ارتعاشات اتفاقی تیر مقطع متغیر یکسرگیردار تحت تحریک تصادفی

رد ﺦﻳرﺎﺗ ﻳ ﺖﻓﺎ 12 / 2 / 91

ﺬﭘ ﺦﻳرﺎﺗ ﻳ شﺮ 4 / 8 / 91

ﻪﺋارا رد ﺎﺳ ﻳ ﺖ 30 / 12 / 91

ﻚﻳﺮﺤﺗ ﺖﺤﺗ ﻲﻓدﺎﺼﺗ

هدﺮ ﺎﺘﺴﻳا ﺎﺑ ﻊﺑﺎﺗ ﻲﻟﺎﮕﭼ لﺎﻤﺘﺣا ﻦﻴﺳﻮﮔ ﻪﺘﺧادﺮﭘ

ﺖﺳﺪ ﻲﻣ

،ﺪﻳآ ﺲﭙﺳ ﺦﺳﺎﭘ و ﻚﻳﺮﺤﺗ ﻢﺘﺴﻴﺳ ﺮﺑ

ﻲ

،ﺦﺳﺎﭘ ﻊﺑﺎﺗ ﻲﻟﺎﮕﭼ ﻲﻔﻴﻃ ﺦﺳﺎﭘ ﺖﺳﺪﺑ ﻲﻣ ﺪﻳآ و

ﻪﺘﻴﺴﻴ و ﻪﺒﺳﺎﺤﻣ ﺲﻧﺎﻳراو ﺶﻨﺗ و ﺶﻧﺮﻛ ﻲﻗﺎﻔﺗا و

ﺖﺳ ﻲﻣ ﺪﻳآ .

Random vibration stochastic excitation

S. Irani

^1*

, S. Sazesh

²

1- Assist. Prof., Aero. Eng., Khaje Nasir Toosi Un 2- Graduated MSc in Aero. Eng., Khaje Nasir Too

* P. O. B. 16765-3381, Tehran, Iran. irani@kntu

Abstract-In this study random vibrati with Gaussian probability density funct shapes of beam in form of Bessel func correlation of response is shaped by c excitation is derived with cooperation o response and taking Fourier integral of using spectral density of displacement achieved. Finally elasticity equation is random stress with yield stress of beam Keywords:Random Vibration, Tapered B

درﻮﻣ تﺎﻋﻮﺿﻮﻣ زا هراﻮﻤﻫ ﺮﻴﻐﺘﻣ صاﻮﺧ يار 1 - 3 .[

ﻪﻟدﺎﻌﻣ ﺐﻳاﺮﺿ ﻪﻛ ﺖﺷاد ﻪﺟﻮﺗ ﺪﻳﺎﺑ ﺎﻫﺮﻴﺗ عﻮ ﺮ ﻲﻣ ﺮﻴﻐﺘﻣ ﻢﺘﺴﻴﺳ ﻦﻳا زا و ﺪﺷﺎﺑ

ور شور ﺎﻫ ي

دادﺮﺧ 1392 هرود ، 13 هرﺎﻤﺷ 3 ص ص 138 - 145

ﻲ ﺮﻴﻐﺘﻣ ﻊﻄﻘﻣ ﺮﻴﺗ ﻚﻳ

ﺮﺳ رادﺮﻴﮔ ﺗ

2

شزﺎﺳ

ﻲﺳﻮﻃ ﻦﻳﺪﻟاﺮﻴﺼﻧ ﻪﺟاﻮﺧ ﻲﺘﻌﻨﺻ

، ناﺮﻬﺗ

ﺎﻀ

، هﺎﮕﺸﻧاد ناﺮﻬﺗ ،ﻲﺳﻮﻃ ﻦﻳﺪﻟاﺮﻴﺼﻧ ﻪﺟاﻮﺧ ﻲﺘﻌﻨﺻ

، irani@kntu.ac.ir

ﻲﺳ تﺎﺷﺎﻌﺗرا ﻲﻗﺎﻔﺗا ﺮﻴﺗ ﻊﻄﻘﻣ ﺮﻴﻐﺘﻣ ﺖﺤﺗ ﻚﻳﺮﺤﺗ ﻲﻓدﺎﺼﺗ ﺮﺘﺴﮔ

تﺎﺷﺎﻌﺗرا دازآ اﺪﺘﺑا ﻞﻜﺷ يﺎﻫدﻮﻣ ﻢﺘﺴﻴﺳ ﺮﺑ ﺐﺴﺣ ﻊﺑاﻮﺗ ﻞﺴﺑ ﺪﺑ

ن ﻲﻣ ﻮﺷ د.

رد ﻠﺣﺮﻣ ﺔ ﺪﻌﺑ ﺎﺑ ﻦﺘﻓﺮﮔ لاﺮﮕﺘﻧا ﻪﻳرﻮﻓ زا ﻲﮕﺘﺴﺒﻤﻫدﻮﺧ

ط ﻒﻠﺘﺨﻣ ﺮﻴﺗ ﻪﺒﺳﺎﺤﻣ ﻲﻣ ﺷ ﻮ د.

رد ﺖﻳﺎﻬﻧ اﺎﺑ هدﺎﻔﺘﺳ زا ﻂﺑاور ﻴﺘﺳﻻا

لﺎﻤﺘ ﻢﻴﻠﺴﺗ ﻢﺘﺴﻴﺳ ﺖﺤﺗ ﻚﻳﺮﺤﺗ ﻲﻓدﺎﺼﺗ رد ﺮﻈﻧ ﻪﺘﻓﺮﮔ

،هﺪﺷ ﺪﺑ

،ﻲﻔﻴﻃ ﻲﻟﺎﮕﭼ ،ﺮﻴﻐﺘﻣ ﻊﻄﻘﻣ لﺎﻤﺘﺣا

ﻢﻴﻠﺴﺗ

.

of cantilever tapered n

niv., Tehran, Iran.

osi Univ., Tehran, Iran.

.ac.ir

ion of a cantilever tapered beam under distributed stati tion is investigated. early free vibration analysis is per ctions, then the response is described in summation considering the mode shapes of tapered beam, also of mode shapes and two dummy variables. in next st autocorrelation of response, spectral density of displac t, variance of random displacements for various pos applied to derive random strain and stress of beam. C leads to obtain probability of beam failure.

eam, Spectral Density, Probability of Yield.

ﺖﺧاﻮﻨﻜﻳ ﺮﻴﻏ ﻪﺑ

ﺮﺘﻬﺑ ﻊﻳزﻮﺗ ﺖﻠﻋ

ﻪﺑ ﺖﺧاﻮﻨﻜﻳ يﺎ هدﺮﺘﺴﮔ رﻮﻃ

يا

ﻲﺸﻤﺧ تﺎﺷﺎﻌﺗرا ﻪﻠﺌﺴﻣ ﻦﻳاﺮﺑﺎﻨ راد يﺎﻫﺮﻴﺗ عﻮﻧ ﻦﻳا

ﺖﺳا هدﻮﺑ ﻲﺳرﺮﺑ ]

ﻮﻧ ﻦﻳا ﻞﻴﻠﺤﺗ رد ﺮﺑ ﻢﻛﺎﺣ ﻞﻴﺴﻧاﺮﻔﻳد

تﺎﺷﺎﻌﺗرا ﻲﻗﺎﻔﺗا

ﻲﻧاﺮﻳا ﺪﻴﻌﺳ

1

،

*

ﺳ ﺪﻴﻌﺳ

1 - ﺎﻀﻓاﻮﻫ ﻲﺳﺪﻨﻬﻣ رﺎﻳدﺎﺘﺳا

، ﻨﺻ هﺎﮕﺸﻧاد

2 - ﺶﻧاد ﻪﺘﺧﻮﻣآ ﺪﺷرا ﻲﺳﺎﻨﺷرﺎﻛ ﻀﻓاﻮﻫ

*

،ناﺮﻬﺗ ﻲﺘﺴﭘ قوﺪﻨﺻ 3381

- 16765

هﺪﻴﻜﭼ

ﻦﻳارد

-

ﻪﻌﻟﺎﻄﻣ ﻪﺑ ﺳرﺮﺑ

ﻲﻣ دﻮﺷ . ﻪﺑ ﻦﻳا رﻮﻈﻨﻣ ﺎﺑ ﻞﻴﻠﺤﺗ

ﺐﺴﺣ ﻞﻜﺷ يﺎﻫدﻮﻣ ﻢﺘﺴﻴﺳ نﺎﻴﺑ

زا يور نآ ﺲﻧﺎﻳراو ﺑﺎﺟ ﻲﻳﺎﺠ طﺎﻘﻧ

ﻪﺴﻳﺎﻘﻣ نآ ﺎﺑ ﺶﻨﺗ ﻢﻴﻠﺴﺗ

،ﺮﻴﺗ ﺘﺣا

ﺪﻴﻠﻛ نﺎﮔژاو : ﺮﻴﺗ ،ﻲﻗﺎﻔﺗا تﺎﺷﺎﻌﺗرا

beam under

ionary stochastic excitation rformed to obtain the mode of mode shapes, and auto spectral density matrix of tep by means of frequency cement is computed and by sitions along the beam are Comparing the variance of

