Modeling and experimental study on dynamics of a gas-driven split nut device

Abstract

Separation devices are widely used in satellites and rockets, where mechanical properties of the devices have significant influences on connection and separation behaviors. Compared to a pyrotechnically actuated separation device, a gas-driven separation device employs preloads, instead of explosions, to actuate separation of the aircrafts, which has advantages of reusability and low impacts. Dynamic modeling of the gas-driven separation device can be used to optimize its separation behaviors, where flexibility is considered since preloads stored in deformations play important roles in separations. Complex interactions of interfaces occur during the separation process in a short time, which raises issues of convergence and computational time when a finite element method is in use. In this work, we firstly develop a simplified model of a gas-driven separation device, where flexibility is introduced only on the interfaces of components, and their bodies are still rigid. We apply the LZB multipoint method to deal with repeated impacts of the interfaces in the separation process and establish a variable-constraint technique to rebuild contact-point pairs when contact areas vary. Simulations of the dynamic model describe detailed interactions of the separation process. The results illustrate that the separation time decreases when the preload increases, and the decreasing rate gets lower due to reactions of split nuts. We develop a test bench of the gas-driven separation device, by which experimental results validate that the proposed dynamic model can accurately predict the separation time and speeds at different preloads.

Appendices

Appendix A: Mass matrices and angular velocity matrices

The mass matrix of the separation ring and separation plate is expressed as

$$ m_{\mathrm{pr}}=\left [ \textstyle\begin{array}{c@{\quad}c} m_{\mathrm{p}} & 0 \\ 0 & m_{\mathrm{r}} \end{array}\displaystyle \right ], $$

(33)

where $m_{\mathrm{p}}$ and $m_{\mathrm{r}}$ are the masses of the separation ring and separation plate, respectively.

The mass matrix of the bolt is expressed as

$$ m_{\mathrm{b}}=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} m_{\mathrm{b}} & 0 & 0 & 0 & 0 & 0 \\ 0 & m_{\mathrm{b}} & 0 & 0 & 0 & 0 \\ 0 & 0 & m_{\mathrm{b}} & 0 & 0 & 0 \\ 0 & 0 & 0 & B_{\mathrm{b}}^{T}J_{\mathrm{bx}}B_{\mathrm{b}} & 0 & 0 \\ 0 & 0 & 0 & 0 & B_{\mathrm{b}}^{T}J_{\mathrm{by}}B_{\mathrm{b}} & 0 \\ 0 & 0 & 0 & 0 & 0 & B_{\mathrm{b}}^{T}J_{\mathrm{bz}}B_{\mathrm{b}} \end{array}\displaystyle \right ], $$

(34)

where $m_{\mathrm{b}}$ is the mass of the bolt, $J_{\mathrm{bx}}$, $J_{\mathrm{by}}$, and $J_{\mathrm{bz}}$ are inertia of the bolt about $\boldsymbol{e}_{{\mathrm{b}},1}$-axis, $\boldsymbol{e}_{{\mathrm{b}},2}$-axis, and $\boldsymbol{e}_{{\mathrm{b}},3}$-axis, respectively. $B_{\mathrm{b}}$ is the angular velocity matrix of the bolt, denoted as

$$ B_{\mathrm{b}}=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} cos(\beta _{\mathrm{b}}) cos(\gamma _{\mathrm{b}}) & sin(\gamma _{\mathrm{b}}) &0 \\ -cos(\beta _{\mathrm{b}}) sin(\gamma _{\mathrm{b}}) & cos(\gamma _{\mathrm{b}}) & 0 \\ sin(\gamma _{\mathrm{b}}) & 0 & 1 \end{array}\displaystyle \right ]. $$

(35)

The mass matrix of the $k$th nut is expressed as

$$ m_{{\mathrm{n}}k}=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} m_{\mathrm{n}} & 0 & 0 & 0 & 0 & 0 \\ 0 & m_{\mathrm{n}} & 0 & 0 & 0 & 0 \\ 0 & 0 & m_{\mathrm{n}} & 0 & 0 & 0 \\ 0 & 0 & 0 & B_{{\mathrm{n}}k}^{T}J_{{\mathrm{n}}k,{\mathrm{x}}}B_{{\mathrm{n}}k} & 0 & 0 \\ 0 & 0 & 0 & 0 & B_{{\mathrm{n}}k}^{T}J_{{\mathrm{n}}k,{\mathrm{y}}}B_{{\mathrm{n}}k} & 0 \\ 0 & 0 & 0 & 0 & 0 & B_{{\mathrm{n}}k}^{T}J_{{\mathrm{n}}k,{\mathrm{z}}}B_{{\mathrm{n}}k} \end{array}\displaystyle \right ],k=1,2,3, $$

