Contact-impact analysis of a rotating geometric nonlinear plate under thermal shock

Abstract

A contact-impact model is developed for the rub-induced vibration analysis of a rotating geometric nonlinear plate under thermal shock. The proposed model takes into account different contact positions in the width direction and is capable of adjusting the form of rubbing forces to different system parameters. A series of discussions is carried out to verify the present contact-impact model and the effects of the friction coefficient, the contact stiffness, and thermal transient are investigated. Numerical results show that the contact stiffness plays an important role in chaotic nonlinear vibrations by changing the friction force. The friction force is the main cause of geometric nonlinear chaos motions of the rotating contact-impact plate. Thermal shock results in the divergence of rub-induced vibrations for the geometric nonlinear plate with an increase in the time $t$. Reducing the time interval before the thermal transient ceases can be an effective way to reduce thermal rub-induced damage.

Appendix

The terms in nonlinear equations of motion

$$\begin{aligned} pa_{mn}^{ij}&= \rho h\int _0^b {\int _0^a {\sin \left( \displaystyle \frac{(2m-1)\pi x}{2a}\right) \varphi _n (y)\sin \left( \displaystyle \frac{(2i-1)\pi x}{2a}\right) \varphi _j (y)\mathrm {d}x\mathrm {d}y}}, \end{aligned}$$

(32)

$$\begin{aligned} pb_{mn}^{ij}&= 2\Omega \rho h\int _0^b {\int _0^a {\phi _m (x)\varphi _n (y)\sin \left( \displaystyle \frac{(2i-1)\pi x}{2a}\right) \varphi _j (y)\mathrm {d}x\mathrm {d}y} },\end{aligned}$$

(33)

$$\begin{aligned} pc_{mn}^{ij}&= \rho h\Omega ^{2}\int _0^b {\int _0^a {\sin \left( \displaystyle \frac{(2m-1)\pi x}{2a}\right) \varphi _n (y) \left( \sin \displaystyle \frac{(2i-1)\pi x}{2a}\right) \varphi _j (y)\mathrm {d}x\mathrm {d}y} }\nonumber \\&+\int _0^b {\int _0^a {\displaystyle \frac{Eh}{1-\upsilon ^{2}}\left( {\displaystyle \frac{d^{2}}{d^{2}x}\sin \left( \displaystyle \frac{(2m-1)\pi x}{2a}\right) } \right) \varphi _n (y)\sin \left( \displaystyle \frac{(2i-1)\pi x}{2a}\right) \varphi _j (y)\mathrm {d}x\mathrm {d}y} }\nonumber \\&+\int _0^b \int _0^a \displaystyle \frac{Eh}{2\left( {1+\upsilon } \right) }\sin \left( \displaystyle \frac{(2m-1)\pi x}{2a}\right) \ddot{\varphi }_n (y)\sin \left( \displaystyle \frac{(2i-1)\pi x}{2a}\right) \varphi _j (y)\mathrm {d}x\mathrm {d}y\nonumber \\&-\int _0^b { {\displaystyle \frac{Eh}{1-\upsilon ^{2}}\left( {\displaystyle \frac{d}{\mathrm {d}x}\sin \displaystyle \left( \frac{(2m-1)\pi x}{2a}\right) } \right) \varphi _n (y)\sin \displaystyle \left( \frac{(2i-1)\pi x}{2a}\right) \varphi _j (y)} \bigg |_0^a \mathrm {d}y}\nonumber \\&-\int _0^a { {\displaystyle \frac{Eh}{2\left( {1+\upsilon } \right) }\sin \displaystyle \left( \frac{(2m-1)\pi x}{2a}\right) \dot{\varphi }_n (y)\sin \displaystyle \left( \frac{(2i-1)\pi x}{2a}\right) \varphi _j (y)} \bigg |_0^b \mathrm {d}x}, \end{aligned}$$

(34)

