Update Content - 2022-04-19

content/zettels/tuned_mass_damper.md

@@ -12,9 +12,99 @@ Review: (<a href="#citeproc_bib_item_1">Elias and Matsagar 2017</a>)
 
 ## Working Principle {#working-principle}
 
+The basic idea is to damp the resonance of a structure (called the primary system) by attaching a resonant system to it, the Tuned Mass Damper (TMD).
+Usually, the resonance frequency of the TMD should match the resonance of the primary system that is to be damped.
+The TMD then has large internal damping such that the energy is dissipated (i.e. the resonance of the primary system is well damped).
+
 {{< youtube qDzGCgLu59A >}}
 
 
+## Tuned Mass Damper Optimization {#tuned-mass-damper-optimization}
+
+The optimal parameters of the tuned mass damper can be roughly estimated as follows:
+
+-   Choose the maximum mass of the TMD \\(m\\) and note:
+    \\[ \mu = m/M \\]
+    where \\(M\\) is the mass of the system to damp
+-   The resonance frequency of the tuned mass damper should be chosen to be
+    \\[ \nu = \frac{1}{1 + \mu} \approx 1 \\]
+    As usually we have \\(\mu \ll 1\\) (i.e. TMD mass small compared to the structure mass, for instance few percent)
+-   This allows to compute the stiffness of the TMD:
+    \\[ k = \nu^2 K \mu = K \frac{\mu}{(1 + \mu)^2} \\]
+-   Finally, the optimal damping of the TMD is:
+    \\[ \xi = \sqrt{\frac{3\mu}{8 (1 + \mu)}} \Longrightarrow c = 2 \xi \sqrt{k m} \\]
+
+
+## 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\\).
+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).
+
+The goal is to limit the peak amplitude of \\(x\_1\\) due to \\(x\_0\\) (or a force affecting \\(m\_1\\) for instance).
+
+<a id="figure--fig:tuned-mass-damper-schematic"></a>
+
+{{< figure src="/ox-hugo/tuned_mass_damper_schematic.png" caption="<span class=\"figure-number\">Figure 1: </span>Mass Spring Damper representation of the Primary System and the Tuned Mass Damper" >}}
+
+The parameter of the primary system are defined as follow:
+
+```matlab
+%% Primary system parameters
+m1 = 100; % Mass [kg]
+k1 = 1e7; % Stiffness [N/m]
+c1 = 300; % Damping [N/(m/s)]
+```
+
+Then, the mass of the TMD is fixed and its optical parameters are computed:
+
+```matlab
+%% Tuned Mass Damper Parameters
+mu = 0.02; % Mass ratio
+
+m2 = mu*m1;
+k2 = k1*mu/(1 + mu)^2;
+xi = sqrt(3*mu/(8*(1 + mu)));
+c2 = 2*xi*sqrt(k2*m2);
+```
+
+<div class="table-caption">
+  <span class="table-number">Table 1</span>:
+  Obtained parameters of the TMD
+</div>
+
+|       | Mass `m2` [kg] | Stiffness `k2` [N/m] | Damping `c2` [N/(m/s)] |
+|-------|----------------|----------------------|------------------------|
+| Value | 2              | 192234               | 106.338                |
+
+The transfer function from \\(x\_0\\) to \\(x\_1\\) with and without the TMD are computed and shown in Figure
+
+```matlab
+%% Transfer function from X0 to X1 without TMD
+G1 = (c1*s + k1)/(m1*s^2 + c1*s + k1);
+
+%% Transfer function from X0 to X1 with TMD
+G2 = (m2*s^2 + c2*s + k2)*(c1*s + k1)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2));
+```
+
+<a id="figure--fig:tuned-mass-damper-effect-tmd"></a>
+
+{{< figure src="/ox-hugo/tuned_mass_damper_effect_tmd.png" caption="<span class=\"figure-number\">Figure 2: </span>Comparison of the transmissibility with and without the TMD" >}}
+
+Let's now see how the mass of the TMD can affect its efficiency.
+
+The following mass ratios are tested:
+
+```matlab
+%% Mass ratios
+mus = [0.01, 0.02, 0.05, 0.1];
+```
+
+<a id="figure--fig:tuned-mass-damper-mass-effect"></a>
+
+{{< figure src="/ox-hugo/tuned_mass_damper_mass_effect.png" caption="<span class=\"figure-number\">Figure 3: </span>Effect of the TMD mass on its efficiency" >}}
+
+
 ## Manufacturers {#manufacturers}
 
 <https://vibratec.se/en/product/high-frequency-tuned-mass-damper/>