 Tuned Mass Damper Design for Attenuating Vibration - Maple Help

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A mass-spring-damper is disturbed by a force that resonates at the natural frequency of the system. This application calculates the optimum spring and damping constant of a parasitic tuned-mass damper that the minimizes the vibration of the system.

The vibration of system with and without the tuned mass-spring-damper is viewed as a frequency response, time-domain simulation and power spectrum. > $\mathrm{restart}:$$\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{with}\left(\mathrm{DynamicSystems}\right):$$\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{with}\left(\mathrm{ColorTools}\right):$ Derive Expressions for the Optimum Spring and Damping Constant of the Tuned Mass Damper

Natural frequency of the tuned mass damper:

 > $\mathrm{ω__2}≔\sqrt{\frac{\mathrm{k__2}}{\mathrm{m__2}}}:$

Natural frequency of the main system:

 > $\mathrm{ω__1}≔\sqrt{\frac{\mathrm{k__1}}{\mathrm{m__1}}}:$

Ratio of the natural frequencies:

 > $\mathrm{α}≔\frac{\mathrm{ω__2}}{\mathrm{ω__1}}$
 ${\mathrm{\alpha }}{≔}\frac{\sqrt{\frac{\mathrm{k__2}}{\mathrm{m__2}}}}{\sqrt{\frac{\mathrm{k__1}}{\mathrm{m__1}}}}$ (2.1)

Optimum ratio of natural frequencies:

 > $\mathrm{α__opt}≔\frac{1}{1+\frac{\mathrm{m__2}}{\mathrm{m__1}}}:$

Hence the optimum spring constant of the tuned mass-spring-damper:

 > $\mathrm{k__2_opt}≔\mathrm{solve}\left(\mathrm{\alpha }=\mathrm{α__opt},\mathrm{k__2}\right)$
 $\mathrm{k__2_opt}{≔}\frac{\mathrm{m__1}{}\mathrm{k__1}{}\mathrm{m__2}}{{\left(\mathrm{m__1}{+}\mathrm{m__2}\right)}^{{2}}}$ (2.2)

Damping ratio:

 >

Optimum damping ratio:

 >

Hence the optimum damping constant of the tuned mass-spring-damper:

 >
 $\mathrm{b__2_opt}{≔}\frac{\sqrt{{6}}{}\sqrt{\frac{\mathrm{m__2}}{\mathrm{m__1}{}{\left({1}{+}\frac{\mathrm{m__2}}{\mathrm{m__1}}\right)}^{{3}}}}{}\sqrt{\frac{\mathrm{m__1}{}\mathrm{k__1}}{{\left(\mathrm{m__1}{+}\mathrm{m__2}\right)}^{{2}}}}{}\mathrm{m__2}}{{2}}$ (2.3) System Parameters

Main spring mass damper parameters:

 >

Mass of the tuned mass damper:

 > $\mathrm{m__TMD}≔8165:$

Optimum spring and damping constants of the tuned mass damper are:

 > $\mathrm{k__2_calc}≔\mathrm{eval}\left(\mathrm{k__2_opt},\left[\mathrm{params__main},\mathrm{m__2}=\mathrm{m__TMD}\right]\right);$
 $\mathrm{k__2_calc}{≔}{1.458730861}{}{{10}}^{{6}}$ (3.1)
 > $\mathrm{b__2_calc}≔\mathrm{evalf}\left(\mathrm{eval}\left(\mathrm{b__2_opt},\left[\mathrm{params__main},\mathrm{m__2}=\mathrm{m__TMD}\right]\right)\right);$
 $\mathrm{b__2_calc}{≔}{26869.77096}$ (3.2)

Parameters for the system with and without a tuned mass damper:

 > $\mathrm{params__TMD}≔\left[\mathrm{params__main},\mathrm{m__2}=\mathrm{m__TMD},\mathrm{k__2}=\mathrm{k__2_calc},\mathrm{b__2}=\mathrm{b__2_calc}\right]:\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{params__noTMD}≔\left[\mathrm{params__main},\mathrm{m__2}=0,\mathrm{k__2}=0,\mathrm{b__2}=0\right]:\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}$ Equations of Motion for the Entire System

 > $\mathrm{ic}:=\mathrm{x__1}\left(0\right)=0,\mathrm{D}\left(\mathrm{x__1}\right)\left(0\right)=0,\mathrm{x__2}\left(0\right)=0,\mathrm{D}\left(\mathrm{x__2}\right)\left(0\right)=0:$
 > $\mathrm{sys}:=\mathrm{DiffEquation}\left(\left[\mathrm{de}\right],\left[\mathrm{F}\left(\mathrm{t}\right)\right],\left[\mathrm{x__1}\left(\mathrm{t}\right)\right]\right):$ Frequency Response

Response with  tuned mass damper:

 > $\mathrm{p1}:=\mathrm{MagnitudePlot}\left(\mathrm{sys},\mathrm{range}=5..30,\mathrm{parameters}=\mathrm{params__TMD},\mathrm{color}=\mathrm{Color}\left("RGB",\left[0/255,79/255,121/255\right]\right),\mathrm{legend}="Tuned"\right):$

Response with no tuned mass damper:

 > $\mathrm{p2}≔\mathrm{MagnitudePlot}\left(\mathrm{sys},\mathrm{range}=5..30,\mathrm{parameters}=\mathrm{params__noTMD},\mathrm{color}=\mathrm{Color}\left("RGB",\left[150/255,40/255,27/255\right]\right),\mathrm{legend}="Not Tuned"\right):$
 > $\mathrm{plots}:-\mathrm{display}\left(\mathrm{p1},\mathrm{p2},\mathrm{size}=\left[800,400\right],\mathrm{thickness}=2,\mathrm{axesfont}=\left[\mathrm{Calibri}\right],\mathrm{labelfont}=\left[\mathrm{Calibri}\right],\mathrm{background}=\mathrm{Color}\left("RGB",\left[218/255,223/255,225/255\right]\right),\mathrm{legendstyle}=\left[\mathrm{font}=\left[\mathrm{Calibri}\right]\right]\right)$  Dynamic Response

Assume that the system is perturbed at the natural frequency of the system.

 > $\mathrm{f__nat}≔\mathrm{eval}\left(\mathrm{ω__1},\left[\mathrm{params__main}\right]\right)$
 $\mathrm{f__nat}{≔}{13.98492872}$ (6.1)
 > $\mathrm{p3}≔\mathrm{ResponsePlot}\left(\mathrm{sys},7500\mathrm{sin}\left(\mathrm{f__nat}\cdot \mathrm{t}\right),\mathrm{parameters}=\mathrm{params__TMD},\mathrm{color}=\mathrm{Color}\left("RGB",\left[0/255.,79/255,121/255\right]\right),\mathrm{legend}="Tuned"\right):$
 >
 > $\mathrm{plots}:-\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{display}\left(\mathrm{p3},\mathrm{p4},\mathrm{axesfont}=\left[\mathrm{Calibri}\right],\mathrm{thickness}=2,\mathrm{size}=\left[800,400\right],\mathrm{gridlines},\mathrm{axesfont}=\left[\mathrm{Calibri}\right],\mathrm{labelfont}=\left[\mathrm{Calibri}\right],\mathrm{background}=\mathrm{Color}\left("RGB",\left[218/255,223/255,225/255\right]\right),\mathrm{legendstyle}=\left[\mathrm{font}=\left[\mathrm{Calibri}\right]\right]\right)$ > 
 >