New Tuned Viscoelastic Damper Design Configuration Using Elastomer Essay

CHAPTER I

New Tuned Viscoelastic Damper Design Configuration Using Elastomer

I. Abstract

This proposal will analyze the behaviors of structures in the presence vibration caused by human activities and tackle on the theoretical and practical concepts on the installation process of viscoelastic damper using elastomer.

This will also study the behavior of the structural damper. This will focused on the conditions where maximum damping is attainable and will analyze the corresponding result.

In addition, the materials that will be used will be examine different factors, including stiffness, the effects of temperature, elasticity and will analyze its effects. This will also lay down possible recommendations to overcome the effects the different factors mentioned.

In addition, it will also tackle the infrastructural factors such as floor mass, foundations’ size, and its effects and will provide possible actions on how to minimize or completely diminished such effects, if possible. The study will also discuss the different theories about damping system and elastomers and will provide practical applications on how to apply it. For its applications, it will provide the process of installation of the elastomer on structures and building as whole.

It will demonstrate and describe the actual happenings of floor vibration. It will also discuss the nature of floor vibration and will offer means on how to avoid it, or in case of existing buildings, reducing it trough the adaptation of damping systems.

This will try to explain the quality of floor vibration while giving alternatives as to how to avoid it. For new ones, there is a specific design that would cater to this. However, when the building is already exiting, floor vibration could be reduce or eliminate through alterations.

II. Introduction

A.        Description of the Study

Nowadays, infrastructural buildings are now in great population density. Human activities become the source of floor vibration. As the building go taller and bigger, the number of population in such also balloons. Hence, floor vibration is now a concerned and become a grater problem. Human activities such as walking, aerobics, and long span floor structures are now very common which is done regularly in buildings. The buildings are more preferred by most of the people as a venue to these activities. They feel comfortable while doing their job inside of it as compared to an open space. This research will provide measurements to reduce the response of structure and probability of pounding due to human activities. However, the installation of these devices is very critical and requires a profound research and well investigation.

B. Background

According to Allen and Pernica ( 1998 ), floor vibration is an up and down movement of floor leading to a shaking of an infrastructure cause by forces such as human and human activities, working machinery and by vibration transmitted from other structures, from other floors or even caused from the ground. Since the vibration has it energy capable to diffuse and travel, any motion happening to other part of a building or structure will be transmitted to the parts. However, magnitude of it is not as greater as the magnitude being experienced by the structure which actually under the influence of motion (as cited in http://irc.nrc-cnrc.gc.ca/pubs/ctus/22_e.html ).

Every building is actually experiencing floor vibration. During the past years, problems involving this are very hard to counteract or to lessen. However, today, it’s not already a new concept and a problem. Tredgold once wrote in 1828, that girders are supposed to be made as deep as it can go so nuisance of shaking everything as someone moves into the room.

(as cited in as cited in http://irc.nrc-cnrc.gc.ca/pubs/ctus/22_e.html ).

Any vibration motion in a building makes people uncomfortable and sometimes create nervous and fear of a possible collapse of a structure which may result to further damage of other structures. The accumulation of little vibration could result into them lessening of stability of a structure because of its magnitude and longer range. The fear made by a low frequency vibration is more often unwarranted because the produce displacements and stresses are just small. However, perceptible vibration is undesirable to some people who are undertaking their works or jobs and losses their concentration. There is a mechanism called the simple floor criterion which is being use to control the excessive motion of one building. This simple floor standard has a deflection of not more than a span or 360 under the distributive live weight. This machine is applicable only to those years since the buildings’ construction are not as the same as today. The buildings today are more complicated in its structure and systems. Thus, a vibration controller is more needed today to assure the peoples’ safety. In addition, today’ building are more in great height and very dangerous when a great vibration occurs. Moreover, the approach mentioned before is only applicable when a building is not a place for human activities and other working machines that contribute to higher vibration effects. But today, buildings are now a placed for everything, such as aerobics and where vibrating machines are installed. In addition, when a building is now composed of longer spans, thinner floor decks, and with less structural damping, the approach is not practical anymore. However, doe to the continuous research and aspiration of many people to have this problem under control and new guidelines that address and deal with these have recently been introduced.

C. Scope of the Study

This study will work on the viscoelastic dampers using elastomers. Viscoelastic dampers is will be used for implementation of damping system or concrete buildings.

The review will reveals that both elastomeric bearings and sliding bearings have been considered for implementation within base isolation systems of wood-framed buildings. In addition, friction dampers, viscoelastic dampers, hysteretic dampers, and fluid viscous dampers for implementation within the framing of wood buildings have been considered. It demonstrates that advanced seismic protection systems offer promise for enabling concrete structures to resist major earthquakes with minimal damage.

It will also present mechanisms in the installation of elastomers in the foundations of structures. This paper will provide a broad tutorial on analytical issues and material behavior.

D. Significance of the Problem

This research results will be used practically in building infrastructure. It could be used in many practical means especially in engineering. This study will provide the theoretical and practical means of proper installation of viscous damping systems and how to eliminate or lessen the internal motion of building due to human activities.

It will provide an application of damping system for seismic protection of concrete structures. The reader will be informed about the importance of using a viscoelastic damper specifically elastomers. Nowadays, destructive phenomenal forces, such as earthquake, wind loading etc. devastate infrastructural buildings including residential and commercial buildings and even lives. These forces cannot be expected to happen. It may happen without a signal and could wreck everything and everyone. This study will provide an idea for its application about how to prevent damages or minimize impairment once these destructive forces occur at a sudden.

The study will also inform about the dynamics of the structure and will identify the key elements of the structure that require stiffness, damping, or mass distribution improvements.

