Пресс-релиз популярных книг

Авторы: 111 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

Книги: 164 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

На сайте 111 авторов, 92 книг, 72 статей, 5913 глав.

20.17 Toward a Universal Model of Damping

Back

20.17.1 Damping Capacity Quadratic in Frequency

The quadratic dependence on frequency of Q (log decrement proportional to period squared) is

equivalent to friction force being frequency-independent. In support for the claim of universality, it was

noted in the Introduction (Section 20.2.2) that three very different systems showed this characteristic:

(i) the vertical seismometer just discussed, (ii) various pendula, and (iii) the rotating rod direct

measurement of internal friction first done by Kimball and Lovell (1927), who measured the transverse

deflection of the end of a rod when it was rotated about a horizontal axis.

3.50

390 K

290 K

T = 27.8 s

T = 18.2 s

bT = 0.45

bT = 0.42

1.00

1.50

2.00

2.50

3.00

3.50

3.00

2.50

2.00

1.50

1.00

FIGURE 20.17 Torsional free-decay records of monofilament nylon at temperatures first below and then above the

glass transition temperature. Although the modulus decreased dramatically at the higher temperature, the damping

did not.

20-48 Vibration and Shock Handbook

20.17.2 Pendula and Universal Damping

An example of one of the author’s experiments that illustrate universal (hysteretic) damping is provided

in Figure 20.14. Other works that illustrate hysteretic damping include those by Peter Saulson of Syracuse

University, who has been frequently cited in the literature (see Saulson et al., 1994).

The pioneering work of Braginsky (important to LIGO) has already been mentioned in the context of

small force measurements and noise. He and his Moscow group members argue that the internal friction

in fused silica may be roughly independent of frequency from 0.1 Hz to 10 kHz (Braginsky et al., 1993).

An oft-cited paper speaking to the issues of hysteretic damping is an article by Quinn et al. (1992)

concerned with material problems in the construction of long-period pendula. (The type of pendulum

on which they based their studies was first described in the scientific literature 2 years earlier (Peters,

1990).) In a follow-on paper, Speake et al. (1999) state the following: “The analogues of anelasticity and

its resultant 1=f noise are seen in a wide range of other processes (for example, dielectric and magnetic

ones) described in terms of frequency-dependent susceptibilities.”

The jerkiness (discontinuous change) that is the hallmark of the Barkhausen effect may have been first

seen mechanically in the experiments that generated the metastable states paper. From a consideration of

the chapter by James Brophy (Brophy, 1965), it was postulated in this 1990 paper that the jerky behavior

of a mesodynamic pendulum is a type of mechanical Barkhausen effect.

20.17.3 Modified Coulomb Model — Background

The results that follow grew naturally out of the

application of fully differential capacitive sensors

to the study of mechanical oscillators. Efforts to

model internal friction influence on long-period

pendula uncovered something surprising to most

— that the foundation for physics laid by Charles

Augustin Coulomb may be much broader than

had been realized. Most individuals in the physics

community do not associate Coulomb’s name with

contributions other than to the laws of electrostatics.

Engineers, however, have long used his

name in the context of sliding friction, since, in

fact, Coulomb gave us the empirical description

which involves static and kinetic coefficients. Because of his interest in the civil engineering of soils

(Heyman, 1997), Coulomb also provided something else — a basis for understanding granular flows and

even some types of fracture. Concerning the latter, the Mohr criterion, applied to the Coulomb failure

envelope, defines a “coefficient of internal friction,” which is used to predict brittle failure (Gere and

Timoshenko, 1996).

Coulomb friction is empirically simple, at least as a first approximation, since it depends only on the

normal force between surfaces and the algebraic sign of the velocity when there is relative motion. Like so

many problems of multibody type, a complete theory of sliding friction is far from being realized.

Simplistic textbook efforts to explain energy loss, by picturing “hills and valleys” of the surface of two solids

in contact, are useless. An example of such naivete can be realized by trying to understand the phenomenon

of optical contacting. Two orthogonally oriented fused silica cylindrical fibers, allowed to touch, can

experience atomic bonding forces that are surprisingly strong, being much greater than the weak attraction

of the van der Waals type. Cleanliness of the surfaces is paramount for success in such a demonstration,

which speaks to another issue — a connection between internal friction and surface physics.

