Tuned Mass Damper Vibration Absorber

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Dynamic Vibration AbsorberTuned Mass Damper

Brad Gassner

10/19/15

Abstract

Dynamic absorbers, or tuned mass dampers, may be used to attenuate vibration of a dynamic system at

a specific frequency. The vibration absorber applied in this lab is meant to simulate adding a spring,

mass, and damper to an existing one-degree-of-freedom spring-mass-damper system. By adjusting the

ratio of the added spring constant to the magnitude of the added mass, the specific attenuated

frequency of the newly created two-degree-of-freedom system may be tuned. The specific hypothesis

that will be tested in this experiment is whether we can see greater vibration amplitude attenuation

correlating positively and linearly with increasing vibration absorber mass. The response characteristics

of the original one-degree-of-freedom system to varying driving frequencies are recorded. We will see a

peak amplitude response at the system's resonant frequency. Adding a spring and mass of appropriate

size and spring constant will result in attenuation of the systems transmissibility by a large amount.

Additionally, the damped natural period of oscillation is used to experimentally determine Young's

Modulus of the added dynamic absorber beam.

Procedure

The experimental apparatus can be seen below, Figure 7.5, obtained from the Laboratory Manual [1].

Accelerometers are attached to mass M1 and to the base. The accelerometers are attached to the

supply signal conditioning and power hardware and connected through the data acquisition hardware to

the LabVIEW program.


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The system's response to a wide range of frequencies is recorded and the frequency at which the system

resonates is noted. It is this peak which we will try to attenuate with the addition of the vibration

absorber to the test apparatus. The dimensionless damping constant is then estimated by finding thehalf power frequencies of the resonance peak. Young's modulus is estimated for the cantilever beam

that is to be added to the apparatus by use of the Euler Bernoulli beam theory. The uncertainty in the

natural frequency of the cantilever beam is calculated, as well as the length of the beam to be used for

the addition of two different absorber masses. The absorbers are applied to the apparatus and the

systems response is noted.

Please refer to Mechanical Engineering Systems Laboratory (Lab Manual)for a step-by-step procedural

breakdown.

Results

To begin, the single degree-of-freedom system was driven at a range of frequencies between 10 and 100

Hertz. The resulting response curve may be seen in the figure below. The plot shows the magnitude of

the mass M1 acceleration and the phase shift detected between the systems vibration and the driving

force.


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Figure 1: Frequency Response of the Mass M1 to the Driving Force

Measurements were taken of the systems response magnitude both at an in -phase and low-frequency

response, and at the resonant frequency. The frequencies selected as maxima and minima for the

software and hardware to sweep through were refined to be able to locate the peak of the resonance

curve more accurately.

Figure 2: A Response Curve of Greater Accuracy

The values recorded for the amplitude of vibration at M1, at resonant and below-resonant frequencies,

along with the peak frequency measured are presented below inTable 1.


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Table 1: Data taken of Single-Degree-of-Freedom System

Signal measured at peak amplitude X_Max 194.1

Base amplitude signal V_Base 1.174

Drive Frequency f_max 27.4694

The maximum displacement transmissibility, the ratio of the peak amplitude to the lowest amplitude,

was calculated to be 165.3, indicating a large amount of the input energy being used to vibrate the mass

M1.

The dimensionless damping constant was estimated using the half-power method. In this method, wecharacterize the spread of the resonance peak by finding the location of the output magnitude of . In this method, described in the lab manual section 6, the spread of the peak, bandwidth, isrelated proportionally to the damping coefficient. Two half-power frequencies were found as below:

Table 2: Half Power Frequency Location

Half Power we're looking for 137.249

Half Power frequencies f_1 27.395 Hz

f_2 27.558 Hz

By this method, the damping coefficient can be found to be described by

=

2 (1)

This approximation is particularly well-suited to small damping coefficients and was estimated to be 0.003. See below for a chart indicating the location of one of the half-power frequencies graphically.


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Figure 3: Locating the Half-Power Frequencies

The beam to be used for the dynamic absorber was then quantified by determining the beams damped

natural period of oscillation and calculating the modulus of elasticity. The beam was placed in a small

bench vise and to it was attached an accelerometer. The beam was placed in the vice vertically, with a

length of six inches exposed out of the top of the vise. The accelerometer was placed halfway up the

beam after a few rounds of trial and error. The beam was plucked with the tip of a finger and allowed to

ring. Data acquisition was started and the following graph was produced:

Figure 4: Free Cantilevered Beam Responding to an Impulse Signal


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The frequency was determined by measuring the time taken between five successive peaks and was

found to be 52.2.The measurement of the frequency as outlined immediately above is used as an approximation for the

natural frequency of the cantilevered beam. The appropriateness of this approximation is governed by

the magnitude of the damping coefficient. If the damping parameter is small, the approximation is

justified. Determination of the damping coefficient for the vibration of this cantilevered beam was

measured through the use of the logarithmic decrement method. It can be shown that for a generalized

underdamped single degree-of-freedom system, the damping coefficient may be given as a function of

the logarithmic decrement of successive peaks in a decaying oscillatory signal.

