CHAPTER 4 PASSIVE VIBRATION CONTROL 4.1. VIBRATION DAMPING iscourses.me.metu.edu.tr/courses/...PASSIVE_VIBRATION_CONTROL_… · o See Tables 4.1 and 4.2 examples for using vibration

4.1 ME 708 LECTURE NOTES, by Dr. Gökhan O. ÖZGEN, Fall 2013‐2014

CHAPTER 4 PASSIVE VIBRATION CONTROL

4.1. VIBRATION DAMPING

What is damping?

Dissipating some of the energy which is added to a dynamic system by exciting forces during each cycle of the response.

Dissipation is achieved through resistive damping forces (may be internal such as molecular level interactions or external such as friction).

Normally damping forces are lot smaller than elastic forces but around resonance elastic forces and balanced out by inertial resistance which leaves only the damping force to resist the motion of the DOF already moving.

Thus, damping forces are of critical importance when the excitation frequency is near or at resonance frequency.

No damping leads to boundless motion, while some damping will lead tolarge oscillations but with bounds (see Figure 4.1).

Figure 4.1. Dependence of the character of a transient decay on loss factor [Nashif et.al. 1985]


The effect of damping on dynamic repsonse of systems makes it an effective parameter to control vibrations only for the following cases:

o For free vibration response (transient response) where the system oscillates at resonant frequencies (the rate at which these oscillations die out is related to system damping).

o Limit steady‐state resonant response. o Attenuate traveling waves. o See Tables 4.1 and 4.2 examples for using vibration damping for vibration control.

Note that damping does little or nothing to reduce the amplitude of the forced responses other than resonance regions.

Typical and very common vibration related problem is the existance of resonances and excitation forces near these system frequencies, making damping‐based passive vibration control techniques very viable.

Table 4.1. Industries and particular mechanical components, and main motivations for which vibration damping treatments are used [Corsaro and Sperling 1990]


Table 4.2. General areas of application of damping in controlling the effects of vibration [Corsaro and Sperling 1990]

Structural fatigue and failure

Damage to equipment

Equipment malfunction

Vibration (and Noise) control for personnel safety and function

Effects on instruments, processes, and precision equipment

When the natural damping in a system is inadequate for its intended function, then an applied damping treatment may be required to achieve the following objectives:

Control of vibration amplitude at resonance

Control excessive resonance vibrations which may cause high stresses, leading to premature failure.

Example: For random excitation, damping in each mode at least exceeds a minimum specified value to keep random stresses within acceptable limits. Sonic fatigue of aircraft fuselage, wing, and control surface panels when they due to excitation by jet noise or boundary layer turbulence‐induced excitation.

Noise control

Control of noise radiation from vibrating surfaces, or the control of noise transmission through a vibrating surface (see Table 4.3).

Decreasing the amplitudes of the vibrating surface.

Example: Oil pan damping increase can drastically decrease structure‐borne noise contribution due structural vibrations (see Figure 4.2).

Table 4.3. Damping levels vs acoustic response [Corsaro and Sperling 1990]


Figure 4.2. Oil sump made of high damping laminated steel and effect of damping on free response of the oil sump.

4.2. MECHANISMS OF DAMPING IN MATERIALS AND MECHANICAL STRUCTURES

4.2.1. Internal Mechanisms of Damping (Material damping) Mechanisms that dissipate vibrational energy in the form of heat within the volume of a

material element as it is deformed. Complex physical effects that convert kinetic and strain energy in a vibrating mechanical

system consisting of a volume of macrocontinuous (solid) matter into heat. Associated with internal atomic or molecular reconstructions of the microstructure or

with thermal effects. o microstructure defects, such as grain boundaries and impurities. o thermoelastic effects caused by local temperature gradients resulting from

nonuniform stresses, as in vibrating beams. o eddy current effects in ferromagnetic. o plastic slip or flow. o magnetomechanical effects. o dislocation movements. o inhomogeneous strain in fibrous materials. o the relaxation and recovery of the molecular chains after deformation

(polymers or viscoelastic materials). Under cyclic stress or strain these mechanisms lead to the formation of a stress‐strain

hysteresis loop of the type shown in Figure 4.3. Most structural metals and alloys have relatively little damping under most

conditions(Steel and alimimum will have exteremely small damping). Some alloy systems, however, have crystal structures specifically selected for their

relatively high damping capability. o Nonlinear and increases as cyclic stress amplitudes increase (see Figure 4.4).


