<h2id="orgea4f0d4"><spanclass="section-number-2">1</span> Variation of the Sample Mass</h2>
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We here study the change of dynamics due to the sample mass.
To see only the effect of the sample mass, we keep the same resonance frequency of the sample, and we set it to 10kHz so it is above the dynamics of interest.
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We initialize all the stages with the default parameters.
We identify the dynamics for the following sample masses, both with a soft and stiff nano-hexapod.
<li>Figure <ahref="#orgfcb2492">1</a>: From actuator forces to force sensors in each nano-hexapod’s leg</li>
<li>Figure <ahref="#orga2c8402">2</a>: From actuator forces to relative displacement of each nano-hexapod’s leg</li>
<li>Figure <ahref="#org389df4f">3</a> (resp. <ahref="#orgb5aaedd">4</a>): From forces applied in the task space by the nano-hexapod to displacement of the sample in the X direction (resp. in the Z direction)</li>
<p><spanclass="figure-number">Figure 1: </span>Variability of the dynamics from actuator force to force sensor with the Sample Mass (<ahref="./figs/dynamics_variability_iff_sample_mass.png">png</a>, <ahref="./figs/dynamics_variability_iff_sample_mass.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 2: </span>Variability of the dynamics from actuator force to relative motion sensor with the Sample Mass (<ahref="./figs/dynamics_variability_dvf_sample_mass.png">png</a>, <ahref="./figs/dynamics_variability_dvf_sample_mass.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 3: </span>Variability of the dynamics from Forces applied in task space (X direction) to the displacement of the sample in the X direction (<ahref="./figs/dynamics_variability_err_x_sample_mass.png">png</a>, <ahref="./figs/dynamics_variability_err_x_sample_mass.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 4: </span>Variability of the dynamics from vertical forces applied in the task space to the displacement of the sample in the vertical direction (<ahref="./figs/dynamics_variability_err_z_sample_mass.png">png</a>, <ahref="./figs/dynamics_variability_err_z_sample_mass.pdf">pdf</a>)</p>
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Let’s note \(\omega_0\) the first resonance which corresponds to the resonance of the payload+nano-hexapod top platform resonating on top of the nano-hexapod base.
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An increase of the payload mass decreases \(\omega_0\).
This is more easily seem with the soft nano-hexapod as the resonance \(\omega_0\) is separated from the resonances of the micro-station.
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<li>For the soft nano-hexapod, the main effect is the change of \(\omega_0\).</li>
<li>For the stiff nano-hexapod, it also affects the others resonances corresponding to the resonances of the micro-station</li>
<p><spanclass="figure-number">Figure 5: </span>Variability of the dynamics from actuator force to force sensor with the Sample Mass (<ahref="./figs/dynamics_variability_iff_sample_w.png">png</a>, <ahref="./figs/dynamics_variability_iff_sample_w.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 6: </span>Variability of the dynamics from actuator force to relative motion sensor with the Sample Mass (<ahref="./figs/dynamics_variability_dvf_sample_w.png">png</a>, <ahref="./figs/dynamics_variability_dvf_sample_w.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 7: </span>Variability of the dynamics from a torque applied on the sample by the nano-hexapod in the X direction to the rotation of the sample around the X axis (<ahref="./figs/dynamics_variability_err_sample_w.png">png</a>, <ahref="./figs/dynamics_variability_err_sample_w.pdf">pdf</a>)</p>
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Let’s note \(\omega_m\) the frequency of the resonance of the Payload.
<p><spanclass="figure-number">Figure 8: </span>Variability of the dynamics from the actuator force to the force sensor with the Spindle Angle (<ahref="./figs/dynamics_variability_iff_spindle_angle.png">png</a>, <ahref="./figs/dynamics_variability_iff_spindle_angle.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 9: </span>Variability of the dynamics from actuator force to absolute velocity with the Spindle Angle (<ahref="./figs/dynamics_variability_err_spindle_angle.png">png</a>, <ahref="./figs/dynamics_variability_err_spindle_angle.pdf">pdf</a>)</p>
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The Spindle angle has no visible effect for the soft nano-hexapod.
