20.5 Damping Models

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20.5.1 Viscous Damped Harmonic Oscillator

As first seen by students in a textbook, the equation of motion for a damped, driven harmonic oscillator

is likely as follows:

mx€ þ cx_ þ kx ¼ FðtÞ ð20:1Þ

where m is mass, k is spring constant, c is a “constant” of viscous damping, and FðtÞ is the external force

driving the oscillator. It is convenient to work with a coefficient of performance, or quality factor Q; and

rewrite Equation 20.1 in canonical form as

x€ þ

v0

Q

x_ þv20

x ¼ v20

FðtÞ

k

; with v20

¼

k

m ð20:2Þ

For FðtÞ ¼ 0 and an assumed solution, x ¼ A expðpuÞ with u ¼ v0t; the differential equation becomes

algebraic (quadratic) in xðuÞ, with the roots given by

p ¼ 2

1

2Q

^

ffiffiffiffiffiffiffiffiffiffiffiffi

1

4Q2 2 1

s

ð20:3Þ

Depending on the value of Q; the motion is either overdamped (nonoscillatory), critically damped, or

underdamped. Here, we restrict our attention to the last case corresponding to Q . 1=2; in which the

square root term of Equation 20.3 is imaginary. Moreover, we are mostly concerned with systems in

which Q .. 1:

20.5.2 Definition of Q

The quality factor Q is in general defined as 2p times the ratio of the energy of the oscillator to the energy

lost to friction per cycle. For viscous damping (and hysteretic damping, later discussed), the Q is independent

of the amplitude of oscillation. For other types of damping, we will see that the Q is not constant.

In the case of the viscous damped oscillator, Q ¼ v0=2b where b appears in the solution as an amplitude

decay “constant.” The parameter b is not really constant, as discussed in Chapter 21, Section 21.9.

x ¼ A e2bt e^jv0 t

ffiffiffiffiffiffiffiffi

121=4Q2 p

ð20:4Þ

20-20 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Since x is real, we use the real part of Equation 20.4 and employ Euler’s identity to obtain

xðtÞ ¼ A e2bt cosðv1t 2 fÞ; with v1 ¼

ffiffiffiffiffiffiffiffiffiffiffi

v20

2 b2

q

ð20:5Þ

where f is a constant determined by the initial conditions.

20.5.3 Damping “Redshift”

It is seen that the frequency of oscillation depends on the damping constant, b; however, the

fractional change Dv=v0 is almost always negligibly small. For example, the reduction in

frequency is only 1.4% for Q ¼ 3, which is close to critical damping of Q ¼ 0:5: At these small

values of Q, the lifetime of a freely decaying oscillator is so short that the frequency is ill-defined

because of the Heisenberg uncertainty principle. At larger Qs, where the frequency is well-defined,

the shift is negligible; i.e., at Q ¼ 100; the fractional shift is only 1.3 £ 1025. In the case of

internal friction (hysteretic) damping, there is no redshift anyway because the oscillator is

isochronous.

20.5.4 Driven System

When FðtÞ is not zero, but rather corresponds to harmonic drive at angular frequency v and amplitude A,

the response involves the sum of Equation 20.5 (transient) and a particular solution (steady state).

xpðtÞ ¼ Ap cosðvt 2 dÞ; with d ¼ tan21 2vb

v20

2 v2

􀁻 !

: ð20:6Þ

The system resonates (amplitude a maximum) at v ! vR ¼

ffiffiffiffiffiffiffiffiffiffiffiffi

v20

2 2b2

q

; and the variation of the

amplitude with v at steady state at any drive frequency v is given by

Ap ¼

Av2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðv20

2 v2Þ2 þ 4v2b2

q ð20:7Þ

The resonance response curve described by Equation 20.7 is called the Lorentzian. More frequently in

physics, the term is used to describe pressure-broadened line widths (Milonni and Eberly, 1988). As

noted previously, Lorentz was never apparently content with the damping term, 2b dx=dt: In his Ph.D.

dissertation concerned with the damping of electron oscillators through electromagnetic radiation, he

was not able to satisfactorily describe the damping from first principles. Although we might be tempted

to say that this failure derived from his classical (prequantum mechanics) description of the problem,

such a viewpoint is an oversimplification.