1 - ﻘﻣ ﻪﻣﺪ

هزﺎﺳ رد يﺎﻫﺮﻴﺗ هﺪﻴﭽﻴﭘ يﺎﻫ ﻏ

ﺎﻫﺮﻴﺗ ﻪﺑ ﺖﺒﺴﻧ نزو و ﻲﺘﺨﺳ ﻲﻣ راﺮﻗ هدﺎﻔﺘﺳا درﻮﻣ دﺮﻴﮔ

ﻨﺑ ،

[ Downloaded from mme.modares.ac.ir on 2022-10-31 ]

ﺋارا ﻞﺋﺎﺴﻣ عﻮﻧ ﻦﻳا ﻲﺳرﺮﺑ ياﺮﺑ ﻲﻔﻠﺘﺨﻣ ﺖﺳا هﺪﺷ ﻪ

. ﻪﻠﻤﺟ زا

ﻦﻳا شور ﺎﻫ ﻲﻣ ﻊﻄﻘﻣ ﺮﻴﺗ ﻚﻳ ﺮﺑ ﻢﻛﺎﺣ ﻪﻟدﺎﻌﻣ ﻪﺘﺴﺑ ﻞﺣ ﻪﺑ ناﻮﺗ

ﻪﻴﻜﺗ ﺎﺑ ﺮﻴﻐﺘﻣ هﺎﮔ

دﻮﻤﻧ هرﺎﺷا هدﺎﺳ يﺎﻫ ]

4 - 7 .[

شور ﺎﻫ يﺮﮕﻳد ي

ﻪﻠﻤﺟ زا ﺰﻴﻧ ﻲﻗﺎﺑ

ﻲﻧزو هﺪﻧﺎﻣ ]

8 [ تاﺮﻴﻴﻐﺗ شور ، ]

9

، 10 [

،ا نﺎﻤﻟا

دوﺪﺤﻣ ] 11 [ دوﺪﺤﻣ ﻞﺿﺎﻔﺗ ، ]

12 [ شور ، يزﺮﻣ نﺎﻤﻟا ]

13 [

،

ﺲﻳﺮﺗﺎﻣ شور لﺎﻘﺘﻧا

] 14 [ ﻲﻜﻴﻣﺎﻨﻳد ﻲﺘﺨﺳ شور ، ]

15 [ شور ،

لادﻮﻣ ﺰﻴﻟﺎﻧآ ﺎﺑ هاﺮﻤﻫ ﻲﻜﻴﻣﺎﻨﻳد ﻲﺘﺨﺳ ]

16

، 17 [ ﻞﻳﺪﺒﺗ و ،

سﻼﭘﻻ ] 18 [ رﺎﻛ ﻪﺑ ﻞﺋﺎﺴﻣ ﻪﻧﻮﮕﻨﻳا ﺮﺑ ﻢﻛﺎﺣ ﻪﻟدﺎﻌﻣ ﻞﺣ ياﺮﺑ

ﺖﺳا ﻪﺘﻓر .

ﻲﺳرﺮﺑ زا ﻲﻀﻌﺑ رد رد ﻦﻳﺮﮔ ﻊﺑاﻮﺗ زا ﺰﻴﻧ ﺮﮕﻳد يﺎﻫ

سﻼﭘﻻ يﺎﻀﻓ ]

19 [ ﺖﺳا هﺪﺷ هدﺎﻔﺘﺳا .

ﻪﻌﻟﺎﻄﻣ ﻪﻣادا رد تاﺮﺛا ،ﺮﻴﻐﺘﻣ ﻊﻄﻘﻣ ﺎﺑ ﻲﻳﺎﻫﺮﻴﺗ يور ﺮﺑ

ﺰﻛﺮﻤﺘﻣ ﺮﻨﻓ دﻮﺟو ]

20 [ ﺮﭙﻣد ، مﺮﺟ و ﺰﻛﺮﻤﺘﻣ ] 21

، 22 [ و

ﻪﻴﻜﺗ هﺎﮔ زﻮﻜﺴﻳو ﺮﭙﻣد ﺎﺑ هاﺮﻤﻫ ﻚﻴﺘﺳﻻا يﺎﻫ مﺮﺟو

ﺰﻛﺮﻤﺘﻣ

] 23 [ ﺖﺳا هﺪﺷ ﻪﺘﻓﺮﮔ ﺮﻈﻧ رد .

يور ﺮﺑ تﺎﻌﻟﺎﻄﻣ ﻦﻳﺮﺗﺪﻳﺪﺟ رد

ﻦﻳا ﻲﺳرﺮﺑ ياﺮﺑ ﺪﻳﺪﺟ ﻲﻠﻜﺷ ﻊﺑﺎﺗ ﻚﻳ ﺰﻴﻧ ﺎﻫﺮﻴﺗ زا ﻪﻧﻮﮔ

تﺎﺷﺎﻌﺗرا ﺖﺳا هﺪﺷ ﻲﻓﺮﻌﻣ ﺎﻬﻧآ ﻲﺿﺮﻋ

] 24 .[

عاﻮﻧا ﺮﺑ دراو يﺎﻫرﺎﺑ زا ﻲﺸﺨﺑ ﻪﻛ ﺎﺠﻧآ زا ﺮﮕﻳد فﺮﻃ زا هزﺎﺳ تﺎﺷﺎﻌﺗرا ﺪﻨﺘﺴﻫ ﻲﻓدﺎﺼﺗ ترﻮﺻ ﻪﺑ ﻲﻌﻗاو ﺖﻟﺎﺣ رد ﺎﻫ يدﺎﻤﺘﻣ يﺎﻬﻟﺎﺳ رد ﻪﻛ ﺖﺳا هدﻮﺑ ﻲﺗﺎﻋﻮﺿﻮﻣ زا ﺰﻴﻧ ﺎﻫﺮﻴﺗ ﻲﻗﺎﻔﺗا

ﺖﺳا هﺪﺷ ﻪﺘﺧادﺮﭘ نآ ﻪﺑ هﺪﺷ ﻪﺘﻓﺮﮔ ﺮﻈﻧ رد ﺮﻴﺗ ﻪﻛ ﻲﻟﺎﺣ رد ،

ﻦﻳا رد ﻲﻣ ﺖﺑﺎﺛ ﻊﻄﻘﻣ ترﻮﺻ ﻪﺑ تﺎﻌﻟﺎﻄﻣ ﺪﺷﺎﺑ

] 25 - 27 [.

ﻲﻣ ﻪﺘﺧادﺮﭘ نآ ﻪﺑ ﻪﻌﻟﺎﻄﻣ ﻦﻳا رد ﻪﭽﻧآ ﻲﺳرﺮﺑ دﻮﺷ

ﻲﻓدﺎﺼﺗ ﻚﻳﺮﺤﺗ ﺖﺤﺗ ﺮﻴﻐﺘﻣ ﻊﻄﻘﻣ ﺮﻴﺗ ﻚﻳ ﻲﻗﺎﻔﺗا تﺎﺷﺎﻌﺗرا لﺎﻤﺘﺣا و ﺖﺳا ﻢﻴﻠﺴﺗ

ﺪﻫاﻮﺧ ﻞﻴﻠﺤﺗ ﺰﻴﻧ نآ ﺪﺷ

.

2 - ﻢﻛﺎﺣ ﺖﻛﺮﺣ تﻻدﺎﻌﻣ

ﻪﻟدﺎﻌﻣ ﻲﻟﻮﻧﺮﺑﺮﻟوا ﺮﻴﺗ ﺖﻛﺮﺣ ار ﺮﻴﻐﺘﻣ ﻊﻄﻘﻣ

ﻚﻳﺮﺤﺗ ﺖﺤﺗ ﻪﻛ

ﻲﻓدﺎﺼﺗ ﻲﻣ ار ﺪﺷﺎﺑ ﻪﺘﻓﺮﮔ راﺮﻗ ترﻮﺻ ﻪﺑ ناﻮﺗ

) 1 ( ﺖﺷﻮﻧ ] 28 .[

( ) ( ) ( ) ( )

( )

2 2

2

2 2 2

w x ,t w x ,t

EI x A x

x x t

p x ,t

 ∂  ρ ∂

∂  −

∂  ∂  ∂

=

(1)

نآ رد ﻪﻛ

،ﮓﻧﺎﻳ لوﺪﻣ

E

،ﺮﻴﺗ ﻊﻄﻘﻣ ﺢﻄﺳ ﻲﺸﻤﺧ نﺎﻤﻣ

I(x)

ﺮﻴﺗ ﻲﻟﺎﮕﭼ

ρ

،

،ﺮﻴﺗ ﻊﻄﻘﻣ ﺢﻄﺳ

A(x) w(x,t)

يدﻮﻤﻋ ﻲﻳﺎﺠﺑﺎﺟ

نﺎﻣز رد ﺮﻴﺗ نﺎﻜﻣ و

t

و

x p(x,t)

يوﺮﻴﻧ ﻲﻓدﺎﺼﺗ هدﺮﺘﺴﮔ

ﺎﺘﺴﻳا ﺮﻴﺗ ﺮﺑ دراو

1

ﻲﻣ ﺪﺷﺎﺑ .