$$\begin{aligned} pd_{mn}^{ij}&= \int _0^b {\int _0^a {\displaystyle \frac{Eh}{1-\upsilon ^{2}}\upsilon \dot{\phi }_m (x)\left( {\displaystyle \frac{d}{\mathrm {d}y}\cos \displaystyle \left( \frac{n\pi y}{b}\right) } \right) \sin \displaystyle \left( \frac{(2i-1)\pi x}{2a}\right) \varphi _j (y)\mathrm {d}x\mathrm {d}y} }\nonumber \\&+\int _0^b {\int _0^a {\displaystyle \frac{Eh}{2\left( {1+\upsilon } \right) }\dot{\phi }_m (x)\left( {\displaystyle \frac{d}{\mathrm {d}y}\cos \displaystyle \left( \frac{n\pi y}{b}\right) } \right) \sin \displaystyle \left( \frac{(2i-1)\pi x}{2a}\right) \varphi _j (y)\mathrm {d}x\mathrm {d}y} }\nonumber \\&-\int _0^b {{\displaystyle \frac{Eh}{1-\upsilon ^{2}}\upsilon \phi _m (x)\left( {\displaystyle \frac{d}{\mathrm {d}y}\cos \displaystyle \left( \frac{n\pi y}{b}\right) } \right) \sin \displaystyle \left( \frac{(2i-1)\pi x}{2a}\right) \varphi _j (y)} \bigg |_0^a \mathrm {d}y} \nonumber \\&-\int _0^a { {\displaystyle \frac{Eh}{2\left( {1+\upsilon } \right) }\dot{\phi }_m (x)\cos \displaystyle \left( \frac{n\pi y}{b}\right) \sin \displaystyle \left( \frac{(2i-1)\pi x}{2a}\right) \varphi _j (y)} \bigg |_0^b \mathrm {d}x}, \end{aligned}$$

(35)

$$\begin{aligned} qa_{mn}^{ij}&= \rho h\int _0^b {\int _0^a {\phi _m (x)\cos \displaystyle \left( \frac{n\pi y}{b}\right) \phi _i (x)\cos \displaystyle \left( \frac{j\pi y}{b}\right) \mathrm {d}x\mathrm {d}y}, } \end{aligned}$$

(36)

$$\begin{aligned} qb_{mn}^{ij}&= \int _0^b {\int _0^a {\displaystyle \frac{Eh}{1-\upsilon ^{2}}\upsilon \left( {\displaystyle \frac{d}{\mathrm {d}x}\sin \displaystyle \left( \frac{(2m-1)\pi x}{2a}\right) } \right) \dot{\varphi }_n (y)\phi _i (x)\cos \displaystyle \left( \frac{j\pi y}{b}\right) \mathrm {d}x\mathrm {d}y} } \nonumber \\&+\displaystyle \frac{Eh}{2\left( {1+\upsilon } \right) }\int _0^b {\int _0^a {\left( {\displaystyle \frac{d}{\mathrm {d}x}\sin \displaystyle \left( \frac{(2m-1)\pi x}{2a}\right) } \right) \dot{\varphi }_n (y)\phi _i (x)\cos \displaystyle \left( \frac{j\pi y}{b}\right) \mathrm {d}x\mathrm {d}y} } \nonumber \\&-\displaystyle \frac{Eh}{1-\upsilon ^{2}}\upsilon \int _0^a {\left( {\displaystyle \frac{d}{\mathrm {d}x}\sin \displaystyle \left( \frac{(2m-1)\pi x}{2a}\right) } \right) \varphi _n (y)\phi _i (x)\cos \displaystyle \left( \frac{j\pi y}{b}\right) } \bigg |_0^b \mathrm {d}x \nonumber \\&-\displaystyle \frac{Eh}{2\left( {1+\upsilon } \right) }\int _0^b { {\sin \displaystyle \left( \frac{(2m-1)\pi x}{2a}\right) \dot{\varphi }_n (y)\phi _i (x)\cos \displaystyle \left( \frac{j\pi y}{b}\right) } \bigg |_0^a \mathrm {d}y}, \end{aligned}$$

(37)

$$\begin{aligned} qc_{mn}^{ij}&= \displaystyle \frac{Eh}{1-\upsilon ^{2}}\int _0^b {\int _0^a {\phi _m (x)\left( {\displaystyle \frac{d^{2}}{\mathrm {d}y^{2}}\cos \displaystyle \left( \frac{n\pi y}{b}\right) } \right) \phi _i (x)\cos \displaystyle \left( \frac{j\pi y}{b}\right) \mathrm {d}x\mathrm {d}y} } \nonumber \\&+\displaystyle \frac{Eh}{2\left( {1+\upsilon } \right) }\int _0^b {\int _0^a {\ddot{\phi }_m (x)\cos \displaystyle \left( \frac{n\pi y}{b}\right) \phi _i (x)\cos \displaystyle \left( \frac{j\pi y}{b}\right) \mathrm {d}x\mathrm {d}y} } \nonumber \\&-\displaystyle \frac{Eh}{1-\upsilon ^{2}}\int _0^a {\phi _m (x)\left( {\displaystyle \frac{d}{\mathrm {d}y}\cos \displaystyle \left( \frac{n\pi y}{b}\right) } \right) \phi _i (x)\cos \displaystyle \left( \frac{j\pi y}{b}\right) } \bigg |_0^b \mathrm {d}x \nonumber \\&-\displaystyle \frac{Eh}{2\left( {1+\upsilon } \right) }\int _0^b { {\dot{\phi }_m (x)\cos \displaystyle \left( \frac{n\pi y}{b}\right) \phi _i (x)\cos \displaystyle \left( \frac{j\pi y}{b}\right) } \bigg |_0^a \mathrm {d}y} , \end{aligned}$$

