Tuned Mass Damper Design for Attenuating Vibration - Maple Programming Help

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Tuned Mass Damper Design for Attenuating Vibration

Introduction

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.

 


restart:withDynamicSystems:withColorTools:

Derive Expressions for the Optimum Spring and Damping Constant of the Tuned Mass Damper


Natural frequency of the tuned mass damper:

ω__2k__2m__2:


Natural frequency of the main system:

ω__1k__1m__1:


Ratio of the natural frequencies:

αω__2ω__1

αk__2m__2k__1m__1

(2.1)


Optimum ratio of natural frequencies:

α__opt11+m__2m__1:


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

k__2_optsolveα=α__opt,k__2

k__2_optm__1k__1m__2m__1+m__22

(2.2)

Damping ratio:

zb__22 m__2 ω__2:


Optimum damping ratio:

z__opt3 m__2m__18 1+m__2m__13:


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

 b__2_optsubsk__2=k__2_opt,solvez=z__opt,b__2

b__2_opt6m__2m__11+m__2m__13m__1k__1m__1+m__22m__22

(2.3)

System Parameters

Main spring mass damper parameters:

params__mainm__1=1.764 105,k__1=3.45 107,b__1=1.531 105:


Mass of the tuned mass damper:

m__TMD8165:

 

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

k__2_calcevalk__2_opt,params__main,m__2=m__TMD;

k__2_calc1.458730861106

(3.1)

b__2_calcevalfevalb__2_opt,params__main,m__2=m__TMD;

b__2_calc26869.77096

(3.2)

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

params__TMDparams__main,m__2=m__TMD,k__2=k__2_calc,b__2=b__2_calc:params__noTMDparams__main,m__2=0,k__2=0,b__2=0:

Equations of Motion for the Entire System

de:=m__2ⅆ2ⅆt2x__2t=k__2 x__2tx__1tb__2 ⅆⅆtx__2tⅆⅆtx__1t,m__1 ⅆ2ⅆt2x__1t=k__1 x__1tb__1 ⅆⅆtx__1tk__2 x__1tx__2tb__2 ⅆⅆtx__1tⅆⅆtx__2t+Ft:ic:=x__10=0,Dx__10=0,x__20=0,Dx__20=0:

sys:=DiffEquationde,Ft,x__1t:

Frequency Response

Response with  tuned mass damper:

p1:=MagnitudePlotsys,range=5..30,parameters=params__TMD,color=ColorRGB,0/255,79/255,121/255,legend=Tuned:


Response with no tuned mass damper:

p2MagnitudePlotsys,range=5..30,parameters=params__noTMD,color=ColorRGB,150/255,40/255,27/255,legend=Not Tuned:

plots:-displayp1,p2,size=800,400,thickness=2,axesfont=Calibri,labelfont=Calibri,background=ColorRGB,218/255,223/255,225/255,legendstyle=font=Calibri

 

Dynamic Response

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

f__natevalω__1,params__main

f__nat13.98492872

(6.1)

p3ResponsePlotsys,7500sinf__natt,parameters=params__TMD,color=ColorRGB,0/255.,79/255,121/255,legend=Tuned:

p4ResponsePlotsys,7500sinf__nat t,parameters=params__noTMD,color=ColorRGB,150/255,40/255,27/255,legend=Not Tuned:

plots:-displayp3,p4,axesfont=Calibri,thickness=2,size=800,400,gridlines,axesfont=Calibri,labelfont=Calibri,background=ColorRGB,218/255,223/255,225/255,legendstyle=font=Calibri