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.

30.3 Pseudoexcitation Method for Structural Random Vibration Analysis

Back

30.3.1 Structures Subjected to Stationary Random Excitations

30.3.1.1 Single Stationary Random Excitations

The basic principle of the PEM for structural stationary random vibration analysis can be explained by

Figure 30.6. Consider a linear system subjected to a zero-mean stationary random excitation xðtÞ

Sxx

(a)

x(t)

y(t)

z(t)

x(t)=

(b)

(c)

Linear

Structure

Linear

Structure

Linear

Structure

Sxx(w)exp(iwt)

Syy(w) = Hy

*(w)Sxx(w)Hy(w)

Syz(w) = Hy

*(w)Sxx(w)Hz(w)

∼ y = Sxx(w)Hy(w)exp(iwt) ∼

∼z = Sxx(w)Hz(w)exp(iwt)

FIGURE 30.6 Basic principle of pseudoexcitation method (stationary analysis).

30-16 Vibration and Shock Handbook

(see Figure 30.6a) and with a given PSD Sxx ðvÞ: Suppose that for two arbitrarily selected responses yðtÞ

and zðtÞ; the auto-PSD Syy ðvÞ and cross-PSD Syz ðvÞ are desired. Figure 30.6b gives the conventional

formulas for computing these PSD functions. Hy ðvÞ and Hz ðvÞ are the frequency-response functions,

that is, if xðtÞ is replaced by a sinusoidal excitation expðivtÞ; the harmonic responses of yðtÞ and zðtÞ

would be Hy ðvÞexpðivtÞ and Hz ðvÞexpðivtÞ; respectively. Thus, if xðtÞ is replaced by a sinusoidal

excitation (Lin et al., 1994b)

x~ ¼

ffiffiffiffiffiffiffiffi

Sxx ðvÞ

expðivtÞ ð30:94Þ

the responses of yðtÞ and zðtÞ would be y~ ¼

ffiffiffiffiffiffiffiffi

pSxxðvÞHyðvÞexpðivtÞ and z~ ¼

ffiffiffiffiffiffiffiffi

pSxxðvÞHzðvÞexpðivtÞ

(see Figure 30.6c). It can be readily verified that

y~p y~ ¼

ffiffiffiffiffiffiffiffi

Sxx ðvÞ

y ðvÞexpð2ivtÞ

ffiffiffiffiffiffiffiffi

Sxx ðvÞ

Hy ðvÞexpðivtÞ ¼ lHy ðvÞl2Sxx ðvÞ ¼ Syy ðvÞ ð30:95Þ

y~p z~ ¼

ffiffiffiffiffiffiffiffi

Sxx ðvÞ

y ðvÞexpð2ivtÞ

ffiffiffiffiffiffiffiffi

Sxx ðvÞ

Hz ðvÞexpðivtÞ ¼ Hp

y ðvÞSxx ðvÞHz ðvÞ ¼ Syz ðvÞ ð30:96Þ

If yðtÞ and zðtÞ are two arbitrarily selected random response vectors of the structure, and y~ ¼ ayexpðivtÞ

and z~ ¼ azexpðivtÞ are the corresponding harmonic response vectors due to the pseudo-excitation

(30.94), it can also be proven that the PSD matrices of yðtÞ and zðtÞ are

SyyðvÞ ¼ y~p y~T ¼ ap

y aT

y ð30:97Þ

SyzðvÞ ¼ y~p z~T ¼ ap

y aT

z ð30:98Þ

This means that the auto- and cross-PSD functions of two arbitrarily selected random responses can be

computed using the corresponding pseudoharmonic responses.

Now, consider a structure subjected to a single seismic random excitation. Its equations of motion are

My€ þ Cy_ þ Ky ¼ 2Mex€gðtÞ ð30:99Þ

in which the ground acceleration x€g ðtÞ is a stationary random process. Its PSD Sx€g ðvÞ is known and e is a

given constant vector, indicating the distribution of inertia forces. In order to solve Equation 30.99, let

the pseudoground acceleration be

x€~gðtÞ ¼

ffiffiffiffiffiffiffiffi

Sx€g ðvÞ

expðivtÞ ð30:100Þ

then Equation 30.99 becomes

My€~ þ Cy_~ þ Ky~ ¼ 2Me

ffiffiffiffiffiffiffiffi

Sx€g ðvÞ

expðivtÞ ð30:101Þ

and its stationary solution is

y~ðtÞ ¼ ayðvÞexpðivtÞ ð30:102Þ

Using its first q normalized modes for mode-superposition, then (Clough and Penzien, 1993)