E. Objectives

The general objectives of the study are:

(1)               To provide measures on how to control or prevent the damages brought about by floor vibrations caused by human activities

(2)               To present way on how to install damping devices in foundations of building.

The specific objectives are:

(1)        To inspect the possible drawbacks of heavy isolation damping in structures subjected to motion

(2)        To identify appropriate combinations of isolation stiffness and damping, which may help to decrease base displacements

(3)        To probe the possible drawbacks of heavy isolation damping

(4)        To design base isolation system with high levels of damping, and

(5)        To know structural behavior under vibrational energy.

II. Review of Related Studies

History of Damping System

During the1990s when the importance of damping system was recognized. Kimbal and Lovell (1927) were the first to present mechanical oscillators. In addition, he according also to him, “for the energy to dissipate, material’s module must be put into consideration to have imaginary components ” ( as cited in http://arxiv.org/html/physics/ 0307016/#01).

When the imaginary part is zero, then the system follows the Hooke’s law (F = ku, where k is the spring’s stiffness and u is the spring stretch). When this happen, there’s no damping occurs since the spring does not experience vibrations or motion. In order to experience the “ frequency dependence ”, it is necessary to use materials with uniform stiffness (Peters, 2003).

Kerwin’s Model

Another person who studied this field was Plass (1957). He did an experiment about putting in a structure circular bars (http://scholar.lib.vt.edu/theses/available/etd-101898-162311/unrestricted/ETD2.PDF). In addition, he assumed the “facesheets” to be very thin.

In his study, it says stiffness only in an equivalent moment of inertia of the sandwich structure rather than modeling them. According to Kerwin (1959), he presented a “general breakdown of viscoelastic material (VEM) constrained by another metal layer, where now the loss mechanism is primarily shear in the VEM ” and his model shows the “condition about the traveling of the wave in a beam but assigning the beam a complex composite stiffness ” (as cited in http://scholar.lib.vt.edu/theses/available/etd-101898

162311/unrestricted/ETD2.PDF#search=%22Influences%20of%20Higher%20Order%20Modeling%20Techniques%20on%20the%20%22 ).

Another treatment was applied but by this time the treatment was consisted of a 10-mil-thick (0.010”) VEM and maximum constraining-layer thickness of 20 mils. Kerwin’s analysis was used by Ross (1959). Ungar (1959 ). The damping tapes used was stiff in shear but weak in extension substrate by a lightweight layer. It was not a single damping state but was a multiple damping tapes and spaced treatments.

This study resulted to a significant number of design charts and from these they concluded that constrained-layer treatments can handle structures and more efficient in most situations.

As mentioned earlier, Kerwin didn’t develop a mathematical model about this study.

The mathematical model was developed by DiTaranto (1965). He called this development the Axial displacements in sixth-order equations of motion. Following DiTaranato’s work was Mead and Markus. They presented the transverse displacements’ sixth order equations of motion in 1965.

However, their equations were based on the works and analysis of Kerwin. They also expanded the idea about it by allowing for general boundary conditions. Due to this significant study, many papers were published since the 1950’s.

Mead and Markus used the knowledge of Kerwin’s assumptions and still further work on the Kerwin’s work by considering “boundary conditions.” By the time of 1950’s there were many published papers about this.

Independent Transverse Displacement Facesheets

Considering all the papers published about the constrained layer damping, there are relatively few who studied another side of Kerwin’s work nevertheless, it is quite related.

Based on the study conducted by Ojalvo (1977), “ Euler beams for the facesheets was used in scrutinizing soft-core sandwiches and h e supposed linear differences in axial and transverse displacements within the VEM core and be able to obtain a closed-form solution for pinned-pinned beam ” (as cited in http://scholar.lib.vt.edu/theses/available/etd-101898-162311/unrestricted/ETD2.PDF#search=%22Influences%20of%20Higher%20Order%20Modeling%20Techniques%20on%20the%20%22 ). However, he didn’t deal with the damping problems.

Poisson’s ratio of the core was formulated by Douglas and Young. This formulation has affected the compressional frequency of a three layer damped beam significantly. This also has a thin relatively incompressible viscoelastic layer and low thickness-to-width ratio. This was stipulation rises from the effects of a nearly incompressible viscoelastic damping core with rotational inertia and shear de formation in the elastic layers.

According to Miles (1984), who introduced the idea of a “beam damper, beam damper utilizes not just shear but also thickness deformation in the VEM” and t he “recognition of the thickness deformation of a soft adhesive can be significant if at least one of the tall layers is stiff in bending” ( http://scholar.lib.vt.edu/theses/available/etd-101898-

162311/unrestricted/ETD2.PDF#search=%22Influences%20of%20Higher%20Order%20Modeling%20Techniques%20on%20the%20%22). He didn’t use mathematical model in presenting this concept. However, mathematics was applied in the work of Miles and Reinhall (1986). This research was quiet different from other’s work. They let their constraining layer float on the VEM core, i.e., it is treated as a free-free beam” (Miles and Reinhall, 1986).

Compressional damping in the VEM was also studied (Sylwan, 1987). He used a 1-kHz mode of a beam and he knew that Poisson’s ratio has a large effect on compressional damping.

Bai and Sun (1995) concentrated on “slip at interfacial bonds between layers of sandwich.” Based on their conducted test, they recognized facesheets must deform independently and that higher order modeling of the core is needed. However, Mead and Markus have a different result for beams with low aspect ratios. They assume the core to be incompressible (Poisson’s ratio = 0.5) and pin all layers at the edges (B ai and Sun, 1995).

Taylor and Nayfeh (1997) developed a “3-D elasticity solution for damping of sandwich beams.” However, their study was restricted by pinning all layers at the boundaries only.