Figure 4.3. Typical stress‐strain (or load‐deflection) hysteresis loop for a material under

cyclic stress [Harris and Pierson 2002].

Figure 4.4. Damping behavior of a copper‐manganese alloy as a function of strain

amplitude and temperature [Nashif et.al. 1985]

The cyclic integral of force with respect to the displacement, which is the area of the hysteresis loop, is equal to the work done against the damping force.

This integral (loop area) is the energy dissipated per cycle of motion. This is the damping capacity which, when divided by the material volume, gives the per‐

unit‐volume damping capacity as before (see Figure 4.5 for damping capacity values for vairous materials).


Figure 4.5. Damping capacity of various materials [Nashif et.al. 1985]

The damping force should be resisting the motion so it should somehow be a function of the velocity of the motion of which it is resisting.

Typical damping ratios for material damping for metals and alloys: 0.0005‐0.005 Typical damping ratios for material damping for viscelastic materials: 0.05‐0.75 Polymeric or elastomeric materials are capable (under certain conditions) of dissipating

far larger amounts of energy per cycle than metals and alloys. We will discuss this further.

4.2.2. External Mechanisms of Damping Structures and machines can be damped by mechanisms which are essentially external to the system or structure itself. Such mechanisms, which can be very useful for vibration control in engineering practice (friction dampers, air‐film dampers), include: Structural damping (friction and impact at joints) Adjacent touching parts of the machine or structure slide cyclically relative to one

another, on a macroscopic or a microscopic scale, dissipating energy (see Figure 4.6).


Figure 4.6. Different mechanisms of structural damping for a clamped beam [Nashif et.al. 1985]

Mechanical energy dissipation caused by friction due to the relative motion between

components. Impacting at the joints in a mechanical system or structure can also be considered under

this type of damping. Difficult to develop a generalized analytical model that would satisfactorily describe

structural damping. Rubbing is usually represented by a Coulomb friction model. Impacting, however, should be determined from the coefficient of restitution of the two

members that are in contact. Most common method of estimating structural damping is by measurementof overall

damping in the mechanical system. o For metals and alloys internal (material) damping is negligible compared to

structural damping. o In tall buildings, bridges, vehicle guideways, and many other civil engineering

structures and in machinery, such as robots and vehicles, dominant damping is the structural damping mechanism.

o A major form of structural damping is the slip damping that results from energy dissipation by interface shear at a structural joint. The degree of slip damping that is directly caused by Coulomb (dry) friction depends on such factors as joint forces (for example, bolt tensions), surface properties and the nature of the materials of the mating surfaces. This is associated with wear, corrosion, and


general deterioration ofthe structural joint. In this sense, slip damping is time‐dependent. It is a common practice to place damping layers at joints to reduce undesirable deterioration of the joints. Sliding causes shear distortions in the damping layers, causing energy dissipation by material damping and also through Coulomb friction. In this way, a high level ofequivalent structural damping can be maintained without causing excessive joint deterioration. These damping layers should have a high stiffness (as well as a high specific‐damping capacity) in order to take the structural loads at the joint. For structural damping at a joint, the damping force varies as slip occurs at the joint.

Impact Impact between imperfectly elastic parts of the system (may also be considered as part

of structural damping). Fluid pumping Vibration of a structure forces the fluid medium within which the structure is immersed

to pass cyclically through narrow paths or leaks between different zones of the system or between the system and the exterior, thereby dissipating energy.

Air‐Based damping Acoustic radiation damping, whereby the vibrational response couples with the

surrounding fluid medium, leading to sound radiation from the structure. Enclosing thin films of air around the vibrating structure allowing leaks from the enclosed

air volume. Pressure gradients are created due to the allowance of leaks which will create net forces resisting the motion of the structure.

Figure 4.6. Various configurations where air film damping is observed [Nashif et.al.

