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91

يدازآ ﻪﺟرد ﻪﺳ لﺪﻣ

، ﻪﻛ

دراد راﺮﻗ ﺐﻠﺻ ﻚﺴﻳد ﻚﻳ ، .

اﺪﺘﺑا

و هﺪﻳدﺮﮔ نﺎﻴﺑ ﻞﻴﻫ

زا ﻢﺘﺴﻴﺳ يﺎﻫﺮﺘﻣارﺎﭘ ﺐﺴﺣ ﺖﻋﺮﺳ ﻞﻴﺒﻗ

ﻒﻠﺘﺨﻣ يﺎﻫﺮﺘﻣارﺎﭘ سﺎﺳا ﻪﺋارا

هﺪﺷ

Stability of a system

Abstract- In rotation axises

a spatial coordinate (three dim Hooke's joints.

are shown to consist of

method. Finally dynamic stability regions have been shown on system parameters such as rotational velocity, misalignment angle of shaft axis, stiffness and rig

It was observed

Keywords: Dynamic Stability,

رﺎﺗ ﻳ ﻪﭽﺨ رد ﻳ ﺖﻓﺎ ﺬﭘ ﻳ شﺮ رد ﻪﺋارا

كﻮﻫ لﺎﺼﺗا ﺎﺑ

ﻲﻔﺠﻧ ﺮﻐﺻا ،

4

ﻪﻠﻴﺳو ﻪﺑ يدازآ ﻪﺟرد ﻪﺳ لﺪﻣ

دراد راﺮﻗ ﺐﻠﺻ ﻚﺴﻳد ﻚﻳ ،

ﻮﻴﺘﻣ تﻻدﺎﻌﻣ هﺎﮕﺘﺳد ترﻮﺻ -

و هﺪﻳدﺮﮔ نﺎﻴﺑ ﻞﻴﻫ

زا ﻢﺘﺴﻴﺳ يﺎﻫﺮﺘﻣارﺎﭘ ﺐﺴﺣ ﻒﻠﺘﺨﻣ يﺎﻫﺮﺘﻣارﺎﭘ سﺎﺳا

ﻲﻣ ﺶﻳاﺰﻓا يراﺪﻳﺎﭘ ،كﻮﻫ لﺎﺼﺗا ﻪﻳواز ﺶﻫﺎﻛ ﺎﺑ ﻦﻴﻨﭽﻤﻫ و ﺎﻫ ﺑﺎﻳ

ﺪ .

tability of a system

M. Soltan

In this study, dynamic stability of a system consisting of three torsionally elastic shafts with different rotation axises is analyzed

spatial coordinate (three dim Hooke's joints. Equations of motion are shown to consist of a set of Mathieu

method. Finally dynamic stability regions have been shown on system parameters such as rotational velocity, misalignment angle of shaft axis, stiffness and rig

It was observed that with increasing inertia disk ratio and decreasing hooke's Dynamic Stability,

كﻮﻫ لﺎﺼﺗا ﺎﺑ

يﺮﻔﻌﺟ ﺮﻐﺻا ﻲﻔﺠﻧ ﺮﻐﺻا ،3

سرﺪﻣ ﺖ ناﺮﻬﺗ ،

ناﺮﻬﺗ ،سرﺪﻣ ﺖﻴﺑﺮﺗ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ

ناﺮﻬﺗ ،

ناﺮﻬﺗ ،سرﺪﻣ ﺖﻴﺑﺮﺗ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ ي

ghazavim@modares.ac.ir

ﻢﻫ ﺮﻴﻏ رﻮﺤﻣ ﻪﻠﻴﺳو ﻪﺑ

ﺮﻫ يﺎﻬﺘﻧا رد و ﺪﻧا ماﺪﻛ

دراد راﺮﻗ ﺐﻠﺻ ﻚﺴﻳد ﻚﻳ ،

ﻮﻴﺘﻣ تﻻدﺎﻌﻣ هﺎﮕﺘﺳد ترﻮﺻ

ﺮﺑ ﻲﻜﻴﻣﺎﻨﻳد يراﺪﻳﺎﭘ زا ﻢﺘﺴﻴﺳ يﺎﻫﺮﺘﻣارﺎﭘ ﺐﺴﺣ

ﺮﺑ يراﺪﻳﺎﭘ يﺎﻫرادﻮﻤﻧ ﻒﻠﺘﺨﻣ يﺎﻫﺮﺘﻣارﺎﭘ سﺎﺳا

ﻲﻣ ﺶﻳاﺰﻓا يراﺪﻳﺎﭘ ،كﻮﻫ لﺎﺼﺗا ﻪﻳواز ﺶﻫﺎﻛ ﺎﺑ ﻦﻴﻨﭽﻤﻫ و ﺎﻫ

tability of a system through

M. Soltan Rezaee

1- MSc. Student, Mech. Eng., 2- Assoc. Prof.,

3- Assoc. Prof., Mech. Eng.,

4- PhD Student, Mech. Eng., Tarbiat Modares Univ., Tehran, Iran

* P. O. B.

this study, dynamic stability of a system consisting of three torsionally elastic shafts with different is analyzed. The system stability

spatial coordinate (three dimensional).

Equations of motion for the a set of Mathieu

method. Finally dynamic stability regions have been shown on system parameters such as rotational velocity, misalignment angle of shaft axis, stiffness and rig

that with increasing inertia disk ratio and decreasing hooke's Dynamic Stability, Shaft System,

http://mechanics.journals.modares.ac.ir

69 - 79

ﻪﺳ زا ﻞﻜﺸﺘﻣ ﻢﺘﺴﻴﺳ ﻚﻳ يراﺪﻳﺎﭘ رﻮﺤﻣ

ﻲﻠﻋ يﺮﻔﻌﺟ ﺮﻐﺻا

ﻴﺑﺮﺗ هﺎﮕﺸﻧاد ، سرﺪﻣ ﺖ

ناﺮﻬﺗ ،سرﺪﻣ ﺖﻴﺑﺮﺗ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ ﺮﻴﺼﻧ ﻪﺟاﻮﺧ ﻲﺘﻌﻨﺻ

ناﺮﻬﺗ ،

ناﺮﻬﺗ ،سرﺪﻣ ﺖﻴﺑﺮﺗ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ ي ghazavim@modares.ac.ir

ﻲﺸﭽﻴﭘ ﻚﻴﺘﺳﻻا ﺖﻓﺎﺷ ﻪﺳ ﻞﻣﺎﺷ ﻢﺘﺴﻴﺳ ﻚﻳ ﻲﻜﻴﻣﺎﻨﻳد يراﺪﻳﺎﭘ ،ﺶﻫوﮋﭘ ﻦﻳا رد ﻢﻫ ﺮﻴﻏ

ﻪﺘﺸﮔ ﻞﺼﺘﻣ كﻮﻫ لﺎﺼﺗا ود ﺎﺑ ﺎﻫ ﺮﻫ يﺎﻬﺘﻧا رد و ﺪﻧا

ﻪﺑ ،ﻞﻴﺴﻧاﺮﻔﻳد تﻻدﺎﻌﻣ ندﺮﻛ ﻮﻴﺘﻣ تﻻدﺎﻌﻣ هﺎﮕﺘﺳد ترﻮﺻ

ﺮﺑ ﻲﻜﻴﻣﺎﻨﻳد يراﺪﻳﺎﭘ ﺖﺳا هﺪﻳدﺮﮔ .