ay ðvÞ ¼

j¼1

gjHjwj

ffiffiffiffiffiffiffiffi

Sxx ðvÞ

ð30:103Þ

in which vj; wj; 6j; Hj and gj are the jth natural angular frequency, mass normalized mode, damping ratio,

frequency-response function, and mode participation factor, respectively. According to PEM, the PSD

matrix of y is

SyyðvÞ ¼ y~p y~T ¼ ap

y ðvÞaT

y ðvÞ ð30:104Þ

Substituting Equation 30.102 into Equation 30.104 and expanding it also gives Equation 30.31. This

means these two equations are mathematically identical. However, the computational effort required by

Equation 30.104 is approximately only 1=q2 of that required by Equation 30.31. Therefore, Equation

30.104 is also known as the fast CQC algorithm (Lin, 1992).

Seismic Random Vibration of Long-Span Structures 30-17

Example 30.1 Derivation of the Kanai – Tajimi PSD Formula

Consider the system shown in Figure 30.7(a). The single-layered homogeneous soil and the super single-

DoF structure can be modeled by the system of Figure 30.7(b). Assume that the horizontal acceleration of

the bedrock x€0 is a stationary random process with white noise spectrum S0: The ground displacement

relative to the bedrock is y, and the displacement of the superstructure (assumed to be an single-DoF

system) relative to the ground is x. The effective horizontal shear-resistant stiffness of the soil layer is kg;

while the corresponding damping coefficient is cg: Assuming that m is very small in comparison with the

equivalent ground mass mg; then the equation of motion of the effective ground mass is

mg y€ þ cg y_ þ kg y ¼ 2mgx€0 ðiÞ

y€ þ 26gvg y_ þv2

g y ¼ 2x€0 ðiiÞ

in which v2

g ¼ kg=mg; 26gvg ¼ cg=mg:

Now, form the horizontal pseudoacceleration for the bedrock

x€~0 ¼

ffiffiffi

expðivtÞ ðiiiÞ

Substituting it into Equation ii gives

y€~ þ 26gvg

y_~ þv2

gy~ ¼ 2

ffiffiffi

expðivtÞ ðivÞ

Its right-hand side is harmonic. Therefore, the stationary solution of y~ is

y~ ¼

ffiffiffi

S0 p

g 2 v2 þ 2i6gvgv

expðivtÞ ðvÞ

The pseudoabsolute displacement of the ground is

x~g ¼ x~0 þ y~ ðviÞ

The corresponding pseudoabsolute acceleration is

x€~g ¼ x€~0 þ y€~ ¼

ffiffiffi

expðivtÞ 1 þ

g 2 v2 þ 2i6gvgv

" #

ffiffiffi

expðivtÞ

g þ 2i6gvgv

g 2 v2 þ 2i6gvgv ðviiÞ

Using PEM, the PSD of x€~g is

Sx€g ðvÞ ¼ x€~pg

x€~g ¼ S0

g þ 462

gv2

ðv2

g 2 v2Þ2 þ 462

gv2

gv2 ðviiiÞ

This is exactly the Kanai – Tajimi filtered white noise spectrum formula (Clough and Penzien, 1993).

(a) x0 (b)

m x

k c

k c y

kg cg

.. x0

wg Vg

FIGURE 30.7 Structural modeling for Kanai – Tajimi PSD formula.

30-18 Vibration and Shock Handbook

30.3.1.2 Multiple Stationary Random Excitations

Consider a linear structure subjected to a number of stationary random excitations, which are denoted

as an m-dimensional stationary random process vector xðtÞ with known PSD matrix Sxx ðvÞ: It is

a Hermitian matrix and so it can be decomposed, for example, by using its eigenpairs cj and

dj ðj ¼ 1; 2; …; rÞ; into

Sxx ðvÞ ¼

j¼1

djcp

j cT

j ðr # mÞ ð30:105Þ

in which r is the rank of Sxx ðvÞ: Next, form the r pseudoharmonic excitations

x~jðtÞ ¼

ffiffiffi

cjexpðivtÞ ð j ¼ 1; 2; …; rÞ ð30:106Þ

By applying each of these pseudoharmonic excitations, two arbitrarily selected response vectors yjðtÞ and

zjðtÞ of the structure, which can be displacements, internal forces, or other linear responses, may be easily

obtained and expressed as

y~jðtÞ ¼ ayjðvÞexpðivtÞ ð30:107Þ

z~jðtÞ ¼ azjðvÞexpðivtÞ ð30:108Þ

The corresponding PSD matrices can be computed by means of the following formulas (Lin et al., 1994a;