1985]


Figure 4.7. A mechanical joint configuration where air film damping is observed

[Jones 2001]

4.3. MEASURES OF DAMPING

How to characterize damping?

Viscous damper:

Hypothetical device that creates mathematically convinient damping forces. If used properly very effective way of representing damping. Not satisfactory way of representing higher levels of damping forces.

To decide on how much damping to introduce to the system , we need to first know how much damping is already present in the system?

4.3.1. Ways of identifying damping in mechanical systems:

As far as passive vibration control applications are concerned most practical measure of damping is the modal damping ratio or modal loss factor.

Damping Ratio: Already discussed (SDOF and MDOF). How to measure damping ratio from measured FRFs? Experimental modal analysis techniques? We will upload a write‐up on this topic on METU‐ONLINE later. Simplest way is to use the half‐power point method.


Loss Factor:


Figure 4.8. a) Effect of loss factor on system response b) Selection of loss factor (adapted

from the 1970 paper by Crandall)

4.3.2. Other measures of Damping

Table 4.4. Common Types of Damping Measures and their interrelationships

)(

)(

F

X

02.0

2.0

(a) /n

() Viscous damper

Actual loss factor

(b)

1.0


4.4. VISCOELASTIC VIBRATION DAMPING

A mechanism commonly known as viscoelastic damping is strongly displayed in many polymeric, elastomeric, and amorphous glassy materials.

Polymeric and Elastomeric materials Polymers Polymers are large molecules consisting of repeated chemical units (`mers') joined

together, usually in a line, like beads on a string. Each `mer' is typically made up of more than 5 and less than 500 atoms; the word

`polymer' is applied when you have more than about 50 `mers' stuck together. Most of the plastics are made of polymers. Molecular chains in a polymer network intertwine and are connected to gether at

various locations (cross‐linked). Specific properties can be tailored by manupulating the size and the atomic composition

of the chains, changing the number of cross‐links, adding fillers such as carbon black to increase stiffness, tear and wear resistance and hysteresis.

Polymers can be made stiff (plastics) or soft (film adhesives). Elastomer: A rubberlike synthetic polymer, as silicone rubber. Elastomers are only slightly cross‐linked. Natural rubber Polymer made up of molecules of great length composed of hydrogen and carbon. Naturally, molecular chains are flexible, independent of each over. In its raw state, natural rubber is a tough material which deforms in part by viscous flow

and in part elastically. It becomes soft and sticky when warms. Its practical uses in the raw state are limited. Vulcanisation Process: The practical potential of rubber is achieved by the process known as vulcanisation.

Vulcanisation with sulphur links the rubber molecules together at intervals along their length by means of short chains of sulphur atoms (other vulcanising systems as peroxides can be used).

Vulcanisation gives strength, elasticity and mechanical behaviour less sensitive to temperature.

Rubbers also contains a filler, generally carbon black, which improves tear and abrasion properties as well as increasing the modulus, hysteresis and creep.

Damping mechanism in polymers

The damping action is caused by the relaxation and recovery of the molecular chains after deformation.

Viscoelasticity is a property which is caused by the molecular rearrangement under stress.

The speed of such processes depend on the speed of molecular motion of which temperature is a measure.


Molecular level explanation for viscoelastic behavior In a polymer, each flexible threadlike molecule is continually changing the shape of its

contour as it wriggles and writhes with its thermal energy (Figure 4.9).

Figure 4.9. Typical polymeric structure network[Nashif et.al. 1985]

Strong dependence between frequency and temperature effects in polymer behavior due to the relationship between temperature and molecular vibrations (We will see it later).

A wide variety of commercial polymeric damping material compositions exist (see Tables 4.5 and 4.6).

o See D.I. Jones’s book for more detailed material and company information. Polymeric damping materials are available commercially in the following categories:

o Mastic treatment materials o Cured polymers o Pressure sensitive adhesives o Damping tapes o Laminates

Damping material data related to the damping performance is provided in many formats. We will discuss this later.

Table 4.5. Typical Damping Material Types [Harris and Pierson 2002]


Table 4.6. List of Damping Materials and systems Manufacturers [Harris and Pierson 2002]

Rubbers (General Information) [Harris and Piersol, 2002]

Elastomers are polymers that can be stretched substantially beyond their original length and will retract rapidly and forcibly to essentially their original dimensions upon release of the force.