ﺮﺑ يراﺪﻳﺎﭘ يﺎﻫرادﻮﻤﻧ

ﻲﻣ ﺶﻳاﺰﻓا يراﺪﻳﺎﭘ ،كﻮﻫ لﺎﺼﺗا ﻪﻳواز ﺶﻫﺎﻛ ﺎﺑ ﻦﻴﻨﭽﻤﻫ و ﺎﻫ

tability of a system consisting of through

Rezaee¹, M. R.

MSc. Student, Mech. Eng.,

. Prof., Mech. Eng., Tarbiat Modares Univ., Tehran, Iran Assoc. Prof., Mech. Eng.,

D Student, Mech. Eng., Tarbiat Modares Univ., Tehran, Iran P. O. B. 14115-111 Tehran

this study, dynamic stability of a system consisting of three torsionally elastic shafts with different . The system stability have been investigated by means of a three degree

ensional). Each shaft carrying a rigid disk at one end and have been for the system

a set of Mathieu-Hill equations. Their

method. Finally dynamic stability regions have been shown on system parameters such as rotational velocity, misalignment angle of shaft axis, stiffness and rigidity of shafts

that with increasing inertia disk ratio and decreasing hooke's Shaft System, Torsional Vibration, Hooke’s

mechanics.journals.modares.ac.ir ﻲﺸﻫوﮋﭘ ﻲﻤﻠﻋ ﻪﻠﺠﻣ

هرﺎﻤﺷ 6

، ﺪﻨﻔﺳا 1391 ص ص 69

ﻪﺳ زا ﻞﻜﺸﺘﻣ ﻢﺘﺴﻴﺳ ﻚﻳ يراﺪﻳﺎﭘ

يوﺎﻀﻗ ﺎﺿرﺪﻤﺤﻣ ،

2

، *

ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ ﻴﺑﺮﺗ هﺎﮕﺸﻧاد ،

ناﺮﻬﺗ ،سرﺪﻣ ﺖﻴﺑﺮﺗ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ رﺎﻴﺸﻧاد ﺮﻴﺼﻧ ﻪﺟاﻮﺧ ﻲﺘﻌﻨﺻ

ناﺮﻬﺗ ،سرﺪﻣ ﺖﻴﺑﺮﺗ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ ي 111 - 14115

، ghazavim@modares.ac.ir

ﻲﺸﭽﻴﭘ ﻚﻴﺘﺳﻻا ﺖﻓﺎﺷ ﻪﺳ ﻞﻣﺎﺷ ﻢﺘﺴﻴﺳ ﻚﻳ ﻲﻜﻴﻣﺎﻨﻳد يراﺪﻳﺎﭘ ،ﺶﻫوﮋﭘ ﻦﻳا رد ﻪﺘﺸﮔ ﻞﺼﺘﻣ كﻮﻫ لﺎﺼﺗا ود ﺎﺑ ﺎﻫ

ﻪﺑ ،ﻞﻴﺴﻧاﺮﻔﻳد تﻻدﺎﻌﻣ ندﺮﻛ رد هدوﺪﺤﻣ ﺎﻬﺘﻧا ﺮﺑ ﻲﻜﻴﻣﺎﻨﻳد يراﺪﻳﺎﭘ

نﺎﻴﺑ ﺎﻫ ﺖﺳا هﺪﻳدﺮﮔ

ﻲﻣ ﺶﻳاﺰﻓا يراﺪﻳﺎﭘ ،كﻮﻫ لﺎﺼﺗا ﻪﻳواز ﺶﻫﺎﻛ ﺎﺑ ﻦﻴﻨﭽﻤﻫ و ﺎﻫ

كﻮﻫ لﺎﺼﺗا ،ﻲﺸﭽﻴﭘ تﺎﺷﺎﻌﺗرا

consisting of

through Hooke’s joints

R. Ghazavi²*

MSc. Student, Mech. Eng., Tarbiat Modares Univ., Tehran, Iran Mech. Eng., Tarbiat Modares Univ., Tehran, Iran Assoc. Prof., Mech. Eng., K. N. Toosi Univ. of Tech.,

D Student, Mech. Eng., Tarbiat Modares Univ., Tehran, Iran 111 Tehran, ghazavim@modares.ac.ir

this study, dynamic stability of a system consisting of three torsionally elastic shafts with different have been investigated by means of a three degree

Each shaft carrying a rigid disk at one end and have been

were derivated and after linearization of the differential equations Hill equations. Their stability

method. Finally dynamic stability regions have been shown on system parameters such as rotational velocity, idity of shafts. The

that with increasing inertia disk ratio and decreasing hooke's Vibration, Hooke’s

mechanics.journals.modares.ac.ir

هرود 12 هرﺎﻤﺷ

ﻪﺳ زا ﻞﻜﺸﺘﻣ ﻢﺘﺴﻴﺳ ﻚﻳ يراﺪﻳﺎﭘ

ﻲﻳﺎﺿر يوﺎﻀﻗ ﺎﺿرﺪﻤﺤﻣ ،1

ﺪﺷرا ﻲﺳﺎﻨﺷرﺎﻛ يﻮﺠﺸﻧاد ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ

رﺎﻴﺸﻧاد ناﺮﻬﺗ ،سرﺪﻣ ﺖﻴﺑﺮﺗ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ

هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ رﺎﻴﺸﻧاد ﺮﺘﻛد يﻮﺠﺸﻧاد ا

ناﺮﻬﺗ ،سرﺪﻣ ﺖﻴﺑﺮﺗ هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ ي

ﻲﺘﺴﭘ قوﺪﻨﺻ ،ناﺮﻬﺗ 111

ﻲﺸﭽﻴﭘ ﻚﻴﺘﺳﻻا ﺖﻓﺎﺷ ﻪﺳ ﻞﻣﺎﺷ ﻢﺘﺴﻴﺳ ﻚﻳ ﻲﻜﻴﻣﺎﻨﻳد يراﺪﻳﺎﭘ ،ﺶﻫوﮋﭘ ﻦﻳا رد ﺖﻓﺎﺷ

ﻪﺘﺸﮔ ﻞﺼﺘﻣ كﻮﻫ لﺎﺼﺗا ود ﺎﺑ ﺎﻫ

ﻲﻄﺧ زا ﺲﭘ و ﺖﺳا هﺪﺷ جاﺮﺨﺘﺳا ﻢﺘﺴﻴﺳ ياﺮﺑ ﻪﺑ ،ﻞﻴﺴﻧاﺮﻔﻳد تﻻدﺎﻌﻣ ندﺮﻛ

ﺖﺳا هﺪﺷ ﻞﻴﻠﺤﺗ ﺲﻳﺮﺗﺎﻣ ﻲﻣرﺪﻨﻣ شور ﻪﺑ ﻪﻋﻮﻤﺠﻣ يراﺪﻳﺎﭘ .