(36)

where $m_{\mathrm{n}}$ is the mass of the nut, $J_{{\mathrm{n}}k,{\mathrm{x}}}$, $J_{{\mathrm{n}}k,{\mathrm{y}}}$, and $J_{{\mathrm{n}}k,{\mathrm{z}}}$ are inertia of the nut about $\boldsymbol{e}_{{\mathrm{n}}k,1}$-axis, $\boldsymbol{e}_{{\mathrm{n}}k,2}$-axis, and $\boldsymbol{e}_{{\mathrm{n}}k,3}$-axis, respectively, and $B_{{\mathrm{n}}k},k=1,2,3$, is the angular velocity matrix of the $k$th nut, denoted as

$$ B_{{\mathrm{n}}k}=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} \cos (\beta _{k}) \cos (\gamma _{k}) & \sin (\gamma _{k}) &0 \\ -\cos (\beta _{k}) \sin (\gamma _{k}) & \cos (\gamma _{k}) & 0 \\ \sin (\gamma _{k}) & 0 & 1 \end{array}\displaystyle \right ],k=1,2,3. $$

(37)

Appendix B: Jacobian matrices and contact force matrices

Jacobian matrices $\boldsymbol{N}_{n}(q)$ and $\boldsymbol{N}_{\tau }(q)$ are defined as

$$ \boldsymbol{N}_{n}(q)=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \frac{\partial \delta _{1,n}}{\partial q_{1}} & \frac{\partial \delta _{1,n}}{\partial q_{2}} & \cdots & \frac{\partial \delta _{1,n}}{\partial q_{26}} \\ \frac{\partial \delta _{2,n}}{\partial q_{1}} & \frac{\partial \delta _{2,n}}{\partial q_{2}} & \cdots & \frac{\partial \delta _{2,n}}{\partial q_{26}} \\ \vdots & \vdots & \vdots \\ \frac{\partial \delta _{n0,n}}{\partial q_{1}} & \frac{\partial \delta _{n0,n}}{\partial q_{2}} & \cdots & \frac{\partial \delta _{n0,n}}{\partial q_{26}} \end{array}\displaystyle \right ], $$

(38)

$$ \boldsymbol{N}_{\tau }(q)=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \frac{\partial \delta _{1,\tau}}{\partial q_{1}} & \frac{\partial \delta _{1,\tau}}{\partial q_{2}} & \cdots & \frac{\partial \delta _{1,\tau}}{\partial q_{26}} \\ \frac{\partial \delta _{2,\tau}}{\partial q_{1}} & \frac{\partial \delta _{2,\tau}}{\partial q_{2}} & \cdots & \frac{\partial \delta _{2,\tau}}{\partial q_{26}} \\ \vdots & \vdots & \vdots \\ \frac{\partial \delta _{n0,\tau}}{\partial q_{1}} & \frac{\partial \delta _{n0,\tau}}{\partial q_{2}} & \cdots & \frac{\partial \delta _{n0,\tau}}{\partial q_{26}} \end{array}\displaystyle \right ], $$

(39)

respectively. The matrices of contact forces $\boldsymbol{F}_{n}$ and $\boldsymbol{F}_{\tau}$ are defined as

$$\begin{aligned} \boldsymbol{F}_{n}(q)&=[F_{1,n},\ F_{2,n},\ \dots ,F_{26,n}], \end{aligned}$$

(40)

$$\begin{aligned} \boldsymbol{F}_{\tau}(q)&=[F_{1,\tau},\ F_{2,\tau},\ \dots ,F_{26,\tau}], \end{aligned}$$

(41)

respectively, where all contact force components are calculated by the method in Sect. 4.2.

