## 2 Torsional Harmonic Oscillator Model  
The motion of continuum joints involve a structural deformation. To accurately describe this deformation, rigorous modeling approaches are required. In the past, different types of application cases for continuum joints and robots have produced a large amount of different modeling techniques to describe their deformation and the overall motion mathematically. The recent work of [22] classified the available models into four categories, Continuum mechanics models, Geometric models, Discrete material models, and Surrogate models.  
The aim of the present paper is the experimental characterization of the resonance behavior of the continuum joint depicted in Figure 3 while torsionally excited. This characterization will be done purely on experimental data by the frequency dependent relation of input \(\theta_{d}(t)\) and output \(\varphi(t)\). Secondly, the damping behavior of the free torsional oscillator shall be characterized using a linear second order mechanical system. Similar to the work from [23], the continuum joint mechanism is thereby approximated as a mass less linear spring, the continuum, with a rigid body on the top end while the bottom end is clamped to the ground. This model approach belongs to third category discrete material models defined by [22].  
The resulting differential equation to describe the experimentally observed damping behavior is approximated as a free torsional oscillator with constant coefficients \(I\), \(d\), and \(k\) for the resulting torsional angle \(\varphi\) at the tip of the continuum (rotation along the \({}_{g}x\)-axis, see Figure 3):  
Figure 2: Illustration of the impulse-loaded spine and coupled motion postulated by the spinal engine theory. The outline of the human was taken from [21]. Pelvic rotation, the curvature of the spine, leads to shoulder rotation and vice versa.  
\[I\bar{\varphi}+d\phi+k\varphi=0\,, \tag{1}\] \[\bar{\varphi}+2\lambda\phi+\omega^{2}\varphi=0\,. \tag{2}\]  
Whereby in Equation (2) the following substitutions have been made:  
\[2\lambda\equiv\frac{d}{I},\quad\omega^{2}\equiv\frac{k}{I}, \tag{3}\]  
to gain a generalized formulation. The inertia \(I\) along the rotation axis due to the structure and additional weights \(m=140\) g employed is:  
\[I=I_{m}+I_{s}=2 mr^{2}+I_{s}\approx 12.74\times 10^{-3}\,\mathrm{kg}\,\mathrm{m}^{ 2}\,, \tag{4}\]  
where \(r\) is the radial distance of the additional weights and \(I_{s}\) is the inertia of the structure, which is calculated by Creo. The damping parameter \(d\) in N m s is used to capture the viscous damping properties of the silicone material and the friction in the tendon kinematics. The stiffness parameter \(k\) in N m captures the torsional stiffness of the continuum, which can be computed straightforwardly from the shear modulus \(G\) of the material, the second moment of area \(J\) of the continuum cross section, and the length \(L\) of the continuum joint:  
\[k=\frac{GJ}{L}=\frac{0.29\,\mathrm{N}\,\mathrm{m}^{2}}{0.095\,\mathrm{m}}=3.0 52\,\mathrm{N}\,\mathrm{m}\,. \tag{5}\]  
The parameters above are taken from a similar continuum joint setup in [24].  
As depicted in Figure 4, the driving oscillation \(\theta_{d}\sin\Omega t\) relative to the \(\mathcal{H}\) frame is induced by an oscillating pendulum motion on the top of the continuum, i.e., the resonance testbed. In addition, the tendons attached to the bottom \(\mathcal{B}\) and head \(\mathcal{H}\) plates allow for a change in the compression state of the continuum by varying simultaneously the length of the tendons which results in a compression \(p_{i}\) of the continuum.  
Figure 3: Sectional view of the test bed.