Design and Analysis of a Tuned Liquid Damper as a

Design and Analysis of a Tuned Liquid Damper as a NonlinearVibration Absorber

ME742 - Nonlinear Mechanical Vibrations (Mann)Final ProjectMay 2, 2019

AuthorsDaniel Connolly

Richard HollenbachEduardo Iturbide

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1 Introduction

Vibrations can be seen everywhere and are caused by a variety of sources. A building may experiencevibrations from an earthquake, a personal walking on a bridge may experience vibrations from a passing car,and a tree can experience vibrations from the wind. When these oscillations occur near a natural frequencyof a structure, the resulting large amplitudes may cause deformation or stress, which could ultimately leadto failure. To counter act these large oscillation amplitudes, vibration absorbers are designed and tuned tosolve this issue.

There are two main categories of vibration absorbers: linear and non-linear. In the linear case, the absorberis tuned so that its natural frequency is at the same as the system it is attached. As a result, the absorber takesthe energy from the system and moves a large amplitude, while the structure’s oscillations are reduced. Anexample would be a skyscraper with a pendulum absorber. As for non-linear absorbers, there are a plethoraof types: hardening restoring force, softening spring, bi-stable, friction oscillators, and more. The non-linearities present in the physics of these absorbers allow for different responses to vibrations. This paperwill focus on one type of non-linear absorber: a tuned liquid damper (TLD), and its non-linear absorbingeffects.

2 System Description

There are two parts to the experimental set-up: the single degree of freedom structure and the TLD. First,the structure was designed to fit a range of parameters. The system was to be built upon an excitation systemwith a cross sectional area less than 10 inches by 10 inches and could not exceed 24 inches in height. Itsnatural frequency was to be between 1 and 4 Hz as was to be comprised of four vertical columns attachedto masses on the top and bottom. Figure 1 and 2 show the final design for the structure. It contains holeson the top plate so that the absorber can be attached. The structure was built using steel for the beams andaluminum for the plates and attached to the excitation system.

Next, the tuned liquid damper is designed and built. The design is based upon work by Gurusamy andKumar (2017). The absorber is a tank that will sit on top of the structure and absorb the vibrations and itwill be tuned to the structures natural frequency. The linear natural frequency of the water depends on thelength of the tank in the direction of the displacement as well as the height of the water. The width of thetank does not affect the frequency, so it will be 15 inches wide so to absorb as much energy as possible andkeep the water from splashing out of the tank. The height will be tuned to the correct level. Figure 3 displaysthe final tank design, which will be 3D printed and coated in a vacuum sealed plastic to seal the inside.

Finally, all papers used to solve for the necessary variables and design parameters were collected andbounded together using a staple, such as the one in Figure 4.

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Figure 1: Two Dimensional Model of Single Degree of Freedom Structure

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Figure 2: Three Dimensional Model of Single Degree of Freedom Structure

Figure 3: Model of the Tuned Liquid Damper

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Figure 4: A stapler used to bound necessary papers together

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3 Math Model

To properly tune the absorber, the natural frequency of the single degree of freedom system must first bedetermined. Beam theory is used to determine this frequency, which is given by:

ω =1

2π∗

√4 ∗ E ∗ I ∗BMC + 4ADρ

(1)

where:B =

22.94

L3(2)

C = 7.42 (3)

D = 1.86L (4)

With the parameters from the system described put into the equation, the model predicts the natural fre-quency to be around 3.6 Hz.

The TLD experiences linear sloshing at a frequency depending on the height of the liquid and the length ofthe tank in the direction of the movement. The natural frequency is described as:

ω =

√gπ

L∗ tanh(

hπ

L) (5)

Since the length of the tank was set to two inches, to tune the liquid damper the height of the water ischanged. Thus, to correctly absorb the vibrations, the natural frequencies of the structure and the dampermust be set to be equal. However, the mass term in the Eq. (1) is located in the denominator; thus, when theplastic tank is added upon the top, the mass must be accounted for in the natural frequency. This lowers itto about 2.8 Hz.

Furthermore, as water is added, this increases the mass more. On the other hand, the height term is locatedin the numerator of the TLD frequency, so increasing the height of the water increases the frequency. Thus,

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Figure 5: Natural Frequency of the structure as water height is increased

there is an optimal crossing of the natural frequencies. Figure 5 shows the model of the predicted structurenatural frequency as water is added to the tank. Even with the added weight, however, the Euler Bucklingload is never exceeded.