1. Stationary

ﺮﻈﻧ رد ﺎﺑ ﻞﻜﺷ ترﻮﺻ ﻪﺑ ﺮﻴﺗ ﻞﻜﺷ ﻦﺘﻓﺮﮔ

1 ﺑ نﺎﻤﻣ ياﺮ

ﻲﻣ ﺖﺳﺪﺑ ﺮﻳز ﻂﺑاور ﻊﻄﻘﻣ ﺢﻄﺳ و ﻲﺸﻤﺧ ﻲﺳﺮﻨﻳا ﺪﻳآ

.

( )

( )

^{3 3 3}

1 1

1 1

12

i i

i i

j i

i

x x

A x r A r bh

bh

x x

I x r I r

h h

r h

   

= +  = + 

   

= +  = + 

= −

(2) (3) (4) ﻻﺎﺑ ﻂﺑاور رد

،ﺮﻴﺗ لﻮﻃ

ℓ Ai

ﻪﻴﻜﺗ رد ﺮﻴﺗ ﻊﻄﻘﻣ ﺢﻄﺳ و هﺎﮔ

Ii

ﻪﻴﻜﺗ رد ﺮﻴﺗ ﻲﺸﻤﺧ نﺎﻤﻣ ﻲﻣ هﺎﮔ

ﺪﺷﺎﺑ .

3 - دازآ تﺎﺷﺎﻌﺗرا ﻞﺣ

،ﺮﻴﺗ دازآ تﺎﺷﺎﻌﺗرا ﻞﺣ ياﺮﺑ اﺪﺘﺑا

ﻚﻳﺮﺤﺗ ﻊﺑﺎﺗ ﺮﻔﺻ

ﺮﻈﻧ رد

ﻪﺘﻓﺮﮔ ﻲﻣ دﻮﺷ و ﻪﻄﺑار زا هدﺎﻔﺘﺳا ﺎﺑ

w(x,t)=W(x)q(t)

و

ﺎﻫﺮﻴﻐﺘﻣ يزﺎﺳاﺪﺟ و

ﻦﻴﻨﭽﻤﻫ ﺮﻴﻐﺘﻣ ﺮﻴﻴﻐﺗ

η=x

ﺮﻳز ﻪﻟدﺎﻌﻣ ،

ﻲﻣ ﻞﺻﺎﺣ ﻮﺷ

د .

( ) ( ) ( ) ( )

2 2

3 4

2 2

4 4 2

1 1 0

i i

d r d W r W

d d

A EI

η η β η η

η η

β ρ ω

 

+ − + =

 

 

=

(5) (6)

ﻞﻜﺷ 1 ﻚﻳ ﺮﻴﻐﺘﻣ ﻊﻄﻘﻣ ﺮﻴﺗ ﻲﻓدﺎﺼﺗ هدﺮﺘﺴﮔ ﻚﻳﺮﺤﺗ ﺖﺤﺗ رادﺮﻴﮔﺮﺳ

ﻪﻟدﺎﻌﻣ ﻞﺣ زا )

5 ( ﺔﻄﺑار ) 7 ( ﻲﻣ ﺖﺳﺪﺑ ﺪﻳآ

.

( )

^{1 1} 2

2

2 1 2 3 1 2

2

4 1 2

1 2 1

1

1 1

2 2

2 1

W C J r

r r

r r

C Y C J

r r

C Y r

r

η β η

η

η η

β β

β η

  + 

= +   

 +   + 

+  +  − 

 + 

+  − 

(7)

رد ﻪﻄﺑار ) 7

J1

(

Y1

و ﻞﺴﺑ ﻊﺑاﻮﺗ ﻚﻳ ﻪﺒﺗﺮﻣ زا مود و لوا عﻮﻧ

و

[ Downloaded from mme.modares.ac.ir on 2022-10-31 ]

C1

C4

ﺎﺗ ﺖﺑﺎﺛ داﺪﻋا ﻲﻣ

ﺪﻨﺷﺎﺑ . نﺎﺸﻧ يزﺮﻣ ﻂﻳاﺮﺷ لﺎﻤﻋا ﺎﺑ

لوﺪﺟ رد هﺪﺷ هداد 1

ﻚﻳ هﺎﮕﺘﺳد 4 ﻪﻟدﺎﻌﻣ 4 ﺮﺑ لﻮﻬﺠﻣ

ﺖﺑاﻮﺛ ﺐﺴﺣ

C1

C4

ﺎﺗ ﻲﻣ ﺖﺳﺪﺑ ﺪﻳآ

.

ﻒﻠﺘﺨﻣ ﺮﻳدﺎﻘﻣ هﺎﮕﺘﺳد ﻦﻳا ﻲﻬﻳﺪﺑ ﺮﻴﻏ ﻞﺣ زا ﻲﻣ ﻞﺻﺎﺣ

β

دﻮﺷ .

ﺮﻴﻏ ﻞﺣ ﻊﺑاﻮﺗ زا ﻞﻜﺸﺘﻣ ﻲﻨﺤﻨﻣ ﻚﻳ دﺎﺠﻳا ﻪﺑ ﺮﺠﻨﻣ ﻲﻬﻳﺪﺑ

هﺪﺷ ﻲﻓﺮﻌﻣ ﻞﺴﺑ

، ﻲﻣ دﻮﺷ رﻮﺤﻣ ﺎﺑ ﻲﻨﺤﻨﻣ ﻦﻳا درﻮﺧﺮﺑ ﻞﺤﻣ ﻪﻛ

ﭘ ،

β

ﺦﺳﺎ ﺮﻴﻏ ﻞﺣ يﺎﻫ

،دﻮﺑ ﺪﻫاﻮﺧ ﻲﻬﻳﺪﺑ ﻞﺤﻣ ﻦﻳا

درﻮﺧﺮﺑ يﺎﻫ

ﻞﻜﺷ رد 2 ﺖﺳا هﺪﺷ هداد نﺎﺸﻧ .

ﻒﻠﺘﺨﻣ ﺮﻳدﺎﻘﻣ يور زا ﻪﺟﻮﺗ ﺎﺑ

β

ﻪﻄﺑار ﻪﺑ )

6 ( ﺲﻧﺎﻛﺮﻓ يﺎﻫ

ﻪﻄﺑار ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ و ﺮﻴﺗ ﻲﻌﻴﺒﻃ )

7 ( ﻢﺘﺴﻴﺳ يﺎﻫدﻮﻣ ﻞﻜﺷ

ﻲﻣ ﻪﺒﺳﺎﺤﻣ دﻮﺷ

.

4 - لادﻮﻣ ﻞﻴﻠﺤﺗ

و مﺮﺟ ﺲﻳﺮﺗﺎﻣ ياﺮﺑ ﺎﻫدﻮﻣ ﻞﻜﺷ ﺪﻣﺎﻌﺗ ﺖﻴﺻﺎﺧ ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ ﺮﻳز ﻂﺑاور ﻢﺘﺴﻴﺳ ﻲﺘﺨﺳ ﻲﻣ راﺮﻗﺮﺑ

ﺪﺷﺎﺑ :

( ) ( ) ( ) ( ) ( ) ( )

0 0

i j i ij

i j i ij

A x W x W x dx EI x W x W x dx K

ρ µ δ

δ

=

′′ ′′ =

∫

∫

(8) (9)

لوﺪﺟ 1 ﺮﻴﻐﺘﻣ ﻊﻄﻘﻣ ﺮﻴﺗ يزﺮﻣ ﻂﻳاﺮﺷ ﻚﻳ

ﺮﺳ رادﺮﻴﮔ

( )

^{0 0}

W =

0 0

dW dη_η= =

2

2 1

d W 0 dη _η= =

( )

²₂

1

d I d W 0

d η d _η

η η ₌

 

=

 

 

 

ﻞﻜﺷ 2 ﺮﻴﺗ يﺎﻫدﻮﻣ ﻞﻜﺷ ﻚﻳ

ﺮﺳ ﺮﻴﻐﺘﻣ ﻊﻄﻘﻣ رادﺮﻴﮔ

نآ رد ﻪﻛ

µi

، ﻪﺘﻓﺎﻳ ﻢﻴﻤﻌﺗ مﺮﺟ

Ki

و

ﻪﺘﻓﺎﻳ ﻢﻴﻤﻌﺗ ﻲﺘﺨﺳ ياﺮﺑ

و دﻮﻣ ﻦﻴﻣا

i δij

ﻲﻣ ﺮﻜﻴﻧوﺮﻛ يﺎﺘﻟد ﻊﺑﺎﺗ ﺪﺷﺎﺑ

.