The optimum properties and/or economics of many rubbers are obtained through formulating with

reinforcing agents,

fillers

extending oils

vulcanizing agents

antioxidants

pigments End‐use markets for formulated rubbers include automotive tire products, adhesives,

cements, caulks, sealants, latex foam products, hose, belting , footwear (, and molded, extruded, and calendered products (athletic goods, flooring, gaskets, household products, O‐rings, blown sponge, thread, and rubber sundries).

A list of general‐purpose elastomers and properties is summarized in Table 4.7.

Table 4.7. Properties of general purpose elastomers [Harris and Piersol]


4.4.1. Theory Of Linear Viscoelasticity

Mathematical modeling of linear viscoelastic behavior. o Small strain‐small stress deformation.

If both strain and rate of strain are infinitesimal, and the time‐dependent stress‐strain relations can be described by linear differential equations with constant coefficients, we have linear viscoelastic behavior.

Then, in a given experiment the ratio of stress to strain is a function of time (or frequency) alone, and not of stress magnitude.

The relations between stress, strain, and their time dependences are in general described by a "constitutive equation" or "rheological equation of state."

Non‐linear viscoelatic behavior (not within the scope of this course): o Hyperelasticity (large strain elasticity) o Stress softening effects o Large strain viscoelasticity

4.4.1.1. Linear Time Domain Behavior of Viscoelastic Materials [Lakes 1998]

Uniaxial, small deformation, linear stress‐strain constitutive relation for an elastic solid is as follows:

Eεσ , (1) where E is Young’s modulus, σ is stress and ε is strain. The same relation can be written in a different form, in which a compliance term is utilized:

Jσε , (2)

where J is the elastic compliance and defined as the inverse of the elastic modulus E,

EJ

1 (3)

On the other hand, for a viscous fluid, shear stress occurring during flow is dependent on strain rate and given as:

dt

dεσ , (4)

where is the viscosity. Viscoelastic materials are the ones that have time dependent relationship between

stress and strain unlike the elastic ones. This can be also defined as a behavior between pure elastic and pure viscous cases. Some of the observed special behaviors for viscoelastic materials (Lakes 1998) are as

follows:


1. For constant stress, strain increases with time (creep behavior). 2. For constant strain, stress decreases with time (relaxation process) 3. Dynamic stiffness depends on the rate of application of the load 4. For cyclic loading, hysteresis is observed between stress and strain, which is an

indication of the energy dissipation. 5. Acoustic waves transmitted are attenuated.

All materials show viscoelastic behavior, but the deviation from elastic characteristics

depends on the type of the material. Metals show strong elastic behavior at moderate temperatures, while polymers show

significant viscoelasticity at same temperature levels. First of the two transient properties of viscoelastic materials is creep. It is a slow, gradual deformation of a material under constant stress. For uniaxial stress case, stress history that will cause pure creep behavior can defined as:

)()( 0 tut , (5)

where u(t) is the unit step function. The strain (t) will increase with time following the relationship

0

)()(

t

tJ , (6)

where J(t) is called the creep compliance. Elastic material is a special case of general viscoelastic model with a creep compliance of

)()( 0 tuJtJ . (7)

Viscous materials (like liquids) is another special case for creep behavior, where the creep compliance is

)()( ttutJ , (8)

where is the viscosity of the liquid. Second transient property of the viscoelastic materials is the stress relaxation, which is the gradual decrease of stress when the material is held at constant strain. If the strain is applied as the given function

)()( 0 tut , (9)

the stress will decrease following the relationship


0

)()(

t

tE , (10)

where E(t) is the relaxation modulus. To simplify the understanding the concepts, only uniaxial stress‐strain case is considered. Relaxation and creep might also be occurring in shear and volumetric deformation as well as extension. Different relaxation and creep functions can be defined for different deformation modes. In order to aid physically realize creep and relaxation process, stress and strain curves are presented in Figure 4.10 (creep) and Figure 4.11 (relaxation). SEE THE POWER POINT SLIDES.