Zhong, 2004):

Syy ðvÞ ¼

j¼1

y~p

j ðtÞy~T

j ðtÞ ¼

j¼1

yjðvÞaT

yjðvÞ ð30:109Þ

Syz ðvÞ ¼

j¼1

y~p

j ðtÞz~T

j ðtÞ ¼

j¼1

yjðvÞaT

zjðvÞ ð30:110Þ

The method used to decompose Sxx ðvÞ into the form of Equation 30.105 is not unique. In fact, the

Cholesky scheme is perhaps the most efficient and convenient way to do it; that is, Sxx ðvÞ is decomposed

into

Sxx ðvÞ ¼ LpDLT ¼

j¼1

djlp

j lT

j ðr # mÞ ð30:111Þ

in which L is a lower triangular matrix with all its diagonal elements equal to unity and D is a real

diagonal matrix with r nonzero diagonal elements dj: The implementation of Cholesky decomposition

for a Hermitian matrix is very similar to that for a real symmetric matrix (Wilkinson and Reinsch, 1971).

Example 30.2 A Massless Trolley Subjected to Excitations with Phase-Lags

The massless trolley shown in Figure 30.8 is connected to two abutments by springs and viscous dashpots

(linear viscous dampers) as shown. The abutments move distances x1ðtÞ and x2ðtÞ with spectral densities

S0 (constant), but x2ðtÞ ¼ x1ðt 2 TÞ; in which T is a fixed time difference. The response spectral

density of the trolley displacement Syy ðvÞ is to be determined. The equation of motion of the trolley is

(Newland, 1975)

ðc1 þ c2Þy_ þ ðk1 þ k2Þy ¼ k1x1 þ c1x_1 þ k2x2 þ c2x_2 ðiÞ

Since the PSD of x1ðtÞ is S0; the pseudoexcitation corresponding to x1ðtÞ is

x~1ðtÞ ¼

ffiffiffi

expðivtÞ ðiiÞ

Seismic Random Vibration of Long-Span Structures 30-19

Because x2ðtÞ ¼ x1ðt 2 TÞ; we have

x~2ðtÞ ¼

ffiffiffi

exp½ivðt 2 TÞ􀀉 ðiiiÞ

Clearly

x_~1ðtÞ ¼ iv

ffiffiffi

expðivtÞ;

x_~2ðtÞ ¼ iv

ffiffiffi

exp½ivðt 2 TÞ􀀉

ðivÞ

Substituting the above equations into Equation i

gives the harmonic equation

ðc1 þ c2Þy_~ þ ðk1 þ k2Þy~

¼ ½ðk1 þ ivc1Þ þ ðk2 þ ivc2Þ

expð2ivTÞ􀀉expðivtÞ ðvÞ

Its solution can be readily obtained as

y~ ¼

k1 þ ivc1 þ ðk2 þ ivc2Þexpð2ivTÞ

k1 þ k2 þ ivðc1 þ c2Þ

expðivtÞ ðviÞ

Hence,

Syy ¼ y~p y~ ¼

k21

þ k22

þ c2

1v2 þ c2

2v2 þ 2ðk1k2 þ c1c2v2Þcos vT þ 2ðk1c2v 2 k2c1vÞsin vT

ðk1 þ k2Þ2 þ ðc1 þ c2Þ2v2 S0 ðviiÞ

This result is identical to that given by Newland (1975). However, the process given here is quite

simple.

