An asymptotic expansion of cable–flexible support coupled nonlinear vibrations using boundary modulations

Abstract

This paper is devoted to cable–flexible support coupled nonlinear vibrations using a asymptotic boundary modulation technique. Asymptotic/reduced cable–support coupled nonlinear models are established first using the boundary modulation concept, after a proper scaling analysis at the cable–support interface. The cable and the support turn out to be coupled through cable-induced and support-induced boundary modulations in a rational way, which are derived analytically by asymptotic approximations and multiple scale expansions. Based upon the reduced models, two prototypical kinds of cable–support coupled dynamics are fully investigated, i.e., one with the support excited and the other with the cable excited. Essentially, they correspond to refined versions of two typical degenerate cable dynamics, i.e., cables excited externally with fixed supports and cables excited by ideal moving supports. Applying numerical continuation algorithms to the reduced models, cable–support typical coupled frequency response diagrams are constructed, with their stabilities, bifurcation characteristics, and the coupling’s effects on the cable determined. All these approximate analytical results are verified by the numerical results from the original full cable–support system using the finite difference method.

Appendices

Appendix 1

Suspended cable’s linear modal analysis can be found in reference [5]. We restrict our attention to cable’s in-plane symmetric modes in this paper, and these modes are given by

$$\begin{aligned}&\phi _i \left( x \right) =c_i \left[ {1-\tan \left( {\frac{\omega _i }{2}} \right) \sin \omega _i x-\cos \omega _i x} \right] ,\nonumber \\&\quad i=1,3,5\cdots \end{aligned}$$

(57)

where $c_{\mathrm{{i}}}$ is the normalization constants. And the associated eigenfrequencies are determined by

$$\begin{aligned} \frac{1}{2}\omega _i -\tan \left( {\frac{1}{2}\omega _i } \right) -\frac{1}{2\lambda ^{2}}\omega _i^3 =0 \end{aligned}$$

(58)

where $\lambda ^{2}=EA/mgl(8b/l)^{3 }$ is the elasto-geometric parameter. The above nonlinear transcendental equations can be solved by the Newton–Raphson method.

The linear and nonlinear operators in cable’s dynamic equations, i.e., Eq. (11), are defined as

$$\begin{aligned} L\left[ w \right]= & {} -{w}''-\alpha {y}''\int _0^1 {\left( {{y}'{w}'} \right) } dx,\quad \nonumber \\ N_2 \left[ w \right]= & {} \alpha {w}''\int _0^1 {\left( {{y}'{w}'} \right) } dx+\alpha {y}''\int _0^1 {\left( {{{w}'^{2}}/2} \right) } dx,\nonumber \\ N_3 \left[ w \right]= & {} \alpha {w}''\int _0^1 {\left( {{{w}'^{2}}/2} \right) } dx \end{aligned}$$

(59)

The cubic nonlinear terms in Eq. (15) are defined as

$$\begin{aligned}&N_3 \left[ {w_1 } \right] =\alpha {w}''_1 \int _0^1 {\left( {{{w'}_1^{2}}/2} \right) } dx \nonumber \\&N_3 \left[ {w_1 ,w_2 } \right] =\alpha {w}''_1 \int _0^1 {\left( {{y}'{w}'_2 } \right) } dx\nonumber \\&\quad +\,\alpha {y}''\int _0^1 {\left( {{w}'_1 {w}'_2 } \right) } dx +\,\alpha {w}''_2 \int _0^1 {\left( {{y}'{w}'_1 } \right) } dx \end{aligned}$$

(60)

Appendix 2

The second-order shape functions in Eq. (19), Eq. (51), and Eq. (55) are illustrated in Fig. 21. They are governed by the following boundary value problem (BVP)

$$\begin{aligned}&4\omega _n^2 \varPsi _1 \left( x \right) +\varPsi _1 ^{\prime \prime }+\alpha {y}''\int _0^1 {{y}' \varPsi _1 ^{\prime }} dx\nonumber \\&={-\alpha }/2\left\langle {{\phi }'_1 ,{\phi }'_1 \;} \right\rangle {y}'' -\alpha \left\langle {{y}',{\phi }'_1 \;} \right\rangle {\phi }''_1 \end{aligned}$$

(61)

$$\begin{aligned}&\varPsi _2 ^{\prime \prime }+\alpha {y}''\int _0^1 {{y}' \varPsi _2 ^{\prime }} dx={-\alpha }/2\left\langle {{\phi }'_1 ,{\phi }'_1 \;} \right\rangle {y}''\nonumber \\&\qquad -\,\alpha \left\langle {{y}',{\phi }'_1 \;} \right\rangle {\phi }''_1 \end{aligned}$$

(62)

with boundary conditions $\varPsi _{\mathrm{{k}}}(0)= \varPsi _{k}(1)=0$, $k=1,2$.