رد

ﺖﻓﺎﺷ مﺎﻜﺤﺘﺳا و ﻲﺘﺨﺳ ،ﺮﮕﻳﺪﻜﻳ ﺎﺑ ﺎﻫ نﺎﻴﺑ ﺎﻫ

ﻚﺴﻳد ﻲﺳﺮﻨﻳا ﺖﺒﺴﻧ ﺶﻳاﺰﻓا ﺎﺑ ﻪﻛ د ﻲﻣ ﺶﻳاﺰﻓا يراﺪﻳﺎﭘ ،كﻮﻫ لﺎﺼﺗا ﻪﻳواز ﺶﻫﺎﻛ ﺎﺑ ﻦﻴﻨﭽﻤﻫ و ﺎﻫ

كﻮﻫ لﺎﺼﺗا ،ﻲﺸﭽﻴﭘ تﺎﺷﺎﻌﺗرا

consisting of three ooke’s joints

*, A. A. Jafari

Tarbiat Modares Univ., Tehran, Iran Mech. Eng., Tarbiat Modares Univ., Tehran, Iran

K. N. Toosi Univ. of Tech.,

D Student, Mech. Eng., Tarbiat Modares Univ., Tehran, Iran ghazavim@modares.ac.ir

this study, dynamic stability of a system consisting of three torsionally elastic shafts with different have been investigated by means of a three degree

Each shaft carrying a rigid disk at one end and have been

derivated and after linearization of the differential equations stability are analyzed by means of a monodromy matrix method. Finally dynamic stability regions have been shown on system parameters such as rotational velocity, . The stability charts constructed on various parameters.

that with increasing inertia disk ratio and decreasing hooke's joint Vibration, Hooke’s Joint

ﻪﺳ زا ﻞﻜﺸﺘﻣ ﻢﺘﺴﻴﺳ ﻚﻳ يراﺪﻳﺎﭘ

نﺎﻄﻠﺳ دﻮﻌﺴﻣ ﻲﻳﺎﺿر

1 - ﺪﺷرا ﻲﺳﺎﻨﺷرﺎﻛ يﻮﺠﺸﻧاد

2 -

3 - هﺎﮕﺸﻧاد ،ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ رﺎﻴﺸﻧاد

4 - ﺮﺘﻛد يﻮﺠﺸﻧاد

* ﻲﺘﺴﭘ قوﺪﻨﺻ ،ناﺮﻬﺗ

ﻲﺸﭽﻴﭘ ﻚﻴﺘﺳﻻا ﺖﻓﺎﺷ ﻪﺳ ﻞﻣﺎﺷ ﻢﺘﺴﻴﺳ ﻚﻳ ﻲﻜﻴﻣﺎﻨﻳد يراﺪﻳﺎﭘ ،ﺶﻫوﮋﭘ ﻦﻳا رد ﺖﺳا هﺪﺷ ﻲﺳرﺮﺑ

.

ﻲﻄﺧ زا ﺲﭘ و ﺖﺳا هﺪﺷ جاﺮﺨﺘﺳا ﻢﺘﺴﻴﺳ ياﺮﺑ

ﺖﺳا هﺪﺷ ﻞﻴﻠﺤﺗ ﺲﻳﺮﺗﺎﻣ ﻲﻣرﺪﻨﻣ شور ﻪﺑ ﻪﻋﻮﻤﺠﻣ يراﺪﻳﺎﭘ ﺖﻓﺎﺷ مﺎﻜﺤﺘﺳا و ﻲﺘﺨﺳ ،ﺮﮕﻳﺪﻜﻳ ﺎﺑ ﺎﻫ

ﻚﺴﻳد ﻲﺳﺮﻨﻳا ﺖﺒﺴﻧ ﺶﻳاﺰﻓا ﺎﺑ ﻪﻛ د

،ﺖﻓﺎﺷ ﻢﺘﺴﻴﺳ كﻮﻫ لﺎﺼﺗا ،ﻲﺸﭽﻴﭘ تﺎﺷﺎﻌﺗرا

hree-axis ooke’s joints

A. Jafari³, A. Najafi

Tarbiat Modares Univ., Tehran, Iran Mech. Eng., Tarbiat Modares Univ., Tehran, Iran

K. N. Toosi Univ. of Tech., Tehran, Iran D Student, Mech. Eng., Tarbiat Modares Univ., Tehran, Iran

ghazavim@modares.ac.ir

this study, dynamic stability of a system consisting of three torsionally elastic shafts with different have been investigated by means of a three degree

Each shaft carrying a rigid disk at one end and have been

derivated and after linearization of the differential equations analyzed by means of a monodromy matrix method. Finally dynamic stability regions have been shown on system parameters such as rotational velocity, stability charts constructed on various parameters.

joint angle cause

ﻪﺳ زا ﻞﻜﺸﺘﻣ ﻢﺘﺴﻴﺳ ﻚﻳ يراﺪﻳﺎﭘ

نﺎﻄﻠﺳ دﻮﻌﺴﻣ

ﻲﺸﭽﻴﭘ ﻚﻴﺘﺳﻻا ﺖﻓﺎﺷ ﻪﺳ ﻞﻣﺎﺷ ﻢﺘﺴﻴﺳ ﻚﻳ ﻲﻜﻴﻣﺎﻨﻳد يراﺪﻳﺎﭘ ،ﺶﻫوﮋﭘ ﻦﻳا رد يﺪﻌﺑ ﻪﺳ (

،دراد ﺖﺳا هﺪﺷ ﻲﺳرﺮﺑ

ﻲﻄﺧ زا ﺲﭘ و ﺖﺳا هﺪﺷ جاﺮﺨﺘﺳا ﻢﺘﺴﻴﺳ ياﺮﺑ

ﺖﺳا هﺪﺷ ﻞﻴﻠﺤﺗ ﺲﻳﺮﺗﺎﻣ ﻲﻣرﺪﻨﻣ شور ﻪﺑ ﻪﻋﻮﻤﺠﻣ يراﺪﻳﺎﭘ ﺎﺷ رﻮﺤﻣ

ﺖﻓ ﺖﻓﺎﺷ مﺎﻜﺤﺘﺳا و ﻲﺘﺨﺳ ،ﺮﮕﻳﺪﻜﻳ ﺎﺑ ﺎﻫ

ﻲﻣ ﻮﺷ ﻚﺴﻳد ﻲﺳﺮﻨﻳا ﺖﺒﺴﻧ ﺶﻳاﺰﻓا ﺎﺑ ﻪﻛ د

،ﻲﻜﻴﻣﺎﻨﻳد يراﺪﻳﺎﭘ

،ﺖﻓﺎﺷ ﻢﺘﺴﻴﺳ

axis connected

, A. Najafi⁴

this study, dynamic stability of a system consisting of three torsionally elastic shafts with different have been investigated by means of a three degree-of-freedom model in Each shaft carrying a rigid disk at one end and have been linked

derivated and after linearization of the differential equations analyzed by means of a monodromy matrix method. Finally dynamic stability regions have been shown on system parameters such as rotational velocity, stability charts constructed on various parameters.