In addition, the experimental systems and theoretical models do not always agree. For example, the naturalfrequency predicted by the MATLAB model and the experimental value did not agree when first tested. InFigure 6 the theoretical model natural frequency was plotted in red and a corrected natural frequency fromexperimentation was plotted in blue. The natural frequency of the tuned liquid damper is plotted in black.By sweeping over a range of water height values, the design space of the absorber can be determined. As aresult, a height of one inch of water was chosen, because the natural frequency of the TLD should be veryclose to the average between the theoretical model and the experimental model. Moreover, it was decidedto sweep the excitation frequencies between 3 Hz and 4.5 Hz, to properly capture this natural frequencylocation.

The non-linearity of the tuned liquid damper comes into play as the water passes the linear sloshing fre-quency and starts slamming. At this point, the water begins to act like a hardening spring, whereby the free,unsheared motion of the fluid is nonlinearly restricted in its motion at the tank wall. Gurusamy also de-scribes a mass transfer between the linear sloshing mode and the nonlinear slamming mode that introducesan additional nonlinear behavior in the system, and is depenedent upon the normalized amplitude of the tankexcitation. Specifically, the system is empirically found to exhibit nonlinear sloshing behavior at an offsetvalue of β from the linear sloshing frequency, determined from Fig. 8 and Eq. ?? for weak-wave breakingand Eq. ?? for strong-wave breaking. The mechanical equivalent model of this non-linear absorber is givenin Figure 7 below.

β = 1.038 ∗ Λ0.034 (6)

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Figure 6: Theoretical and Experimental Models of the Structure and the Natural Frequency of the TunedLiquid Damper

β = 1.59 ∗ Λ0.125 (7)

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Figure 7: Mechanical Equivalent model of the sloshing-slamming TLD (Gurusamy)

Figure 8: Jump Phenomenon in a Tuned Liquid Damper (Gurusamy)

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4 Analysis and Experimental Comparisons

After design and assembly of the single degree of freedom system, the team was able to confirm the naturalfrequency of the system using a modal hammer test. The following figure shows that the system’s naturalfrequency calculations were relatively close to the range that was desired (1-4 Hz).

Figure 9: Modal Test w/o TLD

Knowing that our TLD design might have had some discrepancies with the single degree of freedom designthat we had assembled, a modal hammer test was also completed with the TLD attached in order to find thenew natural frequency of 0.05 Hz, which can be observed in Figure 10.

Figure 10: Modal Test w/ TLD in Run 1-2 Configuration

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Three total runs were conducted in the experimental phase - two with the long end of the tank oriented par-allel to the direction of oscillation with varying methods of tank sealing, and one with the long end orientedperpendicular to the oscillation direction with significantly more water in the tank in an attempt to exam-ine the nonlinear effects of additional sloshing/slamming mass in the system. The individual accelerometerexcitation responses are shown in Appendix B. A summary of each of the three runs and the propertiesassociated is given in Table 1. The focus of Runs 1 and 2 was to examine the response of the system atthe original first linear natural frequency of the SDOF oscillator (=̃5 Hz) with the TLD attached and trulyobserve the TLD as a vibration absorber, while the focus of the Run 3 was to excite the system at and aroundthe sloshing frequency of the TLD as determined from Gurusamy.

Figure 11: Experimental setup off shaker, in Run 1/2 Configuration

Run No. Tank Length [in.] Water Depth [in.] Calculated Linear Sloshing Frequency [Hz] Λ β

1 15 0.1 0.2 0.078 1.1552 15 0.1 0.2 0.078 1.1553 2.25 1 3.9 0.525 1.470

Table 1: A summary of the experiments conducted and the vibration properties of each.

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Figure 12 shows an example of an experimental response where it is very clear that the difference in ampli-tude from the bottom of the single degree of freedom system to the top is substantial.

Figure 12: Run 1 Vibration Response to 5.0 Hertz Excitation

Figure 13-15 show FFT’s of each of the different runs that we executed with the TLD attached to the system.One can see the difference in amplitude at all of the desired frequencies for each run.

Figure 13: Amplitude of Responses for Run 1 Configuration (Dashed lines represent the response amplitudeof the top of the structure)

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5 Summary and Conclusions

5.1 Discussion of Results

The primary result of these experiments, and the plots given in Appendix B, is that a TLD has significantcapability in eliminating vibratory amplitude response to single degree of freedom base excitation of asimple structure. From Figs. 13 and 14, it is apparent that despite seemingly large mistuning of the TLD inthese cases to the ”un-absorbed” structure, the response of the top of the structure was reduced nearly to zero.Visual observations of the TLD during these experiments also corresponded to non-uniform wave breakingand mixing of the fluid, specifically around areas of the tank area that showed wrinkles and irregularitiesin and on the surface of the tank walls. While the data shown here does not necessarily indicate the exactsloshing/slamming fluid mechanism of this absorbing effect, it clearly indicates a nonlinear absorbing effectat the first natural frequency of the structure (response peaks are notably more diminished on the lowerfrequency side of linear resonance than the higher side).

