159 lines
6.5 KiB
Markdown
159 lines
6.5 KiB
Markdown
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title = "Tuned Mass Damper"
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author = ["Dehaeze Thomas"]
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draft = false
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Tags
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: [Passive Damping]({{< relref "passive_damping.md" >}}), [Mass Spring Damper Systems]({{< relref "mass_spring_damper_systems.md" >}})
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Review: (<a href="#citeproc_bib_item_1">Elias and Matsagar 2017</a>), (<a href="#citeproc_bib_item_2">Verbaan 2015</a>)
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## Working Principle {#working-principle}
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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).
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Usually, the resonance frequency of the TMD should match the resonance of the primary system that is to be damped.
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The TMD then has large internal damping such that the energy is dissipated (i.e. the resonance of the primary system is well damped).
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Below is a lecture about tuned mass damper.
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{{< youtube qDzGCgLu59A >}}
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And a simple experiment showing how a tuned mass damper works with a vibration cantilever beam.
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{{< youtube HDa1VO1VDpc >}}
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## How to properly apply a TMD? {#how-to-properly-apply-a-tmd}
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Few questions:
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- What damping mechanism to use?
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Eddy current damping?
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Viscous damping?
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- How to optimize parameters of the TMD (i.e. mass, stiffness and damping)?
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- Where to fix the TMD to the structure?
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## Tuned Mass Damper Optimization {#tuned-mass-damper-optimization}
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The optimal parameters of the tuned mass damper can be roughly estimated as follows:
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- Choose the maximum acceptable mass of the TMD \\(m\_2\\) and note:
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\\[ \mu = m\_2/m\_1 \\]
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where \\(m\_1\\) is the mass of the system to damp
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- The resonance frequency of the tuned mass damper should be chosen to be
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\\[ \nu = \frac{1}{1 + \mu} \approx 1 \\]
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As usually we have \\(\mu \ll 1\\) (i.e. TMD mass small compared to the structure mass, for instance few percent)
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- This allows to compute the stiffness of the TMD:
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\\[ k\_2 = \nu^2 k\_1 \mu = k\_1 \frac{\mu}{(1 + \mu)^2} \\]
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- Finally, the optimal damping of the TMD is:
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\\[ \xi\_2 = \sqrt{\frac{3 \mu}{8 (1 + \mu)}} \Longrightarrow c\_2 = 2 \xi\_2 \sqrt{k\_2 m\_2} \\]
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## Simple TMD model {#simple-tmd-model}
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### Model {#model}
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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\\).
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The TMD is also represented by a mass-spring-damper system with parameters \\(m\_2\\), \\(k\_2\\), \\(c\_2\\).
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The system is schematically represented in Figure [1](#figure--fig:tuned-mass-damper-schematic).
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The goal is to limit the peak amplitude of \\(x\_1\\) due to \\(x\_0\\) (or a force affecting \\(m\_1\\) for instance).
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<a id="figure--fig:tuned-mass-damper-schematic"></a>
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{{< 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" >}}
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The parameter of the primary system are defined as follow:
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```matlab
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%% Primary system parameters
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m1 = 100; % Mass [kg]
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k1 = 1e7; % Stiffness [N/m]
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c1 = 300; % Damping [N/(m/s)]
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```
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Then, the mass of the TMD is fixed and its optical parameters are computed:
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```matlab
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%% Tuned Mass Damper Parameters
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mu = 0.02; % Mass ratio
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m2 = mu*m1;
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k2 = k1*mu/(1 + mu)^2;
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xi = sqrt(3*mu/(8*(1 + mu)));
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c2 = 2*xi*sqrt(k2*m2);
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```
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<div class="table-caption">
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<span class="table-number">Table 1:</span>
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Obtained parameters of the TMD
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</div>
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| | Mass `m2` [kg] | Stiffness `k2` [N/m] | Damping `c2` [N/(m/s)] |
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|-------|----------------|----------------------|------------------------|
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| Value | 2 | 192234 | 106.338 |
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The transfer function from \\(x\_0\\) to \\(x\_1\\) with and without the TMD are computed and shown in Figure
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```matlab
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%% Transfer function from X0 to X1 without TMD
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G1 = (c1*s + k1)/(m1*s^2 + c1*s + k1);
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%% Transfer function from X0 to X1 with TMD
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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));
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```
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<a id="figure--fig:tuned-mass-damper-effect-tmd"></a>
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{{< 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" >}}
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Let's now see how the mass of the TMD can affect its efficiency.
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The following mass ratios are tested:
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```matlab
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%% Mass ratios
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mus = [0.01, 0.02, 0.05, 0.1];
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```
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The obtained transfer functions are shown in Figure [1](#figure--fig:tuned-mass-damper-mass-effect).
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<a id="figure--fig:tuned-mass-damper-mass-effect"></a>
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{{< figure src="/ox-hugo/tuned_mass_damper_mass_effect.png" caption="<span class=\"figure-number\">Figure 1: </span>Effect of the TMD mass on its efficiency" >}}
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The maximum amplification (i.e. \\(\mathcal{H}\_\infty\\) norm) of the transmissibility as a function of the mass ratio is shown in Figure [1](#figure--fig:tuned-mass-damper-effect-mass-ratio).
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This relation can help to determine the minimum mass of the TMD that will give acceptable results.
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<a id="figure--fig:tuned-mass-damper-effect-mass-ratio"></a>
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{{< figure src="/ox-hugo/tuned_mass_damper_effect_mass_ratio.png" caption="<span class=\"figure-number\">Figure 1: </span>Maximum amplification due to resonance as a function of the mass ratio" >}}
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## Manufacturers {#manufacturers}
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<https://vibratec.se/en/product/high-frequency-tuned-mass-damper/>
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<https://micromega-dynamics.com/products/vibration-control/passive-damping-devices/>
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<https://www.csaengineering.com/products-services/tuned-dampers-absorbers.html>
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## Ways to add damping {#ways-to-add-damping}
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Possible damping sources:
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- Magnetic (eddy current)
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- Viscous fluid
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| Fuild | Reference |
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|----------------------|---------------------------------------------------|
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| Rocol Kilopoise 0868 | (<a href="#citeproc_bib_item_2">Verbaan 2015</a>) |
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<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Elias, Said, and Vasant Matsagar. 2017. “Research Developments in Vibration Control of Structures Using Passive Tuned Mass Dampers.” <i>Annual Reviews in Control</i> 44 (nil): 129–56. doi:<a href="https://doi.org/10.1016/j.arcontrol.2017.09.015">10.1016/j.arcontrol.2017.09.015</a>.</div>
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<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Verbaan, C.A.M. 2015. “Robust mass damper design for bandwidth increase of motion stages.” Mechanical Engineering; Technische Universiteit Eindhoven.</div>
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</div>
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