Willem Coudron absorbers for seismic activity Comparison between linear and nonlinear vibration

82 Pages·2016·3.12 MB·English

Comparison between linear and nonlinear vibration
absorbers for seismic activity
Willem Coudron
Supervisor: Prof. dr. ir. Mia Loccufier
Counsellor: Kevin Dekemele
Master's dissertation submitted in order to obtain the academic degree of
Master of Science in Civil Engineering
Department of Electrical Energy, Systems and Automation
Chair: Prof. dr. ir. Jan Melkebeek
Faculty of Engineering and Architecture
Academic year 2015-2016

Comparison between linear and nonlinear vibration
absorbers for seismic activity
Willem Coudron
Supervisor: Prof. dr. ir. Mia Loccufier
Counsellor: Kevin Dekemele
Master's dissertation submitted in order to obtain the academic degree of
Master of Science in Civil Engineering
Department of Electrical Energy, Systems and Automation
Chair: Prof. dr. ir. Jan Melkebeek
Faculty of Engineering and Architecture
Academic year 2015-2016
Permission for use of content

The author gives permission to make this master dissertation available for consultation and
to copy parts of this master dissertation for personal use. In the case of any other use, the
copyright terms have to be respected, in particular with regard to the obligation to state
expressly the source when quoting results from this master dissertation.

June 15, 2016
Acknowledgements

I would like to thank my promotor prof. dr. ir. Mia Loccufier for the assistance during the
academic year and the provision of great research material for this dissertation. I also want to
thank assistants Kevin Dekemele and Bram Vervisch for their tips and help in tackling many
problems on the way.
Abstract

Comparison between linear and nonlinear vibration absorbers
for seismic activity
Willem Coudron

Supervisor: Prof. dr. ir. Mia Loccufier
Counsellor: Kevin Dekemele

Master's dissertation submitted in order to obtain the academic degree of
Master of Science in Civil Engineering

Department of Electrical Energy, Systems and Automation
Chair: Prof. dr. ir. Jan Melkebeek
Faculty of Engineering and Architecture
Academic year 2015-2016

Summary
This dissertation considers the implementation of a linear and nonlinear vibration absorber
on a building model to reduce the mechanical vibrations under dynamic loading. A multi-story
building model was designed in aluminum and placed on a horizontal shaker. This shaker is
able to perform predefined displacement signals to load the building model via base
excitation. Two excitation types are considered: harmonic excitation and impulse excitation.
The harmonic excitation was applied by moving the shaker with a multisine displacement
signal and the impulse excitation was applied by giving the shaker a step displacement. Both
a linear and nonlinear vibration absorber were designed and tuned to these excitations to
reduce the vibrations of the building model. The performance of both absorbers is evaluated
by comparing the response of the modified building model to the response of the original
building model.
Keywords
Multi-story building model, vibration absorber, linear, nonlinear, NES, harmonic excitation,
impulse excitation
Comparison between linear and nonlinear vibration
absorbers for seismic activity
Willem Coudron
Supervisor: prof. dr. ir. Mia Loccufier
Abstract - This dissertation considers the design, which the building model is mounted. The mass of the
implementation and performance assessment of a linear building is concentrated on the floor elements, which will
and nonlinear vibration absorber. A three-story building attract inertial forces under base excitation. The
model was designed and placed on an existing shaker to concerning vibrations are thus horizontal vibrations of
serve as main system for the absorber implementation.
the floor elements. These are reduced due to the action
The shaker was used to load the building model with a
of the absorbers. The performance of both absorbers is
harmonic and impulse excitation. Both absorber types
evaluated by comparing the response of the modified
were tuned to these excitations and installed to reduce the
building to the original one with several criteria. The
vibrations of the building model. The performance of both
absorbers was evaluated by looking at the reduction of interstory drifts and floor accelerations are considered
the interstory drifts and floor accelerations in case of for the harmonic excitation while the 10 % settling time
harmonic excitation, and the reduction of the settling time and maximum floor displacements are checked for the
and maximum displacement in case of impulse excitation. impulse excitation.
Keywords - Multi-story building model, vibration
absorber, linear, nonlinear, NES, harmonic excitation, II. VIBRATION ABSORBER DESIGN
impulse excitation