(38)

$$\begin{aligned} ra_{mn}^{ij}&= \rho h\int _0^b {\int _0^a {\phi _m (x)\varphi _n (y)\phi _i (x)\varphi _j (y)\mathrm {d}x\mathrm {d}y} }, \end{aligned}$$

(39)

$$\begin{aligned} rb_{mn}^{ij}&= D\zeta \int _0^b {\int _0^a {\left[ {\phi ^{(4)}_m (x)\varphi _n (y)+2\ddot{\phi }_m (x)\ddot{\varphi }_n (y)+\phi _m (x)\varphi ^{(4)}_n (y)} \right] \phi _i (x)\varphi _j (y)\mathrm {d}x\mathrm {d}y} } ,\end{aligned}$$

(40)

$$\begin{aligned} rc_{mn}^{ij}&= 2\Omega \rho h\int _0^b {\int _0^a {\sin \displaystyle \left( \frac{(2m-1)\pi x}{2a}\right) \varphi _n (y)\phi _i (x)\varphi _j (y)\mathrm {d}x\mathrm {d}y} },\end{aligned}$$

(41)

$$\begin{aligned} rd_{mn}^{ij}&= \rho h\Omega ^{2}\int _0^b {\int _0^a {\phi _m (x)\varphi _n (y)\phi _i (x)\varphi _j (y)\mathrm {d}x\mathrm {d}y} }\nonumber \\&-D\int _0^b {\int _0^a {\left[ {\phi ^{(4)}_m (x)\varphi _n (y)+2\ddot{\phi }_m (x)\ddot{\varphi }_n (y)+\phi _m (x)\varphi ^{(4)}_n (y)} \right] \phi _i (x)\varphi _j (y)\mathrm {d}x\mathrm {d}y} } \nonumber \\&-\rho h\Omega ^{2}\int _0^b {\int _0^a {\left[ {\left( {R+x} \right) {\phi }'_m (x)\varphi _n (y)-\left( {R(a-x)+\displaystyle \frac{1}{2}(a^{2}-x^{2})} \right) \ddot{\phi }_m (x)\varphi _n (y)} \right] \phi _i (x)\varphi _j (y)\mathrm {d}x\mathrm {d}y} } \nonumber \\&-D{\int _0^b {\left[ {\ddot{\phi }_m (x)\varphi _n (y)+\upsilon \phi _m (x)\ddot{\varphi }_n (y)} \right] \dot{\phi }_i (x)\varphi _j (y)\mathrm {d}y} } \bigg |_0^a \nonumber \\&-D{\int _0^a {\left[ {\phi _m (x)\ddot{\varphi }_n (y)+\upsilon \ddot{\phi }_m (x)\varphi _n (y)} \right] \phi _i (x)\dot{\varphi }_j (y)\mathrm {d}x} } \bigg |_0^b \nonumber \\&+D{\int _0^b {\left[ {\phi ^{(3)}_m (x)\varphi _n (y)+(2-\upsilon )\dot{\phi }_m (x)\ddot{\varphi }_n (y)} \right] } \phi _i (x)\varphi _j (y)\mathrm {d}y} \bigg |_0^a \nonumber \\&+D{\int _0^a {\left[ {\phi _m (x)\varphi ^{(3)}_n (y)+(2-\upsilon )\ddot{\phi }_m (x)\dot{\varphi }_n (y)} \right] \phi _i (x)\varphi _j (y)\mathrm {d}x} } \bigg |_0^b \nonumber \\&-2D(1-\upsilon ){{\dot{\phi }_m (x)\dot{\varphi }_n (y)\phi _i (x)\varphi _j (y)}} \bigg |_{(0,0)}^{(a,b)}\nonumber \\&-\rho h\Omega ^{2}{\int _0^b {\left[ {R(a-x)+\displaystyle \frac{1}{2}(a^{2}-x^{2})} \right] \dot{\phi }_m (x)\varphi _n (y)\phi _i (x)\varphi _j (y)\mathrm {d}y} } \bigg |_0^a \nonumber \\&-\displaystyle \frac{Eh}{(1-\upsilon ^{2})}\left( {1+\upsilon } \right) a_T N_T \int _0^b {\int _0^a {{\phi }''_m (x)\varphi _n (y)\phi _i (x)\varphi _j (y)\mathrm {d}x\mathrm {d}y} } \nonumber \\&-\displaystyle \frac{Eh}{(1-\upsilon ^{2})}\left( {1+\upsilon } \right) a_T N_T \int _0^b {\int _0^a {\phi _m (x){\varphi }''_n (y)\phi _i (x)\varphi _j (y)\mathrm {d}x\mathrm {d}y} } \nonumber \\&+\displaystyle \frac{Eh}{1-\upsilon ^{2}}\left( {1+\upsilon } \right) a_T N_T {\int _0^b {{\phi }'_m (x)\varphi _n (y)\phi _i (x)\varphi _j (y)\mathrm {d}y} } \bigg |_0^a\nonumber \\&+\displaystyle \frac{Eh}{1-\upsilon ^{2}}\left( {1+\upsilon } \right) a_T N_T {\int _0^a {\phi _m (x){\varphi }'_n (y)\phi _i (x)\varphi _j (y)\mathrm {d}x} } \bigg |_0^b \,, \end{aligned}$$