According to Kimball and Lovell (1927), like Taylor and Nayfeh the study, it was also “restricted to the boundaries only to pin but they posed important concepts like Poisson’s ratio and the stiffness of the constraining layer relative to the base layer, but it was very limit ed to the engineering purposes” (as cited in http://scholar.lib.vt.edu/theses/available/etd-101898-162311/unrestricted/ETD2.PDF#search=%22Influences%20of%20Higher%

20Order%20Modeling%20Techniques%20on%20the%20%22).

Austin and Inman (1998) expand the wok of Lee et. al. This was about optimization of the span to control viscoelastic tier damping. They presented the use of cylindrical twisting in which MM assumptions done inadequately on a full coverage, tiny aspect ratio beam where the restricting layer is permitted to float.

(ETD2.PDF#search=%22Influences%20of%20Higher%20Order%20Modeling%20Techniques%20on%20the%20%22).

Damping Treatments

According to Nokes and Nelson (1968), t he “ cover of the central portion of beams used the bas ic Kerwin and Ungar (MM) model and t he “ analysis structure is a steel, free-free beam 20” A 2” A 1 8 ” with a 0.050-inch-thick layer of VEM and a1 32-inch-thick steel constraining layer. For various values of VEM shear modulus they plot loss versus percent cove rage where the results “ show optimum coverage less than 100% for some combinations of VEM shear stiffness divided by constraining layer membrane stiffness ” (as cited in http://scholar.lib.vt.edu/theses/available/etd-101898-162311/unrestricted/ETD2.PDF #search=%22Influences%20of%20Higher%20Order%20Modeling%20Techniques%20on%20the%20%22).

The concept for partial coverage was quite different in the study conducted by Sylwan ( 1978). Sylwan studied “full-coverage constraining layer attached with discrete patches of VEM in the context of civil s tructures, eg, stairs, floors ” and also Lal l, Asnani, and Nakra (1988) proposed another solution for a model using the standard MM assumptions (as cited in http://scholar.lib.vt.edu/theses/available/etd-101898-162311/unrestricted/ETD2.PDF#search

=%22Influences%20of%20Higher%20Order%20Modeling%20Techniques%20on%20the%20%22).

According to Marcelin et. al. (1992), the optimization techniques modify the location and length of partial coverage damping patches and the analyses account on “standard MM assumptions on a 3-mm-thick cantilevered beam with a 1.5-mm-thick, stiff (125 MPa = 18,415 psi) VEM and a 0 .02-mm-thick constraining layer ”) (as cited in http://scholar.lib.vt.edu/theses/available/etd-101898-162311/unrestricted/ETD2.PDF

#search=%22Influences%20of%20Higher%20Order%20Modeling%20Techniques%20on%20the%20%22 ).

Damping’s High Order Modeling

The displacement fields of order is greater than the linear variations of Euler-Bernoulli beam and and Kirchhoff plate theories.

Composite materials were the main subject of most of the papers published. Many authors explicitly presented the advantages of higher order modeling over the others. “When a detailed discussion was done on higher order modeling for the purpose of damping analysis, there are relatively few papers published and it will be discussed (http://scholar.lib.vt.edu/theses/available/etd-101898-162311/unrestricted/ETD2.PDF).

Imaino and Harrison (1991) developed “ higher order analysis via p-version finite elements” and they imparted some “ good feature a bout whether or not the constraining layer follows the base layer for various stiffness of VEM core” (as cited in http://scholar.lib.vt. edu/

theses/available/etd-101898-162311/unrestricted/ETD2.PDF#search=%22Influences%20of

%20Higher%20Order%20Modeling%20Techniques%20on%20the%20%22). They worked and successfully reproduced an example done by Lall and others.

Damping and Damping System

“Damping” refers to the “dissipation of vibration energy” (http://www.csaengineering

.com/vibdamp/vibdamp.shtml). “All physical systems have some inherent damping, but the level of damping can be augmented to increase energy dissipation in particular vibration modes where the response of a structure driven at a resonant frequency can be greatly decreased where, in turn, can significantly reduce overall motion or acceleration of a system” (http://www.csaengineering.com/vibdamp/vibdamp.shtml).

The ways vibration and noise can be minimized in a dynamic system could be categorized into active, passive, and semi-active methods ( Shen and Soong, 1995). Among the three ways, the simplest to implement is passive damping. It is also cost-effective than active and semi-active systems because these two requires on-line control, which the other one doesn’t need to.

Active dampers are force generators that actively push on the structure to counteract a disturbance. They are fully controllable and require a great deal of power. Semi-Active dampers combine features of passive and active damping (Bonsor, 2006).

“ Damping system refers to the collection of damping devices, all structural elements or bracing required to transfer forces from damping devices to the base of the structure, and all structural elements required to transfer forces from damping devices to the seismic-force-resisting system and unlike seismic-force-resisting system which refers to the collection of lateral-force-resisting elements of the structure, the damping system may be external or internal to the structure and may have no shared elements, some shared elements, or all elements in common with the seismic-force-resisting system ”

(http://www.bssconline.org/NEHRP2003/comments/C15.pdf#search=%22%20history%20damping%20system%22).

Damping Devices

“Damping devices are device that dispel energy due to the relative movement of each end of the device and may include all pins, bolts, gusset plates, brace extensions, and other components required to connect damping devices to other elements of the structure and be categorized as either displacement-dependent or velocity-dependent, or a combination thereof, and may be configured to act in either a linear or nonlinear manner” (http://www.bssconline.org/NEHRP2003/comments/C15.pdf).