A. Linear vibration absorber
The linear absorber is designed as a translational
I. INTRODUCTION
spring-mass-damper system that is attached to the
Mechanical vibrations are a common disturbance desired floor element of the building (figure 1). A four
occurring in buildings and civil structures. They are wheel cart is used for the oscillating mass that is
excited by dynamic loads (e.g. earthquakes, wind, etc.) connected to the building with tension springs at both
and range from some minor oscillations to very large sides. There is some unwanted damping due to the
and destructive vibrations depending on the load slippage of the wheels, but the main damping is installed
intensity. Earthquakes in particular are well known for as magnetic damping. The latter is based on the energy
their destructive power. Buildings in seismic zones are dissipation of motional eddy currents [1]. An array of
therefore constructed according to specific design rules magnets is attached to the cart and a hollow aluminum
and often special devices are installed to reduce the tube is installed on the building so that the magnet array
mechanical vibrations. A vibration absorber is a very moves back and forth within the tube. The movement of
effective device to control mechanical vibrations. the magnets cause eddy currents in the tube, which
Both a linear and nonlinear absorber are considered in produce a repulsive force to the magnets proportional to
this dissertation. Their difference is situated in the their velocity. The magnetic damping is thus a viscous
resilient member. The spring of a linear absorber obeys type of damping, which is convenient to model.
Hooke's law and shows a linear relation between the
force in the spring and the displacement of the absorber
mass. The spring of a nonlinear absorber, on the other
c
hand, results in a nonlinear relation between
displacement and force. This leads to the disappearance
of a preferential frequency. Unlike the linear absorber, m
k k
the nonlinear absorber does not oscillate at one natural
frequency, but it is able to oscillate at a range of
frequencies depending on the energy level in the
nonlinear absorber. Another, less convenient,
consequence is the existence of an energy threshold. Figure 1: Schematic image of the linear absorber

Efficient vibration reduction of the nonlinear absorber is
The mass of the absorber is adjusted with additional
only possible when the input energy of the excitation
steel plates and washers on the cart. It can be
exceeds the energy threshold of the absorber.
measured accurately with a scale. The stiffness is
Both absorber types are designed and they are
adjusted with the number of tension springs and the
implemented for harmonic and impulse excitation on a
amount of damping is altered by the number and/or
three-story building model. The excitation is introduced
strength of the magnets. Both the stiffness and damping
at the base via horizontal movements of the shaker on
are measured experimentally by considering the free
vibration of the linear absorber after an initial
W. Coudron. E-mail: [email protected] . displacement. The stiffness is obtained from the damped
natural frequency, which is equal to the inverse time shown in figure 3; the first mode is clearly excited as the
period between two consecutive peaks. The damping three floors of the building move in phase constantly.
factor is determined with the logarithmic decrement
method. B. Impulse excitation
A second loading type is an impulse excitation. This is
B. Nonlinear vibration absorber
also applied at the base by giving the shaker a step
The nonlinear vibration absorber is also designed as a displacement. Different intensities are again considered
translational spring-mass-damper system. The mass is by varying the step displacement (5, 10, 15 or 20 mm).
realized with a slider mounted on a rail which is attached The response of the building to an impulse of 15 mm is
to the concerning floor element (figure 2). The stiffness shown in figure 4; each floor gets a large, initial
is provided with a piano-wire concept [2]. A steel wire is displacement immediately after the impulse, which is
installed in the direction perpendicular to the rail and the followed by a free vibration. This vibration lasts relatively
slider is attached to the center of the wire. The long due to the low damping factors of the aluminum
displacement of the slider deflects the wire transversely building model. The impulse mainly excites the first
and the elastic elongation of the wire results in a mode, but second and third mode vibrations are also
restoring force F according to equation 1. This is the present to a lesser extent during the first oscillations.
nonlinear spring characteristic of a so-called NES
(nonlinear energy sink).

(1)

E, A and L are respectively the young's modulus, the
cross-section and the length of the wire. The nonlinear
spring constant can thus be altered by adjusting the
diameter and length of the steel wire.
Figure 3: Response of the building to the multisine excitation