angle cause more stability

ﻪﺳ زا ﻞﻜﺸﺘﻣ ﻢﺘﺴﻴﺳ ﻚﻳ يراﺪﻳﺎﭘ

هﺪﻴﻜﭼ ﻲﺸﭽﻴﭘ ﻚﻴﺘﺳﻻا ﺖﻓﺎﺷ ﻪﺳ ﻞﻣﺎﺷ ﻢﺘﺴﻴﺳ ﻚﻳ ﻲﻜﻴﻣﺎﻨﻳد يراﺪﻳﺎﭘ ،ﺶﻫوﮋﭘ ﻦﻳا رد -

ﺖﻟﺎﺣ ﻲﻳﺎﻀﻓ ) يﺪﻌﺑ ﻪﺳ

ﺖﻛﺮﺣ تﻻدﺎﻌﻣ ﻲﻄﺧ زا ﺲﭘ و ﺖﺳا هﺪﺷ جاﺮﺨﺘﺳا ﻢﺘﺴﻴﺳ ياﺮﺑ

ﺖﺳا هﺪﺷ ﻞﻴﻠﺤﺗ ﺲﻳﺮﺗﺎﻣ ﻲﻣرﺪﻨﻣ شور ﻪﺑ ﻪﻋﻮﻤﺠﻣ يراﺪﻳﺎﭘ ﻪﻳواز ،ﻲﻧارود ﺎﺷ رﻮﺤﻣ

ﺖﺳا و هﺪﻫﺎﺸﻣ ﻲﻣ

ﺪﻴﻠﻛ نﺎﮔژاو

،ﻲﻜﻴﻣﺎﻨﻳد يراﺪﻳﺎﭘ:

connected

this study, dynamic stability of a system consisting of three torsionally elastic shafts with different freedom model in linked through two derivated and after linearization of the differential equations analyzed by means of a monodromy matrix method. Finally dynamic stability regions have been shown on system parameters such as rotational velocity, stability charts constructed on various parameters.

stability.

هﺪﻴﻜﭼ ﺖﻟﺎﺣ ﺖﻛﺮﺣ تﻻدﺎﻌﻣ ﺖﺳا هﺪﺷ ﻞﻴﻠﺤﺗ ﺲﻳﺮﺗﺎﻣ ﻲﻣرﺪﻨﻣ شور ﻪﺑ ﻪﻋﻮﻤﺠﻣ يراﺪﻳﺎﭘ ﻪﻳواز ،ﻲﻧارود ﺖﺳا ﺪﻴﻠﻛ

this study, dynamic stability of a system consisting of three torsionally elastic shafts with different freedom model in through two derivated and after linearization of the differential equations analyzed by means of a monodromy matrix method. Finally dynamic stability regions have been shown on system parameters such as rotational velocity, stability charts constructed on various parameters.

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

ﻚﻳ يراﺪﻳﺎﭘ كﻮﻫ لﺎﺼﺗا ﺎﺑ رﻮﺤﻣ ﻪﺳ زا ﻞﻜﺸﺘﻣ ﻢﺘﺴﻴﺳ

نﺎﻄﻠﺳ دﻮﻌﺴﻣ ﻲﻳﺎﺿر

رﺎﻜﻤﻫ و نا

70

سرﺪﻣ ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ

ةرود 12 ةرﺎﻤﺷ 6 ، ﺪﻨﻔﺳا 1391

1 - ﻪﻣﺪﻘﻣ

ﻢﺘﺴﻴﺳ عاﻮﻧا زا ﻲﻜﻳ ﺖﻓﺎﺷ ﻢﺘﺴﻴﺳ ﻪﻛ ﺖﺳا ترﺪﻗ لﺎﻘﺘﻧا يﺎﻫ

ﻪﺑ دراد يدﺎﻳز يﺎﻫدﺮﺑرﺎﻛ ﻢﻛ نزو و ﻻﺎﺑ ﺖﻋﺮﺳ ﻦﺘﺷاد ﺮﻃﺎﺧ .