In making damping devices, it must be considered all possible details on how the device might be lost its strength. The design must be appropriate in case of earthquake displacements, velocities, and other phenomenal forces. For assurance of the device, the device may be tested to demonstrate adequacy and proper functioning of the device.

The figure below shows the design and arrangement of both damping system and seismic-force-resisting system (See Figure 1 ).

Figure 1. Damping System Configurations.

Effective Damping

In “effective damping” , it considers the “ response of structure with a damping system by the damping coefficient, B, based on the effective damping , I , of the mode of interest” and the “ effective damping of the fundamental-mode of a damped structure rooted from the nonlinear force-deflect ion properties of the structure” and for “ nonlinear analysis methods, properties of the structure would be linked on explicit modeling of the post-yield behavior of elements ” (http://www.bssconline.org/NEHRP2000/comments/C13.pdf). Nonlinear properties of the structure are deduced from over strength, I(c) 0.

Figure 2 shows the relationship of the spectral acceleration versus the spectral displacement. It exemplifies the “effect of effective damping coefficient in design earthquake response of the fundamental mode” (http://www.bssconline.org/NEHRP2003 /comments

/C15.pdf ). In general, effective damping is a combination of three components, namely: (1) Inherent Damping I I. (2) Hysteretic Damping I H, and. (3) Added Viscous Damping I V ( http://www.bssconline.org/NEHRP2003/comments/C15.pdf ).

Figure 2. Effective Damping Reduction.

The first component is an inherent damping of structure at or just below yield, excluding added viscous damping (http://www.bssconline.org/NEHRP2003/comments/

C15.pdf#search=%22%20history%20damping%20system%22). However, the second component refers to the “post-yield hysteretic damping of the seismic-force-resisting system at the amplitude of interest (taken as 0 percent of critical at or below yield) and the last is viscous component of the damping system (taken as 0 percent for hysteretic or friction-based damping systems and both hysteretic damping and the effects of added viscous damping are being characterized as are amplitude-dependent and the relative contributions to total effective damping is dependent with the amount of post-yield response of the structure (http://www.bssconline.org/NEHRP200/comments/C15.pdf)).

Elastomer

Elastomers are polymers capable of recovering substantially in size and shape after removal of a load. Rubbers, both natural and synthetic, are capable of recovering fast from large deformations. A natural rubber is a hydrocarbon polymer of isoprene units. Synthetic rubbers are generally hydrocarbon polymers with sulphur or other additives. For example, fillers such as carbon black or zinc oxide improve the abrasion resistance and the shear strength.

The use of elastomers and rubbers in engineering practice has increased in recent years. A wide group of products has these materials as a main/important component. Rubber and rubber-like materials are frequently used, for example, in the automotive industry, mechanical, civil, electrical and electronics engineering. The finite-element analysis for elastomers/rubbers is not easy. At high deformations the stress-strain relationship for these materials is nonlinear and is affected by dynamic and thermal effects. Rubber and rubberlike materials are usually modeled as incompressible hyperelastic materials, as well as elastoplastic materials under plastic dominant deformations and the assumption of isochoric plastic flow.

Viscoelasticity

Rubber show signs of a rate-dependent behavior and can be considered as a viscoelastic material, with its properties depending on both temperature and time. Forces are removed from it, it eventually returns to the original, undeformed state. When subjected to a constant stress, it creeps. When given a prescribed strain, the stress decreases with time. This phenomenon is called “stress relaxation” (http://www.axelproducts.com/downloads/MARC_

FEA_ELASTOMERS_2000.pdf).

Pushing on a piece of putty will make it into a new shape and after removing the hand tat pushes it, the putty may not return to its original shape. This is called Hysteresis, the physical property of a system that is characterize by not instantly follow the forces applied to them but with a reaction that is slow and may not return wholly to its original state.

It refers to the different stress-strain relationship during in such materials when the material is subjected to cyclic loading. Collectively, these features of hysteresis, creep, and relaxation – all dependent upon temperature – are often called features of “viscoelasticity” (Christensen, 1982).

Linear Viscoelasticity

“Linear viscoelasticity” refers to a theory which abides the linear superposition principle, where the relaxation rate is proportional to the instantaneous stress, meaning as the relaxation rate increases the instantaneous stress also increases and vice-versa. Experiments point up that “classical” linear viscoelasticity (applicable to a few percent strain) represents the behavior of many materials at small strains (http://www.axelproducts.com/d ownloads/

MARC_FEA_ELASTOMERS_2000.pdf). In this case, the instantaneous stress is also proportional to the strain. Mechanical models are often utilized to talk about the viscoelastic behavior of materials. The first is the Maxwell model, which composed of a spring and a viscous dashpot (damper) in series (See Figure 3). The sudden application of a load produces an immediate deflection of the elastic spring, which is followed. Figure 3. Maxwell and Kelvin Model. by “creep” of the dashpot. On the other hand, a sudden deformation produces an immediate reaction by the spring, which is followed by stress relaxation according to an exponential law. The second is the Kelvin (also called Voigt or Kelvin-Voigt) model, which consists of a spring and dashpot in parallel. A sudden application of force produces no immediate deflection, because the dashpot (arranged in parallel with the spring) will not move instantaneously. Instead, a deformation builds up gradually, while the spring assumes an increasing share of the load. The dashpot displacement relaxes exponentially. A third model is the standard linear solid, which is a combination of two springs and a dashpot as shown. Its behavior is a combination of the Maxwell and Kelvin models. Creep functions and relaxation functions for these three models are also shown in Figure 4.

The Marc program features a more comprehensive mechanical model called the Generalized Maxwell model, which is an exponential or Prony series representation of the

Figure 4. Creep and Relaxation Functions.

stress relaxation function. This model contains, as special cases, the Maxwell, Kelvin, and standard linear solid models (Fung, 1981).