Appendix C: Contact points on the other interfaces of nuts

(1) Contact points on the surface of the separation ring

The contact-point pair with the order of $\xi \eta$ between nuts and the separating ring is $(P_{{\mathrm{r}},\xi \eta},P_{{\mathrm{r}},\xi \eta}^{\prime})$ with $\xi =1,2,\dots ,n_{{\mathrm{r}}z}$ in the axial direction and $\eta =1,2,\dots ,n_{{\mathrm{r}}h}$ in the circumferential direction, where $n_{{\mathrm{r}}z}$ and $n_{{\mathrm{r}}h}$ are numbers of discrete points along their directions, respectively. As shown in Fig. 31, the normal unit vector of the contact pair is $\boldsymbol{e}_{{\mathrm{nr}},\xi \eta ,3}=[\cos \theta _{{\mathrm{r}},\xi \eta},\ - \sin \theta _{{\mathrm{r}},\xi \eta},\ 0]$, where $\xi =1,2,\dots ,n_{rz}$, $\eta =1,2,\dots ,n_{{\mathrm{r}}h}$, and $\theta _{{\mathrm{r}},\xi \eta}$ is the turning angle of separation ring contact point in circumferential direction. The coordinates of $P_{{\mathrm{r}},\xi \eta}$ and $P_{{\mathrm{r}},\xi \eta}^{\prime}$ are $\boldsymbol{r}_{OP_{{\mathrm{r}},\xi \eta}}(q) = \boldsymbol{r}_{OO_{{\mathrm{n}}k}}(q) + \boldsymbol{r}_{OO_{{\mathrm{n}}k}}(q) \boldsymbol{r}_{O_{{\mathrm{n}}k}P_{{\mathrm{r}},\xi \eta}}$ and $\boldsymbol{r}_{OP_{{\mathrm{r}},\xi \eta}^{\prime}}(q) = \boldsymbol{r}_{OO_{\mathrm{r}}}(q) + \boldsymbol{r}_{O_{\mathrm{r}}P_{{\mathrm{r}},\xi \eta}^{\prime}}$, respectively.

Fig. 31

Contact points on separation ring and nuts

(2) Contact points on the surface of separation plate and supporting base

The contact-point pair with the order of $\xi \eta$ between nuts and the separating plate is $(P_{i,\xi \eta},P_{i,\xi \eta}^{\prime})$, where $i={\mathrm{p}}$ and $i={\mathrm{m}}$ are contact pairs of the separation plate and the supporting base, respectively, $\xi =1,2,\dots ,n_{iz}$ is in the axial direction, and $\eta =1,2,\dots ,n_{ih}$ is in the circumferential direction, where $n_{iz}$ and $n_{ih}$ are the numbers of discrete points along the two directions, respectively. As shown in Figs. 32 and 33, the normal unit vector of the $\xi \eta$th contact pair is $\boldsymbol{e}_{{\mathrm{n}}i,\xi \eta ,3}=[\cos (\theta _{i,\xi \eta})\sin ( \gamma _{i}),\ \sin (\theta _{i,\xi \eta})\sin (\gamma _{i}),\ \cos ( \gamma _{i})]$, where $\xi =1,2,\dots ,n_{iz}$, $\eta =1,2,\dots ,n_{ih}$, and $\theta _{i,\xi \eta}$ is the turning angle of contact point in circumferential direction, $\gamma _{\mathrm{p}}=\gamma$, $\gamma _{\mathrm{m}}=\pi -\gamma$. The coordinates of $P_{{\mathrm{p}},\xi \eta}$ and $P_{{\mathrm{p}},\xi \eta}^{\prime}$ are $\boldsymbol{r}_{OP_{{\mathrm{p}},\xi \eta}}(q) = \boldsymbol{r}_{OO_{\mathrm{r}}}(q) + \boldsymbol{r}_{O_{ \mathrm{r}}P_{{\mathrm{p}},\xi \eta}}$ and $\boldsymbol{r}_{OP_{{\mathrm{p}},\xi \eta}^{\prime}}(q) = \boldsymbol{r}_{OO_{{\mathrm{n}}k}}(q) + \boldsymbol{r}_{O_{{\mathrm{n}}k}P_{{\mathrm{p}},\xi \eta}^{\prime}}$, respectively, and the coordinates of $P_{{\mathrm{m}},\xi \eta}$ and $P_{{\mathrm{m}},\xi \eta}^{\prime}$ are $\boldsymbol{r}_{O_{{\mathrm{n}}k}P_{{\mathrm{m}},\xi \eta}^{\prime}}$ and $\boldsymbol{r}_{OP_{{\mathrm{m}},\xi \eta}^{\prime}}(q) = \boldsymbol{r}_{OO_{{\mathrm{n}}k}}(q)+ \boldsymbol{r}_{O_{{\mathrm{n}}k}P_{{\mathrm{m}},\xi \eta}^{\prime}}$, respectively.

Fig. 32

Contact points on separation plate and supporting base

Fig. 33

Contact points on separation plate of a cross-section of $\boldsymbol{e}_{1}\boldsymbol{e}_{2}$

Appendix D: Parameters

Table 1 shows the geometric parameters of the static model. Table 2 shows the mass and inertia parameters of the bolt, nuts, the separation ring, and the separation plate as well as the friction and restitution coefficients of interfaces. Table 3 shows the kinematic parameters of the initial configuration. Table 4 shows the geometric and gas parameters of the gas pressure model.

Table 1 Parameters of static model

Table 2 Material parameters

Table 3 Geometry parameters of initial configuration

Table 4 Parameters of gas actuator