ﻢﺘﺴﻴﺳ ﻦﻳا ﻪﻋﻮﻤﺠﻣ

ﻢﻫ ﻪﺑ ﻞﺼﺘﻣ ﺖﻓﺎﺷ ﺪﻨﭼ زا ﻪﻛ ﺖﺳا يا

ﺖﺳا هﺪﺷ ﻞﻴﻜﺸﺗ نآ رد و

ﺖﻛﺮﺣ ﻲﻧارود ﻪﺑ هﺪﻨﻧار ﺖﻓﺎﺷ زا

ﺖﻓﺎﺷ ) يﺎﻫ ( ﻲﻣ ﻞﻘﺘﻨﻣ وﺮﻴﭘ دﻮﺷ

. ﺮﺑ ،ﻪﻋﻮﻤﺠﻣ ﻦﻳا دﺮﺑرﺎﻛ سﺎﺳا

ﺖﻓﺎﺷ رﻮﺤﻣ ﻲﻣ ﺎﻫ

ﻢﻫﺎﻧ ﺪﻨﻧاﻮﺗ ﺪﻨﺷﺎﺑ ﺎﺘﺳار

. لﺎﺼﺗا ياﺮﺑ

ﺖﻓﺎﺷ يﺎﻫ ﺮﻴﻏ ﻢﻫ شور ،رﻮﺤﻣ دراد دﻮﺟو يدﺎﻳز يﺎﻫ

] 1 .[

ﻚﻳ

كﻮﻫ لﺎﺼﺗا زا هدﺎﻔﺘﺳا لواﺪﺘﻣ هار مﺎﻧ ﺎﺑ ﻪﻛ ﺖﺳا 1

يﺮﮕﻳد يﺎﻫ

لﺎﺼﺗا ﻪﻠﻤﺟ زا نادرﺎﻛ

لﺎﺳرﻮﻴﻧﻮﻳ لﺎﺼﺗا ،2

لﺎﺼﺗا و 3

U4

ﺰﻴﻧ

ﻲﻣ ﻪﺘﺧﺎﻨﺷ دﻮﺷ

. و تﻻﺎﺼﺗا ﻪﻴﻘﺑ ﻪﺑ ﺖﺒﺴﻧ لﺎﺼﺗا ﻦﻳا ﺖﻳﺰﻣ

ﮓﻨﻴﻠﭘﻮﻛ ﺖﻓﺎﺷ ﻦﻴﺑ يﺎﻫ

5ﺎﻫ ﻞﻤﺤﺗ روﺎﺘﺸﮔ يرﻮﺤﻣ يوﺮﻴﻧ و

ﻪﻳواز رد رﺎﻛ ،ﻻﺎﺑ يﺎﻫ

ﻢﻫﺎﻧ ﻲﻳﺎﺘﺳار ًﺎﺘﺒﺴﻧ و ﻦﻴﻳﺎﭘ ﺖﻤﻴﻗ ،دﺎﻳز

ﻲﻣ نﺎﺳآ ﺮﻴﻤﻌﺗ ﺪﺷﺎﺑ

. لﺎﺼﺗا ﻦﻳا زا هدﺎﻔﺘﺳا ﻪﺘﺒﻟا ﺰﻴﻧ ﻲﺗﻼﻜﺸﻣ

ﻢﻫﺎﻧ ؛دراد ﻲﻣ ﺐﺒﺳ ﻲﻳﺎﺘﺳار

ﺖﻓﺎﺷ دﻮﺷ )

يﺎﻫ ( نﺎﺳﻮﻧ وﺮﻴﭘ

ﺪﺷﺎﺑ ﻪﺘﺷاد بوﺎﻨﺘﻣ )

ﺪﺷﺎﺑ ﺖﺑﺎﺛ هﺪﻨﻧار ﺖﻓﺎﺷ ﺖﻋﺮﺳ ﺮﮔا ﻲﺘﺣ (

ﻪﺑ هﺪﻨﻧار ﺖﻓﺎﺷ زا ار ﻲﺸﭽﻴﭘ تﺎﺷﺎﻌﺗرا نﺎﺳﻮﻧ ﻦﻳا ﻪﻛ ﺖﻓﺎﺷ ) يﺎﻫ ( ﻲﻣ ﻞﻘﺘﻨﻣ وﺮﻴﭘ ﻲﻣ ﺮﻣا ﻦﻳا و ﺪﻨﻛ

ﺐﺟﻮﻣ ﺪﻧاﻮﺗ

دﻮﺷ ﻢﺘﺴﻴﺳ يراﺪﻳﺎﭘﺎﻧ .

ﺮﺗﺮﭘ ] 2 [ لﺪﻣ ﻚﻳ ﻲﻄﺧ يدازآ ﻪﺟرد ﻚﻳ

ﺖﻓﺎﺷ ﻢﺘﺴﻴﺳ زا

ار و ﺖﻓﺮﮔ ﺮﻈﻧرد درﻮﻣ

ﻲﺳرﺮﺑ داد راﺮﻗ . يرﻮﺌﺗ زا هدﺎﻔﺘﺳا ﺎﺑ وا

ﻪﻛﻮﻠﻓ ﺶﻴﭘ ياﺮﺑ ﻲﻟوﺪﺟ ﻪﻴﻬﺗ ﻪﺑ ﻖﻓﻮﻣ 6

ﺪﻳﺪﺸﺗ ﻲﺣاﻮﻧ ﻲﻨﻴﺑ

ﺪﺷ ﻪﻴﻟوا يﺮﺘﻣارﺎﭘ .

يﺮﮔﺮﮔ و ﺮﺗﺮﭘ ﺲﭙﺳ ]

3 [ ﻚﻳ لﺪﻣ ﻚﻳ

ﺮﻴﻏ يدازآ ﻪﺟرد رد ار ﻲﻄﺧ

ﺪﻨﺘﻓﺮﮔ ﺮﻈﻧ .

نآ زا هدﺎﻔﺘﺳا ﺎﺑ ﺎﻫ

شور نﻮﭘ هرﺎﻛ - فﻮﻧﺎﭘﺎﻴﻟ ﻢﺘﺴﻴﺳ رﺎﺘﻓر 7

راﺮﻗ ﻪﻌﻟﺎﻄﻣ درﻮﻣ ار

ﺪﻧداد . ﻨﻳﺪﻳا ﺶﻧارﺎﻜﻤﻫ و ﻒ ]

4 [ ﻲﻠﻴﺧ ﻪﻛ ﻲﻤﺘﺴﻴﺳ ﻲﺳرﺮﺑ ﻪﺑ

ﺪﻨﺘﺧادﺮﭘ شور نﺎﻤﻫ ﺎﺑ دﻮﺑ ﻞﺒﻗ لﺪﻣ ﻪﻴﺒﺷ .

ﮓﻧﺎﭼ ] 5 [،

ياﺮﺑ

ﺐﻳﺮﻘﺗ ﻪﺑ ﻲﺑﺎﻴﺘﺳد ﻪﺟرد ﻚﻳ لﺪﻣ ود ﺮﻫ ،ﺮﺗﻻﺎﺑ ﻪﺒﺗﺮﻣ يﺎﻫ

ﻪﻌﻟﺎﻄﻣ ار ﻲﻄﺧﺮﻴﻏ و ﻲﻄﺧ يدازآ دﻮﻤﻧ

. ﻳﺎﭘ ﻞﻴﻠﺤﺗ وا يراﺪ

ﻦﺸﻴﺑﺮﺗﺮﭘ شور ﻪﺑ ار ﻢﺘﺴﻴﺳ داد مﺎﺠﻧا8

. ﮓﻧاﻮﻫ و نﺎﺘﻛﻮﺳآ ]

6 [

شور ﻚﻳ زا و ﺪﻨﺘﺧادﺮﭘ ﻲﻄﺧ يدازآ ﻪﺟرد ود لﺪﻣ ﻚﻳ ﻪﺑ

1. Hooke’s joint 2. Cardan joint 3. Universal joint 4. U joint 5. Shaft couplings 6. Floquet theory 7. Poincare-lyapunov 8. Perturbation method

ﻦﻴﮕﻧﺎﻴﻣ ياﺮﺑ يﺮﻴﮔ ﻴﻴﻌﺗ

ﻪﻴﻟوا يﺮﺘﻣارﺎﭘ ﺪﻳﺪﺸﺗ ﻲﺣاﻮﻧ ﻦ هﺮﻬﺑ

ﺮﺑ ﺪﻧد . ﮓﻧاو و نﺎﺘﻛﻮﺳآ ﻦﻴﻨﭽﻤﻫ ]

7 [ ﻪﺑﺎﺸﻣ ﻲﻟﺪﻣ ]

6 [ ار ﻪﺑ

فﻮﻧﺎﭘﺎﻴﻟ ناﻮﺗ شور ﻲﺳرﺮﺑ 9

ﻮﻤﻧ ﺪﻧد . ﺪﻌﺑ و نﺎﺘﻛﻮﺳآ نآ زا

ﻦﻬﻴﻣ ] 8 [ ﺮﻴﻏ يدازآ ﻪﺟرد ود لﺪﻣ ﻚﻳ ﻪﻌﻟﺎﻄﻣ درﻮﻣ ار ﻲﻄﺧ

ﺎﺑ و ﺪﻧداد راﺮﻗ شور

ار ﻢﺘﺴﻴﺳ ﻢﻈﻨﻣﺎﻧ رﺎﺘﻓر ﻲﻀﻌﺑ ،يدﺪﻋ

ﺑ ﻪ ﺪﻧدروآ ﺖﺳد .