Nonlinear Viscoelasticity

“Nonlinear viscoelastic” behavior is caused when strain is large. A finite strain viscoelastic model may be obtained by generalizing linear viscoelasticity in the sense that the 2nd Piola-Kirchhoff stress is substituted for engineering stress, and Green-Lagrange strain is used instead of engineering strain. The viscoelasticity can be “isotropic or anisotropic. In practice, modified forms of the Mooney-Rivlin, Ogden, and other polynomial strain energy functions are implemented in nonlinear Finite Element Analysis (FEA) codes. The finite strain viscoelastic model with damage has been implemented in Marc (Simo, 1987).

Temperature Effects

Effects of the temperature are very significant in the analysis of elastomers. Temperature can cause (1) thermal strains. (2) material moduli, and. (3) can cause heat flow.

(http://www.axelproducts.com/downloads/MARC_FEA_ELASTOMERS_2000.pdf).

Modern nonlinear FEA accounts for heat flow and can offer opportunity to conduct coupled thermo-mechanical analysis. Material constants associated with the strain rate independent mechanical response, such as Mooney-Rivlin, Ogden, and rubber foam constants, dependent with temperature, as do the coefficient of thermal expansion, Poisson’s ratio, thermal conductivity, etc. The time-dependent phenomena of creep and relaxation also depend on temperature. The viscoelastic analysis is thus temperature-dependent. In contact problems, friction produces heat, which would be included in the analysis. Another important consideration is the heat generation of rubber components in dynamic applications, since

after each deformation cycle some fraction of the elastic energy is dissipated as heat due to viscoelasticity (http://www.axelproducts.com/downloads/MARC_FEA_ELASTOMERS

_2000.pdf).

Large classes of materials exhibit a particular type of viscoelastic behavior which is classified as thermo-rheologically simple (TRS). TRS materials are plastics or glass which exhibit a logarithmic translational property change with a shift in temperature (as shown

in Figure 5). This shift in time t as a function of temperature T is described by the so-called “shift iX?function.” An example of such a shift function is the Williams-Landel-Ferry shift. The WLF-shift function depends on the glass transition temperature of the polymer (Williams et. al., 1955). Another well-known shift function is the BKZ-shift (Bernstein, Kearsley, and Zapa, 1963). Note that with TRS materials, the relaxation function is independent of the temperature at very small times–which implies that the instantaneous properties are not temperature dependent.

iX? Figure 5. Thermo-Rheologically Simple Behavior in Polymers

Material Behaviour

iX? In linear FEA, a simple linear relation exists between force and deflection (Hooke’s Law). For a steel spring under small strain, this means that the force F is the product of the stiffness K (N/m) and the deflection u(m) which is equal to (l-l o ), where l and l o is the deformed and undefoemd length, respectively (Hermann, 1965). See Figure 6.

Figure 6. Hooke’s Law

iX? For this linear spring problem, a typical force-displacement (or stress-strain) plot is thus a straight line, where the stiffness K represents the slope. See Figure 7.

Figure 7. Linear Force-Displacement Relation

In large deformation analysis of elastomers, most nonlinear FEA codes such as Marc use a strain measure called the Green-Lagrange strain, E, which for uniaxial behavior is defined as:

E = 1/2 (IX2 -1)

and a corresponding “work conjugate” stress called the 2nd Piola-Kirchhoff stress, S2:

S2 = F/A (L0/L)2

(Fung, 1965). Therefore, the engineer resorts to either the Cauchy (true) stress, I :

I = F/A

with energetically conjugate iX?strain measure the logarithmic (true) strain, I :

I = ln (L/L0)

or the familiar engineering (Biot) stress, S1:

S1 = F/A0

with energetically conjugate strain measure being engineering strain, e (or deformation gradient, AE in large deformation theory). that is,

e = I L/L0 or AE = a x/a X,

where x and X refer to the deformed and original coordinates of the body (iX?Gent, et. al., 1993). Marc provides all of these strain and stress measures to the analyst.It is vital to take into account that the differences between various measures of stresses and strains, if just small, are negligible.

Time-Independent Nonlinear Elasticity

In the FEA of elastomers, material models are characterized by different forms of their strain energy (density) functions. Such a material is also called “hyperelastic” (iX?Gent, et. al., 1993). The commonly available strain energy functions have been represented either in terms of the strain invariants which are functions of the stretch ratios or directly in terms of the stretch ratios themselves. The three strain invariants can be expressed as:

I1 = IX2 1 + IX2 2 + IX2 3

I2 = IX2 1 IX2 2 + IX2 2 IX2 3 + IX2 3 IX2 1

I3 = IX2 1 IX2 2 IX2 3

(Herman, 1965). In case of perfectly incompressible material, I3 = 1. The Neo-Hookean model is the simplest representation of rubber elasticity that appeals to the concept of statistical mechanics and thermodynamics principles. The Neo-Hookean model can be represented as

W=C10 (I1 -3)

This demonstrates a steady sheer modules and gives a worthy correlation with the experimental data that reaches experimental data up to 40% strain in uniaxial tension and up to 90% strains in simple shear

The earliest phenomenological theory of nonlinear elasticity was propounded by Mooney as: W=C10 (I1 – 3) + C01(I2 – 3)

Although, it shows a good agreement with tensile test data up to 100% strains, it has been found inadequate in describing the compression mode of deformation (Tschoegl, 1971). Moreover, the Mooney-Rivlin model fails to account for the stiffening of the material at large strains. Tschoegl’s investigations underscored the fact that the retention of higher order terms in the generalized Mooney-Rivlin polynomial function of strain energy led to a better agreement with test data for both unfilled as well as filled rubbers.

