Chapter 21: Design Consideration in Material Selection: Design for enhanced

material damping

INTRODUCTION:

Material damping is a name for the complex physical effects that convert kinetic and
strain energy in a vibrating mechanical system consisting of a volume of
macrocontinuous (solid) matter into heat. Studies of material damping are employed
in solid-state physics as guides to the internal structure of solids. The damping
capacity of materials is also a significant property in the design of structures and
mechanical devices; for example, in problems involving mechanical resonance and
fatigue, shaft whirl, instrument hysteresis, and heating under cyclic stress. Three types
of material that have been studied in detail are:
 Viscoelastic materials. The idealized linear behavior generally assumed for
this class of materials is amenable to the laws of superposition and other
conventional rheological treatments including model analog analysis. In most
cases linear (Newtonian) viscosity is considered to be the principal form of
energy dissipation. Many polymeric materials, as well as some other types of
materials, may be treated under this heading.
 Structural metals and nonmetals. The linear dissipation functions generally
assumed for the analysis of viscoelastic materials are not, as a rule, appropriate
for structural materials. Significant nonlinearity characterizes structural
materials, particularly at high levels of stress. A further complication arises
from the fact that the stress and temperature histories may affect the material
damping properties markedly; therefore, the concept of a stable material
assumed in viscoelastic treatments may not be realistic for structural materials.
 Surface coatings. The application of coatings to flat and curved surfaces to
enhance energy dissipation by increasing the losses associated with fluid flow
is a common device in acoustic noise control. These coatings also take
advantage of material and interface damping through their bond with a
structural material.
Material damping of macrocontinuous media may be associated with such
mechanisms as plastic slip or flow, magnetomechanical effects, dislocation
movements, and inhomogeneous strain in fibrous materials. Under cyclic stress or
strain these mechanisms lead to the formation of a stress-strain hysteresis loop of the
type shown in Fig. 6.2. Since a variety of inelastic and anelastic mechanisms can be
operative during cyclic stress, the unloading branch AB of the stress-strain curve falls
below the initial loading branch OPA. Curves OPA and AB coincide only for a
perfectly elastic material; such a material is never encountered in actual practice, even
at very low stresses. The damping energy dissipated per unit volume during one stress
cycle (between stress limits σd or strain limits d is equal to the area within the
hysteresis loop ABCDA.

Fig. 6.2 Typical stress-strain (or load deflection) hysteresis loop for a material under
cyclic stress
When an engineering structure is subjected to a harmonic exciting force Fg sin ωt, an
induced force Fd sin (ωt − ) appears at the support. The ratio of the amplitudes,
Fd/Fg, is a function of the exciting frequency ω. It is known as the vibration
amplification factor. At resonance, when  = 90°, this ratio becomes the resonance
amplification factor Ar:

= (6.25)

This condition is pictured schematically in Fig. 6.3 for low, intermediate, and high
damping (curves 1, 2, 3, respectively). The magnitude of the resonance amplification
factor varies over a wide range in engineering practice. In actual engineering parts
under high stress, a range of 500 to 10 is reasonably inclusive. These limits are
exemplified by an airplane propeller, cyclically stressed in the fatigue range, which
displayed a resonance amplification factor of 490, and a double leaf spring with
optimum interface slip damping which was observed to have a resonance
amplification factor of 10. Because of the wide range of possible values of Ar, each
case must be considered individually.
 Fig. 6.3 Effect of material and slip damping on vibration amplification, Curve (1)
illustrates case of small material and slip damping; (2) one damping is large while
other is small; (3) both material and slip damping are large [2]

In defining the various energy ratio units, it is important to distinguish between loss
factor ηs of a specimen or part (having a variable stress distribution) and the loss
factor η for a material (having a uniform stress distribution). By definition the loss
factor of a specimen (identified by subscript s) is:

 = (6.26)

where the total damping D0 in the specimen is given by Eq. The total strain energy in
the part is of the form:

(6.27)
where E denotes a modulus of elasticity and β is a dimensionless integral whose value
depends upon the volume-stress function and the stress distribution:

(6.28)
On substituting Eq., it follows that:

(6.29)
If the specimen has a uniform stress distribution, α = β = 1 and the specimen loss
factor ηs becomes the material loss factor η; in general, however,

(6.30)
Other energy ratio (or relative energy) damping units in common use are defined
below:
 For specimens with variable stress distribution:

(6.31)
 For materials or specimens with uniform stress distribution:

(6.32)
where η = loss factor of material = dissipation factor (high loss factor signifies high
damping)
tan φ = loss angle, where φ is phase angle by which strain lags stress in sinusoidal
loading
ψ = πη = specific damping capacity
δω/ωn = (bandwidth at half-power point)/(natural frequency)
Ar = resonance amplification factor
Q = 1/η = measure of the sharpness of a resonance peak and amplification produced
by resonance
The material properties are related to the specimen properties as follows:
Thus, the various energy ratio units, as conventionally expressed for specimens,
depend not only on the basic material properties D and E but also on β/α. The ratio
β/α depends on the form of the damping-stress function and the stress distribution in
the specimen. As in the case of average damping energy, Da, the loss factor or the
logarithmic decrement for specimens made from exactly the same material and
exposed to the same stress range, frequency, temperature, and other test variables may
vary significantly if the shape and stress distribution of the specimen are varied. Since
data expressed as logarithmic decrement and similar energy ratio units reported in the
technical literature have been obtained on a variety of specimen types and stress
distributions, any comparison of such data must be considered carefully. The ratio β/α
may vary for specimens of exactly the same shape if made from materials having
different damping-stress functions.