30.3.2 Structures Subjected to Nonstationary Random Vibration

30.3.2.1 Structures Subjected to Uniformly Modulated Evolutionary Random Excitations

30.3.2.1.1 Single Excitation Problems

The basic principle of the PEM for nonstationary random vibration analyses can be described by

Figure 30.9. Consider a linear system subjected to an evolutionary random excitation (see Figure 30.9a)

f ðtÞ ¼ gðtÞxðtÞ ð30:112Þ

in which gðtÞ is a slowly varying modulation function, while xðtÞ is a zero-mean stationary random

process with auto-PSD Sxx ðvÞ: The deterministic functions gðtÞ and Sxx ðvÞ are both assumed to be given.

y(t)

z(t)

y(w, t)

z(w, t)

f(t) = g(t)x(t)

~ ~

Sff (w,t) = g2(t)Sxx(w)

Syy(w, t) = y*yT

S ~ ~ yz(w, t) = y*zT

Linear

Structure

Linear

Structure

Linear

Structure

(a)

(b)

(c)

f(t) = Sxx(w)g(t)exp(iwt)

∼

FIGURE 30.9 Basic principle of pseudoexcitation method (nonstationary analysis).

x1(t)

x2(t)

y(t)

FIGURE 30.8 Two-phase input trolley.

30-20 Vibration and Shock Handbook

The auto-PSD of f ðtÞ (see Figure 30.9b) is

Sff ðv; tÞ ¼ g2ðtÞSxx ðvÞ ð30:113Þ

In order to compute the PSD functions of various linear responses due to the action of f ðtÞ; we note

that the pseudoexcitation has the form

f~ðv; tÞ ¼ gðtÞ

ffiffiffiffiffiffiffiffi

Sxx ðvÞ

expðivtÞ ð30:114Þ

Now suppose that yðtÞ and zðtÞ are two arbitrarily selected response vectors (see Figure 30.9a) and y~ðv; tÞ

and z~ðv; tÞ are the corresponding transient responses due to the pseudoexcitation f~ðv; tÞ with the

structure initially at rest (see Figure 30.9b). It has been proven (Lin et al., 1994a; Lin et al., 2004) that the

desired PSD matrices of yðtÞ and zðtÞ are

Syyðv; tÞ ¼ y~p ðv; tÞy~Tðv; tÞ ð30:115Þ

Syzðv; tÞ ¼ y~pðv; tÞz~Tðv; tÞ ð30:116Þ

as shown in Figure 30.9c.

Now, consider the Equation 30.99 of a linear structure subjected to the evolutionary random excitation

x€g ðtÞ ¼ gðtÞxðtÞ: The pseudoground acceleration is now x€~g ðtÞ ¼ gðtÞ

ffiffiffiffiffiffiffiffi

pSxxðvÞexpðivtÞ: Substituting this

into the right-hand side of Equation 30.99 gives the deterministic equations

My€~ þ Cy_~ þ Ky~ ¼ 2MegðtÞ

ffiffiffiffiffiffiffiffi

Sxx ðvÞ

expðivtÞ ð30:117Þ

For seismic problems, the structure is initially at rest, that is, y~ ¼ y_~ ¼ 0 when t ¼ 0: Thus, the time

history of y~ðv; tÞ can be computed using the Newmark or Wilson-u schemes. Furthermore, any other

linear pseudo response vectors, denoted as u~ ðv; tÞ and v~ðv; tÞ; can be computed from y~ðv; tÞ: The

response PSD matrices of uðtÞ and vðtÞ can also be accurately computed by using their pseudoresponses

u~ ðv; tÞ and v~ðv; tÞ

Suuðv; tÞ ¼ u~ pðv; tÞu~ Tðv; tÞ ð30:118Þ

Suvðv; tÞ ¼ u~ pðv; tÞv~Tðv; tÞ ð30:119Þ

Example 30.3 Single-Degree of Freedom System Subjected to Suddenly

Applied Stationary Random Excitation

Consider the following single-DoF example, which was first given by Caughey and Stumpf (1961) and

has been widely used to compare the efficiency and precision of various methods:

y€ þ 26v0y_ þv20

y ¼ f ðtÞ ¼ gðtÞxðtÞ; yð0Þ ¼ y_ð0Þ ¼ 0 ðiÞ

in which

gðtÞ ¼

1:0 t $ 0

0 t , 0

(

ðiiÞ

and xðtÞ is a zero-mean-valued stationary random process with its PSD Sxx ðvÞ given. Now, constitute a

