diff --git a/content/phdthesis/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md b/content/phdthesis/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md index 8d9e8c4..88731c0 100644 --- a/content/phdthesis/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md +++ b/content/phdthesis/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md @@ -1,13 +1,12 @@ +++ title = "Element and system design for active and passive vibration isolation" -author = ["Dehaeze Thomas"] draft = false ref_author = "Zuo, L." ref_year = 2004 +++ Tags -: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}) +: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Eddy Current Damping]({{< relref "eddy_current_damping.md" >}}) Reference : (Zuo 2004) @@ -28,21 +27,47 @@ Year > They found that coupling from flexible modes is much smaller than in soft active mounts in the load (force) feedback. > Note that reaction force actuators can also work with soft mounts or hard mounts. + +## Passive Vibration Isolation {#passive-vibration-isolation} + + +### The Role of damping and its practical constructions {#the-role-of-damping-and-its-practical-constructions} + + +#### Viscous damping {#viscous-damping} + + +#### Eddy-current damper {#eddy-current-damper} + + + +{{< figure src="/ox-hugo/zuo04_eddy_current_magnets.png" caption="Figure 1: (left) Magnetic field and conductor plates assemblies, (right) magnet arrays" >}} + + + +{{< figure src="/ox-hugo/zuo04_eddy_current_setup.png" caption="Figure 2: Single DoF system damped by eddy current damper" >}} + + +## Elements and configurations for active vibration systems {#elements-and-configurations-for-active-vibration-systems} + + +### System architectures {#system-architectures} + {{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="Figure 1: PZT actuator and spring in series" >}} -{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="Figure 2: Voice coil actuator and spring in parallel" >}} +{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="Figure 1: Voice coil actuator and spring in parallel" >}} -{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="Figure 3: Transmission from PZT voltage to geophone output" >}} +{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="Figure 1: Transmission from PZT voltage to geophone output" >}} -{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="Figure 4: Transmission from voice coil voltage to geophone output" >}} +{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="Figure 1: Transmission from voice coil voltage to geophone output" >}} ## Bibliography {#bibliography} diff --git a/content/zettels/eddy_current_damping.md b/content/zettels/eddy_current_damping.md index 572e64d..8151b25 100644 --- a/content/zettels/eddy_current_damping.md +++ b/content/zettels/eddy_current_damping.md @@ -1,6 +1,5 @@ +++ title = "Eddy Current Damping" -author = ["Dehaeze Thomas"] draft = false +++ @@ -15,7 +14,96 @@ Tags +## Estimate the damping {#estimate-the-damping} + + +### Formulas {#formulas} + +From (Zuo 2004): +The empirical formula for damping coefficient (Ns/m) of an eddy current damper is: + +\begin{equation} \label{eq:damping\_formula} +C = C\_0 B^2 t A \sigma +\end{equation} + +with: + +- \\(B\\) is the magnetic flux density in [T] or in [Vs/m2] +- \\(t\\) is the thickness of the conductor plate in [m] +- \\(A\\) is the area of the conductor intersected by the magnetic field in [m2] +- \\(\sigma\\) is the electrical conductivity of the conductor material [S/m] +- \\(C\_0\\) is a