(2)

Where

is the logarithmic decrement of the amplitude over

periods and

(3)

Using this method, the dimensionless damping coefficient was determined to be 0.0025 1. Thisis indeed small, indicating that the use of the approximation that the damped period of oscillation is

equal to the natural period of oscillation of the beam was indeed justified.

Application of Euler-Bernoulli Small Displacement Beam Theory suggests the ability to compute an

estimate for Youngs Modulus of the beam as

(4)

In our case, the modulus was determined to be 173 .In any data acquisition task, we must be aware of the uncertainties introduced due to the sampling

frequency. Using the time increment between consecutive samples and the uncertainty in the

determination of the time between two signal peaks, we can determine the magnitude of the

uncertainty and the fundamental frequency measurement. The uncertainty in our cantilever beams

fundamental frequency is given as

(5)

It was determined to be 0.03Hz. Therefore, the fundamental frequency is 52.190.03.


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The required length of the dynamic absorber, as measured from clamp to clamp, is given as

(6)

This length was calculated for both supplied beam-end masses and is shown inTable 3.

Table 3: Calculated Absorber Lengths

Calculated absorber length

with smaller mass L 0.036 m

L 1.428 in

with larger mass L 0.030 m

L 1.180 in

The two correction masses and the beam were sequentially applied to the experimental apparatus. The

amplitudes of the base and tower excitation were recorded for each. Specific values may be found in the

Appendix 2: Lab Worksheet.

Answers to Lab Manual Questions

1.

The values of L calculated as shown inTable 3 did in fact yield the greatest reduction in

transmissibility. In fact, the magnitude of L was modified up and down from its original value by

moving the mass up and down by a few millimeters and re-measuring the system's responseamplitude. It was discovered that the response was relatively stable despite changes in L.

2.

As mentioned immediately above, an increase or decrease in the length of the beam yielded

little change in the recorded amplitude for small values of L. It is inferred from this result that

the system is robust to variations in the specific placement of the mass.

3.

The transmissibility reduction ratio for the smaller and larger masses was calculated to be 108.5

and 192.5 respectively. It is noted in the Lab Manual that values of over 1000 are to be

expected.

4.

The larger mass did in fact use a greater reduction in the displacement transmissibility than the

smaller one. The transmissibility reduction achieved with the larger mass is approximately 77%

percent larger than the transmissibility reduction obtained with the smaller mass.

5.

The value of Young's modulus for the cantilever beam that we calculated was 173 gigapascals.The nominal published value in the lab manual is given as 200 gigapascals. The percent

difference between the two quantities is 14.5%. The extended combined uncertainties for thiscalculation, as given in the procedure used in the pump experiment, is given by

=[+ + + + ] (7)


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When carried through, the combined standard uncertainties are 8.9%. SeeTable 4 developed

below to propagate the errors through the equation.

Table 4: Error Propagation

Varying

Parameter E E + error (one-sided)

One Sided

Error

Two-Sided

Error

Error

Squaredrho 1.726E+11 1.738E+11 1.187E+09 2.375E+09 5.640E+18

l 1.726E+11 1.799E+11 7.304E+09 1.461E+10 2.134E+20

f_1 1.726E+11 1.728E+11 1.985E+08 3.969E+08 1.576E+17

h 1.726E+11 1.704E+11 -2.237E+09 -4.473E+09 2.001E+19

B_1*L 1.726E+11 1.726E+11 -1.841E+04 -3.682E+04 1.355E+09

Sum of the Squares 2.392E+20

Square Root 1.547E+10

Error Fraction of Whole 0.0896

6.

The characteristic spring parameter k may be derived from Equation 6.4 from [1]:

Or

= 4 = 7897/ (8)

Conclusion

This laboratory served as a brief introduction to the field of vibrations analysis and the concepts of

tuned mass dampers. The experimental apparatus was designed to provide a lumped-parameter

reasonable facsimile model of a structure such as a building. Single degree-of-freedom systems and a

double degree-of-freedom system are explored as are their equations of motion. The parameters of

spring stiffness, point mass magnitudes, and damping coefficient are investigated for further

understanding into lightly damped resonant systems. The response of a two-degree-of-freedom

resonant system to a sinusoidal driving force was investigated. It was shown that the transmissibility

between the driving force and the amplitude of the resonant mass was a function of the driving

frequency. Phase lag between driving force and driven dynamic system was plotted and shown to be a

function of driving frequency. A brief aside in to Euler-Bernoulli beam theory was completed todetermine the modulus of elasticity for a vibrating cantilever beam.

Two methods for determining the dimensionless damping coefficient were discussed and performed. In

the first, we used the half power method to quantify the band width of a resonant peak of a 2 degree-

of-freedom system. It was shown that the width of this band is proportional to the damping coefficient.


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In the second method to determine the magnitude of the damping coefficient, the fixed-free cantilever

beam was struck with an impulse force and the resulting oscillation was recorded. By using the

logarithmic decrement method, the damping coefficient was shown to be a function of the logarithm of

the ratio of two amplitude peak magnitudes.

