{{\text{f}}} = \frac{{EI_{1} }}{{l^{3} }}\left[ {\begin{array}{*{20}c} {12} & 0 & 0 & {6l} & { - 12} & 0 & 0 & {6l} \\ 0 & {12} & { - 6l} & 0 & 0 & { - 12} & { - 6l} & 0 \\ 0 & { - 6l} & {4l^{2} } & 0 & 0 & {6l} & {2l^{2} } & 0 \\ {6l} & 0 & 0 & {4l^{2} } & { - 6l} & 0 & 0 & {2l^{2} } \\ { - 12} & 0 & 0 & { - 6l} & {12} & 0 & 0 & { - 6l} \\ 0 & { - 12} & {6l} & 0 & 0 & {12} & {6l} & 0 \\ 0 & { - 6l} & {2l^{2} } & 0 & 0 & {6l} & {4l^{2} } & 0 \\ {6l} & 0 & 0 & {2l^{2} } & { - 6l} & 0 & 0 & {4l^{2} } \\ \end{array} } \right]$$

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$$K_{1}^{{\text{f}}} = \frac{{EI_{2} }}{{l^{3} }}\left[ {\begin{array}{*{20}c} {12} & 0 & 0 & {6l} & { - 12} & 0 & 0 & {6l} \\ 0 & { - 12} & {6l} & 0 & 0 & {12} & {6l} & 0 \\ 0 & {6l} & { - 4l^{2} } & 0 & 0 & { - 6l} & { - 2l^{2} } & 0 \\ {6l} & 0 & 0 & {4l^{2} } & { - 6l} & 0 & 0 & {2l^{2} } \\ { - 12} & 0 & 0 & { - 6l} & {12} & 0 & 0 & { - 6l} \\ 0 & {12} & { - 6l} & 0 & 0 & { - 12} & { - 6l} & 0 \\ 0 & {6l} & { - 2l^{2} } & 0 & 0 & { - 6l} & { - 4l^{2} } & 0 \\ {6l} & 0 & 0 & {2l^{2} } & { - 6l} & 0 & 0 & {4l^{2} } \\ \end{array} } \right]$$

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$$K_{2}^{{\text{f}}} = \frac{{EI_{2} }}{{l^{3} }}\left[ {\begin{array}{*{20}c} 0 & {12} & { - 6l} & 0 & 0 & { - 12} & { - 6l} & 0 \\ {12} & 0 & 0 & {6l} & { - 12} & 0 & 0 & {6l} \\ { - 6l} & 0 & 0 & { - 4l^{2} } & {6l} & 0 & 0 & { - 2l^{2} } \\ 0 & {6l} & { - 4l^{2} } & 0 & 0 & { - 6l} & { - 2l^{2} } & 0 \\ 0 & { - 12} & {6l} & 0 & 0 & {12} & {6l} & 0 \\ { - 12} & 0 & 0 & { - 6l} & {12} & 0 & 0 & { - 6l} \\ { - 6l} & 0 & 0 & { - 2l^{2} } & {6l} & 0 & 0 & { - 4l^{2} } \\ 0 & {6l} & { - 2l^{2} } & 0 & 0 & { - 6l} & { - 4l^{2} } & 0 \\ \end{array} } \right]$$

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$$G_{{\text{b}}} = \frac{{\rho I_{1} }}{15l}\left[ {\begin{array}{*{20}c} 0 & {36} & { - 3l} & 0 & 0 & { - 36} & { - 3l} & 0 \\ { - 36} & 0 & 0 & { - 3l} & {36} & 0 & 0 & { - 3l} \\ {3l} & 0 & 0 & {4l^{2} } & { - 3l} & 0 & 0 & { - l^{2} } \\ 0 & {3l} & { - 4l^{2} } & 0 & 0 & { - 3l} & {l^{2} } & 0 \\ 0 & { - 36} & {3l} & 0 & 0 & {36} & {3l} & 0 \\ {36} & 0 & 0 & {3l} & { - 36} & 0 & 0 & {3l} \\ {3l} & 0 & 0 & { - l^{2} } & { - 3l} & 0 & 0 & {4l^{2} } \\ 0 & {3l} & {l^{2} } & 0 & 0 & { - 3l} & { - 4l^{2} } & 0 \\ \end{array} } \right]$$

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