The conversion of mechanical energy to thermal energy must involve nonlinear (avalanche or cascade)

processes. A heuristic description of defect structural interactions that generate heat and eventual failure

is the phonon triangle of Tom Erber (Illinois Tech University) shown in Figure 20.18. The author has

Virgin

state

Cascades

from cycling

(Tidal forcing)

Phonons

(Free-earth modes)

Micro-level

Macro-level

Defect

Organization

Failed

state

(Earthquake)

FIGURE 20.18 Heuristic description of how materials

fail — processes connected with damping.

Damping Theory 20-49

extended Erber’s triangle to include the larger-scale Earth in an attempt to explain earthquakes.

Everybody recognizes that the bending of a wire does not take it from the virgin initial state to the failed

state along the macroscopic upper leg.

There must first be a downward path to the microlevel, through cascading. These cascades can

cause Barkhausen noise in the case of ferromagnetic samples, and acoustic emission in nonmagnetic

metallic alloys (PLC effect). Failure requires the upward path of defect organization, the mechanisms

of which are not yet understood. One of the first theories with possible implications to the

organization leg is that of self-organized criticality. In the magnetic case, Erber has used a fluxgate

magnetometer to improve failure predictions, since magnetic hysteresis is proving to be a sensitive

indicator of mesoscale structure changes during cycling toward failure. Inferred from these studies is

some yet-to-be discovered connectivity between noise, damping, and failure.

Surface friction is expected somehow to be connected with internal friction, the biggest difference

being that the surface has many more defect states with which to redistribute energy. The larger density of

states of the surface (reduced order) is probably an important factor in the difference between surface

friction and the modified internal friction model which follows.

20.17.4 Modified Coulomb Damping Model — Equations of Motion

In the following damping model for internal friction, Coulomb’s law of sliding friction is modified by

assuming that the coefficient of friction is not constant, but rather involves the energy of oscillation E in a

power law; i.e.

mx€ þ cm

􀀒 􀀓l

sgnðx_Þ þ kx ¼ 0; E ¼

mx_2 þ

kx2 ð20:51Þ

where c ¼ constant that is different for each l: For Coulomb (sliding) friction l ¼ 0: For amplitudeindependent

damping of hysteretic type, l ¼ 1

: For amplitude-dependent (such as large Reynolds

number fluid) damping, l ¼ 1: In all cases, if c ,, 1 (small damping), the damping capacity is quadratic

in the frequency, so that the internal friction Q21 , v22. Equation 20.51 is easily implemented, in spite

of its nonlinearity, which we will see later to be a cause for harmonics in the decay.

It is convenient to rewrite Equation 20.51 in canonical form so as to involve the Q of the oscillator. For

the case of hysteretic damping ðl ¼ 1

2 Þ; the equation becomes

x€ þ

4Qh

ffiffiffiffiffiffiffiffiffiffiffiffi

v2x2 þ x_2

sgnðx_Þ þv2x ¼ 0 ð20:52Þ

Similarly, for amplitude-dependent damping ðl ¼ 1Þ

x€ þ

4y0Qf 0 ðv2x2 þ x_2Þ sgnðx_Þ þv2x ¼ 0 ð20:53Þ

where y0 is the initial value of the amplitude of x (largest maximum of x ), and Qf is found not to be

constant, as in the case of hysteretic damping. Rather, in this case, the Q increases as the amplitude

decreases. On the other hand, the Q of an oscillator influenced only by Coulomb (l ¼ 0; sliding) friction

decreases with the amplitude, and the equation of motion in canonical form is given by

x€ þ

pv2y0

4Qc0

sgnðx_Þ þv2x ¼ 0 ð20:54Þ

In Equation 20.53 and Equation 20.54, the subscript 0 is used to identify the initial value of the time

varying Q: Equation 20.54 is equivalent to equation 20.12 with Qc0=y0 ¼ p=ð4Dx Þ:

20-50 Vibration and Shock Handbook

As will be illustrated with some examples, it is possible for an oscillator to be influenced

simultaneously by all three types of friction. One may treat such a system with the following equation of

motion

x€ þ

pv2y0

4Qc0 þ

4Qh

ffiffiffiffiffiffiffiffiffiffiffiffi

v2x2 þ x_2

4y0Qf 0 ðv2x2 þ x_2Þ

" #

sgnðx_Þ þv2x ¼ 0 ð20:55Þ

At any instant during the decay, the total (time-dependent Q) is given by

QðtÞ ¼

Qc þ

Qh þ

Qf ð20:56Þ

in which it is seen that the smallest Q in the set (largest damping term) is dominant in a manner

reminiscent of capacitors connected in series.