يد ﺶﻧارﺎﻜﻤﻫ و ﺖﻴﻤﺳا ]

9 [ ود لﺪﻣ ﻚﻳ ﻪﺟرد

رد زﻮﻜﺴﻳو كﻼﻬﺘﺳا ﺎﺑ يدازآ ار ﻢﺘﺴﻴﺳ يراﺪﻳﺎﭘﺎﻧ و ﺪﻨﺘﻓﺮﮔ ﺮﻈﻧ

ﻪﻛﻮﻠﻓ يرﻮﺌﺗ ﺎﺑ ﺪﻧدﺮﻛ ﻲﺳرﺮﺑ

.

ﺶﻧارﺎﻜﻤﻫ و يآزﺎﻣ ]

10 [ ﺎﺑ ﺖﻓﺎﺷ ﻢﺘﺴﻴﺳ ﻚﻳ ﻪﻌﻟﺎﻄﻣ ﻪﺑ

شور ﻪﺑ ار يراﺪﻳﺎﭘﺎﻧ ﻲﺣاﻮﻧ و ﺪﻨﺘﺧادﺮﭘ كﻮﻫ لﺎﺼﺗا ﻚﻳ ﺲﻳﺮﺗﺎﻣ ﻲﻣرﺪﻨﻣ ﺑ 10

ﻪ ﺪﻧدروآ ﺖﺳد .

نآ ﺎﻫ ] 11 [ دﻮﺟو ﺮﻴﺛﺄﺗ

ﻚﻠﻬﺘﺴﻣ فﺎﻄﻌﻧا لﺪﻣ رد ار ﻢﺘﺴﻴﺳ يراﺪﻳﺎﭘ رد هﺪﻨﻨﻛ ﺮﻳﺬﭘ

ﺮﻴﻏ و ﻲﻄﺧ ﺐﻠﺻ لﺪﻣ ﺰﻴﻧ و ﻲﻄﺧ ﻲﺳرﺮﺑ ﻲﻄﺧ

ﺪﻧدﺮﻛ .

يﺮﮕﻳد ﻪﻟﺎﻘﻣ رد ﻦﻴﻨﭽﻤﻫ ]

12 [ درﻮﻣ كﻼﻬﺘﺳا راﺪﻘﻣ ﻦﻳﺮﺘﻤﻛ

راﺪﻳﺎﭘ ﻪﺑ ﻚﻳﺮﺘﻣارﺎﭘ راﺪﻳﺎﭘﺎﻧ ﻪﻴﺣﺎﻧ زا لﺎﻘﺘﻧا ياﺮﺑ ار ﻢﺘﺴﻴﺳ زﺎﻴﻧ ﺑ ﻪ ﺪﻧدروآ ﺖﺳد .

نآ ﺮﺑ ﺐﻠﺻ لﺪﻣ ﻚﻳ رﻮﻈﻨﻣ ﻦﻳا ياﺮﺑ ﺎﻫ سﺎﺳا

ﻮﮕﻳﺎﺳ لﺪﻣ ]

13 [ رد شور ﻪﺑ ار يراﺪﻳﺎﭘﺎﻧ ﻲﺣاﻮﻧ و ﺪﻨﺘﻓﺮﮔ ﺮﻈﻧ

ﻞﻴﻫ دوﺪﺤﻣﺎﻧ ﺪﻧدز ﻦﻴﻤﺨﺗ11

. تﺎﻜﺳا و يازﺎﻣ ]

14 [ ﻲﺳرﺮﺑ ﻪﺑ

يدورو ﺖﻋﺮﺳ ﺎﺑ يدازآ ﻪﺟرد ود لﺪﻣ ﻚﻳ راﺪﺑﺎﺘﺷ

ﺪﻨﺘﺧادﺮﭘ .

ﻪﻳواز ﺖﻋﺮﺳ ﺐﺟﻮﻣ ﻪﻛ يراﺪﻘﻣ زا ﺮﺘﻤﻛ زا هﺪﻨﻧار ﺖﻓﺎﺷ يا

ﻲﻣ عوﺮﺷ دﻮﺷ ﻢﺘﺴﻴﺳ ﺪﻳﺪﺸﺗ ﺶﻳاﺰﻓا ﻲﻄﺧ ترﻮﺻ ﻪﺑ و ﺪﺷ

ﻲﻣ ﺖﻓﺎﻳ . نآ ﻦﻴﻴﻌﺗ و ﺲﻳﺮﺗﺎﻣ ﻲﻣرﺪﻨﻣ شور ﻪﺑ ار ﻪﻠﺌﺴﻣ ﻦﻳا ﺎﻫ

ﻞﻴﻠﺤﺗ ﻞﻴﻫ دوﺪﺤﻣﺎﻧ ﺪﻧدﺮﻛ

. يازﺎﻣ ﺲﭙﺳ ]

15 [ هرﺎﺑرد ﻪﻌﻟﺎﻄﻣ ﻪﺑ

ﺖﺧادﺮﭘ ﺖﻟﺎﺣ نآ زا رﻮﺒﻋ و ترﺪﻗ لﺎﻘﺘﻧا ﻢﺘﺴﻴﺳ رد ﺪﻳﺪﺸﺗ .

وا

ﺪﻨﻧﺎﻣ ﻲﻟﺪﻣ ]

14 [ رد ﺖﻓﺮﮔ ﺮﻈﻧ .

ﻞﻴﺴﻧاﺮﻔﻳد تﻻدﺎﻌﻣ يازﺎﻣ

دﻮﺑ ﺺﺨﺸﻣ بوﺎﻨﺗ هرود ﺎﺑ ﻲﺗﻻدﺎﻌﻣ ﻪﻛ ار ﻢﺘﺴﻴﺳ ﻦﮕﻤﻫﺎﻧ ﻪﭼ ﻪﺑ ﻢﺘﺴﻴﺳ ﻚﻳ ﻪﻛ دﺮﻛ ﻲﺳرﺮﺑ يژﺮﻧا شور ﻪﺑ و جاﺮﺨﺘﺳا ﻣ ناﺰﻴﻣ ﻲ ﻲﺒﻴﺳآ نآ ﻪﺑ ﻪﻜﻧآ نوﺪﺑ ،ﺪﻧﺎﻤﺑ ﺪﻳﺪﺸﺗ ﺖﻟﺎﺣ رد ﺪﻧاﻮﺗ

دﻮﺷ دراو .