The Yeoh (1995) model differs from the above higher order models in that it depends on the first strain invariant only:

W=C10 (I1 – 3) + C20(I1 – 3)2 +C30(I1 – 3)3

This model is more versatile than the others since it has been demonstrated to fit various modes of deformation using the data obtained from a uniaxial tension test only. This leads to reduced requirements on material testing. However, caution needs to be exercised when applying this model for deformations involving low strains (Yeoh, 1995). The Arruda-Boyce model ameliorates this defect and is unique since the standard tensile test data provides sufficient accuracy for multiple modes of deformation at all strain levels. In the Arruda-Boyce and Gent strain energy models, the underlying molecular structure of elastomer is represented to simulate the non-Gaussian behavior of individual chains in the network thus representing the physics of network deformation (Tschoegl 1971). The Arruda-Boyce model can be described as:

Ogden (1982) proposed the energy function as separable functions of principal stretches, which is implemented in Marc in its generalized form as:

where J is the Jacobian measuring dilatancy, defined as the determinant of deformation gradient AE. The Neo-Hookean, Mooney-Rivlin, and Varga material models can be recovered as special cases from the Ogden model. The model gives a good correlation with test data in simple tension up to 700%. The model accommodates nonconstant shear modulus and slightly compressible material behavior. Also, for I+- &lt.2 or &gt.2, the material softens or stiffens respectively with increasing strain.

Viscoelastic Damper

Viscoelastic damper are devices that serve as a joints to the bottom cords of the web trusses that support the floor diaphragms of a building (http://911research.wtc7.net/index.

html). Web trusses are structural member commonly used in buildings as floor joists. The trusses are made up of two separated parallel bars. Most of the buildings, especially the taller ones were supported by interlocking single or double web trusses. Web trusses are used for support in buildings that experiencing bad phenomena such as earthquake or heavy winds. The floor diaphragm is also connected to the spandrel plates, horizontal steel plates that connect the perimeter columns. The device is used as an absorber of oscillations caused by wind loading or other phenomena that cause the building to shake. In this system, the seismic response of multi-storied base-isolated structure to various types of isolation systems connected using viscoelastic dampers.

Viscoelastic dampers that absorbed vibration were first used in the Twin Trade Center in New York in 1969 for protection against wind loading and on 1980’s the SeaFirst and Two Union Square Buildings in Seattle utilized dampers for wind and a 13 story steel moment frame building in Santa Clara County was retrofitted with viscoelastic dampers in 1994 to reduce seismic vibrations (http://www.spie.org/web/abstracts/2700/2720.html). Today, the use of viscoelastic dampers demonstrated how it is very significant for steel and concrete structures.

“When a polymeric material is subjected to vibrations, energy is dissipated through relaxation processes in the long chain molecular networks and the properties of viscoelastic materials vary with frequency and temperature (shown in figure 8) (http://www.md-utc.group.shef.ac.uk/visco/visco.html).

Figure 8. Viscoelastic Materials’ Property

`Different combinations of properties is offered by many different types of polymeric materials (including elastomers, acrylics, epoxies, polyurethanes and PVC) offer and to incorporate a viscoelastic damper in a component it is necessary to match these properties to the expected conditions and to achieve high levels of damping, the device must also be designed to maximize the strain energy in the viscoelastic material” (http://www.md-utc.group.shef.ac.uk/visco/visco.html).

Passive Control of Vibration

A comprehensive review of the literature on passive supplemental damping devices

for civil engineering structures can be found in Soong and Dargush (1997) and Soong and Spencer (2001). Passive supplemental damping devices for structural applications have been researched and developed for the past thirty years. Recently, efforts to develop these concepts into a workable technology have increased significantly, and a number of these devices have now been installed in structures throughout the world. For example, metallic yield dampers were used to retrofit a two-story concrete building in San Francisco (Perry, et al. 1993), and three reinforced concrete buildings in Mexico City (Martinez-Romero, 1993). Taylor viscous fluid dampers were installed in the newly-constructed San Bernardino County Medical Center in California as components in the rubber bearing seismic isolation system (Soong and Dargush 1997). TMDs were installed in the Citicorp Center in New York City, the John Hancock Tower in Boston, the main towers of the Akashi-Kaikyo Bridge in Japan (Koshimura et al. 1994), and the Sydney Tower in Australia (Kwok and Denoon 2000). tuned liquid dampers were installed in the Nagasaki Airport Tower, the Yokohama Marine Tower, and the Shin-Yokohama Prince Hotel (Soong and Dargush 1997). Passive control systems alleviate energy dissipation demand on the primary structure by reflecting or absorbing part of the input energy, thereby reducing possible structural damage (Housner et al. 1997). Passive devices are those whose operation does not depend on an external power source. Passive control of vibration may employ either vibration absorbers (otherwise known as dynamic absorbers or Frahm absorbers) or dampers. Both of them operate on common principles: vibrations are sensed implicitly and controlled through a force that the device generates in response to the vibrations. Vibration absorbers are mass-spring type devices that offer little or no damping. Instead, they “absorb” the vibration through energy transfer into the mechanism, thereby reducing the vibration of the system that contains it. The energy it absorbs is dispelled over time by its own damping. On the other hand, a damper immediately dispels the energy rather than retaining it for any amount of time. Thus, it is more wasteful than the absorber [you might want to elucidate why this is] and more prone to problems stemming from wear and tear and thermal effects. These disadvantages may be somewhat offset by some advantages it offers over absorbers – eg, it is effective throughout a wider frequency range.