It is instructive to look at the analytical solution for the time dependence of the amplitude (turning

points, yðtÞ ¼ lxmaxl), when all the Qs .. 1: Such a solution is obtained from energy considerations by

noting first that the time rate of change of the energy is zero in the absence of friction, i.e.

E_ ¼

mx_2 þ

kx2

􀀏 􀀐

¼ x_ðmx€ þ kxÞ ¼ 0; no friction ð20:57Þ

With friction, dE=dt is determined by the rate of doing work against the friction force; i.e., dE=dt is

proportional to vyf ; where f is the friction force. In the case of Coulomb friction, f is constant

(determined by y0;) so dE=dt is proportional to E1=2: For hysteretic damping, f is proportional to y; so

dE=dt is proportional to E: For fluid damping, f is proportional to y2; so dE=dt is proportional to E3=2:

Thus, the general case is described by

E_ ¼ 2

􀀍

c1 þ c2

ffiffi

E p þ c3E

􀀎 ffiffi

E p ð20:58Þ

Because the energy is proportional to y2, we can write down the equation for the time varying

amplitude as

y_ ¼ 2c 2 by 2 ay2 ð20:59Þ

where a; b, and c are constants. The solution to this first-order equation can be found in integral tables,

and the result depends on the size of c relative to the product ab: For present purposes, we will restrict

ourselves to the case where Coulomb damping is not dominant, in which the solution involves an

exponential. (For large c; one may develop the corresponding general case in terms of the tangent or its

inverse.) The present result is as follows, using r ¼ ðb2 2 4acÞ1=2; where 4ac , b2

with a ¼ 2ay0 þ b 2 r; b ¼ 2ay0 þ b þ r; p ¼

e2rt

y ¼

bðp 2 1Þ þ rðp þ 1Þ

2að1 2 pÞ ð20:60Þ

In the case where c ¼ 0; Equation 20.60 can be simplified to the following form, which is useful for curve

fitting:

y ¼

b þ

􀀏 􀀐

ebt 2

b ð20:61Þ

For the case where a ¼ 0; the better form for curve fitting is

y ¼ y0 þ

􀀏 􀀐

e2bt 2

b ðuntil y ¼ 0Þ ð20:62Þ

Curve-fits based on the modified Coulomb damping model are summarized in Box 20.2.

Damping Theory 20-51

20.17.5 Model Output

Shown in Figure 20.19 is a case in which the decay is influenced by all three types of friction. Notice how

the Q rises initially, peaks at a value less than what would be true for hysteretic damping alone (constant

Q case), and then later declines. The initial rise is due to the amplitude-dependent damping term (size

determined by coefficient a), and the later decline is due to the Coulomb damping term (determined by

coefficient b).

The code in Table 20.1 that was used to generate Figure 20.19 has been reproduced here for two

reasons: (i) to show the ease with which the modified Coulomb model may be numerically applied in

general to a damping problem, and (ii) to illustrate an integration algorithm that has proven to be

intuitive, simple, and powerful — the Cromer– Euler technique, which Alan Cromer first described as the

“last point approximation (LPA)” (Cromer, 1981) in contrast to the unstable “first point approximation”

given to us by Euler. Over the last 20 years, the author has employed the LPA in a host of applications that

span from the generation of satellite ephemerides in the U.S. antisatellite program to both simple and

several-body nonlinear problems of deterministic chaos type.

Box 20.2

CURVE-FIT TO THE TURNING POINTS

If no damping

E_ ¼

mx_2 þ

kx2

􀀏 􀀐

¼ x_ðmx€ þ kxÞ ¼ 0; no friction

with damping (E prop. to y2; E_ prop. to vy · friction force)

E_ ¼ 2

􀀍

c1 þ c2

ffiffi

E p þ c3E

􀀎 ffiffi

E p

equivalent to (c for Coulomb, b for hysteretic, a for fluid)

y_ ¼ 2c 2 by 2 ay2

general solution

with a ¼ 2ay0 þ b 2 r; b ¼ 2ay0 þ b þ r; p ¼

e2rt

y ¼

bðp 2 1Þ þ rðp þ 1Þ

2að1 2 pÞ

special case, c ¼ 0

y ¼

b þ

􀀏 􀀐

ebt 2

special case, a ¼ 0

y ¼ y0 þ

􀀏 􀀐

e2bt 2

b ðuntil y ¼ 0Þ

20-52 Vibration and Shock Handbook

20.17.6 Experimental Examples

The code of Table 20.1 is useful in determining the nature of a given experimental case. Too frequently in

the past, it has been naively assumed that the entire decay record was exponential. Particularly when

longer records are collected, it is found that most damping is nonlinear. In the two experimental

examples that follow to backup this claim, one is a near-perfect (nonlinear) particular case of amplitudedependent