Fig. 21

Second-order shape functions $\varPsi _{1}(x)$ and $\varPsi _{2}(x)$ a for cable’s symmetric dynamics, b for cable’s anti-symmetric dynamics

Appendix 3

To validate our reduced models, numerical full/discrete cable–support coupled models are derived by using the finite difference method. Briefly, using the second-order finite difference scheme,

$$\begin{aligned} \frac{\partial \left( \right) }{\partial x}\approx & {} \frac{\left( \right) _{i+1} -\left( \right) _{i-1} }{2\Delta x},\nonumber \\ \frac{\partial ^{2}\left( \right) }{\partial x^{2}}\approx & {} \frac{\left( \right) _{i+1} -2\left( \right) _i +\left( \right) _{i-1} }{\Delta x^{2}}, \nonumber \\ \frac{\partial \left( \right) }{\partial t}\approx & {} \frac{\left( \right) _{j+1} -\left( \right) _{j-1} }{2\Delta t},\nonumber \\ \frac{\partial ^{2}\left( \right) }{\partial t^{2}}\approx & {} \frac{\left( \right) _{j+1} -2\left( \right) _j +\left( \right) _{j-1} }{\Delta t^{2}} \end{aligned}$$

(63)

The cable dynamics in Eq. (1) is discretized as

$$\begin{aligned} w_{i,j+1}= & {} \frac{1}{\left( {1+c\Delta t} \right) \Delta x^{2}}\left\{ \left( {1+\alpha S\Delta t^{2}} \right) w_{i-1,j}\right. \nonumber \\&\left. -\,\left( {1-c\Delta t} \right) \Delta x^{2}w_{i,j-1} \right. \nonumber \\&+\,\left( {2\Delta x^{2}-2\Delta t^{2}-2\alpha S\Delta t^{2}} \right) w_{i,j} \nonumber \\&+\,\left( {\Delta t^{2}+\alpha S\Delta t^{2}} \right) w_{i+1,j} \nonumber \\&-\,8\alpha fS\Delta t^{2}\Delta x^{2}\nonumber \\&\left. +\,F_c \left( {i\Delta x} \right) \cos \left( {\Omega j\Delta t} \right) \Delta t^{2}\Delta x^{2} \right\} \end{aligned}$$

(64)

and the support dynamics in Eq. (3) is discretized as

(65)

where $\Delta x$, $\Delta t$ are space/time steps, and $x_i =i\Delta x,\;t_j =j \Delta t$ are the discrete grids for time marching. The distributed excitation amplitude $F_c$ (in physics space) is related to the modal excitation amplitude in Eq. (41) by $f_c =\left\langle {F_c \left( x \right) ,\phi _n } \right\rangle$.

The integral term in the cable equation (64), i.e., S, is obtained by the Simpson’s integral rule

$$\begin{aligned} S= & {} \int _0^1 {\left( {{y}'{w}'+\frac{1}{2}{w}'^{2}} \right) dx} \approx \frac{\Delta x}{3}\nonumber \\&\times \left\{ s_0 +\,2\sum _{i=1}^{n/2-1} {s_{2i} } +4\sum _{i=1}^{n/2} {s_{2i-1} } +s_n \right\} \nonumber \\ s_i\triangleq & {} 4f\left( {1-2\Delta x i} \right) \frac{w_{i+1,j} -w_{i-1,j} }{2\Delta x}\nonumber \\&+\,\frac{1}{2}\left( {\frac{w_{i+1,j} -w_{i-1,j} }{2\Delta x}} \right) ,\quad i=0,1,\cdots n \end{aligned}$$

(66)

The support-induced boundary excitation, i.e., Eq. (2), is discretized as

$$\begin{aligned} w_{0,j} =0,\quad w_{n,j} =-z\left( {t_j } \right) ,\quad t_j =j \Delta t \end{aligned}$$

(67)

and the dynamic tension $T_\mathrm{{d}} \left( {t_j } \right)$ in Eq. (65) is calculated by

$$\begin{aligned} T_d \left( {t_j } \right)= & {} \left. {{w}'\left( 1 \right) } \right| _{t=t_j } +\left( {y^{\prime }\left( 1 \right) +\left. {{w}'\left( 1 \right) } \right| _{t=t_j } } \right) \nonumber \\&\times \,\alpha \int _0^1 {\left( {y^{\prime }{w}'+\frac{1}{2}{w}'^{2}} \right) dx} \nonumber \\= & {} \frac{3w_{N,j} -4w_{N-1,j} +w_{N-2,j} }{2\Delta x}\\&+\,\left( {-4f+\frac{3w_{N,j} -4w_{N-1,j} +w_{N-2,j} }{2\Delta x}} \right) \alpha S \nonumber \end{aligned}$$

(68)

We split the cable into 1000 segments, i.e., the space step $\Delta x=0.001$, and set the time step $\Delta t=0.0001(2)$. Based upon this discrete model, we use a time-stepping program coded by C++ to simulate the dynamic responses of cable–support coupled system directly. After the cable–support coupled steady responses are obtained, we compare these numerical results from the numerical full model to the approximate analytical ones.