تﻮﻟﻮﺑ ] 16 [ ﺖﻓﺎﺷ ﻲﺴﻧﺎﻛﺮﻓ ﺦﺳﺎﭘ ﻲﺳرﺮﺑ ﻪﺑ راود يﺎﻫ

ﻲﻄﺧﺮﻴﻏ يدازآ ﻪﺟرد ود رﻮﺤﻣ ﻢﺘﺴﻴﺳ رد

ﺖﺧادﺮﭘ و

ﻪﺑ

هاﺮﻤﻫ رﻻرﺎﭘ ] 17 [ ود لﺪﻣ ﻚﻳ يﺪﻌﺑ

ﺮﻴﻏ ﻲﻄﺧ ار ﻲﻄﺧ يزﺎﺳ

،هدﺮﻛ ﻞﻴﻠﺤﺗ شور ﻚﻤﻛ ﻪﺑ ار نآ يراﺪﻳﺎﭘ ﺲﻳﺮﺗﺎﻣ ﻲﻣرﺪﻨﻣ

اد مﺎﺠﻧا ﺪﻧد .

9. Lyapunov exponent 10. Monodromy matrix method 11. Hill’s infinite method

[ DOR: 20.1001.1.10275940.1391.12.6.2.7 ] [ Downloaded from mme.modares.ac.ir on 2022-10-31 ]

كﻮﻫ لﺎﺼﺗا ﺎﺑ رﻮﺤﻣ ﻪﺳ زا ﻞﻜﺸﺘﻣ ﻢﺘﺴﻴﺳ ﻚﻳ يراﺪﻳﺎﭘ نﺎﻄﻠﺳ دﻮﻌﺴﻣ

ﻲﻳﺎﺿر رﺎﻜﻤﻫ و نا

سرﺪﻣ ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ

ةرود 12 ةرﺎﻤﺷ 6 ، ﺪﻨﻔﺳا

1391

71

ﻞﻜﺸﺘﻣ ﻢﺘﺴﻴﺳ ﻚﻳ يراﺪﻳﺎﭘ ﻞﻴﻠﺤﺗ ،ﺶﻫوﮋﭘ ﻦﻳا زا فﺪﻫ ﺖﺳا كﻮﻫ لﺎﺼﺗا ﺎﺑ رﻮﺤﻣ ﻪﺳ زا .

ﺎﻫرﺎﻛ رد ،ﻪﺘﺷﺬﮔ ي

ﻪﻌﻟﺎﻄﻣ ﻪﺑ ناﺮﮕﺸﻫوﮋﭘ ﻢﺘﺴﻴﺳ

ﻪﺘﺧادﺮﭘ كﻮﻫ لﺎﺼﺗا ﻚﻳ ﺎﺑ ﻲﻳﺎﻫ

ﺪﻧدﻮﺑ . ﻢﺘﺴﻴﺳ ﻦﻳا ﻲﻣ يدازآ ﻪﺟرد ود ياراد ﺎﻫ

ﺪﻨﺷﺎﺑ . رد

ﻢﺘﺴﻴﺳ زا يرﺎﻴﺴﺑ ﻗ لﺎﻘﺘﻧا يﺎﻫ

ﺖﺳا مزﻻ ترﺪ و ﺖﻋﺮﺳ لﺎﻘﺘﻧا

رد و ﻪﺤﻔﺻ زا جرﺎﺧ رد روﺎﺘﺸﮔ دﻮﺷ مﺎﺠﻧا ﻲﻳﺎﻀﻓ ﻂﻴﺤﻣ

. رد

نآ رد ﻪﻛ ﺖﺳا يدازآ ﻪﺟرد ﻪﺳ لﺪﻣ ﻪﺑ زﺎﻴﻧ ﻲﻄﻳاﺮﺷ ﻦﻴﻨﭼ زا

هدﺎﻔﺘﺳا ﻲﻧارود ﺖﻛﺮﺣ لﺎﻘﺘﻧا ياﺮﺑ كﻮﻫ لﺎﺼﺗا ود ﻲﻣ

دﻮﺷ . رد

يﺎﻫﺪﻳﺪﺸﺗ عاﻮﻧا ﻲﺳرﺮﺑ و يراﺪﻳﺎﭘﺎﻧ ﻲﺣاﻮﻧ ﻞﻴﻠﺤﺗ ،دراﻮﻣ ﻦﻳا ﺰﺋﺎﺣ ﻢﺘﺴﻴﺳ ﺖﺳا ﺖﻴﻤﻫا

.

ﻢﺘﺴﻴﺳ رد نآ زا هدﺎﻔﺘﺳا ،لﺪﻣ ﻦﻳا يﺎﻫدﺮﺑرﺎﻛ ﻪﻠﻤﺟ زا نﻮﻴﻣﺎﻛ ترﺪﻗ لﺎﻘﺘﻧا ﻪﺳ ﻦﻴﮕﻨﺳ يﺎﻫوردﻮﺧ و ﺎﻫ

ﻲﻣ هرﻮﺤﻣ ﺪﺷﺎﺑ

.

ﻦﻴﻨﭽﻤﻫ

، نﺎﻜﻣ رد و دراد دﻮﺟو ﻲﻜﻳﺰﻴﻓ ﺖﻳدوﺪﺤﻣ ﻪﻛ ﻲﻳﺎﻫ

ﻲﻤﻧ ﺖﻓﺎﺷ ود ﺎﻬﻨﺗ زا هدﺎﻔﺘﺳا ﺎﺑ ناﻮﺗ )

ﺎﺑ و هدوﺪﺤﻣ ﻪﺑ ﻪﺟﻮﺗ

ﺑ ﻪﻳواز ﻪ كﻮﻫ لﺎﺼﺗا يﺮﻴﮔرﺎﻛ (

ﻪﺑ ﻲﻧارود ﺖﻛﺮﺣ ﺪﻴﻟﻮﺗ اﺪﺒﻣ زا

ﻪﻄﻘﻧ ﺑ ،ﺖﺳا زﺎﻴﻧ درﻮﻣ روﺎﺘﺸﮔ و ناﻮﺗ ﻪﻛ ﺪﻴﺳر يا ﻪ

ﻲﻣ رﺎﻛ دور .

