Cable’s non-planar coupled vibrations under asynchronous out-of-plane support motions: travelling wave effect

Abstract

Cable’s non-planar coupled dynamics under asynchronous out-of-plane support motions is investigated in this paper. The moving boundary difficulty is coped by transforming the small support motions into two resonant boundary modulation terms, and cable’s one-to-one resonant coupled non-planar dynamics is reduced through attacking directly the continuous dynamic equations by the multiple scale method. Two boundary dynamic coefficients are derived, characterizing the boundary modulation effects, which are equal for symmetric out-of-plane modes, while opposite for asymmetric ones. Both cable’s frequency and amplitude (of support deflection) response diagrams, with phase lags between supports, are constructed using numerical continuation algorithms, and the steady-state solutions’ stability and bifurcation properties are determined, based upon which the phase lags’ effects on dynamic responses, or travelling wave effects, are thoroughly investigated, through varying the phase lags.

Appendices

Appendix 1

The suspended cable’s linear modal analysis can be found in references [3, 30]. The in-plane symmetric modes are given by

$$\begin{aligned} \phi _n \left( x \right) =c_n \left[ {1-\tan \left( {\frac{ \omega _n^{\left( \mathrm{in} \right) } }{2}} \right) \sin \omega _n^{\left( \mathrm{in} \right) } x-\cos \omega _n^{\left( \mathrm{in} \right) } x} \right] ,\;n=1,3,5,\ldots \end{aligned}$$

(53)

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

$$\begin{aligned} \frac{1}{2}\omega _n^{\left( \mathrm{in} \right) } -\tan \left( {\frac{1}{2}\omega _n^{\left( \mathrm{in} \right) } } \right) -\frac{1}{2\lambda ^{2}}\left( {\omega _n^{\left( \mathrm{in} \right) } } \right) ^{3}=0,\;n=1,3,5,\ldots \end{aligned}$$

(54)

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 in-plane anti-symmetric modes are

$$\begin{aligned} \phi _n \left( x \right) =\sqrt{2}\sin \left( {n\pi x} \right) ,\;n=2,4,6, \ldots \end{aligned}$$

(55)

with the associated eigenfrequencies as

$$\begin{aligned} \omega _n^{\left( \mathrm{in} \right) } =n\pi ,\;n=2,4,6, \ldots \end{aligned}$$

(56)

And the out-of-plane modes are

$$\begin{aligned} \varphi _m \left( x \right) =\sqrt{2}\sin \left( {m\pi x} \right) ,\;m=1,2,3,\ldots \end{aligned}$$

(57)

with the associated eigenfrequencies as

$$\begin{aligned} \omega _m^{\left( \mathrm{out} \right) } =m\pi ,\;m=1,2,3, \ldots \end{aligned}$$

(58)

Fig. 10

Illustrations of the shape functions ${\varPsi }_\mathrm{k}(x)$, a cable model 1 with $\sigma _{1}=0.0505$, b cable model 2 with $\sigma _{1}=0.3026$

Appendix 2

$$\begin{aligned}&\begin{array}{l} {\varPi }_1 \left( x \right) =\alpha /2\left\langle {{\phi }'_n ,{\phi }'_n \;} \right\rangle {y}''+\alpha \left\langle {{y}',{\phi }'_n \;} \right\rangle {\phi }''_n ,\quad {\varPi }_3 \left( x \right) =\alpha /2\left\langle {{\varphi }'_m ,{\varphi }'_m \;} \right\rangle {y}'' \\ {\varPi }_2 \left( x \right) =\alpha /2\left\langle {{\phi }'_n ,{\phi }'_n \;} \right\rangle {y}''+\alpha \left\langle {{y}',{\phi }'_n \;} \right\rangle {\phi }''_n ,\quad {\varPi }_4 \left( x \right) =\alpha /2\left\langle {{\varphi }'_m ,{\varphi }'_m \;} \right\rangle {y}'' \\ \end{array} \end{aligned}$$

(59)

$$\begin{aligned}&{\varPi }_5 \left( x \right) =\alpha \left\langle {{y}',{\phi }'_n \;} \right\rangle {\varphi }''_m ,\quad {\varPi }_6 \left( x \right) =\alpha \left\langle {{y}',{\phi }'_n \;} \right\rangle {\varphi }''_m \end{aligned}$$

