Stability analysis of a rotor system with electromechanically coupled boundary conditions under periodic axial load

Abstract

The parametric instability of a rotor system with electromechanically coupled boundary conditions under periodic axial loads is studied. Based on the current flowing piezoelectric shunt damping technique, the detailed rotor model is established by the finite element (FE) method. In the matrix assembly procedure, a novel simple process is proposed to make the equations of shunt circuits more conveniently to be introduced into the global FE equations. The discrete state transition matrix method which is used for determining the influence of circuit parameters on instability regions in this paper has also been presented. The numerical simulation shows that only the combination instability regions exist when the shaft is rotating. The mechanical damping has different effect on the simple and combined instability regions. These two points are consistent with the previous references, which verifies the obtained FE model. In addition, the simulated results also reveal that the introduction of shunt circuits has little influence on the rotor’s original whirling frequencies. It gives rise to the appearance of new synchronous whirl modes. The new whirling frequencies are combined with the original ones to form the new combination instability regions. Furthermore, the resistance of shunt circuits has the same performance as the mechanical damping has. That is, moving up the start points of instability regions and expanding its width.

Appendix

Note that the second equation in Eq. (7)–(10) are similar and compact. What their final forms like depends on what shunted circuit the vibration ring connected. Hence, it is necessary to derive their specific forms in the case of using current flowing shunt circuits. Before giving the general FE model using N branch shunt circuits, the case of using single branch is discussed as follows. In which case, the second equation in Eq. (7) is analyzed here.

For a shunted piezo stack, it is equivalent to an ideal voltage source U_p in series with a capacitor $C_{p}^{S}$ or an ideal current source in parallel with such capacitor. When the stack is shunted with a single resonant circuit, its circuit diagram is shown in Fig. 

Fig. 14

The circuit diagram for a piezo stack shunted with a single resonant circuit

14. In this case, the electrical impedance Z_e is equal to 1/sC_f1 + sL₁ + R_t1, where s is the Laplace transform variable. According to Eq. (1), the second equation in Eq. (7) can be written as

$$ 2\cot^{2} \beta \frac{{\theta_{p}^{2} }}{{C_{p}^{S} }}\left( {a_{lz} - \frac{{q_{lz} }}{{\theta_{p} }}} \right) = 2\theta_{p}^{2} \cot^{2} \beta \left( {\frac{1}{{sC_{f1} }} + sL_{1} + R_{t1} } \right)\frac{{\dot{q}_{lz} }}{{\theta_{p} }} $$

(A1)

Equation (A1) can be straightforwardly transformed into the following time domain expression:

$$ L_{1} \ddot{q}_{lz} + R_{t1} \dot{q}_{lz} + \left( {\frac{1}{{C_{p}^{S} }} + \frac{1}{{C_{f1} }}} \right)q_{lz} - \frac{{\theta_{p} }}{{C_{p}^{S} }}a_{lz} = 0 $$

(A2)

Actually Eq. (A2) can be directly derived using Kirchhoff’s Voltage Law (KVL) through Fig. 14. It is obvious that the voltage generated by the piezo stack due to the electromechanical effect is equal to $(\theta_{p} /C_{p}^{S} )a_{lz}$. This inspired us that for the multi-resonant shunts, its circuit diagram can be drawn as Fig. 

Fig. 15

The circuit diagram of multi-resonant shunt circuit

15. From Fig. 15 the following equations can be obtained using KVL

$$ \left\{ \begin{gathered} \dot{\tilde{q}}_{lz} = \dot{\tilde{q}}_{lz1} + \dot{\tilde{q}}_{lz2} + \cdots + \dot{\tilde{q}}_{lzN} \Rightarrow \tilde{q}_{lz} = \tilde{q}_{lz1} + \tilde{q}_{lz2} + \cdots + \tilde{q}_{lzN} \hfill \\ L_{1} \ddot{\tilde{q}}_{lz1} + R_{t1} \dot{\tilde{q}}_{lz1} + \frac{{\tilde{q}_{lz1} }}{{C_{f1} }} + \frac{{\tilde{q}_{lz} }}{{C_{p}^{S} }} - \frac{{a_{lz} }}{{C_{p}^{S} }} = 0 \hfill \\ L_{2} \ddot{\tilde{q}}_{lz2} + R_{t2} \dot{\tilde{q}}_{lz2} + \frac{{\tilde{q}_{lz2} }}{{C_{f2} }} + \frac{{\tilde{q}_{lz} }}{{C_{p}^{S} }} - \frac{{a_{lz} }}{{C_{p}^{S} }} = 0 \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \hfill \\ L_{N} \ddot{\tilde{q}}_{lzN} + R_{tN} \dot{\tilde{q}}_{lzN} + \frac{{\tilde{q}_{lzN} }}{{C_{fN} }} + \frac{{\tilde{q}_{lz} }}{{C_{p}^{S} }} - \frac{{a_{lz} }}{{C_{p}^{S} }} = 0 \hfill \\ \end{gathered} \right. $$