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It has little effect on the dynamics when a stiff nano-hexapod is used.
This is seem between 50Hz and 100Hz.
This is probably due to the fact that the micro-station compliance is not uniform in the X and Y directions.
<li>Figure <ahref="#org82ab7aa">10</a>: From actuator forces to force sensors in each nano-hexapod’s leg</li>
<li>Figure <ahref="#org3f98487">11</a>: From actuator forces to relative displacement of each nano-hexapod’s leg</li>
<li>Figure <ahref="#org0d347dd">12</a>: From forces applied in the task space by the nano-hexapod to displacement of the sample in the X direction</li>
<li>Figure <ahref="#org61e9acd">13</a>: From forces applied in the task space in the X direction by the nano-hexapod to displacement of the sample in the Y direction (coupling)</li>
<p><spanclass="figure-number">Figure 10: </span>Variability of the dynamics from the actuator force to the force sensor with the Spindle rotation speed (<ahref="./figs/dynamics_variability_iff_spindle_speed.png">png</a>, <ahref="./figs/dynamics_variability_iff_spindle_speed.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 11: </span>Variability of the dynamics from the actuator force to the relative motion sensor with the Spindle rotation speed (<ahref="./figs/dynamics_variability_dvf_spindle_speed.png">png</a>, <ahref="./figs/dynamics_variability_dvf_spindle_speed.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 12: </span>Variability of the dynamics from the actuator force in the task force to the position error of the sample (<ahref="./figs/dynamics_variability_err_spindle_speed.png">png</a>, <ahref="./figs/dynamics_variability_err_spindle_speed.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 13: </span>Variability of the coupling from the actuator force in the task force to the position error of the sample (<ahref="./figs/dynamics_variability_err_spindle_speed_coupling.png">png</a>, <ahref="./figs/dynamics_variability_err_spindle_speed_coupling.pdf">pdf</a>)</p>
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For the stiff nano-hexapod, the rotation speed of the Spindle does not affect the (main) dynamics.
It only affects the coupling due to Coriolis forces.
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For the soft nano-hexapod, it greatly affects the obtained dynamics around the main resonance which corresponds to the payload vibrating on top of the nano-hexapod.
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This effect is similar to the one described in rotating machinery, the c.c. poles is separated into two sets of c.c. poles, one going to decreasing frequencies while the other going to positive frequencies.
This effect is due to centrifugal forces that can be modeled as negative stiffness.
At some point, one of the pair of c.c. pole becomes unstable.
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Also, the coupling from forces applied in the X direction to induced displacement in the Y direction becomes very high when the rotating speed is increased.
<li>Figure <ahref="#org8eda147">14</a>: From actuator forces to force sensors in each nano-hexapod’s leg</li>
<li>Figure <ahref="#orgfacbfbb">15</a>: From forces applied in the task space by the nano-hexapod to displacement of the sample in the X direction</li>
<p><spanclass="figure-number">Figure 14: </span>Variability of the dynamics from the actuator force to the force sensor with the Tilt stage Angle (<ahref="./figs/dynamics_variability_iff_tilt_angle.png">png</a>, <ahref="./figs/dynamics_variability_iff_tilt_angle.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 15: </span>Variability of the dynamics from the actuator force in the task space to the displacement of the sample (<ahref="./figs/dynamics_variability_err_tilt_angle.png">png</a>, <ahref="./figs/dynamics_variability_err_tilt_angle.pdf">pdf</a>)</p>
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The tilt angle has no visible effect on the dynamics.
<p><spanclass="figure-number">Figure 16: </span>Variability of the dynamics from the actuator force to the force sensor with the Tilt stage Angle (<ahref="./figs/dynamics_variability_iff_micro_hexapod_x.png">png</a>, <ahref="./figs/dynamics_variability_iff_micro_hexapod_x.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 17: </span>Variability of the dynamics from the actuator force in the task space to the displacement of the sample (<ahref="./figs/dynamics_variability_err_micro_hexapod_x.png">png</a>, <ahref="./figs/dynamics_variability_err_micro_hexapod_x.pdf">pdf</a>)</p>
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The pose of the micro-hexapod has negligible effect on the dynamics.