pseudo excitation

f~ðtÞ ¼

ffiffiffiffiffiffiffiffi

Sxx ðvÞ

expðivtÞ ðt $ 0Þ ðiiiÞ

Thus, the motion equation that should be solved is

y€~ þ 26v0 y_~ þv20

y~ ¼

ffiffiffiffiffiffiffiffi

Sxx ðvÞ

expðivtÞ; y~ð0Þ ¼ y_~ð0Þ ¼ 0 ðivÞ

Seismic Random Vibration of Long-Span Structures 30-21

The solution of this simple equation is

y~ðv; tÞ ¼ H expðivtÞ 2 expð21tÞ cos v1t þ

iv þ 1

sin v1t

􀀘 􀀒 􀀓􀀙 ffiffiffiffiffiffiffiffi

Sxx ðvÞ

ðvÞ

in which

H ¼ ðv20

2 v2 þ 2i6v0vÞ21; 1 ¼ 6v0; v1 ¼ v0

ffiffiffiffiffiffiffiffi

1 2 62

ðviÞ

Therefore, the auto-PSD of y is

Syyðv; tÞ ¼ y~p y~ ¼ ly~l2 ¼ lHl2 expð21tÞ

sin v1t þ cos v1t

􀀏 􀀐

2 cos vt

􀀒 􀀓2 􀀘

þ expð21tÞ

sin v1t 2 sin vt

􀀒 􀀓2􀀙

Sxx ðvÞ

ðviiÞ

This result is identical to the exact solution given by Caughey and Stumpf (1961), who used a more

complicated process and expressed it less concisely as

Syy ðv; tÞ ¼ lHl2½1 þ expð221tÞ􀀉

1 þ

2v0

6 sin v1t cos v1t 2 expð1tÞ 2 cos v1 t þ

2v0

6 sin v1t

􀀏 􀀐

cos vt:

2expð1tÞ

sin v1 t sin vt þ ðv06Þ2 2 v2

1 þv2

sin2v1t

Sxx ðvÞ

ðviiiÞ

30.3.2.1.2 Multiple Excitation Problems

Fully coherent excitations. In order to include the phase-lags between ground excitations, that is, the

wave passage effect, the evolutionary random excitation vector fðtÞ to which the structure is subjected

should be

f ðtÞ ¼

F1ðtÞ

F2ðtÞ

.. .

FnðtÞ

8>>>>>><

>>>>>>:

9>>>>>>=

>>>>>>;

a1gðt 2 t1ÞFðt 2 t1Þ

a2gðt 2 t2ÞFðt 2 t2Þ

.. .

angðt 2 tnÞFðt 2 tnÞ

8>>>>>><

>>>>>>:

9>>>>>>=

>>>>>>;

¼ GðtÞf􀀊ðtÞ ð30:120Þ

in which

GðtÞ ¼

a1gðt 2 t1Þ

a2gðt 2 t2Þ

. .

angðt 2 tnÞ

66666664

77777775

; f􀀊ðtÞ ¼

Fðt 2 t1Þ

Fðt 2 t2Þ

.. .

Fðt 2 tnÞ

8>>>>>><

>>>>>>:

9>>>>>>=

>>>>>>;

ð30:121Þ

Here, all components of f􀀊ðtÞ clearly have the same form, although there are time lags tjðj ¼ 1; 2;…; nÞ

between them; gðtÞ is the modulation function; ajðj ¼ 1; 2; …; nÞ are given real numbers; and tj are given

constants. FðtÞ is a stationary random process, of which the auto-PSD SFF ðvÞ is known. The pseudoexcitations

corresponding to {fðtÞ} are (Lin and Zhang, 2004)

f~ðtÞ ¼ GðtÞVq0

ffiffiffiffiffiffiffiffi

SFF ðvÞ

expðivtÞ ð30:122Þ

30-22 Vibration and Shock Handbook

in which

R0 ¼ q0qT

0 ¼

1 1 · · · 1

.. .

. .

. .. .

1 1 · · · 1

6666664

7777775

V ¼ diag½expð2ivt1Þ; expð2ivt2Þ; …; expð2ivtnÞ􀀉

0 ¼ {1; 1; …; 1}

ð30:123Þ

The excitation PSD matrix can be written as

Sff ðv; tÞ¼f~pðtÞf~TðtÞ ¼ SFF ðvÞGðtÞVpq0qT

0 VTGTðtÞ ¼ SFF ðvÞ

􀀐

g2ðt 2 t1Þ gðt 2 t1Þgðt 2 t2Þexpðivðt12 t2ÞÞ · ·· gðt 2 t1Þgðt 2 tnÞexpðivðt12 tnÞÞ

gðt 2 t2Þgðt 2 t1Þexpðivðt22 t1ÞÞ g2ðt 2 t2Þ · ·· gðt 2 t2Þgðt 2 tnÞexpðivðt22 tnÞÞ

.. .