Figure 4.10. Stress and strain curves vs. time (Creep case) [adapted from Lakes 1998] Figure 4.11. Stress and strain curves vs. time (Relaxation case) [adapted from Lakes 1998]

0

ViscoelasticElastic

Viscous

Relaxation Recovery

t

t

0

Viscoelastic

Elastic

Viscous

Creep Recovery

t

t


Viscoelastic materials can be categorized into two groups based on their phase. They

might be in solid or liquid state.

Although the transient behavior of viscoelastic solids and liquids are similar, there are some generic differences.

For example the relaxation modulus for viscoelastic converges to zero as time goes to

infinity, i.e. 0)( tEliquids (Figure 4.12).

For viscoelastic solids on the other hand, relaxation modulus converges to a positive

finite value as the time goes to infinity, i.e. esolids EtE )( (Figure 4.13).

In case of creep, viscoelastic liquids achieve a steady flow – or linearly increasing deformation – with time as shown in the creep compliance plot given in Figure 4.14.

For the solids, the compliance approaches to an equilibrium value which represents zero strain rate (Figure 4.15).

Figure 4.12 Typical Relaxation Modulus distribution for viscoelastic liquids

Figure 4.13 Typical Relaxation Modulus distribution for viscoelastic solids

Figure 4.14. Typical Creep Compliance distribution for viscoelastic liquids

J(t)

t

0

E(t)

t

Ee

E(t)

t


Figure 4.15. Typical Creep Compliance distribution for viscoelastic solids

4.4.1.2. Viscoelastic Response due to an Arbitrary Input A very powerful concept in polymer physics is the Boltzmann’s Superposition Principle. Simply Boltzmann’s Superposition Principle states that each loading state of a

viscoelastic material makes independent contribution to total loading history, and total final deformation is the sum of each contribution.

For linear viscoelastic behavior, Boltzmann’s superposition principle can be applied.

For consecutive step stress inputs as shown in Figure 4.16, the strain can be written as a combination of the strain caused by each individual step input, i.e (see Figure 4.17);

210 )()()()( tJtJtJt (11)

Figure 4.16. Superposed step stress inputs

(t)

t

0

1

2

3

t0 t1 t2

J(t)

t

0 Je


Figure 4.17. Strain response to multiple step stress inputs As 0t , the strain becomes:

d

d

dtJt

t )()()(

(12)

where )(t is an arbitrary function of time. Similary for an arbitrary function of strain, the

stress can be obtained as:

d

d

dtEt

t )()()(

(13)

4.4.1.3. Viscoelastic Constitutive Relation and Representation of Material Damping by Frequency Dependent Complex Modulus

The standard linear viscoleastic constitutive relation for a uniaxial state of stress is [Christensen 1971, Bagley 1979]:

J

j j

j

jK

k k

k

kdt

εdEε(t)E

dt

σdbσ(t)

10

1 (14)

or

J

j j

j

jK

k k

k

kdt

εdq

dt

σdp

00.(Gaul 1991) (15)

(t)

t


For harmonic response analysis, stress and strain can be defined as tiet 0)( , tiet 0)( . Substituting these into the differential form of constitutive equation gives

[Nashif et al 1985]:

K

k

kk

J

j

jj

K

k

kk

J

j

jj

ib

iaE

ib

iEE

1

100

1

100

0

)(1

)(1

)(1

)(

(16)

This relation between stress and strain amplitudes can be written in a more compact form as:

00 )( EiE (17)

where E and E are storage modulus and loss modulus respectively.Both are functions of frequency for the general viscoelastic behavior. For metals, storage modulus E is constant over the frequency domain and loss modulus

E is a weak function of frequency. For elastomeric materials both moduli are functions of frequency. Besides the frequency dependence, most viscoelastic materials have also strong

temperature dependence. The relation between harmonic response amplitudes of stress and strain can be written in

terms of a complex frequency dependent modulus )(~* jE , which is defined as:

(18)

Frequency domain distributions of storage and loss moduli of a specific material can be experimentally obtained and viscoelastic model terms can be used to represent the real material behavior. Complex moduli can be written in another form:

)(1)()(~* iEiE (19)

where a loss factor term () is defined as

)(

)()(

E

E

(20)

00*

0 ))()(()(~ EjEjE


What does loss factor represent?