dimensionless coefficient to account for the shapes and sizes of the conductor and magnetic field + +\\(C\_0 = 1\\) corresponds to a conductor with conductivity \\(\sigma\\) inside a uniform magnetic field and conductivity infinite outside this field. +A typical value of \\(C\_0\\) is about 0.25-0.4 for a conductor plate with area 2 to 5 times that of the magnetic field. + +From , we see that the damping coefficient is proportional to: + +- the square of the magnetic flux density \\(B\\). Therefore it is very important to have large magnetic field strengh +- the thickness \\(t\\) of the conductor. However due to **skin depth effect**, the benefit of increasing the thickness is limited. + The apparent conductivity \\(\sigma\_e\\) is: + + \begin{equation} + \sigma\_e = \frac{2\delta\_s}{t}(1 - e^{-\frac{t}{2\delta\_s}})\sigma + \end{equation} + + where \\(\delta\_s\\) is the skin depth in [m] of the conductor with permeability \\(\mu\\) in [H/m] at frequency \\(f\\) in [Hz]: + + \begin{equation} + \delta\_s = \sqrt{\frac{2}{2 \pi f \cdot \mu \cdot \sigma}} + \end{equation} + +An eddy current damper is developed in (Zuo 2004). +The magnets have alternating poles to optimize the eddy current damping (stronger varying magnetic field). +See Figures [1](#figure--fig:zuo04-eddy-current-magnets) and [2](#figure--fig:zuo04-eddy-current-setup). + + + +{{< figure src="/ox-hugo/zuo04_eddy_current_magnets.png" caption="Figure 1: (left) Magnetic field and conductor plates assemblies, (right) magnet arrays" >}} + + + +{{< figure src="/ox-hugo/zuo04_eddy_current_setup.png" caption="Figure 1: Single DoF system damped by eddy current damper" >}} + + +### Numerical Simulation {#numerical-simulation} + +It is possible to estimate that with FEM simulation: + +An approximation is done bellow. + +```matlab +B = 1.0; % Magnetic Flux Density [T] +t = 5e-3; % Thickness [m] +A = 50e-3*50e-3; % Area [m2] +sigma = 6e7; % Copper conductivity [S/m] +C0 = 0.5; % [-] +``` + +```matlab +C = C0*B^2*t*A*sigma; % Damping in [N/(m/s)] +``` + +```text +C = 375 [N/(m/s)] +``` + +```matlab +m = 10; % [kg] +k = m*(2*pi*10)^2; % [N/m] +``` + +```matlab +xi = 1/2*C/sqrt(k*m); +``` + +```text +xi = 0.298 +``` + + ## Bibliography {#bibliography}
+
Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.
diff --git a/content/zettels/encoders.md b/content/zettels/encoders.md index 24311c3..da65a6e 100644 --- a/content/zettels/encoders.md +++ b/content/zettels/encoders.md @@ -22,6 +22,7 @@ There are two main types of encoders: optical encoders, and magnetic encoders. | [RLS](https://www.rls.si/eng/products/rotary-magnetic-encoders) | Slovenia | | [AMO](https://www.amo-gmbh.com/en/) | Australia | | [NumerikJena](https://www.numerikjena.de/en/) | Germany | +| [RSF Elektronik](https://www.rsf.at/en/) | Austria | ## Bibliography {#bibliography} diff --git a/content/zettels/mass_spring_damper_systems.md b/content/zettels/mass_spring_damper_systems.md index 06aca22..deef32f 100644 --- a/content/zettels/mass_spring_damper_systems.md +++ b/content/zettels/mass_spring_damper_systems.md @@ -7,7 +7,10 @@ Tags : -## Actuated Mass Spring Damper System {#actuated-mass-spring-damper-system} +## One Degree of Freedom {#one-degree-of-freedom} + + +### Model and equation of motion {#model-and-equation-of-motion} Let's consider Figure [1](#figure--fig:mass-spring-damper-system) where: @@ -23,57 +26,150 @@ Let's consider Figure [1](#figure--fig:mass-spring-damper-system) where: {{< figure src="/ox-hugo/mass_spring_damper_system.png" caption="Figure 