. .

. .. .

gðt 2 tnÞgðt 2 t1Þexpðivðtn2 t1ÞÞ gðt 2 tnÞgðt 2 t2Þexpðivðtn2 t2ÞÞ · ·· g2ðt 2 tnÞ

6666666664

7777777775

ð30:124Þ

An arbitrarily chosen response y~kðv; tkÞ excited by f~ðtÞ can be computed using a numerical integration

scheme and the results can be expressed as

y~kðv; tkÞ¼aykðv; tÞ

ffiffiffiffiffiffiffiffi

SFF ðvÞ

ð30:125Þ

in which

aykðv; tkÞ ¼

ðtk

Hðtk 2 tkÞGðtkÞVq0expðivtkÞdtk ð30:126Þ

Here, aykðv; tÞ is the response excited by GðtÞVq0expðivtÞ and HðtÞ is the impulse-response function

matrix. The cross-PSD matrix between ykðtÞ and ylðtÞ is

Sykyl ðv; tÞ¼y~p

k ðv; tÞy~T

l ðv; tÞ ð30:127Þ

By letting k ¼ l; the auto-PSD matrix of ykðtÞ can be directly computed by Equation 30.127.

Partially coherent excitations. In addition to the time lags between excitations, if the arbitrary coherence

between such nonstationary excitations is also taken into account, then the problem is more difficult.

According to PEM, however, it is only required that the matrix R0 in Equation 30.123 is changed into

R ¼

1 r12 · · · r1n

r21 1 · · · r2n

.. .

. .

. .. .

rn1 rn2 · · · 1

66666664

77777775

ð30:128Þ

in which rij reflects the coherence between the excitations at points i and j, and matrix R is usually

symmetric and positive definite or semipositive definite. Denoting its rank as rðr $ 1Þ; R can be

Seismic Random Vibration of Long-Span Structures 30-23

expressed as

R ¼

j¼1

ajcp

j cT

j ð30:129Þ

This means that the global excitation PSD matrix is decomposed into r matrices with rank unity where

the jth of them corresponds to the pseudoexcitation

f~jðtÞ ¼ GðtÞVcj

ffiffiffiffiffiffiffiffiffiffiffi

ajSFF ðvÞ

expðivtÞ ð30:130Þ

If y~kjðtÞ and y~ljðtÞ are two arbitrary responses due to f~jðtÞ; then

Syk yl ðv; tÞ ¼

j¼1

y~p

kjðtÞy~T

lj ðtÞ ð30:131Þ

By letting k ¼ l; the auto-PSD matrix of ykðtÞ can be directly computed by Equation 30.131.

30.3.2.2 Structures Subjected to Nonuniformly Modulated Evolutionary

Random Excitations

30.3.2.2.1 Single Excitation Problems

Consider a nonuniformly modulated evolutionary random excitation f ðtÞ (Priestly, 1967)

f ðtÞ ¼

ð1

Aðv; tÞexpðivtÞdaðvÞ ð30:132Þ

in which Aðv; tÞ is a given nonuniform modulation function and a satisfies the equation

E ½dapðv1Þdaðv2Þ􀀉 ¼ Sxx ðvÞdðv2 2 v1Þdv1 dv2 ð30:133Þ

Here, Sxx ðv1Þ is the auto-PSD of the stationary random process xðtÞ:

The Riemann – Stieltjes integration in Equation 30.132 causes difficulties in conventional

computations. However, this problem can be conveniently solved using PEM as follows. First, constitute

the following pseudoexcitation (Lin et al., 1997a, 1997b):

f~ðv; tÞ ¼ Aðv; tÞ

ffiffiffiffiffiffiffiffi

Sxx ðvÞ

expðivtÞ ð30:134Þ

Second, replace gðtÞ in Equation 30.117 by Aðv; tÞ to yield

My€~ þ Cy_~ þ Ky~ ¼ 2MeAðv; tÞ

ffiffiffiffiffiffiffiffi

Sxx ðvÞ

expðivtÞ ð30:135Þ

Third, with the structure initially at rest, compute the time history, y~ðv; tÞ; for any arbitrary frequency v;

and the required time histories u~ ðv; tÞ and v~ðv; tÞ: Finally, the PSD matrices of uðtÞ and vðtÞ can be

computed from

Suuðv; tÞ ¼ u~ pðv; tÞu~ Tðv; tÞ ð30:136Þ

Suvðv; tÞ ¼ u~ pðv; tÞv~Tðv; tÞ ð30:137Þ

Evidently, the PEM is nearly identical for uniformly or nonuniformly modulated evolutionary random

excitations. The unique difference is the use of either Aðv; tÞ or gðtÞ in the pseudoexcitation expressions,

see Equation 30.114 and Equation 30.134.