In an effective passive vibration control technique based on the use of high damping viscoelastic materials, the viscoelastic components or elements must participate significantly in the energy of vibration.

Typical values of moduli for viscoelastic materials are given in Table

Table 4.8. Typical Moduli of Viscoelastic Materials [Harris and Pierson 2002]


4.4.1.4. Complex Modulus in terms of Relaxation Modulus Using Equation (12), the complex modulus can be obtained in terms of the relaxation modulus as follows:

dtetEiiE ti

)()()(

~* (21)

or

)()(~* tEiiE (22)

where )(~* iE is the complex modulus and is the Fourier transform symbol.

4.4.1.5. Generalized Maxwell Model Representation of Linear Viscoelastic Behavior The simplest version of the Maxwell model for viscoelastic materials is composed of an elastic and viscous components connected in series as shown in Figure 4.18.

Figure 4.18. Single element Maxwell model [adapted from Lakes 1998]

The spring represents the elastic component and E is the elastic modulus (unit is Pa).

The dashpot is the viscous component with being the viscosity(unit is N.s/m2).

The total strain can be written as the summation of the strain in the elastic and viscous components:

21 (29)

The strain rate is:

dt

d

dt

d

dt

d 21 (30)

E

E

2

1


The stress is same in both components:

2 E (31)

and

dt

d 1 (32)

From Equations (31) and (32):

Edt

d

dt

d 12 (33)

and

dt

d 1 (34)

Substituting Equations (33) and (34) into Equation (30), the constitutive equation for a single Maxwell model element is obtained:

Edt

d

dt

d 1 (35)

For a step input )()( 0 tut , the stress response for the single Maxwell element can be

derived as:

EtEt

eEet 00)( (36)

The relaxation modulus )(tE can be written as:

Et

Eet

tE0

)()( (37)

A new parameter, relaxation time, is defined in terms Eand :

E

(38)

Now the relaxation modulus can be written in terms of the relaxation time:

t

EetE )( (39)


Complex shear modulus for the single element Maxwell model can be obtained from the Laplace transform of the relaxation modulus:

)()(~

tEssE (40)

s

EssE

1)(

~ (41)

The complex modulus can be obtained by replacing Laplace variable s with i :

i

EiisEiE

1)(

~)(

~* (42)

Some common simple viscoelastic models – including the single element Maxwell model – are given in Table 4.9 with corresponding constitutive equation and the complex modulus

expressions. The units of the parameters E and are Pa and N.s/m2 respectively. Table 4.9. Some common, simple viscoelastic models

Model Constitutive equation [Lakes 1998]

Complex Modulus

)(~* iE

[Jones 2001] Maxwell model

dt

d

dt

d

E

1

jE

Ej

Kelvin-Voigt model

dt

dE

jE

Standar linear model

dt

dEE

EE

dt

dE

)( 211

21

1

1 12

11

11

jE

jE

Ej

E

E

E1 1

E2


A generalized Maxwell model can be constructed by connecting N single Maxwell elements as shown in Figure 4.19. The relaxation modulus for the generalized Maxwell model can written as:

N

i

t

iieEtE

1

)( (43)

Figure 4.19. Generalized Maxwell model with N elements [adapted from Lakes 1998]

Similarly, the complex modulus for the generalized Maxwell model can be obtained as:

N

n n

nn

i

EiiE

1

*

1)(

~ (44)

Real and imaginary parts )(~ E and )(

~ E can written separately as:

N

n n

nn

i

EE

122

22

1)(

~

(45)

and

N

n n

nn

i

EE

1221

)(~

(45)

E1

1

E2

2

En

n

EN

N

……


The generalized Maxwell can be modified by adding a pure elastic element with an

elastic parameter eE in parallel as shown in Figure 4.20.

This modified model is essentially for representing viscoelastic solids which has a non‐

zero relaxation modulus as time approaches infinity, i.e. eEtE )( .

Note that for the original Maxwell model, 0)( tE , which is typical liquid behavior.