Finally, the length of the cantilever spring was calculated and the beam and mass were applied to the

apparatus. The resulting two degree-of-freedom system was shown to have an anti-resonance peak at

the previous resonance peak, indicating that we have successfully split the resonance of the system into

two smaller peaks. In addition, a short review of error propagation was completed.

The use of a tuned mass damper is a viable dynamic vibration absorber was successfully explored.

Conservation of energy predicts a greater vibration amplitude attenuation with increasing mass, as

discussed in the lab manual. It was indeed shown that the larger mass produced a much greater

transmissibility reduction ratio. Certainly, the model and associated approximation used in the

performance of this lab are just that: approximations. The treatment of masses as point masses and

other such simplifications will certainly decrease the accuracy of this analysis. However, judicious use of

scale models would be infinitely helpful when implementing such a system in the real world.

Reference

Mechanical Engineering Systems Laboratory (Lab Manual). Stutts, Daniel S. et al., Missouri University of

Science and Technology. 2015. Obtained from

http://web.mst.edu/~stutts/ME242/LABMANUAL/ME242Text.pdf . Accessed September 2015.
http://web.mst.edu/~stutts/ME242/LABMANUAL/ME242Text.pdf%20.%20Accessed%20September%202015http://web.mst.edu/~stutts/ME242/LABMANUAL/ME242Text.pdf%20.%20Accessed%20September%202015http://web.mst.edu/~stutts/ME242/LABMANUAL/ME242Text.pdf%20.%20Accessed%20September%202015


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Appendix 1: Sample Calculations

Transmissibility

= =194.11.174= 165.3Half-Power

=2 =194.12 = 137.25Damping Coefficient by Half-Power

= 2 =27.55827.395227.469 = 0.002966Frequency of Plucked Beam

= =0.45170.35595 = 52.19Damping Coefficient by Logarithmic Decrement

= + 4( 1) = .06202

.06202+ 4(5 1) = .002467Modulus of Elasticity

=48() =48 7565 . 1524 52.19.0015711.875104 = 1.72611Required Length of Dynamic Absorber

=3 =31.726118.100310.206727.4694 = 0.036

System Spring Constant

= 4 = 427.4694 .2651 = 7897.1


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Appendix 2: Lab Worksheet

Dynamic Absorber Lab Worksheet - Mechanical Systems Laboratory

Step Variable Description Symbol Value Units

Absorber Masses Mass 1 116.48 g

Mass 2 206.7 g

1 Signal measured at peak amplitude X_Max 194.1

Base amplitude signal V_Base 1.174

Drive Frequency f_max 27.4694 Hz

Maximum Displacement Transmissability T_dmax 165.332

2 Half Power we're looking for 137.249

Half Power frequencies f_1 27.395 Hz

3 f_2 27.558 Hz

4 Zeta estimated by half power Eqn 6.72 Zeta 0.002966938

5a Length of absorber beam L 6 inches

L 152.4 mm

L 0.1524 m

5d Time of first zero crossing t_0_1 0.3559 s

Time of eleventh zero crossing t_0_2 0.4517 s

Time for 5 periods t5 0.0958 s

Time for one period T_d1 0.01916 s

f_d1 52.19206681 Hz

Number of Peaks Measured Over k 5 peaks

First peak amplitude x_1 2.09043

kth peak amplitude x_k 1.96472

Logarithmic decrement over k-1 periods delta_k 0.062020045

Zeta, dimensionless damping parameter Zeta_1 0.002467692

Table 7.1 M_1 0.2651 kg

b 25.07 mm

b 0.02507 m

h 1.571 mm

h 0.001571 m


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rho 7565 kg/m^3

beta_1*L 1.875104

Area moment of inertia about bending axis I 8.10031E-12

5e Young's Modulus E 1.72601E+11

uncertainty in fundamental beam natural frequency

sampling frequency f_s 10000 Hz

number of periods n 5

u_f 0.030182717 Hz

6 Calculated absorber length

with smaller mass L 0.036 m

L 1.428 in

7 with larger mass L 0.030 m

L 1.180 in

8 Base amplitude signal V_Base 0.84

small

M

M1 accelerometer signal peak V_acc_max 1.28

Maximum Displacement Transmissability T_dmax 1.523809524

transmissibility reduction ratio Rt 108.4992547

9 Base amplitude signal V_Base 0.78

Large

M

M1 accelerometer signal peak V_acc_max 0.67

Appendix 3:

Table of Tables

Table 1: Data taken of Single-Degree-of-Freedom System........................................................................... 4

Table 2: Half Power Frequency Location ...................................................................................................... 4

Table 3: Calculated Absorber Lengths .......................................................................................................... 7

Table 4: Error Propagation ............................................................................................................................ 8

Table of Figures

Figure 1: Frequency Response of the Mass M1 to the Driving Force ........................................................... 3

Figure 2: A Response Curve of Greater Accuracy ......................................................................................... 3

Figure 3: Locating the Half-Power Frequencies ............................................................................................ 5

Figure 4: Free Cantilevered Beam Responding to an Impulse Signal ........................................................... 5


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Tuned Mass Damper Vibration Absorber

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