20.5.5 Damping Capacity

20.5.5.1 Viscous Damping

The loss per cycle, called the damping capacity, is computed for the viscous damping case as follows (per

unit mass):

dv ¼ 2b

þ

x_ dx ¼ 2bvA2

ð2p

0

sin2 u du ¼ 2pbvA2 ð20:8Þ

where A is the amplitude of the oscillation. Because the total energy per unit mass is v2A2=2; we see

that Q ¼ v=ð2bÞ:

Damping Theory 20-21

© 2005 by Taylor & Francis Group, LLC

20.5.5.2 Hysteretic Damping, Linear Approximation

The equation of motion in this case is given by mx€ þ h=vx_ þ kx ¼ 0 where h is a constant. The energy

loss in one cycle is given by

2DE ¼ mdh ¼

h

v

ðT

0

x_2 dt ¼

h

v

vA2

ð2p

0

cos2 u du ¼ phA2 ! dh ¼

p

m

hA2 ð20:9Þ

so that Q ¼ mv2=h:

20.5.5.3 Hysteretic Damping, Modified Coulomb Model

The nonlinear equation of motion introduced in this chapter to describe hysteretic damping is as follows:

x€þc

ffiffiffiffiffi

2E

k

r

sgnðx_Þþv2x¼0¼x€þ

pv

4Qh

ffiffiffiffiffiffiffiffiffiffiffiffi

v2x2 þx_2

p

sgnðx_Þþv2x¼x€þcAprevsgnðx_Þþv2x ð20:10Þ

where, in the last expression, the subscript “prev” implies amplitude at the last (previous) turning point

of the motion. This particular form for the damping term (Peters, 2002a, 2002b, 2002c), thought to result

from secondary as opposed to primary creep (Peters, 2001a, 2001b), is not as computationally useful as

the middle expression involving the Q. The damping capacity is given by

2DE ¼ mdh ¼ 4cmA

ðp=2

0

A cos u du !dh ¼ 4cA2 ð20:11Þ

yielding Q ¼pv2=ð4cÞ; so that the constant in the nonlinear model is related to the linear approximation

constant through

c ¼ph=ð4mÞ

20.5.6 Coulomb Damping

One of the simplest friction models is that in which a Hooke’s Law spring is connected on one end to a

mass that slides on a level table. The other end of the spring is connected to a stationary wall. The friction

force of the mass against the table is of the type first described quantitatively by Charles Augustin

Coulomb (1736 – 1806), although Leonardo da Vinci is probably the first to consider it scientifically. The

equation of motion and its solution, for the free-decay of an oscillator damped by Coulomb friction, is

given by

mx€ þ f sgnðx_Þ þ kx ¼ 0

Solution

xðtÞ ¼ ½x0 2 ð2n þ 1ÞDx 􀀉cos vt þ ð21ÞnDx

ð20:12Þ

The equation is nonlinear because of the sign of the velocity term, but it is easily integrated numerically;

additionally, it is one of the few nonlinear equations for which an analytic solution is known and is given

above (for more details, the reader is referred to Peters and Pritchett, 1997). The integer, n; specifies the

number of half-cycle turning points from t ¼ 0; and Dx is the decrement (linear, not logarithmic, having

units of m) per half-cycle. There are occasions to use Equation 20.12; for example, problems in civil

engineering where relative motion of members (slipping) occurs at a structural joint. The work against

friction in one cycle can be obtained from energy considerations and is given by

f ð4x0 2 8DxÞ ¼

1

2

kx2

0 2

1

2

kðx0 2 4Dx Þ2 ð20:13Þ

which, for small decrement, yields

Dx ¼

f

k ¼

f

mv2 ð20:14Þ

Damping characteristics for the models presently treated are summarized in Box 20.1.

20-22 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

20.5.7 Thermoelastic Damping

A microphone with Labview was used to analyze vibratory data of an aluminum rod. A rod of 1 m length

can be excited to ear-piercing intensities by holding it at its center between thumb and finger of one hand,

and stroking along the length with the other hand that is coated with violin-bow rosin. The decay of this

“singing rod”, which is a common part of physics demonstration equipment, was found to be in agreement

with the following theoretical expression for thermoeleastic damping (Landau and Lifshitz, 1965):

1

QTh:d ¼

kTa2rv

9C2 ð20:15Þ

where v is the vibrational angular frequency, T is the temperature, r is the density of the bar, C is the

heat capacity per unit volume, a is its thermal expansion coefficient, and k is the thermal

conductivity. The expression assumes adiabatic vibrations and there is no thermoelastic dissipation in

pure shear oscillations (e.g., torsional oscillations of a bar) because the volume does not change and

hence there is no local oscillation of the temperature. Notice, in particular, that the Q is inversely

proportional to frequency, unlike viscous damping that is proportional to the frequency, or hysteretic

damping that is proportional to the square of the frequency. Thermoelastic damping is important for

high-frequency compressional oscillations in materials with significant thermal coefficients, and

especially for metals because of their large thermal conductivity.

The demonstration of comparable behavior in polymers (entropy spring, but opposite sign compared

with metals) is quite easy. Stretch a rubber band between the hands and immediately touch it to the

forehead. The increase in temperature is easily sensed. Conversely, releasing the tension in the band cools

it enough to be sensed by placing the band to a part of the face that is sensitive to temperature change.

Equation 20.15 does not apply to polymers.

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