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 othe 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 : with DynamicSystems : with plots : with SignalProcessing :575> params := m1 = 1.764 10 , k1 = 3.45 10 , b1 = 1.531 10 , m2 = 8165 :Derive Expressions for the Optimum Spring and Damping Constant of the Tuned
Mass Damper
Mass ratio
m2> md:m1
Natural frequency of tuned mass damper

k2 _calc> w2 dm2:

Natural frequency of main systemk1> w1 d

m1:

Hence the natural frequency in rad s-1 is> eval w1, params
13.98492872(2.1

Ratio of natural frequencies> adw2
w1:

Optimum ratio of natural frequencies
1> aopt d:
1 Cm
Hence the optimum spring constant of the tuned mass-spring-damper> k2 _calc d solve a = aopt, k2 _calc
k2 _calc :=m1 k1 m2
m1 Cm2(2.22Damping Ratio
b2 _calc> zd:
2 m2 w2
Optimum damping ratio> zopt d3m8 1 Cm

3:Hence the optimum damping constant of the tuned mass-spring-damper> b2 _calc d solve z = zopt, b2 _calc
1b2 _calc :=26m2
m1 1 C

m23

m1 k1

m2

m1 Cm22(2.3m1> k2 _calc d eval k2 _calc, params ;6> b2 _calc d evalf eval b2 _calc, params

k2 _calc := 1.458730861 10(2.4b2 _calc := 26869.77094(2.5;

Full list of parameters575> paramstuned d m1 = 1.764 10 , k1 = 3.45 10 , b1 = 1.531 10 , m2 = 8165, k2 = k2 _calc, b2 = b2 _calc :575> paramsnottuned d m1 = 1.764 10 , k1 = 3.45 10 , b1 = 1.531 10 , m2 = 0, k2 = k2 _calc, b2 = b2 _calc :Equations of Motion for the Entire System
Equation of motion for the whole system
2> de := m2ddt

2 x2t2m1d2 x1 t

=Kk2 x2 t Kx1 t

Kb2dx t Kk2 x1 t Kx2 t
dt 1
0 = 0, x2 0 = 0, D x2 0 = 0 :=Kk1 x1 t Kb1

dt
ic := x1 0 = 0, D x1
> sys := DiffEquation

ddx2 t Kx t
dt
dt 1

de , F t, x1 tKb2,ddx1 t Kx t
dt
dt 2

CF t ::

Frequency Response
Response with Tuned Mass Damper> p1 := MagnitudePlot sys, range = 5 ..30, parameters = paramstuned, color = blue, legend = "Tuned" :Response with no Tuned Mass Damper> p2 d MagnitudePlot sys, range = 5 ..30, parameters = paramsnottuned, color = red, axesfont = Calibri , labelfont= Calibri , legend = "Not Tuned" :> display p1, p2

Freq [rad/s]

5 5.5 66.57

8 9 1015TunedNot Tuned

202530K130

Magnitude [dB]

K135
K140
K145
K150
K155
K160

Dynamic Response
Assuming that the main system is perturbed at its natural frequency> f d eval w1, params
f := 13.98492872(5.110> p1 := ResponsePlot sys, 7500 sin f$t , parameters = paramstuned, color = blue, numpoints = 2 , legend = "Tuned" :10> p2 d ResponsePlot sys, 7500 sin f t , parameters = paramsnottuned, color = red, numpoints = 2 , legend= "Not Tuned" :> display p1, p2, axesfont = Calibri , labelfont = Calibri , axesfont = Calibri , gridlines

0.0030.002

x10.001
02

46810

t

K0.001
K0.002
K0.003
Tuned

Not Tuned

Power Spectrum> tunedResponseData d plottools:-getdata p1 3 :> notTunedResponseData d plottools:-getdata p2 3 :1> samplingRate d:tunedResponseData 2, 1 KtunedResponseData 1, 1
> psTuned d PowerSpectrum FFT tunedResponseData .., 2:> psNotTuned d PowerSpectrum FFT notTunedResponseData .., 2:> psPlot1 d pointplot

seq

i$samplingRate
102

10

2$2$Pi, psTuned i , i = 1 ..
2, connect = true, legend = "Tuned",

gridlines :psPlot2 d pointplot

seq

i$samplingRate
102102$2$Pi, psNotTuned i , i = 1 ..
2, connect = true, color = red, legend= "Not Tuned" :> display psPlot1, psPlot2, axis 1 = mode = log , axesfont = Calibri , labels = "Frequency [rad/s]", "Power" ,labeldirections = horizontal, vertical , labelfont = Calibri0.0015

Power0.0010

0.00050151050

Frequency [rad/s]

Tuned>

Not Tuned100