(fluid) damping, and the other is a mixture of amplitude-dependent and hysteretic damping,

but devoid of any Coulombic influence. Coulomb friction frequently tends to be either “all or nothing,”

depending on whether there is an unwanted mechanical contact that involves slippage. A notable

exception is found in the case where a pendulum is influenced by eddy current damping in a narrow

region of its total motion (Singh et al., 2002).

The pendulum that was used to generate the data displayed in Figure 20.14 was also used as follows. A

large flat piece of plastic was attached to the bottom of the pendulum, so that its movement (normal

vector to the surface in the direction of motion) would disturb a great amount of air in turbulent manner.

As expected, there was a dominant initial (large level) amplitude-dependent damping, as shown in

Figure 20.20.

The speed (maximum) was measured with a photogate as previously discussed. The expressions

shown in the figure are consistent with Equation 20.59 and Equation 20.61. The data, which were

collected by hand and typed into Excel, produced the “jagged” curve, and the computerized fit

according to Equation 20.61 is the smooth curve of the pair. It is noteworthy that the quadratic drag of

the air (determined by a ¼ 0:036) is 40% greater than the viscous drag at the start of the decay. By cycle

37, the quadratic part has become much less significant than the constant Q viscous part, having

become roughly 60% smaller.

The fluid damping “soup-can” pendulum data of Figure 20.21 was generated with a can of Bush’s

black-eye peas. The container with enclosed unbroken contents, being a right circular cylinder of length

11 cm £ diameter 7.4 cm, was suspended horizontally by a pair of knife edges under opposing end-lips

x(t)

Time

Q total

y(t) Q due to b only (63)

DAMPING COEFFICIENTS: a = 0.1, b = 0.1, c = 0.01

initial Q = 12, initial amplitude = 4, period = 0.5, final Q = 22

FIGURE 20.19 Model generated results based on Equation 20.55 and Equation 20.60.

Damping Theory 20-53

(Peters, 2002a, 2002b, 2002c). The motion of the can was measured with an SDC sensor connected to a

Dataq A/D converter. Whereas experiments of similar type, with homogeneous fluid contents, have

produced viscous decay records, the present case involved only friction of so-called “fluid” type; i.e.,

quadratic in the “velocity.” To generate the figure, the A/D record was exported to the Microsoft software

package, Excel. Fits to the data were then obtained by adjusting, through trial and error, the a, b; and c

TABLE 20.1 QuickBasic Code to Calculate Amplitude History yðtÞ and Integrate Equation of Motion to Obtain

xðtÞ; Accommodates Three Common Forms of Friction

CLS

REM: setup display

SCREEN 12: VIEW (0, 0) 2 (600, 470): WINDOW (2 .2, 2 5) 2 (1, 5)

REM: assign constants and initialize variables

pi ¼ 3.1416: dt ¼ 0:002: t ¼ 0

x0 ¼ 4: x ¼ x0 : y0 ¼ x0 : xd ¼ 0

Period ¼ .5: omega ¼ 2ppi/period: b ¼ :1: a ¼ :1: c ¼ :01

REM: print damping coefficients

PRINT “DAMPING COEFFICIENTS: a ¼ ”; a; “, b ¼ ”; b; “, c ¼ ”; c

r ¼ SQRðb^2 2 4p ap cÞ: alpha ¼ 2p ap x0 þ b 2 r

Beta ¼ 2p ap x0 þ b þ r

REM: Use a, b and c — set Q’s to dampen (quadratic, linear, and constant resp.)

qf ¼ omega/2/a/y0: qh ¼ omega/2/b: qc ¼ y0

p omega/2/c

REM: start integration loop

LOOP0:

t ¼ t þ dt

REM: analytically compute amplitude (y ¼ magnitude of x) at each time point

p ¼ alphapEXPð2r p tÞ=beta

y ¼ ðbp ðp 2 1Þ þ r p ðp þ 1ÞÞ=2=a=ð1 2 pÞ

REM: integr. the eq. of motion to get xðtÞ; using 3 fric. force/mass terms

REM: The coeff.’s ff, fh & fc correspond to: quadratic in speed (fluid),

REM: linear in speed (hysteretic), and independ. of speed (Coulomb) resp.

