## 1. Introduction

## 2. Nonlinear Dynamic Characterization

#### 2.1. Kinematic Descriptors

#### 2.2. Nonlinear Equations of Motion

#### 2.3. Nondimensional Vector Form of the Equations of Motion

## 3. Modal Characterization

## 4. Asymptotic Solution of the Equations of Motion

#### 4.1. Nonlinearity of the Lowest Normal Modes

#### One-Mode Projection of the Nonlinear Equations of Motion

## 5. Cable Motion-Induced Resonances

#### 5.1. Primary Resonance

#### 5.2. Two-to-One Internal Resonance

#### 5.2.1. The Case of $\mathsf{\Omega}\approx {\omega}_{m}$

#### 5.2.2. The Case of $\mathsf{\Omega}\approx {\omega}_{n}$

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**Top**) Geometric characteristics of the roller battery system in compression towers. (

**Bottom**) Degrees-of-freedom and FE discretization (i.e., diamonds symbols).

**Figure 3.**Ratios of the lowest eight frequencies of the system with respect to (

**a**) the lowest frequency ${\omega}_{1}$ and (

**b**) the second frequency ${\omega}_{2}$. Red and blue bars and dashed lines indicate 2:1 and 3:1 ratios, respectively.

**Figure 4.**(

**a**) Backbones of the lowest four nonlinear normal modes. (

**b**) Shape of the first and (

**c**) second normal modes at $\sigma =-0.01$; one-mode projection (dashed lines) vs. full modal basis discretization (solid lines).

**Figure 5.**Coefficients of the inertial (${c}_{i2}$ and ${c}_{i3}$), velocity-proportional (${c}_{d2}$ and ${c}_{d3}$), and stiffness nonlinear terms (${c}_{k2}$ and ${c}_{k3}$), of the lowest four modes.

**Figure 6.**Primary resonance: frequency response curves of the vertical displacements of points (

**a**) ${P}_{8}$, (

**b**) ${A}_{2}$, (

**c**) ${B}_{1}$, and (

**d**) C. Solid and dashed lines refer to the stable and unstable branches, respectively, evaluated via asymptotic approaches, while dots indicate the results of the numerical time integration. Configurations at $\mathsf{\Omega}=0.3623$ in the multistability range: (

**e**) low-amplitude response, (

**f**) high-amplitude response.

**Figure 7.**(

**a**) FRCs of the modal amplitude ${a}_{m}$ (case $m=1$) and stability regions for two different values of the coefficient $\xi $ which rescales the damping; the dotted line indicates the backbone of the mode $m=1$. (

**b**) Loci of the fold bifurcation points for $\xi =(0,0.2,0.4,0.6,0.8,1)$.

**Figure 8.**(

**a**) Force response curves of the amplitude ${a}_{m}$ (black line) and ${a}_{n}$ (gray line) when ${\sigma}_{1}=0$ and ${\sigma}_{2}=0$. (

**b**) Frequency response curves of the amplitude ${a}_{m}$ (black line) and ${a}_{n}$ (gray line) when ${\sigma}_{2}=0$ and $\lambda =2.5$. Solid lines indicate stable responses while dashed lines indicate unstable responses.

**Figure 9.**(

**a**) Force response curves of the amplitude ${a}_{m}$ (black line) and ${a}_{n}$ (gray line), when ${\sigma}_{1}=0$ and ${\sigma}_{2}=-0.018$ in a lower range of force amplitudes $\lambda $. (

**b**) Ratio between the amplitudes ${a}_{m}$ and ${a}_{n}$ varying $\lambda $. The dashed vertical line indicates the value of $\lambda $ adopted in Section 5.1.

**Figure 10.**(

**a**) Force response curves of the amplitude ${a}_{m}$ (black line) and ${a}_{n}$ (gray line) when ${\sigma}_{1}=0$ and ${\sigma}_{2}=-0.018$. (

**b**) Frequency response curves of the amplitude ${a}_{m}$ (black line) and ${a}_{n}$ (gray line) when ${\sigma}_{2}=-0.018$ and $\lambda =2.5$. Solid lines indicate stable responses while dashed lines indicate unstable responses.

**Figure 11.**(

**a**) Force response curves of the amplitude ${a}_{m}$ (black line) and ${a}_{n}$ (gray line) when ${\sigma}_{1}=0$ and ${\sigma}_{2}=0$. Frequency response curves of the amplitude ${a}_{m}$ (black line) and ${a}_{n}$ (gray line) when ${\sigma}_{2}=0$ and (

**b**) at high excitation amplitude $\lambda =0.21$ (

**c**) at a low excitation amplitude $\lambda =8.27\times {10}^{-3}$.

**Figure 12.**(

**a**) Force response curves of the amplitude ${a}_{m}$ (black line) and ${a}_{n}$ (gray line) when ${\sigma}_{1}=0$ and ${\sigma}_{2}=-0.018$. (

**b**) Frequency response curves of the amplitude ${a}_{m}$ (black line) and ${a}_{n}$ (gray line) when ${\sigma}_{2}=0$ and at high excitation amplitude $\lambda =0.21$.

Mode s | 1 | 2 | 3 | 4 |

${\omega}_{s}$ | 0.376 | 0.581 | 0.734 | 0.96 |

Mode s | 5 | 6 | 7 | 8 |

${\omega}_{s}$ | 1.02 | 1.109 | 1.163 | 1.757 |

**Table 2.**Nonlinearity coefficient ${\mathsf{\Gamma}}_{m}$ of the lowest four nonlinear normal modes ($m=1,\dots ,4$): effect of the nonlinearity type.

All | Inertia | Velocity | Stiffness | |
---|---|---|---|---|

${\mathsf{\Gamma}}_{1}$ | $-8.58\times {10}^{-3}$ | $-2.12\times {10}^{-4}$ | $7.00\times {10}^{-5}$ | $-8.44\times {10}^{-3}$ |

${\mathsf{\Gamma}}_{2}$ | $-6.19\times {10}^{-2}$ | $-1.52\times {10}^{-3}$ | $5.04\times {10}^{-4}$ | $-6.09\times {10}^{-2}$ |

${\mathsf{\Gamma}}_{3}$ | $-1.08\times {10}^{-1}$ | $-1.36\times {10}^{-3}$ | $4.52\times {10}^{-4}$ | $-1.07\times {10}^{-1}$ |

${\mathsf{\Gamma}}_{4}$ | $-8.46\times {10}^{-1}$ | $-2.29\times {10}^{-2}$ | $7.55\times {10}^{-3}$ | $-8.31\times {10}^{-1}$ |

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