VISCOELASTIC MATERIALS
Some materials respond to load in a way that shows a pronounced influence of the
rate of loading. Generally the strain is larger if the stress varies slowly than it is if the
stress reaches its peak value swiftly. Among materials that exhibit this viscoelastic
behavior are high polymers and metals at elevated temperatures, as well as many
glasses, rubbers, and plastics. As might be expected, these materials usually also
exhibit creep, an increasing deformation under constant applied load. When a
sinusoidal exciting force is applied to a viscoelastic solid, the strain is observed to lag
behind the stress. The phase angle between them, denoted by , is the loss angle. The
stress may be separated into two components, one in phase with the strain and one
leading it by a quarter cycles. The magnitudes of these components depend upon the
material and upon the exciting frequency, ω. For a specimen subject to homogeneous
shear (α = β = 1),
γ = γ0 sin ωt (6.33)

σ = γ0 [G′(ω) sin ωt + G″(ω) cos ω t] (6.34)

This is a linear viscoelastic stress-strain law. The theory of linear viscoelasticity is the
most thoroughly developed of viscoelastic theories. In Eq., G′(ω) is known as the
“storage modulus in shear” and G″(ω) is the “loss modulus in shear” (the symbols G1
and G2 are also widely used in the literature). The stiffness of the material depends on
G′ and the damping capacity on G″. In terms of these quantities the loss angle  =
tan−1 (G″/G′).The complex, or resultant, modulus in shear is G* = G′ + iG″. In
questions of stress analysis, complex moduli have the advantage that the form of
Hooke’s law is the same as in the elastic case except that the elastic constants are
replaced by the corresponding complex moduli. Then a correspondence principle
often makes it possible to adapt an existing elastic solution to the viscoelastic case.
The moduli of linear viscoelasticity are readily related to the specific damping energy
D introduced previously. For a specimen in homogeneous shear of peak magnitude
γ0, the energy dissipated per cycle and per unit volume is

(6.35)
Also,

Controlling damping

Damping can be controlled by two major methods – passive control and active
control. Passive control can involve several strategies for damping, all of which
involve some mechanical characteristic of the system, either inherent or added, to
control vibrations. Once the characteristic becomes part of the system, no further
action is taken; hence, the system is passive. All of the treatments which have been
implied in the previous examples, such as adding damping materials, weight or
stiffness are passive. Passive control also includes the use of discrete devices, such as
shock absorbers, and the addition of materials that have high inherent energy loss.
Materials with highly mobile molecules, such as elastomers, have long been known as
highly damped materials. Therefore, damping control can be achieved by making the
part out of an elastomer. A part could also be damped by adding elastomers to the
normal material that the part is made of. In both of these cases, the internal molecular
nature of the part furnishes the desired damping. Changing the shape of a vibrating
system by joining system components together with elastomeric adhesives would also
increase damping. Damping could also be achieved by mounting the vibrating part on
an elastomer, such as would be done by using a damping pad for a motor. You might
also wrap the part in an elastomer. These solutions reflect the general methods of
damping that were discussed previously in the discussion of damping fundamentals.
Active damping is a much more recent development in damping engineering. This
strategy involves the addition of elements to the part that sense the amount of
vibration and trigger some remedial action to dampen the movement. The most
common system of this type can be achieved by embedding sensors in a part to detect
vibrations and piezo-electric devices which extend and retract in response to the
sensor signals in such a way as to counteract the vibrations. This system requires that
electric power be supplied to the actuators.
Active systems of the type described above have been used in aircraft. These systems
drastically reduce the vibrations associated with flight, especially at times such as
breaking the sound barrier. They are able to control these vibrations without the
penalty of reducing the stiffness of the aircraft parts or changing their shape. Some
advanced active systems also use the signals from piezo-electric devices to drive
actuator motors which can make minor adjustments to the shape (geometry) of the
airplane components. For instance, the wing shape can be changed during high
turbulence to optimize flight control.
Another method of active control is through the use of embedded fiber optics. These
fibers can sense gross vibrations much like electronic sensors. The fiber optics can be
monitored for changes in cross-sectional area of the fiber which will cause a change in
the light transmission and, therefore, indicate that vibrations are occurring. These
changes might even be able to pinpoint the actual location of the vibration, thus
giving tighter control than is usually possible with electronic sensors.