ff ¼ (pi/4)p(1/y0)p(1/qf)p(omega^2 p x^2 þ xdot^2)

fh ¼ (pi/4)p (omega/qh)p SQR(omega^2 p x^2 þ xdot^2)

fc ¼ (pi/4)p omega^2p y0/qc

REM: check algebraic sign – USE SIGN BUT NOT MAGNITUDE OF VELOCITY

IF xdot . 0 THEN GOTO SKIP

ff ¼ 2 ff: fh ¼ 2 fh: fc ¼ 2 fc

SKIP: xdoubledot ¼ 2 ff 2 fh 2 fc 2 omega^2 p x

xdot ¼ xdot þ xdoubledot p dt: x ¼ x þ xdot p dt

REM: calculate the energy and then the amplitude to evaluate Q

REM: could instead use analytical result q ¼ ðpi=4Þpomega2p x=absðff þ fh þ fcÞ

Energy ¼ .5 p xdot^2 þ .5 p omega^2 p x^2

Amplitude ¼ SQR(2 p energy)/omega

REM: calculate loss per period due to friction

loss ¼ ABS(ff þ fh þ fc) p 4 p amplitude

q ¼ 2 p pi p energy/loss

IF t , 1:2p dt THEN PRINT “initial Q ¼ ”; 10pINTðqÞ=10;

IF t , 20 THEN GOTO SKIP2

PRINT “, initial Amplitude ¼ ”; x0; “, Period ¼ ”; period;

PRINT “, final Q ¼ ”; 10pINTðqÞ=10

REM: DO GRAPH

SKIP2: PSET(.04 p t, .5 p q/omega): PSET (.04 p t, .5 p qh/omega), 4

PSET (.04 p t, 4 p x/y0): PSET (.04 p t, 0): PSET (.04 p t, 48 p y/y0)

IF t . 20 OR y , 0 THEN GOTO pause

GOTO LOOP0

Pause: GOTO pause

RETURN: END: STOP

20-54 Vibration and Shock Handbook

coefficients of a “fit” to the amplitude. For this case, the fit was easily accomplished because both b and c

proved to be essentially zero.

The second case, involving an evacuated pendulum, was not a single pure type of damping, but can be

seen in Figure 20.22 to have both hysteretic and amplitude-dependent contributions. Although fluid

damping is amplitude-dependent in the same manner, with the damping term being proportional to the

square of the amplitude, the word “fluid” is not used to describe this case since the system involved

exclusively solid materials.

Not all decay records of this pendulum in vacuum yielded a mix of friction types as displayed in the

figure. The effect was observed to be transient, and it is speculated that outgassing of components may

have been a factor.

vmax vs cycle number n

1/y−(a/b+1/y0) exp(bn)−a/b

a = 0.036, b = 0.025, y0 = 7.69

−dy/dn = by+ay*y

speed (cm/s)

0 10 20

30 40

FIGURE 20.20 Decay of an air-damped pendulum as a function of cycle number n:

4.00

3.00

2.00

1.00

0.00

–1.00

Period = 0.5 s

–2.00

–3.00

–4.00

Sensor output (V)

0 5 10 15 20

Time (s)

b = 0, c = 0, a = 0.22 s/v

FIGURE 20.21 Example of fluid damping of a “soup-can” pendulum. The granular contents (black-eye peas and

water) result in a friction force that is quadratic in the velocity.

Damping Theory 20-55

20.17.6.1 Numerical Integration

Instead of integrating the second-order equation of motion twice — first the acceleration, followed by the

resulting velocity — more accurate results are obtained by integrating the equivalent pair of first-order

equations (also see Chapter 6).

For example, the equation of the simple harmonic oscillator with viscous damping is expressible as

p_ ¼ 2q 2 kp q_ ¼ p ð20:63Þ

where the position variable has been represented by the generalized coordinate q (x elsewhere), and for

the momentum p ¼ m dq=dt; and here the mass, m; has been set to unity. Likewise, the spring constant

has been set to unity. It is generally useful to distill a given problem to its most basic form when

attempting to understand the physics. Constants that provide no useful information for trend analysis

purposes are conveniently “normalized.” Such is common practice, for example, in modeling chaotic

systems.