L

Figure 4: Response of the building to the impulse excitation
Figure 2: Schematic top view of the nonlinear absorber
installed on a floor element of the building
IV. ABSORBER IMPLEMENTATION
The mass and damping are again determined with a
scale and the logarithmic decrement method A. Tuning to the harmonic excitation
respectively. The damping originates from the friction The linear absorber is tuned to the first mode and it is
between the slider and the rail and an additional amount placed on the third floor, which is the antinode, for an
of magnetic damping. The rail friction is assumed to be efficient vibration reduction. The tuning procedure of
viscous by applying sewing machine oil on the rail. An Rana & Soong [3] for harmonic base excitation is
amount of magnetic damping is added to increase the applied. This approach considers the MDOF building
dissipative capacity of the NES, which was found to be model as a SDOF system and applies the formulas of
beneficial for the performance of the nonlinear absorber. Den Hartog by using the modal mass ratio μ of the first
1
The nonlinear spring constant of the wire construction is mode. The formulas to determine the optimal frequency
measured with a static loading test. The results are and damping factor of the absorber are given in
close to the theoretical prediction of equation 1, which equations 2 and 3; f represents the optimal ratio of the
opt
can thus be used for the estimation of knl. absorber's frequency to the first eigenfrequency of the
building, while ζ is the optimal damping factor of the
opt
III. DYNAMIC LOADS absorber.

A. Harmonic excitation
μ (2)
In a first phase, the building is loaded with a harmonic μ

base excitation. This is applied by moving the shaker
with a multisine displacement signal. This multisine is (3)
the sum of ten sines with frequencies ranging from 3 to
5 Hz to excite the first mode (f = 4,2Hz). The amplitude
1
is limited to 1, 2, 3 or 4 mm to consider the absorber Based on the spring choice, the parameters of the
implementation for different load intensities. The absorber were selected to satisfy these equations. The
response of the building to an excitation of 2 mm is response of the modified building with a linear absorber
to the multisine excitation of 2 mm is shown in figure 5. linear absorber is also tuned with the mentioned
Compared to the original case, the floor vibrations are procedure of Rana and Soong; this is an approximate
much more reduced while the linear absorber oscillates approach as this method is optimized for harmonic base
heavily to dissipate the vibration energy. excitation. The response of the modified building with a
linear absorber to an impulse of 15 mm is shown in
figure 7. The decay of the free vibration happens much
faster and the duration of the transient vibrations is
seriously reduced. Just after the impulse the linear
absorber reacts heavily and dissipates a lot of vibration
energy while the first mode vibrations of the building
floors decays rapidly. The second and third mode
vibrations go on relatively undisturbed and are mainly
damped by the building's inherent damping.

Figure 5: Response of the modified building with a linear
absorber to the multisine excitation

The nonlinear absorber is also placed on the third floor
and tuned to the first mode according to a tuning method
optimized for NES in case of harmonic excitation [4].
This method optimizes the nonlinear spring constant
based on the chosen absorber mass and damping, the
main system's properties and the input energy of the
Figure 7: Response of the modified building with a linear
harmonic excitation. The latter is an important parameter
absorber to the impulse excitation
to optimize the absorber with respect to its energy

threshold; the tuning procedure actually comes down to
the optimal placement of the energy threshold so that The nonlinear absorber is tuned to the impulse
the action of the NES is triggered for the concerning excitation with an optimization method designed for NES
excitation. The nonlinear spring constant k should be in impulse load conditions [4]. The nonlinear spring
nl
larger than or equal to the optimal value knl,opt according constant should satisfy equation 5 to obtain an optimal
to equation 4.
placement of the energy threshold.

ζ (4) ζ (5)
Parameters ζ and - depend on the absorber's
The energy of the impulse is taken into account with
damping. Parameter X takes the harmonic excitation
the parameter ; this depends on the start velocities of
frequency into account, which was taken equal to the 00
the building floors imposed by the impulse. The
first eigenfrequency, and depends on the energy
0
response of the modified building with a nonlinear
introduced by the multisine excitation. The optimal
absorber to an impulse of 15 mm is shown in figure 8.
spring constant is thus different for each load intensity.
The first mode vibrations are again heavily reduced
The modified response to a multisine excitation of 2 mm
while the nonlinear absorber dissipates a large amount
is shown in figure 6. The vibrations of the building floors
of energy through some wide oscillations. The vibrations
are again reduced while the nonlinear absorber
of the second and third mode are also left unaltered by
undergoes wide oscillations. The action of the nonlinear
the nonlinear absorber. The frequency-energy
absorber is, however, smaller than the linear absorber.
dependence of the NES is visible in figure 8: the

frequency decreases with decreasing energy in the
NES.

