# Experimental Testing of Passive Linear TMD for Postural Tremor Attenuation
## 2 Materials and Methods
### Passive Tuned Mass Damper  
In the current work, an experiment is conducted to study the ability of passive TMD in reducing the amplitude of complex types of signals, the PT signal, which can be modeled as a second-order non-linear Stochastic oscillator [16], Gaussian noise signal [17], etc. The passive TMD having mass, stiffness, and damper as the effective elements, is fabricated in the form of a Cantilever-type TMD, as the design suggested by [PERSON] et al. [18]. However, in the current design, the beam has a circular cross-section as shown in Figure 3. The TMD is designed to operate within the frequency range of PT. The proof mass (\(m_{a}\)) of the TMD is represented as a screw attached along a beam of length \(l_{a}\). This TMD can provide different operating frequencies depending on the position (\(a\)) of the screw. Figure 3: Manufactured cantilever-type TMD. Figure 2: (**a**) Signal measured using the EMG for the ECR muscle of a patient suffering from PT, used as input signal of the shaker (**b**) Normalized PSD for the input and output signals of the mechanical shaker, the input signal corresponding to the measured ECR signal of the patient. The measurement point of the vibrometer, on the beam of the TMD, is located at a distance of 15.0 mm from its clamped end. The used measurement full-scale range is 20 mm/s with a corresponding resolution of <0.02 \(\mu m/s/(\sqrt{Hz})\). The measured signals are downsampled to obtain a sampling frequency of 170 Hz, which is greater than twice the fundamental frequency of the cantilever beam alone, obtained as 67.30 Hz. Measurements using the vibrometer also allow the identification of the modal damping ratio of the TMD, which is estimated in this study using the half-power bandwidth method [19], and is also calculated to obtain reliable values using the continuous wavelet method [20]. A comparison is performed between both methods to test the accuracy level of the half-power bandwidth method. The mean value \(\pm\) standard deviation (SD) is calculated for data obtained using each method, in addition to the root mean square (RMS) error between both. Measured signals are carried out for different positions of the screw along the beam. The position of the center of gravity of the screw, from the clamped end of the beam, is calculated and represented by the position \(a\). The measured frequency (\(f_{a}^{meas}\)) and the modal damping ratio estimated using the half-power bandwidth method are obtained using the fast Fourier transform (FFT) of the measured velocity signal for each position \(a\), whereas the modal damping ratio calculated using the continuous wavelets method is obtained using the velocity signal of the measurements. Three TMDs are manufactured to have approximately the same size and mass. These TMDs are placed at the free end of the experimental arm, on the TMD holder (Figure 4a), according to the following arrangement: a single TMD is placed in position '0', two TMDs in positions '1' and '2', and three TMDs in positions '0', '1', and '2', as shown in Figure 4b. The beam of the TMDs has a diameter of 0.79 mm. A 13.80 g screw is attached to a 91.0 mm long beam at position '0', a 13.69 g screw to a 91.0 mm long beam at position '1', and a 13.74 g screw to a 92.5 mm long beam at position '2'. For one, two, and three TMDs, the mass ratio is equal to 5.7%, 11.4%, and 17.0%, respectively, which represents the ratio of the total mass of the TMD(s) relative to the mass of the experimental arm. The position \(a\) of the screw for each TMD is adjusted using the vibrometer, to approach the desired frequency, before it is fixed to the experimental arm. IMU of Hikob Fox type is added to the experimental arm to obtain the acceleration and angular velocity signals. IMU is added at the position of the measurement point of the vibrometer. The measurements of the full-scale ranges of the accelerometer and gyrometer are \(\pm\)2 g and \(\pm\)500\({}^{\circ}\)/s with a corresponding resolution of 12 mg and 17.5 m\({}^{\circ}\)/s, respectively.