1: Mass Spring Damper System" >}} -Let's write the transfer function from \\(F\\) to \\(x\\): +Transmissibility: \begin{equation} - \frac{x}{F}(s) = \frac{1}{m s^2 + c s + k} + \frac{x}{w}(s) = \frac{c s + k}{m s^2 + c s + k} = \frac{2 \xi \frac{s}{\omega\_0} + 1}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1} \end{equation} -This can be re-written as: +Compliance: \begin{equation} - \frac{x}{F}(s) = \frac{1/k}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1} + \frac{x}{F}(s) = \frac{x}{F\_d}(s) = \frac{1}{m s^2 + c s + k} = \frac{1/k}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1} \end{equation} with: -- \\(\omega\_0\\) the natural frequency in [rad/s] -- \\(\xi\\) the damping ratio +- \\(\omega\_0 = \sqrt{k/m}\\) the natural frequency in [rad/s] +- \\(\xi = \frac{1}{2} \frac{c}{\sqrt{km}}\\) the damping ratio [unit-less] -## Transfer function {#transfer-function} - - -### Voice Coil Actuator with flexible guiding {#voice-coil-actuator-with-flexible-guiding} +### Matlab model {#matlab-model} ```matlab %% Mechanical properties m = 1; % Mobile mass [kg] k = 1e6; % stiffness [N/m] -xi = 0.01; % Modal Damping +xi = 0.1; % Modal Damping c = 2*xi*sqrt(k*m); ``` ```matlab -%% Transfer function from F [N] to x [m] -G = 1/(m*s^2 + c*s + k); +%% Compliance: Transfer function from F [N] to x [m] +Gf = 1/(m*s^2 + c*s + k); + +%% Transmissibility: Transfer function from w [m] to x [m] +Gw = (c*s + k)/(m*s^2 + c*s + k); ``` + -### Transmissibility {#transmissibility} +{{< figure src="/ox-hugo/mass_spring_damper_1dof_compliance.png" caption="Figure 2: 1dof Mass spring damper system - Compliance" >}} + + + +{{< figure src="/ox-hugo/mass_spring_damper_1dof_transmissibility.png" caption="Figure 1: 1dof Mass spring damper system - Transmissibility" >}} + + +## Two Degrees of Freedom {#two-degrees-of-freedom} + + +### Model and equation of motion {#model-and-equation-of-motion} + +Consider the two degrees of freedom mass spring damper system of Figure [1](#figure--fig:mass-spring-damper-2dof). + + + +{{< figure src="/ox-hugo/mass_spring_damper_2dof.png" caption="Figure 1: 2 DoF Mass Spring Damper system" >}} + +We can write the Newton's second law of motion to the two masses: + +\begin{align} +m\_2 s^2 x\_2 &= F\_2 + (k\_2 + c\_2 s) (x\_1 - x\_2) \\\\ +m\_1 s^2 x\_1 &= F\_1 + (k\_1 + c\_1 s) (x\_0 - x\_1) + (k\_2 + c\_2 s) (x\_2 - x\_1) +\end{align} + +The goal is to have \\(x\_1\\) and \\(x\_2\\) as a function of \\(F\_1\\), \\(F\_2\\) and \\(x\_0\\). + +When, we have: \begin{equation} - \frac{x}{w}(s) = \frac{1}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1} +\boxed{x\_1 = \frac{(m\_2 s^2 + c\_2 s + k\_2) F\_1 + (k\_1 + c\_1 s) (m\_2 s^2 + c\_2 s + k\_2) x\_0 + (k\_2 + c\_2 s) F\_2}{(m\_1 s^2 + c\_1 s + k\_1)(m\_2 s^2 + c\_2 s + k\_2) + m\_2 s^2 (c\_2 s + k\_2)}} \end{equation} - -### Compliance {#compliance} - \begin{equation} - \frac{x}{F\_d}(s) = \frac{1/k}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1} +\boxed{x\_2 = \frac{(c\_2s + k\_2)F\_1 + (c\_2s + k\_2)(k\_1 + c\_1 s) x\_0 + (m\_1 s^2 + c\_1 s + k\_1 + c\_2 s + k\_2) F\_2}{(m\_1 s^2 + c\_1 s + k\_1)(m\_2 s^2 + c\_2 s + k\_2) + m\_2 s^2 (c\_2 s + k\_2)}} \end{equation} +We can see that the effects of \\(x\_0\\) and \\(F\_1\\) are related with a factor \\((c\_1 s + k\_1)\\). + +If we are interested by \\(x\_2-x\_1\\): + +\begin{equation} +(x\_2 - x1) = \frac{- m\_2 s^2 F\_1 - (m\_2 s^2)(k\_1 + c\_1 s) x\_0 + (m\_1 s^2 + c\_1 s + k\_1) F\_2}{(m\_1 s^2 + c\_1 