ﻪﻄﺑار سﺎﺳا ﺮﺑ ﺦﺳﺎﭘ ﻂﺴﺑ ﺎﺑ

( ) ( ) ( )

1 i i

i

w x ,t ^∞W x q t

=

=

∑

ﻪﻄﺑار رد ﻪﻟدﺎﻌﻣ ﻦﻳا نداد راﺮﻗ و )

1 ( ﻪﻟدﺎﻌﻣ ﻦﻴﻓﺮﻃ بﺮﺿ و

رد ﻞﺻﺎﺣ

Wj(x)

ﺎﺗ ﺮﻔﺻ زا يﺮﻴﮔ لاﺮﮕﺘﻧا و

،

ℓ

ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ

ﻣ ﻞﻜﺷ ﺪﻣﺎﻌﺗ ﺖﻴﺻﺎﺧ ﻲﻣ ﺖﺳﺪﺑ ﺮﻳز ﻪﻟدﺎﻌﻣ ،ﺎﻫدﻮ

ﺪﻳآ .

( )

²

( )

¹ ₀

( ) ( ) ( )

i i i i i

i

q t ω q t p x ,t W x dx Q t

+ =µ

∫

=

) 10 ( ﻪﻄﺑار ترﻮﺻ ﻪﺑ ﻢﺘﺴﻴﺳ ﻲﺴﻧﺎﻛﺮﻓ ﺦﺳﺎﭘ ﻊﺑﺎﺗ ﻪﻛ ﻲﺗرﻮﺻ رد

) 11 ( ﻒﻳﺮﻌﺗ ﻮﺷ

،د ﻢﺘﺴﻴﺳ ﻪﻴﻟوا ﻂﻳاﺮﺷ ندﻮﺑ ﺮﻔﺻ ضﺮﻓ ﺎﺑ و

،

ﻪﺘﻓﺎﻳ ﻢﻴﻤﻌﺗ تﺎﺼﺘﺨﻣ ﺦﺳﺎﭘ

qi(t)

ترﻮﺻ ﻪﺑ ﻪﻄﺑار

) 12 (

ﻲﻣ ﻪﺒﺳﺎﺤﻣ دﻮﺷ

.

( )

2 2

i 1

i

H ω

ω ω

=− +

(11)

( ) ( ) ( )

^{i t}

i i i

q t ^∞ H ω Q ω e d^ω ω

=

∫

−∞

⁽¹²⁾

5 - ﻲﻗﺎﻔﺗا تﺎﺷﺎﻌﺗرا

ﻪﻛ ﺮﻈﻧ درﻮﻣ ﻢﺘﺴﻴﺳ ﻲﻗﺎﻔﺗا تﺎﺷﺎﻌﺗرا ﻞﻴﻠﺤﺗ ﻪﺑ ﺶﺨﺑ ﻦﻳا رد ﻚﻳﺮﺤﺗ ﻚﻳ ﺖﺤﺗ ﻲﻓدﺎﺼﺗ

راﺮﻗ ﺮﻔﺻ ﻦﻴﮕﻧﺎﻴﻣ راﺪﻘﻣ ﺎﺑ ﺎﺘﺴﻳا

ﻲﻣ ﻪﺘﺧادﺮﭘ ،ﺖﺳا ﻪﺘﻓﺮﮔ دﻮﺷ

. ﻪﻛ دﻮﺷ ﻪﺟﻮﺗ ﻪﻴﻟوا ﻂﻳاﺮﺷ ضﺮﻓ

ﺖﺳا راﺮﻗﺮﺑ نﺎﻛﺎﻤﻛ ﺮﻔﺻ .

نﻮﭼ ﻚﻳﺮﺤﺗ ﻊﺑﺎﺗ

P(x,t)

ﻪﺘﻓﺮﮔ ﺮﻈﻧ رد ﺎﺘﺴﻳا ترﻮﺻ ﻪﺑ

ﻪﺘﻓﺎﻳ ﻢﻴﻤﻌﺗ يﺎﻫوﺮﻴﻧ ﻦﻳاﺮﺑﺎﻨﺑ ،ﺖﺳا هﺪﺷ

Qi(t)

ﺎﺘﺴﻳا ﺰﻴﻧ

ﺪﻨﺘﺴﻫ . ﺮﮕﻳد ﻪﻄﺑار ﻲﮕﺘﺴﺒﻤﻫ

يﺎﻫوﺮﻴﻧ

1

Qi(t) Qj(t)

و

ﻪﺑ

ترﻮﺻ ﻪﻄﺑار ) 13 ( ﻲﻣ ﻪﺘﺷﻮﻧ دﻮﺷ

.

( ) { ( ) ( ) }

( ) ( ) ( ) ( )

( ) ( )

( ) }

1 2

1 1 1

0

2 2 2

0

1 2

0 0

1 2 1 2

1 ,

1 ,

1 1

, ,

i j

x x

Q Q i j

i i

j j

i j

i j

P P

R E Q t Q t

E p x t W x dx

p x t W x dx

E W x W x

R x x dx dx

τ τ

µ µ

µ µ τ

= +

= 

 

× 

 

= 

×

∫

∫

∫ ∫

(13)

1.Cross-Correlation

[ Downloaded from mme.modares.ac.ir on 2022-10-31 ]

ﻪﻄﺑار ﻦﻳا رد ﻦﻴﮕﻧﺎﻴﻣ ﻊﺑﺎﺗ ﺮﮕﻧﺎﻳﺎﻤﻧ

E{.}

و ﺖﺳا يﺮﻴﮔ

x1

x2

و

بذﺎﻛ ﺮﻴﻐﺘﻣ ود ﻪﻠﺻﺎﻓ رد ار اﺰﺠﻣ ﻪﻄﻘﻧ ود و ﺪﻧا

0<x<ℓ

ﻲﻣ ﺺﺨﺸﻣ ﺪﻨﻳﺎﻤﻧ

( ) .

1 2 1 2

x x

RP P x ,x ,τ

ﺮﮕﻳد ﻊﺑﺎﺗ

ﻲﮕﺘﺴﺒﻤﻫ

ﻚﻳﺮﺤﺗ ﻦﻴﺑ ﻓدﺎﺼﺗ

هدﺮﺘﺴﮔ ﻲ

p(x1,t)

و

p(x2,t)

ﻲﻣ ﺪﺷﺎﺑ .

ﺮﮕﻳد ﻲﻟﺎﮕﭼ

1

ﻲﻔﻴﻃ

i j

( )

SQ Q ω

ﻪﺘﻓﺎﻳ ﻢﻴﻤﻌﺗ يﺎﻫوﺮﻴﻧ ﻦﻴﺑ

Qi(t)

و

Qj(t)

زا ﻪﻳرﻮﻓ لاﺮﮕﺘﻧا ﺎﺑ

i j

( )

RQ Q τ

ﻲﻣ ﺖﺳﺪﺑ

ﺪﻳآ .

( ) ( )

i j i j

Q Q Q Q i

S ω ^+∞R τ e^−ωτdτ

=

∫

−∞

⁽¹⁴⁾

نﻮﻨﻛا ﻪﻄﺑار ) 13 ( ﻪﻄﺑار ﻞﺧاد رد )

14 ( ﻲﻣ يراﺬﮕﻳﺎﺟ ﻮﺷ

د،

( ) ( ) ( )

( )

1 2

1 2

1 2 1 2

1 1

i j

x x

Q Q i j

i j

p p

S W x W x

S x ,x , dx dx ω µ µ

ω

+∞ +∞

−∞ −∞

=

×

∫ ∫

(15) ﻪﻄﺑار ﻦﻳا رد ﻪﻛ

( )

1 2 1 2

x x

Sp p x ,x ,ω

ﻊﺑﺎﺗ ﻪﻳرﻮﻓ لاﺮﮕﺘﻧا ،

( )

1 2 1 2

x x

RP P x ,x ,τ

ﻲﻣ

ﺪﺷﺎﺑ .

ﺮﮕﻳد ﻊﺑﺎﺗ ﻪﻄﻘﻧ ود ﻦﻴﺑ ﺦﺳﺎﭘ ﻲﮕﺘﺴﺒﻤﻫ

x1

x2

و ترﻮﺻ ﻪﺑ

ﻪﻄﺑار ) 16 ( ﻲﻣ ﻪﺘﺷﻮﻧ دﻮﺷ

.