30.3.2.2.2 Multiple Excitation Problems

For multiple nonstationary random excitation problems, the PEM-based analysis process for uniformly

modulated evolutionary random excitations can be immediately extended to that for nonuniformly

modulated evolutionary random excitations if the modulation function gðtÞ is replaced by Aðv; tÞ:

30-24 Vibration and Shock Handbook

All other formulas remain exactly the same. The difference in computational effort from using different

modulation functions is negligible.

30.3.3 Precise Integration Method

30.3.3.1 Precise Integration of Exponential Matrices

Structural equations of motion, for example, Equation 30.99, can be written as

My€ þ Gy_ þ Ky ¼ f ðtÞ ð30:138Þ

in which M, G, and K are given time-invariant n £ n matrices, respectively, and f ðtÞ is the given external

force vector. The initial displacement yðtÞ and the initial velocity y_ðtÞ of the system are specified. The

equation of motion, Equation 30.138, combined with the identity y_ ¼ y_ leads to the first-order equations

of motion in the state space being

v_ ¼ Hv þ r ð30:139Þ

in which

H ¼

0 I

B D

" #

; v ¼

( )

; r ¼

M21fðtÞ

( )

; B ¼ 2M21K; D ¼ 2M21G ð30:140Þ

The homogeneous solution of Equation 30.139 is

vhðtÞ ¼ TðtÞc ð30:141Þ

in which

TðtÞ ¼ exp ðHtÞ ð30:142Þ

Consider the current integration interval t [ ½tk; tkþ1􀀉; t ¼ t 2 tk: When t ¼ 0 or t ¼ tk; TðtÞ ¼ I and,

therefore, c is a constant vector. If the particular solution to Equation 30.139, vpðtÞ; is temporarily

assumed to have been found, then the general solution of Equation 30.139 is

vðtÞ ¼ TðtÞðvðtkÞ 2 vpðtkÞÞ þ vpðtÞ ð30:143Þ

In order to compute TðtÞ accurately, it is desirable to subdivide the step t into m ¼ 2N equal intervals,

that is,

Dt ¼ t=m ¼ 22N t ð30:144Þ

For application purposes, the use of N ¼ 20 is sufficient, because it leads to Dt < 1026t: Such a small Dt

is, in general, much smaller than the highest natural period of any practical discretized system.

Using the Taylor series expansion, we have

expðH £ DtÞ < I þ Ta0 ð30:145Þ

in which

Ta0 ¼ ðH £ DtÞ þ ðH £ DtÞ2􀀋

2! þ ðH £ DtÞ3􀀋

3! þ ðH £ DtÞ4􀀋

4! ð30:146Þ

Substituting Equation 30.145 into Equation 30.142 gives

TðtÞ ¼ ðexpðH £ DtÞÞm ¼ ðI þ Ta0Þm ð30:147Þ

Note that

I þ Tai ¼ ðI þ Ta;i21Þ2 ¼ ðI þ 2 £ Ta;i21 þ Ta;i21 £ Ta;i21Þ; ði ¼ 1; 2; …; NÞ ð30:148Þ

so that

I þ TaN ¼ ðI þ Ta;N21Þ2 ¼ ðI þ Ta;N22Þ4 ¼ · · · ¼ ðI þ Ta0Þm ¼ TðtÞ ð30:149Þ

Seismic Random Vibration of Long-Span Structures 30-25

Equation 30.148 and Equation 30.149 suggest the following computing strategy. In order to avoid the

loss of significant digits in the matrix TðtÞ; it is necessary to compute Ta1 directly from Ta0; Ta2 directly

from Ta1; and so on, by using

Tai ¼ 2 £ Ta;i21 þ Ta;i21 £ Ta;i21; ði ¼ 1; 2; …; NÞ ð30:150Þ

Then TðtÞ should be computed from

TðtÞ < I þ TaN ð30:151Þ

In Equation 30.151, the approximation is caused by the truncation of the Taylor expansion of Equation

30.146. It is generally negligibly small because when N ¼ 20; the first term ignored by the truncation is of

the order OðDt5Þ ¼ 10230Oðt5Þ; which is on the order of the round-off errors of a typical computer.