s + k\_1)(m\_2 s^2 + c\_2 s + k\_2) + m\_2 s^2 (c\_2 s + k\_2)} +\end{equation} + +| | x1 | x2 | x2-x1 | +|----|-----------------------------|----------------------------|--------------------| +| x0 | (c1s + k1)(m2s2 + c2s + k2) | (c1s + k1)(c2s + k2) | - m2s2\*(k1 + c1s) | +| F1 | m2s2 + c2s + k2 | c2s + k2 | - m2s2 | +| F2 | c2s + k2 | m1s2 + c1s + k1 + c2s + k2 | m1s2 + c1s + k1 | + + +### Matlab model {#matlab-model} + +```matlab +%% Values for the 2dof Mass-Spring-Damper system +m1 = 5e2; % [kg] +k1 = 2e6; % [N/m] +c1 = 2*0.01*sqrt(m1*k1); % [N/(m/s)] + +m2 = 10; % [kg] +k2 = 1e6; % [N/m] +c2 = 2*0.01*sqrt(m2*k2); % [N/(m/s)] +``` + +```matlab +%% Transfer functions +G_x0_to_x1 = (c1*s + k1)*(m2*s^2 + c2*s + k2)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2)); +G_F1_to_x1 = (m2*s^2 + c2*s + k2)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2)); +G_F2_to_x1 = (c2*s + k2)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2)); + +G_x0_to_x2 = (c1*s + k1)*(c2*s + k2)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2)); +G_F1_to_x2 = (c2*s + k2)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2)); +G_F2_to_x2 = (m1*s^2 + c1*s + k1 + c2*s + k2)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2)); + +G_x0_to_d2 = -m2*s^2*(c1*s + k1)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2)); +G_F1_to_d2 = -m2*s^2/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2)); +G_F2_to_d2 = (m1*s^2 + c1*s + k1)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2)); +``` + +From Figure [1](#figure--fig:mass-spring-damper-2dof-x0-bode-plots), we can see that: + +- the low frequency transmissibility is equal to one +- the high frequency transmissibility to the second mass is smaller than to the first mass + + + +{{< figure src="/ox-hugo/mass_spring_damper_2dof_x0_bode_plots.png" caption="Figure 1: Transfer functions from x0 to x1 and x2 (Transmissibility)" >}} + +The transfer function from \\(F\_1\\) to the mass displacements (Figure [1](#figure--fig:mass-spring-damper-2dof-F1-bode-plots)) has the same shape than the transmissibility (Figure [1](#figure--fig:mass-spring-damper-2dof-x0-bode-plots)). + +However, the low frequency gain is now equal to \\(1/k\_1\\). + + + +{{< figure src="/ox-hugo/mass_spring_damper_2dof_F1_bode_plots.png" caption="Figure 1: Transfer functions from F1 to x1 and x2" >}} + +The transfer functions from \\(F\_2\\) to the mass displacements are shown in Figure [1](#figure--fig:mass-spring-damper-2dof-F2-bode-plots): + +- the motion \\(x\_1\\) is smaller than \\(x\_2\\) + + + +{{< figure src="/ox-hugo/mass_spring_damper_2dof_F2_bode_plots.png" caption="Figure 1: Transfer functions from F2 to x1 and x2" >}} + ## Bibliography {#bibliography} diff --git a/content/zettels/tuned_mass_damper.md b/content/zettels/tuned_mass_damper.md index 64d3f85..6ded695 100644 --- a/content/zettels/tuned_mass_damper.md +++ b/content/zettels/tuned_mass_damper.md @@ -1,11 +1,10 @@ +++ title = "Tuned Mass Damper" -author = ["Dehaeze Thomas"] draft = false +++ Tags -: [Passive Damping]({{< relref "passive_damping.md" >}}) +: [Passive Damping]({{< relref "passive_damping.md" >}}), [Mass Spring Damper Systems]({{< relref "mass_spring_damper_systems.md" >}}) Review: (Elias and Matsagar 2017), (Verbaan 2015) @@ -48,7 +47,10 @@ The optimal parameters of the tuned mass damper can be roughly estimated as foll ## Simple TMD model {#simple-tmd-model} -Let's consider a primary system that is represented by a mass-spring-damper system with the following