Figure 15 represents a further interesting case that again appears to represent a nonlinear absorbing effectacross a spectrum of frequencies around the linear sloshing frequency of the reconfigured TLD. Assumingthat each peak is sharp, the relative decrease in response amplitude is not as strong as that around thestructural resonant frequency, but still appears to remove a comparatively large amount (about 40-60%) ofthe base excitation amplitude. Further, the amount of absorption looks to continue to grow as the excitationfrequency approaches that of the linear structure. As discussed below, future experiments would capturethis behavior in full. Visually, this effect was again apparent, as wave sizes continued to grow and break asfrequency approached both the TLD linear frequency and the the structural first linear frequency. The freesurface of the liquid in fact struck the top covering of the tank around the TLD sloshing frequency and wasgrowing towards siilar behavior as the frequency again approached stuctural resonance.

5.2 Experimental Limitations

Under the time and project constraints, the full implications and possible benefits and behaviors of a non-linear TLD were simply unable to be explored. While this lacking is certainly present in many design andoptimization problems, a couple here that affect the results are important to note. First, from a practicalstandpoint, TLD systems are well-suited to extremely low-frequency applications. While the approximate5 Hz natural frequency of the original project system is certainly not entirely unrealistic for a building orfloor model, it is definitely on the upper end of the spectrum for a) typical frequencies of interest to such astructure, and b) the feasibility of implementing a TLD. From Equation 1, and Figure 3 of Gurusamy, it isclear that h

l dependence is strong, and that large first natural frequencies of the absorber are only availablethrough very longitudinally short, shallow tanks. However, for lower frequencies of interest, the dependenceis less strong, and the design space opens up considerably, allowing for shallower depths and wider basesthat are more practical to construction and physical implementation in a residential or commercial structure.

Second, the quality of the data collection and tank construction in this experiment was not necessarily ofthe best quality to compare to theoretical analyses. Namely, the sealing process that was required to holdwater in the additively-manufactured tank used here (various approaches to impermeable plastic wrappingthe interior of the tank) may have introduced additional nonlinearities that were not accounted for in the

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analytical model. These nonlinearities would have come primarily in the form of wrinkles in the sealingplastic and the non-square edges of the tank. Such features could easily have acted as baffles and flowobstructions that would have precluded the application of a linear model in the ”non-sloshing” regime -i.e., these obstructions could easily have moved the ”non-linear” sloshing frequency that accounts for theabsorbing behavior observed.

5.3 Future Work and Prospects

To extend the TLD work done here, a natural progression would be the exploration of additional tank de-signs and improved tuning to a more closely matched natural frequency of the base structure. Differinggeometries, specifically variation of the h

l parameter, as well as exploring the specific influence of bafflestructures within the liquid would extend the scope and usefulness of this study. These studies would alsobe accompanied by larger frequency sweeps of each design and configuration that would capture all thelinear frequency of the structure, the linear sloshing frequency of the tank, the nonlinear sloshing frequencyof the tank for the given excitation amplitude, and the linear response frequency of the sum system.

Another parameter that could be explored would be the excitation amplitude. As discussed in in Section3, Gurusamy describes a jump phenomenon for a Tuned Liquid Damper, which occurs at specific valuesof non-dimensional amplitude. A sweep over excitation amplitudes would provide additional on this jumpphenomenon (described empirically in Figure 8), where the waves transition from weak wave breaking tostrong wave breaking. This study will describe more about the shallow water sloshing and give more insightinto the nonlinearities of the absorber.

Additionally, specific measurement of the free surface of the liquid would be useful data to correlate withthe theoretical analysis. With the use of optical and digital video measurement equipment, the exact motionsof the liquid could be captured, as well as the onset of nonlinear behavior in the system. Where the analysishere assumes that nonlinear ”sloshing” behavior is restricted to the regime surrounding the single ”sloshing”frequency, that is, at least intuitively, a rather dubious assumption. Dependence of the nonlinear absorbingeffect on the amplitude of excitation as well as the size of the frequency band over which the liquid responseactually presents a nonlinear benefit would be best collected with this type of supplementary measurement.

5.4 Conclusion

In summary, the experiments performed here indicate the strong, visibly apparent, and complex vibrationabsorbing ability of a Tuned Liquid Damper. While a much more rigorous analysis would reveal the exactnature of the nonlinear nature of this response behavior, the data shown in this study points towards highpotential for structural applications for this type of absorber. Even with mistuned tanks, the relative responseof the single degree-of-freedom is reduced nearly to zero at the linear natural frequency of the structure, andsignificant benefit is also observed in a rather large spectrum around this frequency of excitation. Furtherstudy to determine the full breadth of this potential as well as models of the nonlinearity in a TLD as theyapply to this case would lead naturally to the implementation of this system, which is notably simpler todesign and construct than vibration absorbers with similar effects.