(A3)

where the charge in each branch shunt has been transformed into generalized charge through dividing it by the generalized electromechanical coupling factor θ_p. Note that the first equation of Eq. (A3) is only valid under the zero initial condition. Combining Eq. (A3) and the first equation in Eq. (7), the following matrix equation can be obtained

$$ {\mathbf{M}}_{{\text{c}}} \ddot{\mathbf{{u}}}_{{{\text{lzc}}}} + {\mathbf{C}}_{{\text{c}}} {\dot{\mathbf{u}}}_{{{\text{lzc}}}} + {\mathbf{K}}_{{\text{c}}} {\mathbf{u}}_{{{\text{lzc}}}} = {\mathbf{f}}_{{{\text{lzc}}}} $$

(A4)

where

$$ \begin{gathered} {\mathbf{M}}_{{\text{c}}} = {\text{diag}}\left[ {m_{b} ,\;L_{1} ,\;L_{2} ,\; \cdots ,\;L_{N} } \right],\;\;{\mathbf{C}}_{{\text{c}}} = {\text{diag}}\left[ {\eta ,\;R_{t1} ,\;R_{t2} ,\; \cdots ,\;R_{tN} } \right] \hfill \\ {\mathbf{K}}_{{\text{c}}} = \left[ {\begin{array}{*{20}c} {k_{sum} } & { - k_{eq2} } & { - k_{eq2} } & \cdots & { - k_{eq2} } \\ { - \frac{1}{{C_{p}^{S} }}} & {\frac{1}{{C_{f1} }} + \frac{1}{{C_{p}^{S} }}} & {\frac{1}{{C_{p}^{S} }}} & \cdots & {\frac{1}{{C_{p}^{S} }}} \\ { - \frac{1}{{C_{p}^{S} }}} & {\frac{1}{{C_{p}^{S} }}} & {\frac{1}{{C_{f2} }} + \frac{1}{{C_{p}^{S} }}} & \ddots & \vdots \\ \vdots & \vdots & \ddots & \ddots & {\frac{1}{{C_{p}^{S} }}} \\ { - \frac{1}{{C_{p}^{S} }}} & {\frac{1}{{C_{p}^{S} }}} & \cdots & {\frac{1}{{C_{p}^{S} }}} & {\frac{1}{{C_{fN} }} + \frac{1}{{C_{p}^{S} }}} \\ \end{array} } \right] \hfill \\ {\mathbf{f}}_{{{\text{lzc}}}} = \left[ {k_{eq} w\left( {0,t} \right) + \eta \dot{w}\left( {0,t} \right),\;0,\; \cdots ,\;0} \right]^{{\text{T}}} = \left[ {k_{eq} w_{1} + \eta \dot{w}_{1} ,\;0,\; \cdots ,\;0} \right]^{{\text{T}}} \hfill \\ {\mathbf{u}}_{{{\text{lzc}}}} = \left[ {a_{lz} ,\;\tilde{q}_{lz1} ,\;\tilde{q}_{lz2} ,\; \cdots ,\;\tilde{q}_{lzN} } \right]^{{\text{T}}} \hfill \\ \end{gathered} $$

(A5)

In the same manner, the matrix equation of Eq. (8)–(10) for the other boundary conditions can be derived, which are given as

$$ \begin{gathered} {\mathbf{M}}_{{\text{c}}} \ddot{\mathbf{{u}}}_{{{\text{rzc}}}} + {\mathbf{C}}_{{\text{c}}} {\dot{\mathbf{u}}}_{{{\text{rzc}}}} + {\mathbf{K}}_{{\text{c}}} {\mathbf{u}}_{{{\text{rzc}}}} = {\mathbf{f}}_{{{\text{rzc}}}} \hfill \\ {\mathbf{M}}_{{\text{c}}} \ddot{\mathbf{{u}}}_{{{\text{lyc}}}} + {\mathbf{C}}_{{\text{c}}} {\dot{\mathbf{u}}}_{{{\text{lyc}}}} + {\mathbf{K}}_{{\text{c}}} {\mathbf{u}}_{{{\text{lyc}}}} = {\mathbf{f}}_{{{\text{lyc}}}} \hfill \\ {\mathbf{M}}_{{\text{c}}} \ddot{\mathbf{{u}}}_{{{\text{ryc}}}} + {\mathbf{C}}_{{\text{c}}} {\dot{\mathbf{u}}}_{{{\text{ryc}}}} + {\mathbf{K}}_{{\text{c}}} {\mathbf{u}}_{{{\text{ryc}}}} = {\mathbf{f}}_{{{\text{ryc}}}} \hfill \\ \end{gathered} $$