Figure 6: Response of the modified building with a nonlinear
absorber to the multisine excitation
Figure 8: Response of the modified building with a nonlinear
absorber to the impulse excitation
B. Tuning to the impulse excitation
Both the linear and nonlinear absorber are again tuned
to the first mode and placed on the third floor because
the impulse excitation mainly triggers the first mode. The
V. PERFORMANCE Based on these results the nonlinear absorber is also
better suited for the impulse excitation.
A. Performance for harmonic excitation
The performance of both absorbers is quantified VI. CONCLUSION
based on two criteria: the interstory drifts and the floor The response of the modified building with both a
accelerations. The interstory drift is a measure that is linear and nonlinear absorber indicated that the
related to the occurring stresses in the bearing elements nonlinear absorber achieves the best performances for
of the floors (walls, columns) and to the damage in non- both the harmonic and impulse excitation. Moreover, the
structural elements like partition walls or glazing units. nonlinear absorber undergoes smaller oscillations during
Hence it is beneficial to reduce the interstory drifts. The its operation so that this absorber type is preferred when
floor accelerations should be reduced as well to the minimization of the absorber action is an objective.
enhance the comfort and safety of building users. Both The tuning and implementation of the nonlinear
the maximum and root mean square value of these absorber, however, requires more effort than the linear
quantities during the multisine excitation were selected absorber. The nonlinear spring constant needs to be
for comparison. optimized separately for the different intensities of the
First of all, it is noticed that both absorbers work well excitations and the energy threshold was only
under harmonic load conditions. Both the interstory drifts estimated roughly with the mentioned formulas. An
and floor accelerations are reduced with ± 40 to 70 %. experimental finetuning was necessary to approximate
Besides there is a trend visible in function of the load the optimal value of the nonlinear spring constant as
intensity: the relative reduction increases for a larger good as possible.
intensity of the multisine excitation. When comparing
both absorbers, the interstory drift criterion reveals that ACKNOWLEDGEMENTS
the nonlinear absorber performs slightly better while
The author would like to acknowledge the support of
both absorbers perform equally well with respect to the
prof. dr. ir. Mia Loccufier, Kevin Dekemele and Bram
reduction of the floor accelerations. The action of the
Vervisch.
nonlinear absorber is lower than that of its linear
counterpart; this is based on the absorber's
REFERENCES
displacement relative to the building and the
acceleration of the absorber. Hence it can be concluded [1] B. Ebrahimi, B. Khamesee and F. Golnaraghi. A novel eddy
current damper: theory and experiment. Journal of physics D:
that the nonlinear absorber performs better for the
applied physics, 2009.
multisine excitation and is especially suited when the [2] M. McFarland, A. Vakakis, L. Bergman and T. Copeland.
action of the absorber should be limited. Characterization of a strongly nonlinear laboratory benchmark
system. Dynamics of civil structures volume 4, 2010.
[3] R. Rana and T. Soong. Parametric study and simplified design
B. Performance for impulse excitation
of tuned mass dampers. Engineering structures, 1998.
The performance criteria for impulse excitation are the [4] B. Vaurigaud, A. T. Savadkoohi and C.-H. Lamarque. Targeted
energy transfer with parallel nonlinear energy sinks. Part I:
10% settling time and maximum value of the floor
theory and numerical results. Nonlinear Dynamics, 2011.
displacements. The 10% settling time is the time after
which the vibration is smaller than 10% of its initial
value. This quantity is related to the duration of the
transient vibrations after the impulse.
The 10% settling time of each building floor is reduced
heavily with values of ± 80 to 90 % for both absorbers so
that the duration of free vibration after the impulse is
significantly reduced. The nonlinear absorber performs
better than the linear absorber with ± 6 %. The addition
of a second linear absorber, tuned to the second mode,
was also considered to tackle the second mode
vibrations occurring after the impulse. This proved to be
beneficial for the nonlinear absorber but detrimental for
the linear absorber. The 10% settling time is reduced
slightly more when the second mode linear absorber is
added to the configuration with a first mode nonlinear
absorber. In the case of a first mode linear absorber,
however, the addition of a second mode linear absorber
has a negative effect; the operation of the first mode
absorber is affected by the second mode absorber.
The maximum displacement, which occurs just after
the impulse is not altered by the linear absorber. The
nonlinear absorber, on the other hand, is able to reduce
this quantity moderately, especially for the second floor.
The nonlinear absorber reduces the maximum
displacement of the first, second and third floor with ± 10
%, ± 30 % and ± 20 % respectively.