The second-order set can always be reduced mathematically to a first-order pair; however, the pair

results naturally from the use of the Hamiltonian as opposed to the Newtonian formulation of

mechanics.

20.17.7 Damping and Harmonic Content

Equation 20.52 to Equation 20.54 are the nonlinear, modified Coulomb damping model forms that

correspond, respectively, to (i) hysteretic, (ii) amplitude-dependent, and (iii) Coulomb damping. The

damping term for each of the three cases can be expressed as follows:

m ¼

Q ½v􀀉 ð20:64Þ

where f is the friction force, and ½v􀀉 is the square wave whose fundamental in a Fourier series expansion is

equal to the velocity of the oscillator times 4/p; i.e., for a square wave ^h; the amplitude of the

fundamental is ^ð4=pÞh: We see that all the damping types that have been considered in this chapter,

when expressed in canonical form, correspond to a fundamental friction force f ¼ mvv=Q: The simplicity

of this result is probably why viscous damping has been viewed by so many physicists as “inviolate.” One

must be careful, however, because (as noted in the previous section) only for the case of hysteretic

damping is Q constant. For amplitude-dependent damping Qf ¼ Qf 0ðy0=yÞ and for Coulomb damping

1.50

1.00

0.50

0.00

−0.50

−1.00

−1.50

0 200 400

Sensor output (V)

Period = 1.0 s

600 800 1000 1200 1400

Qi = 3000, Qf = 8000

b = 0.0004 s−1, a = 0.00120 s/v, c = 0

a = 0.00141 s/v, b = 0

time (s)

.0011s−1, a = 0

.0017s− 1 ,a=0

FIGURE 20.22 Example of a mix of two damping types, hysteretic and amplitude-dependent.

20-56 Vibration and Shock Handbook

Qc ¼ Qc0ðy=y0Þ: The time-dependent Q of nonexponential cases will have significant influence on mode

development in many-body systems because of elastic nonlinearity (necessary for mode coupling).

There is another important subtlety of Equation 20.64. When only the fundamental of ½v􀀉 is retained,

equivalent to viscous damping, Q is proportional to frequency. When all odd harmonics are included

(full square wave), Q becomes proportional to frequency squared. This means that harmonics in the

friction force are responsible for the primary difference between hysteretic damping and viscous damping.

Something being presently considered is how, in an algorithmic sense, to modify Equation 20.52 to

Equation 20.54 to provide for “dispersion,” i.e., means for providing Q dependence other than frequency

squared. We posit the following: that hysteretic (exponential) damping is the idealized universal form of

damping due to secondary creep. When there is an activation process of Zener (Debye) type, such as

dislocation relaxation, then additional terms must be added to the hysteretic “background.” It may be

that this can be accommodated by a suitable removal of harmonics from the square wave of the hysteretic

case, and it may happen that Q is constant for systems that vary continuously. It is conjectured that the

PLC effect, responsible for discontinuous changes, plays a role in those cases where Q is not constant.

Equations of motion based on the modified Coulomb damping model are summarized in Box 20.3.

Box 20.3

EQUATIONS OF MOTION BASED ON

NONLINEAR DAMPING

Equation of motion in terms of energy

mx€ þ cm

􀀒 􀀓2

sgnðx_Þ þ kx ¼ 0; E ¼

mx_2 þ

kx2

Hysteretic-only damping (exponential)

x€ þ

4Qk

ffiffiffiffiffiffiffiffiffiffiffiffi

v2x2 þ x_2

sgnðx_Þ þv2x ¼ 0

Velocity-square (fluid) damping

x€ þ

4y0Qf 0 ðv2x2 þ x_2Þ sgnðx_Þ þv2x ¼ 0

Coulomb damping

x€ þ

pv2y0

4Qc0

sgnðx_Þ þv2x ¼ 0

All three damping types simultaneously active

x€ þ

pv2y0

4Qc0 þ

4Qk

ffiffiffiffiffiffiffiffiffiffiffiffi

v2x2 þ x_2

4y0Qf 0 ðv2x2 þ x_2Þ

" #

sgnðx_Þ þv2x ¼ 0

Quality factor

QðtÞ ¼

Qc þ

Qk þ

Damping Theory 20-57

Пресс-релиз популярных книг

20.17 Toward a Universal Model of Damping

Популярные книги

Популярные статьи

Навигация