( ) { ( ) ( ) }

( ) ( ) ( ) ( ) ( ) ( ) ( )

1 2 1 2 1 2

1 2 2

1 1

1 2

1 1

, , , ,

x x

i j

w w

i i j

i j

i j q q

i j

R x x E w x t w x t E W x q t W x q t

W x W x R

τ τ

τ

τ

∞ ∞

= =

∞ ∞

= =

= +

 

 

=  + 

 

 

=

∑ ∑

∑∑ ⁽¹⁶⁾

ﻪﻄﺑار ﻦﺘﻓﺮﮔ ﺮﻈﻧ رد ﺎﺑ )

12 ( ﻪﻄﺑار و ) 16 ( رد هﺪﺷ هداد ترﺎﺒﻋ ،

ﻪﻄﺑار ) 17 ( ﻲﻔﻴﻃ ﻲﻟﺎﮕﭼ ﺐﺴﺣ ﺮﺑ ﺦﺳﺎﭘ ﻲﮕﺘﺴﺒﻤﻫ ﺮﮕﻳد ياﺮﺑ

ﻲﻣ ﺖﺳﺪﺑ ﻪﺘﻓﺎﻳ ﻢﻴﻤﻌﺗ يوﺮﻴﻧ ﺪﻳآ

.

( ) ( ) ( )

( ) ( ) ( )

1 2 1 2 1 2

1 1

1 2

x x

i j

w w i j

i j

i j Q Q i

R x ,x , W x W x

H H S e d^ωτ τ π

ω ω ω ω

∞ ∞

= =

+∞ ∗

−∞

=

×

∑∑

∫ ⁽¹⁷⁾

،ﻪﻄﺑار ﻦﻳا رد

i j

( )

SQ Q ω

ﻪﻟدﺎﻌﻣ زا

) 15 ( ﻲﻣ ﻪﺒﺳﺎﺤﻣ دﻮﺷ

. ﻊﺑﺎﺗ

( )

Hj^∗ ω

ﻊﺑﺎﺗ ﻂﻠﺘﺨﻣ جودﺰﻣ ( )

Hj ω

ﻲﻣ ﺪﺷﺎﺑ .

ﻪﻛ ﻲﺗرﻮﺻ رد

x1=x2

ﻲﮕﺘﺴﺒﻤﻫدﻮﺧ ﻊﺑﺎﺗ ﺪﺷﺎﺑ ترﻮﺻ ﻪﺑ ﺦﺳﺎﭘ

2

ﻲﻣ ﻞﺻﺎﺣ ﺮﻳز دﻮﺷ

.

( ) ( ) ( )

( ) ( ) ( )

1 1

1 2

i j

w i j

i j

i j Q Q i

R W x W x

H H S e d^ωτ τ π

ω ω ω ω

∞ ∞

= =

+∞ ∗

−∞

=

×

∑∑

∫ ⁽¹⁸⁾

1. Cross-Spectral Density 2. Auto-Correlation

نداد راﺮﻗ ﺎﺑ

0

=

،

τ

ﻪﻄﺑار رد )

18 ( روﺬﺠﻣ ﻦﻴﮕﻧﺎﻴﻣ ﻲﻳﺎﺠﺑﺎﺟ

3

ﻪﻄﻘﻧ رد ﻲﻣ ﻞﺻﺎﺣ ﺮﻴﺗ زا

x

ﻮﺷ د .

{ ( ) } ^{( ) ( )}

( ) ( ) ( )

2

1 1

1 2

i j

i j

i j

i j Q Q

w x W x W x

H H S d

π

ω ω ω ω

∞ ∞

= =

+∞ ∗

−∞

=

×

∑∑

∫ ⁽¹⁹⁾

E

ﻪﺑ طﻮﺑﺮﻣ يرﺎﻣآ صاﻮﺧ ﺮﮕﻳد ﻪﺘﻴﺴﻴﺘﺳﻻا ﻂﺑاور ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ لﺎﺣ ﺮﻴﺗ ﺶﻧﺮﻛ و ﻲﺸﻤﺧ نﺎﻤﻣ ﻲﻣ ﻪﺒﺳﺎﺤﻣ

دﻮﺷ . نﺎﻤﻣ ﻪﻄﺑار

ترﻮﺻ ﻪﺑ ﺮﻴﺗ ﻲﺸﻤﺧ ﻪﻄﺑار

) 20 ( ﺖﺳا . ( )

²

( )

b w x ,t2

M EI x

x

= − ∂

∂

(20)

ﻄﻘﻧ رد ﻲﺸﻤﺧ نﺎﻤﻣ روﺬﺠﻣ ﻦﻴﮕﻧﺎﻴﻣ ﻦﻳاﺮﺑﺎﻨﺑ

x

ﺔ

زا ﺮﻴﺗ زا

ﺮﻳز ﻪﻄﺑار ﻲﻣ ﺖﺳﺪﺑ

ﺪﻳآ

،

{ } ^{( ( ) ) ( ) ( )}

( ) ( ) ( )

2 2

1 1

1 2

i j

b i j

i j

i j Q Q

M EI x W x W x

H H S d

π

ω ω ω ω

∞ ∞

= =

+∞ ∗

−∞

′′ ′′

=

×

∑∑

∫ ⁽²¹⁾

E

ﻪﻄﺑار زا ﺮﻴﺗ ﻲﻟﻮﻃ يﺎﺘﺳار رد ﺶﻧﺮﻛ ﻪﻨﻴﺸﻴﺑ )

22 ( ﻪﺒﺳﺎﺤﻣ

ﻲﻣ ﻮﺷ د . د ﻪﻟدﺎﻌﻣ ﻦﻳا ر

( )

ﺮﻴﺗ ﻊﻄﻘﻣ عﺎﻔﺗرا

h x

ﺮﺑاﺮﺑ و

1 rx h_i

 + 

 

 

ﻲﻣ ﺪﺷﺎﺑ . ( )

²

( )

2 2

h x w x ,t

ε ∂ x

= − ∂

(22)

ﻪﻄﻘﻧ رد ﻪﻨﻴﺸﻴﺑ ﺶﻧﺮﻛ روﺬﺠﻣ ﻦﻴﮕﻧﺎﻴﻣ ﺮﻳز ﻪﻄﺑار زا ﺮﻴﺗ

x

ﻪﺒﺳﺎﺤﻣ ﻲﻣ دﻮﺷ .

{ ( ) } ^{( ) ( ) ( )}

( ) ( ) ( )

2 2

1 1

1

2 2

i j

i j

i j

i j Q Q

E x h x W x W x

H H S d

ε π

ω ω ω ω

∞ ∞

= =

+∞ ∗

−∞

  ′′ ′′

=  

 

×

∑∑

∫ ⁽²³⁾

ﻪﻄﺑار زا ﺮﻴﺗ لﻮﻃ رد ﺶﻧﺮﻛ و ﺶﻨﺗ ﻦﻴﺑ ﻪﻄﺑار ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ )

24 (

،

ﺮﻴﺗ رد ﻪﻨﻴﺸﻴﺑ ﺶﻨﺗ روﺬﺠﻣ ﻦﻴﮕﻧﺎﻴﻣ ياﺮﺑ ﻪﻄﺑار،

) 25 ( ﻞﺻﺎﺣ

ﻲﻣ دﻮﺷ .

S E= ε

(24)

{

^{S x2}

( ) }

^=E2

{

^ε2

^{( )}

^x

} ⁽²⁵⁾

E E

6 - لﺎﻤﺘﺣا ﻢﻴﻠﺴﺗ

ﻚﻳﺮﺤﺗ لﺎﻤﺘﺣا ﻊﻳزﻮﺗ ﻪﻛ ﺎﺠﻧآ زا ﻦﻴﺳﻮﮔ

ا

4

ا و ﺖﺳ ﺎﺘﺴﻳ ﺰﻴﻧ

3. Mean Square 4 . Gaussian

[ Downloaded from mme.modares.ac.ir on 2022-10-31 ]

ﻲﻣ

،ﺪﺷﺎﺑ دﻮﺑ ﺪﻫاﻮﺧ ﻦﻴﺳﻮﮔ ﺰﻴﻧ ﻢﺘﺴﻴﺳ ﺦﺳﺎﭘ .

ﺎﺑ ﻦﻳاﺮﺑﺎﻨﺑ

ﺲﻧﺎﻳراو ندروآ ﺖﺳﺪﺑ رﺎﻴﻌﻣ فاﺮﺤﻧا و

1

ﻲﻣ ﺦﺳﺎﭘ

2

لﺎﻤﺘﺣا ناﻮﺗ

دﺮﻛ ﺺﺨﺸﻣ ار هزﺎﺳ ﻢﻴﻠﺴﺗ .

