From e4c613f43829e6872cbe001d4e958e12e7c3dec2 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 19 Apr 2022 15:02:29 +0200 Subject: [PATCH] Update Content - 2022-04-19 --- content/zettels/tuned_mass_damper.md | 90 ++++++++++++++++++ .../ox-hugo/tuned_mass_damper_effect_tmd.png | Bin 0 -> 45227 bytes .../ox-hugo/tuned_mass_damper_mass_effect.png | Bin 0 -> 49679 bytes .../ox-hugo/tuned_mass_damper_schematic.png | Bin 0 -> 12234 bytes 4 files changed, 90 insertions(+) create mode 100644 static/ox-hugo/tuned_mass_damper_effect_tmd.png create mode 100644 static/ox-hugo/tuned_mass_damper_mass_effect.png create mode 100644 static/ox-hugo/tuned_mass_damper_schematic.png diff --git a/content/zettels/tuned_mass_damper.md b/content/zettels/tuned_mass_damper.md index 0592497..10ee42c 100644 --- a/content/zettels/tuned_mass_damper.md +++ b/content/zettels/tuned_mass_damper.md @@ -12,9 +12,99 @@ Review: (Elias and Matsagar 2017) ## 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). + + + +{{< figure src="/ox-hugo/tuned_mass_damper_schematic.png" caption="Figure 1: 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); +``` + +
+ Table 1: + Obtained parameters of the TMD +
+ +| | 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)); +``` + + + +{{< figure src="/ox-hugo/tuned_mass_damper_effect_tmd.png" caption="Figure 2: 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]; +``` + + + +{{< figure src="/ox-hugo/tuned_mass_damper_mass_effect.png" caption="Figure 3: Effect of the TMD mass on its efficiency" >}} + + ## Manufacturers {#manufacturers} diff --git a/static/ox-hugo/tuned_mass_damper_effect_tmd.png b/static/ox-hugo/tuned_mass_damper_effect_tmd.png new file mode 100644 index 0000000000000000000000000000000000000000..f7728985e5a18bf374f44a32e75d5658cb271f24 GIT binary patch literal 45227 zcmZsD1yof*+wP{jB&0z?k?xT0k`gKDk}eTJT3S*-5b5sjkdhXqyWya8ci%z3|Nh@y z>$;AMIIz#`Iq%H-)bLSNSr!YO6di&fEO|Mp7Z3z*13_@ksL0?u#*dVvzy~yw#@#w?}E;h!3d6k|3(vD z^EHF}z{f0k@OxH^v(Pa$;$;3_iVV02?^08eDE3l6METue@D}4~05vKNUwF-G4I|7m z_wlXDBEPuvCN$b2&yN-eoxZ5>@;|9MWv z^h!-Glns8*B)#xXL+CDh&*{Tyz0%2ExJ;I_r-)&ge(6)vh=F(f3%a2yS$bSEzs!yBj z`SoDb(N{KsNQx7=PIfHp;H>>*=K&oo$lTnVfpKKCio$iy$+1CiEbjHEx&?QPLE$s< z(uO~Yq$9{f6B9wY_6Fu_eF_K7fve`1mE}?L`gfXe*v>Z+ZcUawF^8sVY#s_x zswi)m?uqHC=%(liA844~T&ml?FS$D1@$dg#8Fx$deie`TRdsKSO$;3xRBk{2Y+uOX zhgsm-1xfRyBR^)4vE|Q4%WD*RE`(LpPl44ba9h~kS_Sgu}r=D5j3(ON`G+qJL~q*a=q(AHUn~5>pL=I8mjX)lyd8F#urk4ffx@> zj{f`-a8Wt>l8U)4Zzql}Wj<@L_3i!9{tu?Ism( zx!;f4_D0tzc55Hr4X@1r5>@zi#_3xE*(+WxutJMS`-;#SLITc`0ptLkPgm6AN~yd1iELt5k;>*^lf+}uFE%ge@Z->#Fr zHK`!RCLhuwc&X7>$!lOiU~$EseCn_$aHXi3|w|NnreszU0?}2f>+{c67EpgjJzmkw)t2}93*aMw-ay}>Egjpj{Q-caS`2`7m}@x z9PYf>cHNqvz%uZ?o3VxZr*=hNYDm*P5H~i?u)B<-lLwb1@6pLD0u>z9W2)f6j_I$> zj9fP>Y|2R>EpBK?^?EQ>$yBhf{J9JRfnQ-Et~B2G_&BtLNNwI!!5eE|KiInGwbuk2 z*D}lzwjsj(D8P z85!-olVg*UDBzW|2{`eUMu%o*Lf;?#`Cv?}e!Mt79@yZ#Hn7e$F*zA5;QVLES8~ea zr%G{Y0~7*dt-=FaB}d+9LCoXg_N|l0w8p6GSFvwc^q;?^yrQKF`BBR~;g(t#mw^mv zXlTUuy(UHmC7QeG^B9ZAubj*(Ha)!gsJiG4b6~sCCy$$x$qKt!-s7_~S|Ua1pp&QAS!^N>t?@}ffP^CBLlr2TIf_jAk;D$%IWi;LKPex+x1w4BfX zCj+C~Tz4Tj*D6Gcrgl>+G?RERl&~?ze7gE{h}$k)h}2bHhV{o(l?6_%-E2_ZqW5h1 zY_E0_V!Zj6orUJvqYV`w(uewQ@ovE`lGWTie=c-W4{ENmShn)w#+x={et*Bb!ioLC zMbknDs;bzxai4}qNBwhibCa!|eHeKEG?0Dlul

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