(A6)

where

$$\begin{aligned} & {\mathbf{f}}_{{{\text{rzc}}}} = \left[ {k_{{eq}} w\left( {L,t} \right) + \eta \dot{w}\left( {L,t} \right),0, \ldots ,0} \right]^{{\text{T}}} = \left[ {k_{{eq}} w_{{n + 1}} + \eta \dot{w}_{{n + 1}} ,0, \ldots ,0} \right]^{{\text{T}}} \\ & {\mathbf{f}}_{{{\text{lyc}}}} = \left[ {k_{{eq}} v\left( {0,t} \right) + \eta \dot{v}\left( {0,t} \right),0, \ldots ,0} \right]^{{\text{T}}} = \left[ {k_{{eq}} v_{1} + \eta \dot{v}_{1} ,0, \ldots ,0} \right]^{{\text{T}}} \\ & {\mathbf{f}}_{{{\text{ryc}}}} = \left[ {k_{{eq}} v\left( {L,t} \right) + \eta \dot{v}\left( {L,t} \right),0, \ldots ,0} \right]^{{\text{T}}} = \left[ {k_{{eq}} v_{{n + 1}} + \eta \dot{v}_{{n + 1}} ,0, \ldots ,0} \right]^{{\text{T}}} \\ & {\mathbf{u}}_{{{\text{rzc}}}} = \left[ {a_{{rz}} ,\tilde{q}_{{rz1}} ,\tilde{q}_{{rz2}} , \ldots ,\tilde{q}_{{rzN}} } \right]^{{\text{T}}} \\ & {\mathbf{u}}_{{{\text{lyc}}}} = \left[ {a_{{ly}} ,\tilde{q}_{{ly1}} ,\tilde{q}_{{ly2}} , \ldots ,\tilde{q}_{{lyN}} } \right]^{{\text{T}}} \\ & {\mathbf{u}}_{{{\text{ryc}}}} = \left[ {a_{{ry}} ,\tilde{q}_{{ry1}} ,\tilde{q}_{{ry2}} , \ldots ,\tilde{q}_{{ryN}} } \right]^{{\text{T}}} \\ \end{aligned}$$

(A7)

Note that the stiffness matrix K_c in Eq. (A5) is asymmetric. To symmetrize it, enlightened by the designation: k_eq2 = 2cot²β ·$\theta_{p}^{2} /C_{p}^{S}$, multiply each row which corresponding to the generalized charge $\tilde{q}_{lzi}$ by 2cot²β ·$\theta_{p}^{2}$, one can obtain the following symmetric expression:

$$ \begin{gathered} M_{c} = diag\left[ {m_{b} ,\;2\cot^{2} \beta \theta_{p}^{2} L_{1} ,\;2\cot^{2} \beta \theta_{p}^{2} L_{2} ,\; \cdots ,\;2\cot^{2} \beta \theta_{p}^{2} L_{N} } \right] \hfill \\ C_{c} = diag\left[ {\eta ,\;2\cot^{2} \beta \theta_{p}^{2} R_{t1} ,\;2\cot^{2} \beta \theta_{p}^{2} R_{t2} ,\; \cdots ,\;2\cot^{2} \beta \theta_{p}^{2} R_{tN} } \right] \hfill \\ K_{c} = \left[ {\begin{array}{*{20}c} {k_{sum} } & { - k_{eq2} } & { - k_{eq2} } & \cdots & { - k_{eq2} } \\ { - k_{eq2} } & {\frac{{2\cot^{2} \beta \theta_{p}^{2} }}{{C_{f1} }} + k_{eq2} } & {k_{eq2} } & \cdots & {k_{eq2} } \\ { - k_{eq2} } & {k_{eq2} } & {\frac{{2\cot^{2} \beta \theta_{p}^{2} }}{{C_{f2} }} + k_{eq2} } & \ddots & \vdots \\ \vdots & \vdots & \ddots & \ddots & {k_{eq2} } \\ { - k_{eq2} } & {k_{eq2} } & \cdots & {k_{eq2} } & {\frac{{2\cot^{2} \beta \theta_{p}^{2} }}{{C_{fN} }} + k_{eq2} } \\ \end{array} } \right] \hfill \\ \end{gathered} $$

(A8)

From above description, one can see that this process is not only applicable to the current flowing shunt circuits, but also for the other kinds of shunt circuits. One can directly use the circuit theory to derive the shunt circuits’ governing equations and assemble them into the global FE equations. The key point is that substituting the voltage source which generated from the piezo stack by the term $(\theta_{p} /C_{p}^{S} )a_{lz}$. This is namely the representative of electromechanical coupling.