(60)

$$\begin{aligned}&\begin{array}{rcl} \chi _1 \left( x \right) &{}=&{}\frac{3\alpha }{2}\phi _n ^{\prime \prime }\left\langle {\phi _n ^{\prime },\phi _n ^{\prime }} \right\rangle +\alpha \phi _n ^{\prime \prime }\left\langle {{y}',{\varPsi }_1 ^{\prime }} \right\rangle +\alpha {y}''\left\langle {\phi _n ^{\prime },{\varPsi }_1 ^{\prime }} \right\rangle +\alpha {\varPsi }_1 ^{\prime \prime }\left\langle {{y}',\phi _n ^{\prime }} \right\rangle \\ &{}&{} \;+2\alpha \phi _n ^{\prime \prime }\left\langle {{y}',{\varPsi }_2 ^{\prime }} \right\rangle +2\alpha {y}''\left\langle {\phi _n ^{\prime },{\varPsi }_2 ^{\prime }} \right\rangle +2\alpha {\varPsi }_2 ^{\prime \prime }\left\langle {{y}',\phi _n ^{\prime }} \right\rangle \\ \end{array} \end{aligned}$$

(61)

$$\begin{aligned}&\begin{array}{rcl} \chi _2 \left( x \right) &{}=&{}\alpha {\phi }''_n \left\langle {{\varphi }'_m ,{\varphi }'_m } \right\rangle +2\alpha \phi _n ^{\prime \prime }\left\langle {{y}',{\varPsi }_4 ^{\prime }} \right\rangle +2\alpha {y}''\left\langle {\phi _n ^{\prime },{\varPsi }_4 ^{\prime }} \right\rangle +2\alpha {\varPsi }_4 ^{\prime \prime }\left\langle {{y}',\phi _n ^{\prime }} \right\rangle \\ &{}&{} +\alpha {y}''\left\langle {{\varphi }'_m ,{\varPsi }_5 ^{\prime }} \right\rangle +\alpha {y}''\left\langle {{\varphi }'_m ,{\varPsi }_6 ^{\prime }} \right\rangle \\ \end{array} \end{aligned}$$

(62)

$$\begin{aligned}&\begin{array}{l} \chi _3 \left( x \right) =\frac{\alpha }{2}{\phi }''_n \left\langle {{\varphi }'_m ,{\varphi }'_m } \right\rangle +\alpha \phi _n ^{\prime \prime }\left\langle {{y}',{\varPsi }_3 ^{\prime }} \right\rangle +\alpha {y}''\left\langle {\phi _n ^{\prime },{\varPsi }_3 ^{\prime }} \right\rangle +\alpha {\varPsi }_3 ^{\prime \prime }\left\langle {{y}',\phi _n ^{\prime }} \right\rangle +\alpha {y}''\left\langle {{\varphi }'_m ,{\varPsi }_6 ^{\prime }} \right\rangle \\ \end{array} \end{aligned}$$

(63)

$$\begin{aligned}&\chi _4 \left( x \right) =\frac{3\alpha }{2}{\varphi }''_m \left\langle {{\varphi }'_m , {\varphi }'_m } \right\rangle +\alpha {\varphi }''_m \left\langle {{y}', {{\varPsi }}'_3 \left( x \right) } \right\rangle +2\alpha {\varphi }''_m \left\langle {{y}', {{\varPsi }}'_4 \left( x \right) } \right\rangle \end{aligned}$$

(64)

$$\begin{aligned}&\chi _5 \left( x \right) =\alpha {\varphi }''_m \left\langle {{\phi }'_n ,{\phi }'_n } \right\rangle +2\alpha {\varphi }''_m \left\langle {{y}',{\varPsi }_2 ^{\prime }} \right\rangle +\alpha {{\varPsi }}''_5 \left\langle {{y}',{\phi }'_n } \right\rangle +\alpha {{\varPsi }}''_6 \left\langle {{y}',{\phi }'_n } \right\rangle \end{aligned}$$

(65)

$$\begin{aligned}&\chi _6 \left( x \right) =\frac{\alpha }{2} {\varphi }''_m \left\langle {{\phi }'_n ,{\phi }'_n } \right\rangle +\alpha {\varphi }''_m \left\langle {{y}',{\varPsi }_1 ^{\prime }} \right\rangle +\alpha {{\varPsi }}''_6 \left\langle {{y}',{\phi }'_n } \right\rangle \end{aligned}$$

(66)

Appendix 3

The shape functions ${\varPsi }_\mathrm{k}(x)$ are illustrated in the following, and they are used to calculate the nonlinear resonant interaction coefficients $\Gamma _\mathrm{k}$ (Fig. 10).