## 3 Nonlinear Tension Model of the DYNAMICA Cable
### Dynamic Modulus of Elasticity  
The modulus of elasticity obtained by the quasi-static method is frequently used to describe the dynamic behavior of CDPR systems. However, the effectiveness of this method is negligible; thus, many researchers have proposed different methods. For example, in [28], a cable model integrating hysteresis effects by applying force was improved; and in [10], a complex model combining a linear damper model and a rate-independent hysteresis model was proposed. In this article, to identify the dynamic modulus under forced oscillation, the dynamic mechanical analysis (DMA) method was used. The DMA method can be simply described as the application of a sinusoidal force to the cable to cause a sinusoidal elongation response [29]. In addition, an advantage of the DMA method is that it is possible to identify a dynamic modulus under various frequencies. To obtain the modulus of elasticity, we conducted experiments. First, the experimental setup included the tensile test using an MTS-810 Material Testing System (Figure 5). A Dynamica cable with polyethylene material was used in the CDPR system. The cable of 6 mm diameter consisted of 12 strands, with a breaking strength at t of 4.20. Then, four samples of the Dynamica cable with a length of 300 mm were prepared and fixed on an MTS-810 material testing system, and four tests were performed at different frequencies of 0.1, 1, 2, and 5 Hz, respectively. A preload was applied to the cable at 1500 N before sinusoidal tensile loading was controlled at different frequencies, each experiment was repeated 3 times and averaged, and the results were then analyzed. By verifying the amplitude of the elongation at the peak of a sine wave and the phase lag between the applied force and the elongation sine waves, the dynamic modulus and the damping could be calculated. Based on the discussion above, it is known that the DMA allows for the calculation of a dynamic modulus from the cable response. \(E^{\prime}\) is a real part of the dynamic modulus described as a recovered part and can be considered as the elastic modulus. Compared with the [PERSON]'s modulus, which is normally calculated based on the resulting stresses and strains and represented as the slope of a line, \(E^{\prime}\) can be a point on the line [29]. In addition, in this study, the tests differed, with the material stretched in the stress-strain test but oscillating in the dynamic test. \(E^{\prime}\) is defined as follows:  
\[E^{\prime}=\frac{\sigma_{\text{max}}}{\varepsilon_{\text{max}}}\cos\varphi \tag{24}\]  
Figure 4: Flowchart of the proposed method. where \(\sigma_{\rm max}\) and \(\varepsilon_{\rm max}\) are the maximum amplitude of the applied stress and the maximum amplitude of the strain response, respectively. The ratio of the maximum amplitude stress-strain is denoted as \(E^{*}\) and called the absolute value of the dynamic modulus, and \(\varphi\) is the phase angle. \(E^{\prime\prime}\) represents the energy dissipation, also called the imaginary modulus. It is calculated as follows:  
\[E^{\prime\prime}=\frac{\sigma_{\rm max}}{\varepsilon_{\rm max}}\sin\varphi \tag{25}\]  
The ratio of the dissipation to the storage modulus is called the damping \(\eta\) and defined as follows:  
\[\eta=\frac{E^{\prime\prime}}{E^{\prime}}=\tan\varphi \tag{26}\]  
Figure 6 shows the resulting sinusoidal elongation when a sinusoidal force at various frequencies was applied. The pre-force exerted on the cable was 1.5 kN before controlling the sinusoidal force waves and the values of the elongation varied depending on the frequencies. The values of the elongation varied depending on the frequencies. The higher the frequency, the smaller the rate of change in the elongation value. In addition, the elongation profile was a time-dependent nonlinear curve. Although the sinusoidal oscillation was constant, the cable elongation increased significantly over time. Table 2 shows the dynamic and elastic modulus, the phase angle, and the damping factor.