Artificial Damping

Every real structure is subjected to damping actions whereby vibration energy is lost into heat within the structure or into acoustic/hydrodynamic energy in the surrounding medium. Artificial damping covers that means of increasing the structural damping by the addition of extra damping materials and mechanisms. The usefulness of the artificial damping arises from increasing the system damping above the initial damping of the system to produce sufficient attenuation. The initial structural damping originates in hysteresis of the structural material, friction at structural joints (Coulomb damping), friction with attached non-structural items (eg cables, pipes, stowed equipment, etc., and viscous damping at lubricated sliding surfaces (as in engines and machinery). Vibration damping is presented by, so the total damping is given by a structural element, which vibrates in bending, can be damped by the appropriate addition of a layer of damping material. The damping material is usually a rubber material, where it has a high capacity for dissipating the energy. This is conveniently quantified by the material loss factor, and the complex cyclic strain and stress are related.

Controlling the direct strain and the shear strain in these damping layers provides the means to control the whole system damping characteristics. The addition of these damping layers is done by two techniques, unconstrained or constrained layer damping.

Damping of B ending Vibrations by the Unconstrained Layer Damping

The unconstrained layer is a layer of damping material added to the surface of the vibrating element with its outer surface perfectly free and unconstrained. Such layers can be glue d in the form of plastic tiles, or sprayed on as a wet plaster, which subsequently dries and hardens.

Figure 12. Unconstrained Layer Damping

Consider a layer of uniform thickness attached to a beam or plate of uniform cross-section (See Figure 12 .). It should be noticed that “ m i ” and “ s ” do not depend upon the mode of vibration when the damping material is uniformly applied over the whole surface of the vibrating structure, see figure above.

Damping of Bending Vibrations b y t he Constrained Layer Damping

Now consider the constrained layer damping where the damping layer, adhering to the outer surface of the structural element, has a stiff layer attached to its outer surface to sandwich the damping layer. This is called the passive constrained layer damping (PCLD), which consists of a constrained damping material attached to the surface of the vibrating structure. When the whole assembly bends, the damping layer is subjected to shear strain, and this cyclic shearing dissipates energy in the layer, as shown in Figure 13.

Figure 13. Constrained Layer Damping

The simple constrained layer system (involving three layers in all) can be extended to “N” multiple layer system, with damping layers and “N+1” constraining elastic layers. In all cases, shearing of the damping layer dissipates energy and damps the bending vibration. Constrained layer damping does depend on the mode of vibration, at very long and very short wavelengths. there is very little shear strain in the damping layer and very little damping. The damping and the bending stiffness both depend upon the bending wave number “K B ”, and there are also two important governing non-dimensional parameter.

“Damping is still present even when there is no interaction of the oscillator with a fluid (for example, an evacuated pendulum) and probably for “reason of the simple mathematics, most have wanted to retain in the equation of motion a damping term that is proportional to the velocity and this has no physical basis and leads to the wrong frequency dependence for the damping coefficient (http://arxiv.org/html/physics/0307016).

The bottom graph of Figure 14 shows internal friction. Instead of insisting on a “ fixed harmonic potential (parabola), one lets the parabola shift back and forth at each turning point and if the amount of shift is constant, then the decay of this `flip-flop’ system is the same as an oscillator influence d by sliding (Coulomb) friction but if the amount of the shift is proportional to the amplitude of the turning point just passed, then exponential decay results (http://arxiv.org/html/physics/0307016).

Figure 14. Diagram of the probable function differences concerning a viscous damping model (top) and the Modified Coulomb Damping model (bottom).

Vibration Limits

Vibration is dependent on the actions of people within or what they are doing. A work that uses more force to execute it will definitely have a greater vibration. For example, a simple walking has a lesser vibration effect as compared when they are doing jog or a man doing jog has a lesser vibration effect as compared to man jumping.

The term commonly use to express acceleration in limits of vibration and suitable thresholds is the magnitude of the acceleration due to gravity (g = 9.81 m/s 2 ) (Allen and Pernica, (1998). The limit is dependent on what the people are doing when they currently experiencing the motion. For instance, a people lying in a bed will experience a greater perceptible vibration effect as compared to those who are doing aerobics.

Generally, those whose activities are in parallel will not be disturbed by a vibration but those doing things adjacent to the activity place will the more disturbed and annoyed.

This was supported by the findings of Allen and Pernica (1998), that if vibration is more than 20% g, a very large one, and that it take place more often the result will be fatigue failure of the floor. A natural frequency of 6 Hz. 2 or less than is needed to prevent collapse done by overloading or fatigue. This could be done by doing a dynamic analysis of the building.

Principal Cause

Vibrations key factor is the resonance that happens if a load is given on a floor at a specific frequency. (Allen and Pernica, 1998) Example of this would be the cyclic force of the feet of a group of people dancing leads to the accumulation of load. (see figure 15).

Figure 15. Repetition of forces as done by human activities

Maximum floor acceleration is being produced by the cyclic force. This depends on the proportion of the innate frequency of the structure of the floor to the cyclic frequency of the applied force. The outcome of a resonance occurring that coincides or is close to the forcing frequency is severe.

In the process of doing this, the cyclic force which cause vibration, is being absorbed and continuously fed into the structure and the magnitude of the vibration also become until such time that the maximum is reached that can already cause a possible collapse. The structure will collapse if the applied force on it exceeds the internal force of the structures. A rhythmic activity, applied repeatedly, distributes a force to the floor that ranges from 2 to 3 Hz. The problem here is when this distributed load is now applied to a single structure which is now has greater magnitude since the single structure absorbs the resultant applied force. This frequency is called the “step frequency.” For aerobics which has the minimum frequency of 9 Hz, resonance occurs.