ﻪﺒﺳﺎﺤﻣ ﺎﺑ اﺬﻟ ﻲﻗﺎﻔﺗا ﺶﻨﺗ

ﻢﺘﺴﻴﺳ

، ﻪﻄﺑار زا )

26 ( زا ﺮﺗﻻﺎﺑ ﺶﻨﺗ ﻪﻨﻴﺸﻴﺑ دﻮﺟو لﺎﻤﺘﺣا

ﻲﻣ ﺺﺨﺸﻣ ﻢﻴﻠﺴﺗ ﺶﻨﺗ ﻮﺷ

د ] 29 .[

( ) 2

2 2 y

y s x

P e

σ

− σ

=

(26)

ﻪﻄﺑار رد )

26

y

( و ﻢﻴﻠﺴﺗ ﺶﻨﺗ ﺮﮕﻧﺎﻳﺎﻤﻧ

σ

2( )

σS x

ﺮﮕﻧﺎﻳﺎﻤﻧ

ﻪﻄﻘﻧ رد ﺶﻨﺗ ﺲﻧﺎﻳراو ﻲﻣ ﺮﻴﺗ زا

x

ﺪﺷﺎﺑ .

7 - يدﺪﻋ لﺎﺜﻣ ﻞﺣ

ﻚﻳﺮﺤﺗ ﺖﺤﺗ ﻪﻛ ﻪﻧﻮﻤﻧ ﺮﻴﺗ ﻚﻳ ﺖﻤﺴﻗ ﻦﻳا رد ﻲﻓدﺎﺼﺗ

ﺎﺑ

ﻲﻔﻴﻃ ﻲﻟﺎﮕﭼ

1 2 2

0 − x x− −

S e ^α e ^{γ ω}

ﻲﻣ راﺮﻗ

دﺮﻴﮔ ] 30 [

، و

لوﺪﺟ رد نآ صاﻮﺧ 2

ﺪﻫاﻮﺧ راﺮﻗ ﻞﻴﻠﺤﺗ درﻮﻣ ،ﺖﺳا هﺪﻣآ

ﺖﻓﺮﮔ . اﺪﺘﺑا رد و ﻲﻌﻴﺒﻃ ﺲﻧﺎﻛﺮﻓ رﺎﻬﭼ لوا دﻮﻣ ﻞﻜﺷ رﺎﻬﭼ

ﻲﻣ ﻪﺒﺳﺎﺤﻣ ﻢﺘﺴﻴﺳ

،ﺮﻴﺗ ﻲﻔﻴﻃ ﻲﻟﺎﮕﭼ ﺐﻴﺗﺮﺗ ﻪﺑ ﺲﭙﺳ ،ددﺮﮔ

ﺶﻨﺗ ﺲﻧﺎﻳراو ،ﺮﻴﺗ ﻒﻠﺘﺨﻣ طﺎﻘﻧ ﺦﺳﺎﭘ ﺲﻧﺎﻳراو

،ﺮﻴﺗ لﻮﻃ رد

ﺮﻴﺗ ﻢﻴﻠﺴﺗ لﺎﻤﺘﺣا ﺖﻳﺎﻬﻧ رد و ﺮﻴﺗ ﺦﺳﺎﭘ لﺎﻤﺘﺣا ﻲﻟﺎﮕﭼ ﻊﺑﺎﺗ ﻲﻣ ﺖﺳﺪﺑ هﺪﺷ ﺺﺨﺸﻣ يراﺬﮔرﺎﺑ ﺖﺤﺗ ﺪﻳآ

.

ﻞﻜﺷ رد ﺮﻴﺗ لوا دﻮﻣ ﻞﻜﺷ رﺎﻬﭼ 3

و 4 لوا ﻲﻌﻴﺒﻃ ﺲﻧﺎﻛﺮﻓ

لوﺪﺟ رد 3

ﺖﺳا هﺪﺷ هداد نﺎﺸﻧ .

لوﺪﺟ 2 ﻲﺳرﺮﺑ درﻮﻣ ﺮﻴﺗ تﺎﺼﺨﺸﻣ

ﺮﺘﻣارﺎﭘ راﺪﻘﻣ

1 α 1 γ

200 E لﺎﻜﺳﺎﭘﺎﮕﻴﮔ

8000 ρ ﺐﻌﻜﻣﺮﺘﻣ ﺮﺑ مﺮﮔﻮﻠﻴﻛ

1 ﺮﺘﻣ

hi

06 / 0 ﺮﺘﻣ

hj

04 / 0 ﺮﺘﻣ

03 b / 0 ﺮﺘﻣ

S0

1×108 نﻮﺗﻮﻴﻧ ﻊﺑﺮﻣﺮﺘﻣ ﺮﺑ ﻪﻴﻧﺎﺛ

σy

25 / 0 لﺎﻜﺳﺎﭘﺎﮕﻴﮔ

1 . Variance

2 . Standard Deviation

ﻲﺒﺴﻧﻲﻳﺎﺠﺑﺎﺟ

ﻞﻜﺷ 3 ﺮﻴﺗ يﺎﻫدﻮﻣ ﻞﻜﺷ ﻚﻳ

ﺮﺳ ﺮﻴﻐﺘﻣ ﻊﻄﻘﻣ رادﺮﻴﮔ

ﻞﻜﺷ 4 ﺮﻴﺗ لﻮﻃ رد ﺦﺳﺎﭘ ﻲﻔﻴﻃ ﻲﻟﺎﮕﭼ

لوﺪﺟ 3 ﺮﻴﻐﺘﻣ ﻊﻄﻘﻣ ﺮﻴﺗ ﻲﻌﻴﺒﻃ ﺲﻧﺎﻛﺮﻓ ﻚﻳ

ﺮﺳ رادﺮﻴﮔ

ﻲﻌﻴﺒﻃ ﺲﻧﺎﻛﺮﻓ راﺪﻘﻣ

) ﻪﻴﻧﺎﺛ ﺮﺑ نﺎﻳدار (

ω1

45 / 319

ω2

84 / 1699

ω3

17 / 4533

ω4

45 / 8764

رد ﺮﻴﺗ لﻮﻃ رد ﻒﻠﺘﺨﻣ طﺎﻘﻧ يازا ﻪﺑ ﺮﻴﺗ ﻲﻔﻴﻃ ﻲﻟﺎﮕﭼ ﻊﺑﺎﺗ ﻞﻜﺷ 4 ﺖﺳا هﺪﺷ هداد نﺎﺸﻧ .

نﺎﻤﻫ ﻞﻜﺷ رد ﻪﻛ رﻮﻃ 4

[ Downloaded from mme.modares.ac.ir on 2022-10-31 ]

ﺮﻴﺗ كﻮﻧ رد ﺦﺳﺎﭘ ﻲﻔﻴﻃ ﻲﻟﺎﮕﭼ ﻊﺑﺎﺗ راﺪﻘﻣ ،ﺖﺳا ﺺﺨﺸﻣ ﻪﻄﺑار ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ ﻦﻳاﺮﺑﺎﻨﺑ ،ﺖﺳاراد ار دﻮﺧ ﻢﻤﻳﺰﻛﺎﻣ راﺪﻘﻣ )

19 (

ﺪﺷ ﺪﻫاﻮﺧ ﻪﻨﻴﺸﻴﺑ ﺮﻴﺗ كﻮﻧ رد ﺰﻴﻧ ﺦﺳﺎﭘ ﺲﻧﺎﻳراو )

ﻞﻜﺷ 5 .(

ﻪﻴﻜﺗ ﻊﻄﻘﻣ رد ﻲﻗﺎﻔﺗا ﺶﻨﺗ لﺎﻤﺘﺣا ﻲﻟﺎﮕﭼ ﻊﺑﺎﺗ ﻪﺑ ﺮﻴﺗ هﺎﮔ

ﻪﻠﻴﻣ رادﻮﻤﻧ ترﻮﺻ ﻞﻜﺷ رد ،يا

6 ﺖﺳا هﺪﺷ هداد نﺎﺸﻧ .

ﻞﻜﺷ 5 ﺮﻴﺗ لﻮﻃ رد ﻲﻳﺎﺠﺑﺎﺟ ﺲﻧﺎﻳراو

ﻞﻜﺷ 6 ﻲﻟﺎﮕﭼ ﻊﺑﺎﺗ ﺮﻴﺗ رد ﻲﻗﺎﻔﺗا ﺶﻨﺗ لﺎﻤﺘﺣا

نﺎﻤﻫ ﻞﻜﺷ رد ﻪﻛ رﻮﻃ 7

ﻲﻗﺎﻔﺗا ﺶﻨﺗ ﺲﻧﺎﻳراو ،ﺖﺳا ﺺﺨﺸﻣ ﻪﻴﻜﺗ رد

ﻲﻣ ﻪﻨﻴﺸﻴﺑ راﺪﻘﻣ ياراد هﺎﮔ رد ﻪﻛ يرﻮﻃ ﻪﺑ ﺪﺷﺎﺑ

ﻪﻴﻜﺗ هداد ﻲﮔﺪﻨﻛاﺮﭘ ﻪﻨﻴﺸﻴﺑ هﺎﮔ ﻲﻣ خر ﺦﺳﺎﭘ ﻲﻓدﺎﺼﺗ يﺎﻫ

ﺪﻫد .