(42)

$$\begin{aligned} f_1^{ij}&= \rho h\Omega ^{2}\int _0^b {\int _0^a {(R+x)\sin \displaystyle \left( \frac{(2i-1)\pi x}{2a}\right) \varphi _j (y)\mathrm {d}x\mathrm {d}y} } -\sin \displaystyle \left( \frac{(2i-1)\pi }{2}\right) \int _0^b {F_n (y,t)\varphi _j (y)\mathrm {d}y} \nonumber \\&-\rho h\int _0^b \int _0^a \left[ {\ddot{x}_D \cos \Omega t+\ddot{z}_D \sin \Omega t} \right] \sin \displaystyle \left( \frac{(2i-1)\pi x}{2a}\right) \varphi _j (y)\mathrm {d}x\mathrm {d}y\nonumber \\&+\displaystyle \frac{Eh\left( {1+\upsilon } \right) a_T N_T }{1-\upsilon ^{2}}{\int _0^b {\sin \displaystyle \left( \frac{(2i-1)\pi x}{2a}\right) \varphi _j (y)\mathrm {d}y} } \bigg |_0^a ,\end{aligned}$$

(43)

$$\begin{aligned} f_2^{ij}&= \displaystyle \frac{Eh}{1-\upsilon ^{2}}\left( {1+\upsilon } \right) a_T N_T {\int _0^a {\phi _i (x)\cos \displaystyle \left( \frac{j\pi y}{b}\right) \mathrm {d}x} } \bigg |_0^b \,, \end{aligned}$$

(44)

$$\begin{aligned} \displaystyle f_3^{ij}&= -\int _0^b {\int _0^a {\mu F_n (y,t)\delta (x-a)\phi _i (x)\varphi _j (y)\mathrm {d}x\mathrm {d}y} } -\rho h\int _0^b {\int _0^a {\left[ {-\ddot{x}_D \sin \Omega t+\ddot{z}_D \cos \Omega t} \right] \phi _i (x)\varphi _j (y)\mathrm {d}x\mathrm {d}y} } \nonumber \\&-D\left( {1+\upsilon } \right) a_T \left\{ {\begin{array}{l} \displaystyle \int _0^b {\int _0^a {\displaystyle \frac{\partial ^{2}M_T }{\partial x^{2}}} \phi _i (x)\varphi _j (y)\mathrm {d}x\mathrm {d}y+\int _0^b {\int _0^a {\displaystyle \frac{\partial ^{2}M_T }{\partial y^{2}}\phi _i (x)\varphi _j (y)\mathrm {d}x\mathrm {d}y } } } \\ \displaystyle +\int _0^b {{M_T {\phi }'_i (x)\varphi _j (y)} \bigg |_0^a \mathrm {d}y}-\int _0^b {{\displaystyle \frac{\partial M_T }{\partial x}\phi _i (x)\varphi _j (y)} \bigg |_0^a \mathrm {d}y}\\ \displaystyle +\int _0^a { {M_T \phi _i (x){\varphi }'_j (y)} \bigg |_0^b \mathrm {d}x} -\int _0^a {{\displaystyle \frac{\partial M_T }{\partial y}\phi _i (x)\varphi _j (y)} \bigg |_0^b \mathrm {d}x}\\ \end{array}} \right\} . \end{aligned}$$

(45)