Figure 4.20. Modified Generalized Maxwell model with N elements (also called the

generalized standard model) [adapted from Lakes 1998] The relaxation modulus for the modified Maxwell model (also called the generalized standard model) becomes:

1

1

)(N

n

t

neneEEtE (46)

The complex modulus for the modified model can also be written as:

1

1

*

1)(

~ N

n n

nne i

EiEiE

(47)

Real and imaginary parts of the complex modulus becomes:

1

122

22

1)(

~ N

nn

nne

EEE

(48)

and

1

1221

)(~ N

nn

nnEE

(49)

Ee

E1

1

En

n

NN-1

N-1

……


4.4.2. Typical Viscoelastic Material Behavior As A Function Of Temperature

Modulus and loss factor distribution as a function of temperature at constant frequency (see Figure 4.21).

Linear behavior: modulus is independent of the strain amplitudes. Nonlinearity occurs above certain treshold strain levels (depending on the type of

polymer structure and composition). The strain levels in vibration control applications are often below linear behavior

treshold. At low temperatures, high modulus and low loss factors are typical (Glassy region). After a particulat temperature, material starts to soften drastically with sharp increase in

loss factor (Transition region). As temperature is increased more beyond the transition region modulus decreases and

reaches stable low value (Rubbery region). Further increase in temperature beyond rubbery region causes the material to

disintegrate (Flow region) (for most rubberlike materials (polymers with cross‐links this region do not exist)).

Modulus values could be as high as 100 GPa in the glassy region and as low as 10000 Pa in the rubbery region.

Width of transition region 20oC to 200oC – 300oC. Encompasses a temperature range of a few hundred degrees. Loss factor usually below 10‐2 and 10‐3 for glassy region. Loss factor as high as 1‐2 in the transition region. Loss factor between 0.1 to 0.3 in the rubbery region for many materials. These properties depend on the composition.

Figure 4.21. Variation of the storage modulus and loss factor with temperature [Nashif

et.al.1985]


Critical Parameters that may be used when evaluating the performance of a viscoelastic material (see Figure 4.22):

o Ts: softening temperature or transition temperature

o T0: peak loss factor temperature

o Peak loss factor

o ΔT: Width of the transition region (required to be large for vibration

control applications). Plastics

o Softening temperature Ts is quite high. o Wide glassy region.

Elastomers o Softening temperature Ts is low below room temperature. o A large transition region is desired for vibration control usage.

Figure 4.22. Variation of the real part of the modulus and loss factor with temperature – constant frequency [Jones 2001]


4.4.3. The Effects Of Frequency On Viscoelastic Material Properties

Inverse effect of temperature: Increasing frequency (at constant temperature) has similar effect on the complex modulus as decreasing temperature.

Several decades of frequency change similar to a change of a few degrees of temperature(basis for temperature‐frequency superposition principle – will be discussed later. This principle will enable characterization of viscoelastic materials in a large frequency range using a single test technique).

Large range of frequency (10 to 20 decades) (compare it to few hundred degrees) o (LOW) 10‐8Hz to 108 Hz (HIGH)

Figure 4.23. Variation of the real part of the modulus and loss factor with frequency –

constant temperature [Jones 2001]


Typical carpet plots of storage modulus and loss factor of viscoelastic materials are given in Figure 4.24.

In Table 4.10, you can find typical values for viscoelastic solid elastic (storage) tensile modulus and loss factors (see Table 4.11 for shear modulus properties).

Figure 4.24. Frequency and temperature dependence of loss factor and storage modulus of viscoelastic materials [Encyclopedia of Vibration]


Table 4.10. Typical viscoelastic solid elastic (storage) tensile modulus and loss factors [Jones 2001 and Nahif et.al. 1985]

Table 4.11. Typical viscoelastic solid elastic (storage) shear modulus and loss factors [Jones 2001 and Nahif et.al. 1985]

Documents

CHAPTER 4 PASSIVE VIBRATION CONTROL 4.1. VIBRATION DAMPING iscourses.me.metu.edu.tr/courses/...PASSIVE_VIBRATION_CONTROL_… · o See Tables 4.1 and 4.2 examples for using vibration