ﻪﻴﻜﺗ رد ﻲﻗﺎﻔﺗا ﺶﻨﺗ ﺲﻧﺎﻳراو ﻪﺒﺳﺎﺤﻣ زا ﺪﻌﺑ ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ هﺎﮔ

ﻞﻜﺷ 6 ﻪﻛ ﻲﻣ ﺦﺳﺎﭘ لﺎﻣﺮﻧ ﻊﻳزﻮﺗ ﺮﮕﻧﺎﻳﺎﻤﻧ ﻲﻣ ،ﺪﺷﺎﺑ

رادﻮﻤﻧ ناﻮﺗ

ﻪﻴﻜﺗ رد ﻲﻗﺎﻔﺗا ﺶﻨﺗ ﻞﻜﺷ ترﻮﺻ ﻪﺑ ار هﺎﮔ

8 ﺖﺷاد .

ﻞﻜﺷ 7 ﺮﻴﺗ لﻮﻃ رد ﺶﻨﺗ ﺲﻧﺎﻳراو

ﻞﻜﺷ 8 ﻪﻴﻜﺗ رد ﺮﻴﻐﺘﻣ ﻊﻄﻘﻣ ﺮﻴﺗ ﻲﻗﺎﻔﺗا ﺶﻨﺗ هﺎﮔ

[ Downloaded from mme.modares.ac.ir on 2022-10-31 ]

نﺎﻤﻫ ﻞﻜﺷ رد ﻪﻛ رﻮﻃ 8

ﺎﺑ ﻚﻳﺮﺤﺗ يازا ﻪﺑ ﺖﺳا ﺺﺨﺸﻣ

ﻪﻴﻜﺗ رد ﺶﻨﺗ هﺪﺷ هداد تﺎﺼﺨﺸﻣ ﻪﺘﻓر ﺮﺗاﺮﻓ ﻢﻴﻠﺴﺗ ﺶﻨﺗ زا هﺎﮔ

،ﺖﺳا اﺬﻟ ﻲﻣ ﺶﻨﺗ ﻂﺧ يﻻﺎﺑ ﻢﻤﻳﺰﻛﺎﻣ ﺶﻨﺗ دﻮﺟو لﺎﻤﺘﺣا ناﻮﺗ

ﻪﻄﺑار زا ار ﻢﻴﻠﺴﺗ )

26 ( دﻮﻤﻧ ﺺﺨﺸﻣ

، ﻪﻛ لﺎﻤﺘﺣا ﻢﻴﻠﺴﺗ رد

ﻚﻳﺮﺤﺗ زا تﺪﺷ ﻦﻳا

،

% 64 / 15 ﻲﻣ ﺪﺷﺎﺑ .

8 - ﻪﺠﻴﺘﻧ يﺮﻴﮔ

رﺮﺑ ﻪﺑ ﻪﻌﻟﺎﻄﻣ ﻦﻳا رد ﺮﻴﻐﺘﻣ ﻊﻄﻘﻣ ﺮﻴﺗ ﻲﻗﺎﻔﺗا تﺎﺷﺎﻌﺗرا ﻲﺳ

يرﺎﻣآ صاﻮﺧ ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ و ﺪﺷ ﻪﺘﺧادﺮﭘ ﻲﻓدﺎﺼﺗ ﻚﻳﺮﺤﺗ ﺖﺤﺗ ﻞﻴﻠﺤﺗ ﺰﻴﻧ ﻲﻓدﺎﺼﺗ رﺎﺑ ﻪﺠﻴﺘﻧ رد ﺮﻴﺗ ﻢﻴﻠﺴﺗ لﺎﻤﺘﺣا ﻲﻗﺎﻔﺗا ﺦﺳﺎﭘ ﺷ ﺪ . ﻪﺤﺻ رﻮﻈﻨﻣ ﻪﺑ ﻲﻓﺎﻛ هﺪﻣآ ﺖﺳﺪﺑ ﺦﺳﺎﭘ ﺮﺑ يراﺬﮔ

ا ﺖﺳ

ﻲﻠﻛ رﻮﻄﺑ ﺎﻳ و ﺮﻴﺗ يﺎﻫدﻮﻣ ﻞﻜﺷ ﻪﺼﺨﺸﻣ ﻪﻟدﺎﻌﻣ ﻞﺣ ،ﺮﺗ

اﺮﺷ رد ﻢﺘﺴﻴﺳ ﺎﺑ ﺪﺷﺎﺒﻧ دﺎﻳز ﻊﻄﻘﻣ ود توﺎﻔﺗ ﻪﻛ ﻲﺻﺎﺧ ﻂﻳ

ﺖﺑﺎﺛ ﻊﻄﻘﻣ ﺮﻴﺗ ﻚﻳ ﻪﺼﺨﺸﻣ ﻪﻟدﺎﻌﻣ ﻞﺣ ﺮﻳدﺎﻘﻣ ﻚﻳ

ﺮﺳ رادﺮﻴﮔ

ﻪﺴﻳﺎﻘﻣ ﻮﺷ د . ﺮﺘﻣارﺎﭘ راﺪﻘﻣ رﻮﻈﻨﻣ ﻦﻳا ياﺮﺑ

hi

06 / 0 راﺪﻘﻣ و

ﺮﺘﻣارﺎﭘ

hj

0599 / 0 ﺮﻈﻧ رد ﻲﻣ ﻪﺘﻓﺮﮔ دﻮﺷ

ﺮﻴﺗ ﺎﺗ ﻪﻌﻟﺎﻄﻣ درﻮﻣ

زا هﺪﻣآ ﺖﺳﺪﺑ ﺞﻳﺎﺘﻧ لﺎﺣ ،دﻮﺷ ﺖﺑﺎﺛ ﻊﻄﻘﻣ ﺮﻴﺗ ﻪﺑ ﻞﻳﺪﺒﺗ ًﺎﺒﻳﺮﻘﺗ ﺮﻴﻏ ﻞﺣ ﻪﻟدﺎﻌﻣ ﻲﻬﻳﺪﺑ )

7 ( ﺎﺑ ﺮﻴﺗ ﻪﺑ طﻮﺑﺮﻣ ﺞﻳﺎﺘﻧ ﻚﻳ

ﺮﺳ رادﺮﻴﮔ

ﺖﺑﺎﺛ ﻊﻄﻘﻣ ]

28 [ ﻣ ﻲﻣ ﻪﺴﻳﺎﻘ ﻮﺷ

د ) لوﺪﺟ 4 (.

ﻪﻌﻟﺎﻄﻣ ﻦﻳا رد ﻢﻤﻳﺰﻛﺎـﻣ ﻪﻄﻘﻧ دﻮﺟو لﺎﻤﺘﺣا

ﺶﻨـﺗ زا ﺮﺗﻻﺎـﺑ

ﻢﻴﻠﺴﺗ ﻪﺒﺳﺎﺤﻣ ﺷ ﺪ . ﻢﻠﺴﻣ ﻪﭽﻧآ ﻚـﻳﺮﺤﺗ تﺪـﺷ ﺶﻳاﺰـﻓا ﺖـﺳا

ﺎـﺑ لﺎـﺜﻣ رﻮـﻃ ﻪـﺑ داد ﺪﻫاﻮﺧ ﺶﻳاﺰﻓا ار ﻢﻴﻠﺴﺗ لﺎﻤﺘﺣا 5

ﺮـﺑاﺮﺑ

ﻪـﺑ ﻢﻴﻠـﺴﺗ لﺎـﻤﺘﺣا ﻚـﻳﺮﺤﺗ تﺪـﺷ ندﺮﻛ

% 69 اﺪـﻴﭘ ﺶﻳاﺰـﻓا

ﻲﻣ ﺪﻨﻛ . دﺎـﻌﺑا ،ﺮﻴﺗ ﻢﻴﻠﺴﺗ لﺎﻤﺘﺣا يور ﺮﺑ راﺬﮔﺮﻴﺛﺄﺗ ﺮﮕﻳد ﻞﻣاﻮﻋ

ﻲﻣ ﺮﻴﺗ تﺎﺼﺨﺸﻣ و ﻲـﻣ هﺪﺷ ﻪﺋارا ﻞﺣ ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ ﻪﻛ ﺪﺷﺎﺑ

ناﻮـﺗ

ماﺪﻛﺮﻫ ﺮﻴﻴﻐﺗ ﺎﺑ ﻢﻴﻠـﺴﺗ لﺎـﻤﺘﺣا يور ﺮﺑ ﺮﻴﺛﺄﺗ ناﺰﻴﻣ ﻲﺳرﺮﺑ ﻪﺑ

لﺎـﻤﺘﺣا ﺮـﻴﺗ لﻮـﻃ ﺶﻳاﺰـﻓا ﺎـﺑ لﺎـﺜﻣ ناﻮـﻨﻋ ﻪـﺑ ﺖـﺧادﺮﭘ ﺮﻴﺗ لﺎﻤﺘﺣا ﺶﻫﺎﻛ ﺐﺟﻮﻣ ﺮﻴﺗ يﺎﻨﻬﭘ ﺶﻳاﺰﻓا و ﻪﺘﻓﺎﻳ ﺶﻳاﺰﻓا ﻢﻴﻠﺴﺗ ﻲﻣ ﻢﻴﻠﺴﺗ ﻮﺷ

د .