According to Allen and Pernica (1998), “If the repeated force has an impact component,

as is the case for aerobics in which everyone jumps at the same time (see Figure 1), the resonance can occur not only at the step frequency, but also at multiples, or harmonics, of this frequency. For instance, an aerobics class that jumps in synchronized with the step frequency of 2.5Hz will make a harmonic vibrations at multiples of 2.5 Hz – i.e., at 2.5 Hz (first harmonic), 5 Hz (second harmonic), and 7.5 Hz (third harmonic).

Since the natural frequencies of most floors are greater than 3 Hz (they often fall between 4 Hz and 8 Hz), problems are most likely to occur as a result of the second and third harmonics. However, the lower the harmonic the larger the vibration produced by resonance”

(http://irc.nrc-cnrc.gc.ca/pubs/ctus/22_e.html).

Rhythmic Activities

Rhythmic activities such as dancing which are usually done in auditoriums, buildings with dance and health clubs, and convention centers.

Figure 16.Floor acceleration that is made of the range of natural frequencies as to cyclic force

Large rhythmic activities can bring a big resonance vibration. To control floor vibration due to these rhythmic activities, the floor structure must be constructed in such a way that it can handle those loads. So that vibration will be restricted, design must be equal to highest significant harmony of the forcing frequency  and not higher than the internal forces that has natural frequency. Vibration will be controlled if the forcing frequency of the highest significant harmonic should be lower than the internal forces having a natural frequency will be incorporated in the design. The simple formula for the natural floor frequency (fn in HZ) is as follow:

fn (Hz) =18 D(mm)

where D is the total deflection of the floor structure because of the weight given by all of the members (joists, girders and columns) (Allen and Pernica 1998).

The plan for the floor structure fort rhythmic activities would include key aspects such as (1) span (2) storey height (3) walking vibration (4) steel/concrete floor construction (5) increased damping (6) increased stiffness (7) light-frame construction (8) stiffening, and (9) deflection criterion.

“Span. The elongated the floor span is the lesser the natural frequency. Convention centers have very long floor spans (approximately 30 m) and it is generally not convenient to design such floor structures to attain the minimum natural frequency shown in Table 1 for rhythmic activities that have impact (eg, aerobics) for this condition, the alternative is to displace either the rhythmic activity (to a stiffer floor) or the vulnerable occupants.

Storey height. The taller the columns supporting the floor on which the periodic movement takes place the subordinate the innate frequency of the floor. An example of this occurred when aerobics on the top storey of a 26-storey building caused second harmonic resonance due to the axial flexibility of the columns. This resonance creates exasperating vibrations of approximately 1% g in the offices beneath. Because the aerobics motion could not be transferred in the building, it had to be terminated. In choosing a place where rhythmic activities could be done (i.e., storey, floor area, etc.) would be concluded as the most important factor in planning the design of the floors. It is also important to properly position touchy tenants relative to rhythmic activities.

Walking Vibration. Walking vibration, which is largely reliant on the type of floor construction, which is another decisive factor in structuring the floor of most buildings.

Steel/Concrete Floor Construction. A steel floor with a solid deck more often has an innate frequency of between 3 Hz and 10 Hz. A person walking across a floor applies a force at a step frequency of approximately 2 Hz, which can result in resonance build-up when the natural floor frequency is around 2, 4, 6 or 8 Hz.  A design criterion has recently been developed in which the acceleration due to harmonic resonance is calculated and compared with a vibration limit, eg, 0.5% g, for office and residential occupancies” (http://irc.nrc-cnrc.gc.ca/pubs/ctus/22_e.html).

In the construction of a building, the design criterion must meet (as shown in Figure 17).

Rhythmic Activity

Steel/Concrete Floor

Light-Frame Floor

Dancing and Dining

5

10

Aerobics

9

13

Table 1. Minimum Floor Frequencies (Hz) for Distinctive Actions and Floor structures

Figure 17. Criterion for light- frame floors design

According to Allen and Rainer (1975), “the panels are large for low-frequency modes (panel length usually corresponds to floor span) and small for high-frequency modes and if the floor is left to vibrate in any mode, the motion will die out (Figure 18) at a rate determined by the damping in the floor (as cited in http://irc.nrc-cnrc.gc.ca/pubs/cbd/cbd173_e.html).

Figure 18. Typical transient vibrations from heel drop (high frequencies filtered out).

Allen and Rainer (1975) found out that “continuous vertical floor oscillation becomes distinctly perceptible to people when peak acceleration reaches approximately 0.5 per cent g, where g is the acceleration due to gravity” (as cited in http://irc.nrc-cnrc.gc.ca/pubs

/cbd/cbd173_e.html). Figure 19 illustrates the criteria for continuous shaking of both short (10 to 30 cycles) and long (8 hours) period.

Figure 19. Annoyance criteria.

Allen (1997) found out that problems resulting form the vibration mainly occur when a “forcing frequency coincides with the natural frequency (shown in figure 20) of the floor or other structural building element” (as cited in http://www.itc.nl/~ingeokri/newsletter

/summer98/newpage11.html).

Figure 20. Natural frequency vs, forcing frequency.

The following equation can be used to calculate natural frequency (http://www.itc.nl/~ingeokri/newsletter/summer98/newpage11.html).

III. Study Methodology

The study will be conducted with human activities in order to analyze and examine the dynamic response of a structure. The buildings will be represented by steel frames with elastomers as a damping device. The behavior of the structure will be observed and it will be analyzed under the presence of human activities. Different frequencies will be used to know the range of frequency values where the stability of the structure falls. The isolation mechanisms will be designed so that seismic energy will absorb by the damping system and be dissipated by the device. It will use different theories and principle formulated by the different authors about damping system.

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