30.3.3.2 Particular Solutions and Precise Integration Formulas for Various Forms

of Loading

30.3.3.2.1 Linear Loading Form (HPD-L Form)

Assume that the loading varies linearly within the time step ðtk; tkþ1Þ; that is,

r ¼ r0 þ r1 £ ðt 2 t0Þ ð30:152Þ

in which r0 and r1 are time-invariant vectors. The particular solution of Equation 30.139 is then (Lin

et al., 1995a, 1995b; Zhong, 2004)

vpðtÞ ¼ ðH21 þ ItÞð2H21r1Þ 2 H21ðr0 2 r1tkÞ ð30:153Þ

Substituting Equation 30.152 into Equation 30.143 gives the HPD-L (High Precision Direct integration-

Linear) integration formula

vðtkþ1Þ ¼ TðtÞðvðtkÞ þ H21ðr0 þ H21r1ÞÞ 2 H21ðr0 þ H21r1 þ r1tÞ ð30:154Þ

30.3.3.2.2 Sinusoidal Loading Form (HPD-S Form)

If the applied loading is sinusoidal within the time region t [ ðtk ; tkþ1Þ; then

rðtÞ ¼ r1 sin vt þ r2 cos vt ð30:155Þ

in which r1 and r2 are time-invariant vectors. Substituting Equation 30.155 into Equation 30.139 enables

the particular solution to be obtained (Lin et al., 1995a, 1995b; Zhong, 2004)

vpðtÞ ¼ v1 sin vt þ v2 cos vt ð30:156Þ

in which

v1 ¼ ðvI þ H2=vÞ21ðr2 2 Hr1=vÞ v2 ¼ ðvI þ H2=vÞ21ð2r1 2 Hr2=vÞ ð30:157Þ

Substituting Equation 30.156 into Equation 30.143 gives the general solution of Equation 30.139, that is,

the HPD-S direct integration formula

vðtkþ1Þ ¼ TðtÞðvðtkÞ 2 v1 sin vtk 2 v2 cos vtkÞ þ v1 sin vtkþ1 þ v2 cos vtkþ1 ð30:158Þ

The time interval t ¼ tkþ1 2 tk can cover an arbitrary segment, or even many periods, of a sinusoidal

wave because, no matter how large the step size may be, exact responses will be obtained provided the

matrix TðtÞ has been generated accurately, without any instability occurring.

30.3.3.2.3 Exponentially Decaying Sinusoidal Loading Form (HPD-E Form)

Suppose that the applied loading varies according to the following exponentially decaying sinusoidal law

within the time region t [ ðtk; tkþ1Þ:

rðtÞ ¼ expðatÞðr1 sin vt þ r2 cos vtÞ ð30:159Þ

30-26 Vibration and Shock Handbook

in which r1 and r2 are time-invariant vectors. Substituting Equation 30.159 into Equation 30.139 enables

the particular solution to be obtained (Lin et al., 1995a, 1995b; Zhong, 2004) as

vpðtÞ ¼ expðatÞðv1 sin vt þ v2 cos vtÞ ð30:160Þ

in which

v1 ¼ ððaI 2 H2Þ þ v2IÞ21ððaI 2 HÞr1 þ vr2Þ

v2 ¼ ððaI 2 H2Þ þ v2IÞ21ððaI 2 HÞr2 2 vr1Þ

ð30:161Þ

Thus, substituting Equation 30.161 into Equation 30.143 gives the general solution of Equation 30.139,

that is, the HPD-E direct integration formula

vðtkþ1Þ ¼ TðtÞðvðtkÞ 2 expðatkÞðv1 sin vtk þ v2 cos vtkÞÞ þ expðatkþ1Þðv1 sin vtkþ1 þ v2 cos vtkþ1Þ

ð30:162Þ

The time interval is t ¼ tkþ1 2 tk:

Пресс-релиз популярных книг

30.3 Pseudoexcitation Method for Structural Random Vibration Analysis

Популярные книги

Популярные статьи

Навигация