لوﺪﺟ 4 ﻪﺤﺻ ﺦﺳﺎﭘ ﺮﺑ يراﺬﮔ ﻪﺼﺨﺸﻣ ﻪﻟدﺎﻌﻣ ﻞﺣ ﻪﺴﻳﺎﻘﻣ ﺎﺑ ﺎﻫ

ﻪﺼﺨﺸﻣ ﻪﻟدﺎﻌﻣ ﻞﺣ هﺪﺷ ﻪﺋارا ﻪﻌﻟﺎﻄﻣ

ﻊﺟﺮﻣ ] 28 [

β1

8752 / 1 8751

/ 1

β2

6928 / 4 6941

/ 4

β3

8519 / 7 8547

/ 7

β4

9912 / 10 9956

/ 10

اﺮﻧآ ﻲﻔﻴﻃ ﻲﻟﺎﮕﭼ ،ﻲﻓدﺎﺼﺗ يراﺬﮔرﺎﺑ ﻚﻳ يرﺎﻣآ يﺎﻫﺰﻴﻟﺎﻧآ ﺎﺷﺎﻌﺗرا ﻞﻴﻠﺤﺗ مﺎﺠﻧا ﺎﺑ نآ يور زا و هدﺮﻛ ﺺﺨﺸﻣ و ﻲﻗﺎﻔﺗا ت

ﻲﻣ ﻢﻴﻠﺴﺗ لﺎﻤﺘﺣا ندروآ ﺖﺳﺪﺑ هزﺎﺳ ﻚﻳ نﺎﻨﻴﻤﻃا ﺖﻴﻠﺑﺎﻗ ناﻮﺗ

ﻲﺣاﺮﻃ ياﺮﺑ مزﻻ رﺎﻴﻌﻣ و دﻮﻤﻧ ﺺﺨﺸﻣ ﻚﻳﺮﺤﺗ نآ ﻪﺑ ﺖﺒﺴﻧ ار دﺮﻛ ﺺﺨﺸﻣ ار هزﺎﺳ .

9 - ﻣ ﺟاﺮ ﻊ

[1] Taleb N. J. and Suppiger E. W., “Vibration of Stepped Beams”, Journal of the Aerospace Sciences, Vol. 28, 1961, pp. 295-298

[2] Lindberg G. M., “Vibrations of Nonuniform Beams”, The Aeronautica Quarterly, Vol. 14, 1963, pp. 387-395.

[3] Wang H. C., “Generalized Hypergeometric Function Solutions on Transverse Vibrations of a Class of Nonuniform Beams”, Journal of Applied Mechanics, Vol. 34, 1968, pp. 702-708.

[4] Mabie H. H. and Rogers C. B., “Transverse Vibrations of Tapered Cantilever Beams with End Supports”, Journal of the Acoustical Society of America, Vol. 44, 1968, pp. 1739-1741.

[5] Mabie H. H. and Rogers C. B., “Transverse Vibrations of Double-Tapered Cantilever Beams”, Journal of the Acoustical Society of America, Vol.

57, 1972, pp. 1771-1774.

[6] Mabie H. H. and Rogers C. B., “Transverse Vibrations of Double-Tapered Cantilever Beams with End Support and with End Mass”, Journal of the Acoustical Society of America, Vol. 55, 1974, pp. 986-991.

[7] Ganga Rao H. V. S. and Spyrakos C. C., “Closed form Series Solutions of Boundary Value Problems with Variable Properties”, Computers and Structures, Vol. 23, 1986, pp. 211-215.

[8] Finlayson B. A., The Method of Weighted Residuals and Variational Principals, New York Academic Press, 1972.

[9] Mikhlin S. G.,Variational Methods in Mathematical Physics, New York Macmillan, 1964.

[10]Laura P. A. A., Valerga De Greco B., Utjes J. C.

and Carnicer R., “Numerical Experiments on Free and Forced Vibrations of beams of nonuniform cross-section”, Journal of Sound and Vibration, Vol. 120, 1988, pp. 587-596.

[11]Bathe K. J., Finite Element Procedures in Engineering Analysis, Englewood Cliffs, New Jersey: Prentice-Hall, 1982.

[12]Ghali A. And Neville A. M., Structural Analysis, Pennsylvania: Intext Educational Publishers, 1972.

[13]Beskos D. E., “Boundary Element Methods in Dynamic Analysis” Applied Mechanics Reviews, Vol. 40, 1987, pp. 1-23.

[ Downloaded from mme.modares.ac.ir on 2022-10-31 ]

[14]Chu F. H. and Pilkeyw. D., “Transient Analysis of Structural Members by the CSDT Riccati Transfer Matrix Method”, Computers and Structures, Vol.

10, 1979, pp. 599-611.

[15]JUST D. J., “Plane Frameworks of Tapered Box and I-Section”,Structural Division, Proceedings of the American Society of Civil Engineers, Vol. 103, 1977, pp. 71-86.

[16]Ovunk B. A., “Dynamics of Frameworks by Continuous Mass Method”, Computers and Structures, Vol. 4, 1974, pp. 1061-1089.

[17]Beskosd. E., “Dynamics and Stability of Plane Trusses with Gusset Plate” Computers and Structures, Vol. 10, 1979, pp. 785-795.

[18]Beskos D. E. and Narayanan G. V., “Dynamic Response of Frameworks by Numerical Laplace Transform”, Computer Methods in Applied Mechanics and Engineering, Vol. 37, 1983, pp. 289-307.

[19]Lee S. Y., Ke H. Y. and Kuo Y. H., “Analysisof Non-Uniform Beam Vibration”, Journal of Sound and Vibration, Vol. 142(1), 1990, pp. 15-29.

[20]Craver W. L., Jampala J.R. and P., “Transverse Vibration of a Linearly Tapered Cantilever Beam with Constraining Springs”, Journal of Sound and Vibration, Vol. 166(3), 1993, pp. 521-529.

[21]De Rosa M.A., Auciello N.M., Maurizi M.J., “The use of Mathematical in the Dynamic Analysis of a Beam with a Concentrated Mass and Dashpot”, Journal of Sound and Vibration, Vol. 263, 2003, pp. 219-226.

[22]De Rosaa M.A., Maurizi M.J., “Damping in Exact Analysis of Tapered Beams”, Journal of Sound and Vibration, Vol. 286, 2005, pp. 1041-1047.

[23]De Rosa M.A., Lippiello M., Maurizi M.J., Martin H.D., “Free Vibration of Elastically Restrained Cantilever Tapered Beams with Concentrated Viscous Damping and Mass”, Journal of Mechanics Research Communications, Vol. 37, 2010, pp. 261-264.

[24]Valipour Hamid R., Bradford Mark A., “A New Shape Function for Tapered Three-Dimensional Beams with Flexible Connections”, Journal of Constructional Steel Research, Vol. 70, 2012, pp. 43-50.

[25]Rzhanitsyn A.R., Probabilistic Calculation of Beams on a Random Load, Stroizdat Publ. House, Moscow, Vol. 12, 1977.

[26] Elishakoff, I.,Probabilistic Methods in the Theory of Structures, John Wiley and Sons, New York, 1983.

[27]Waleed F., Faris J., DaoudI. and Sutjipto A.G.E.,

“Random Vibrations of a Beam on Elastic Foundation”, Australian Journal of Basic and Applied Sciences, Vol. 5(6), 2011, pp. 456-460.

[28]Singiresu S. Rao,Vibration of Continuous Systents, John Wiley and Sons, New York, 2007.

[29]Newland, D.E., An Introduction to Random Vibrations, Spectral & Wavelet Analysis, Dover publication, Mineola New York, 3th Edition, 2005.

[30]Chang, T.P. and Chang, C.Y., “Vibration Analysis of Beams with a Two Degree-of-Freedom Spring- ass System”,lnt. J. Solids Structures, Vol. 35, No.

5, 1998, pp. 383-401.

[ Downloaded from mme.modares.ac.ir on 2022-10-31 ]