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6 Acknowledgements

This work could not be completed without guide and direction led by Dr. Brian Mann. Extra acknowledg-ments for their assistance goes to Dr. Dane Sequeira and Andrew Hutchins. Finally, for her help in buildingthe tank, Bethany Davidson.

7 References

Gurusamy, Saravanan and Kumar, Deepak. Numerical Modeling of Nonlinear Sloshing in Tuned LiquidDamper (TLD). International Ocean and Polar Engineering Conference. June 25-30, 2017.

Mann, Brian. Non-Linear Mechanical Vibrations Lecture Notes. 2019.

Meirovitch, Leonard. Fundamentals of Vibrations. 2001

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8 Appendix A: Code

The following scripts in MATLAB were used to design the system and process the data.

clc;

clear;

close all;

n = 10;

a = 0.5;

b = 0.125;

L = 15.0;

h = linspace(0,3.0,n);

M = 0.3407 + (15 * 2 * h) * (2.542) ∗ (1.225e− 3);

c = 8;

d = 8;

t = 1.00;

freq = zeros(1,n);

freqsimple = zeros(1, n);

P = zeros(1,n);

W = zeros(1,n);

for i=1:n

[freq(1,i),P(1,i),W(1,i)] = freqEval(a,b,c,d,t,L,M(1,i),1);

[freqsimple(1, i), P (1, i),W (1, i)] = freqEval(a, b, c, d, t, L,M(1, i), 2);

end

time = linspace(0,10,10);

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figure(1);

subplot(1,2,1);

plot(h,freq,’k’);

title(’Design Recommendation’);

xlabel(’height of water h (in)’);

ylabel(’frequency (Hz)’);

legend(’Full Mode Shape’);

subplot(1,2,2);

plot(h,P,’k’,h,W,’kx’);

title(’Euler Buckling’);


ylabel(’Force (N)’);

legend(’Critical Load’,’Weight of the Top Plate’);

L = 2.0 * 0.0254;

H = h * 0.0254;

g = 9.81;

w = sqrt(pi * g * (1/L) * tanh(pi * H * (1/L)));

f = w / (2 * pi);

figure(2);

newfreq = freq + 2.2470;

plot(h,freq,’r’,h,newfreq,’b’,h,f,’k’);


ylabel(’natural frequency f (Hz)’);

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title(’Design Space of a Tuned Liquid Damper’);

legend(’Original 1 DOF Model Natural Freuquency’,’Experimental 1 DOF Natural Frequency’,’Tuned Liq-uid Absorber’);

figure (3);

plot(h,f,’k’);

title(’Natural Frequency of the Tuned Liquid Damper’);

xlabel(’Height of Water h (in)’);

ylabel(’Natural Frequency f (Hz)’);

function [f,P,W] = freqEval(a,b,c,d,t,L,M,flag)

a = a * 0.0254;

b = b * 0.0254;

c = c * 0.0254;

d = d * 0.0254;

t = t * 0.0254;

L = L * 0.0254;

psi2 = 7.422445231;

ddpsi = 22.93633631/(L3);

psi2I = 1.855335876*L;

psisimple = 1.0;

ddpsisimple = 2/L;

psi2Isimple = L/5;

E = 200.0 * 10 9;

rhos = 8050.0;

rhoa = 2700.0;

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if flag == 1

f = (1/(2*pi))*sqrt(((4*E*(1/12))*a*(b3)∗ddpsi)/((1.5∗rhoa∗c∗d∗t+M)∗psi2+4∗rhos∗a∗b∗psi2I));

P = (E*(1/12)*a*b3)/(4 ∗ L ∗ L);

W = 1.5 * rhoa ∗ c ∗ d ∗ t ∗ 9.81/4;

end

if flag == 2

f = (1/(2*pi))*sqrt(((4*E*(1/12))*a*(b3) ∗ ddpsisimple)/(rhoa ∗ c ∗ d ∗ t ∗ psisimple+ 4 ∗ rhos ∗ a ∗ b ∗psi2Isimple));

P = (E*(1/12)*a*b3)/(4 ∗ L ∗ L);

W = rhoa ∗ c ∗ d ∗ t ∗ 9.81/4;

end

end

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9 Appendix B: Accelerometer Data

9.1 Run 1 Data

Figure 16: Vibration Response 4.0 Hertz Figure 17: Vibration Response 4.5 Hertz



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Figure 22: Vibration Response 7.0 Hertz

9.2 Run 2 Data



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9.3 Run 3 Data


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Design and Analysis of a Tuned Liquid Damper as a