parameters: \\(m\_1\\), \\(k\_1\\), \\(c\_1\\). + +### Model {#model} + +Let's consider a primary system that is represented by a [Mass Spring Damper Systems]({{< relref "mass_spring_damper_systems.md" >}}) with the following parameters: \\(m\_1\\), \\(k\_1\\), \\(c\_1\\). The TMD is also represented by a mass-spring-damper system with parameters \\(m\_2\\), \\(k\_2\\), \\(c\_2\\). The system is schematically represented in Figure [1](#figure--fig:tuned-mass-damper-schematic). diff --git a/content/zettels/voice_coil_actuators.md b/content/zettels/voice_coil_actuators.md index a86b956..6cf7543 100644 --- a/content/zettels/voice_coil_actuators.md +++ b/content/zettels/voice_coil_actuators.md @@ -1,6 +1,5 @@ +++ title = "Voice Coil Actuators" -author = ["Dehaeze Thomas"] draft = false category = "equipment" +++ @@ -37,7 +36,6 @@ As the force is proportional to the current, a [Transconductance Amplifiers]({{< | [H2tech](https://www.h2wtech.com/) | USA | | [Beikimco](http://www.beikimco.com/) | USA | | [Monticont](http://www.moticont.com/) | USA | -| [Thorlabs](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=14116) | USA | | [Celera](https://www.celeramotion.com/applimotion/products/direct-drive-frameless-linear-motors/voice-coil/juke-series-round-body/) | | diff --git a/static/ox-hugo/mass_spring_damper_1dof_compliance.png b/static/ox-hugo/mass_spring_damper_1dof_compliance.png new file mode 100644 index 0000000..934586a Binary files /dev/null and b/static/ox-hugo/mass_spring_damper_1dof_compliance.png differ diff --git a/static/ox-hugo/mass_spring_damper_1dof_transmissibility.png b/static/ox-hugo/mass_spring_damper_1dof_transmissibility.png new file mode 100644 index 0000000..52a1c1f Binary files /dev/null and b/static/ox-hugo/mass_spring_damper_1dof_transmissibility.png differ diff --git a/static/ox-hugo/mass_spring_damper_2dof.png b/static/ox-hugo/mass_spring_damper_2dof.png new file mode 100644 index 0000000..2e210b3 Binary files /dev/null and b/static/ox-hugo/mass_spring_damper_2dof.png differ diff --git a/static/ox-hugo/mass_spring_damper_2dof_F1_bode_plots.png b/static/ox-hugo/mass_spring_damper_2dof_F1_bode_plots.png new file mode 100644 index 0000000..c24d1df Binary files /dev/null and b/static/ox-hugo/mass_spring_damper_2dof_F1_bode_plots.png differ diff --git a/static/ox-hugo/mass_spring_damper_2dof_F2_bode_plots.png b/static/ox-hugo/mass_spring_damper_2dof_F2_bode_plots.png new file mode 100644 index 0000000..5e92974 Binary files /dev/null and b/static/ox-hugo/mass_spring_damper_2dof_F2_bode_plots.png differ diff --git a/static/ox-hugo/mass_spring_damper_2dof_x0_bode_plots.png b/static/ox-hugo/mass_spring_damper_2dof_x0_bode_plots.png new file mode 100644 index 0000000..e9f1c19 Binary files /dev/null and b/static/ox-hugo/mass_spring_damper_2dof_x0_bode_plots.png differ diff --git a/static/ox-hugo/mass_spring_damper_3dof.png b/static/ox-hugo/mass_spring_damper_3dof.png new file mode 100644 index 0000000..6b0f5f4 Binary files /dev/null and b/static/ox-hugo/mass_spring_damper_3dof.png differ diff --git a/static/ox-hugo/zuo04_eddy_current_magnets.png b/static/ox-hugo/zuo04_eddy_current_magnets.png new file mode 100644 index 0000000..5908f5b Binary files /dev/null and b/static/ox-hugo/zuo04_eddy_current_magnets.png differ diff --git a/static/ox-hugo/zuo04_eddy_current_setup.png b/static/ox-hugo/zuo04_eddy_current_setup.png new file mode 100644 index 0000000..bd370bb Binary files /dev/null and b/static/ox-hugo/zuo04_eddy_current_setup.png differ