2 - ﻪﻠﺌﺴﻣ حﺮﺷ

ﻳاﺮﺿ ﺮﮔا ياراد ﻢﺘﺴﻴﺳ ﺖﻛﺮﺣ ﻞﻴﺴﻧاﺮﻔﻳد ﻪﻟدﺎﻌﻣ رد دﻮﺟﻮﻣ ﺐ ﺲﻧﺎﻛﺮﻓ ﻪﻛ ﻲﺗرﻮﺻ رد ،ﺪﻨﺷﺎﺑ ﻲﺟرﺎﺧ ﻚﻳﺮﺤﺗ يﺎﻫﺮﺘﻣارﺎﭘ

دﻮﻣ ﺲﻧﺎﻛﺮﻓ زا ﻲﺤﻴﺤﺻ ﺖﺒﺴﻧ ﻲﺟرﺎﺧ ﻚﻳﺮﺤﺗ ﻲﺷﺎﻌﺗرا يﺎﻫ

ﻳاﺮﺿ ﻦﻳا ،ﺪﺷﺎﺑ ﻢﺘﺴﻴﺳ ﺮﺑ ﻚﻳﺮﺘﻣارﺎﭘ ﻚﻳﺮﺤﺗ ترﻮﺻ ﻪﺑ ﺐ

يا

ﻲﻣ ﻞﻤﻋ ﻢﺘﺴﻴﺳ ﻪﺠﻴﺘﻧ رد و ﺪﻨﻨﻛ

رد ﻚﻳﺮﺘﻣارﺎﭘ يراﺪﻳﺎﭘﺎﻧ ،

ﻢﺘﺴﻴﺳ ﺑ ﻪ دﻮﺟو ﻲﻣ ﺪﻳآ ] 18 [.

ﻦﻳا عﻮﻧ ،يراﺪﻳﺎﭘﺎﻧ رد

ﻢﺘﺴﻴﺳ

ﺮﺑ ﻲﻟﺎﻤﻋا ﻲﺸﭽﻴﭘ ﮓﻨﻴﻠﭘﻮﻛ زا ﻲﺷﺎﻧ و هﺪﺷ ﻲﺳرﺮﺑ ﺮﺘﻤﻛ رﻮﺤﻣ ﻲﻣ ﮓﻨﻴﻠﭘﻮﻛ ﺎﻳ لﺎﺼﺗا ﻂﺳﻮﺗ ﺖﻓﺎﺷ ﺪﺷﺎﺑ

.

ﺖﻓﺎﺷ ﻢﺘﺴﻴﺳ ﻚﻳ ﻲﻜﻴﻣﺎﻨﻳد يراﺪﻳﺎﭘ ﻪﻟﺎﻘﻣ ﻦﻳا رد

، ﺎﺑ ﻪﻛ

ﻪﺘﺸﮔ ﻞﺼﺘﻣ كﻮﻫ لﺎﺼﺗا ﻲﺳرﺮﺑ ﺲﻳﺮﺗﺎﻣ ﻲﻣرﺪﻨﻣ شور ﺎﺑ ،

ﺖﺳا هﺪﺷ .

رﻮﻈﻨﻣ ﻦﻳا ياﺮﺑ

، ﻞﻣﺎﺷ يدازآ ﻪﺟرد ﻪﺳ لﺪﻣ ﻚﻳ

ا ﺖﻓﺎﺷ ﻪﺳ نوﺪﺑ ﻪﻛ ﻲﺸﭽﻴﭘ ﻚﻴﺘﺳﻻ

ﺮﻴﻏ و مﺮﺟ ﻢﻫ

هدﻮﺑ رﻮﺤﻣ

ﺮﻈﻧرد ،ﻪﺘﺸﮔ ﻞﺼﺘﻣ ﺐﻠﺻ ﻲﻜﺴﻳد ﻪﺑ ﺮﺳ ﻚﻳ رد ماﺪﻛ ﺮﻫ و ﺘﻓﺮﮔ ﺖﺳا هﺪﺷ ﻪ )

ﻞﻜﺷ 1 .(

ﻻدﺎﻌﻣ زا ﺲﭘ ﻢﺘﺴﻴﺳ ﺖﻛﺮﺣ ت

ﻲﻄﺧ ،جاﺮﺨﺘﺳا ﻲﺑ و ندﺮﻛ

ﺲﻳﺮﺗﺎﻣ ﻲﻣرﺪﻨﻣ شور ﺎﺑ يزﺎﺳﺪﻌﺑ

ﺖﺳا ﻪﺘﻓﺮﮔ راﺮﻗ ﻞﻴﻠﺤﺗ درﻮﻣ .

ﻪﺑ ﺞﻳﺎﺘﻧ ﺖﻳﺎﻬﻧ رد لواﺪﺟ ترﻮﺻ

ﻪﻳواز ﻪﻠﻤﺟ زا ﻢﺘﺴﻴﺳ ﻒﻠﺘﺨﻣ يﺎﻫﺮﺘﻣارﺎﭘ ﺐﺴﺣ ﺮﺑ يراﺪﻳﺎﭘ ﺖﻓﺎﺷ رﻮﺤﻣ ﺖﺳا هﺪﺷ شراﺰﮔ يدورو نارود ﺖﻋﺮﺳ و ﺎﻫ

. ﻞﻜﺷ

ﺪﻌﺑ ﻪﺳ رد ﺖﻓﺎﺷ ﻢﺘﺴﻴﺳ ﻚﻳ ﻚﻴﺗﺎﻤﺷ 1 )

و هﺪﻨﻧار ﺖﻓﺎﺷ

ﻲﻣ مود وﺮﻴﭘ ﺖﻓﺎﺷ ﺪﻨﺷﺎﺑ ﻒﻠﺘﺨﻣ تﺎﺤﻔﺻ رد ﺪﻨﻧاﻮﺗ

(.

3 - ﻲﺿﺎﻳر لﺪﻣ

ﻳا رد ﺖﻓﺎﺷ ﻪﻛ دراد دﻮﺟو ﺖﻓﺎﺷ ﻪﺳ لﺪﻣ ﻦ وﺮﻴﭘ ،هﺪﻨﻧار يﺎﻫ

هﺪﺷ هﺪﻴﻣﺎﻧ مود وﺮﻴﭘ و لوا ﺪﻧا

. ﺖﻓﺎﺷ رﻮﺤﻣ ﻢﻫ ﺎﻫ

ﺪﻨﺘﺴﻴﻧ ﺎﺘﺳار

ﻪﺑ و ﻪﺑ كﻮﻫ لﺎﺼﺗا ﻪﻠﻴﺳو ﺮﮕﻳﺪﻜﻳ

هﺪﺷ ﻂﺒﺗﺮﻣ ﺪﻧا

. ﻪﻳواز

ﻢﻫﺎﻧ ﻲﻳﺎﺘﺳار )

يﺎﻫرﻮﺤﻣ ﻪﻳواز فﻼﺘﺧا (

ﺖﻓﺎﺷ و هﺪﻨﻧار ﺖﻓﺎﺷ

لوا وﺮﻴﭘ β1

ﻢﻫﺎﻧ ﻪﻳواز و وﺮﻴﭘ ﺖﻓﺎﺷ و لوا وﺮﻴﭘ ﺖﻓﺎﺷ ﻲﻳﺎﺘﺳار

2 مود ﺖﺳا هﺪﺷ هﺪﻴﻣﺎﻧ β .

يدازآ ﻪﺟرد ﻚﻳ ياراد ﺖﻓﺎﺷ ﺮﻫ

رد ﻪﻛ ﺖﺳا عﻮﻤﺠﻣ

و هدﻮﺑ يدازآ ﻪﺟرد ﻪﺳ ياراد ﻢﺘﺴﻴﺳ

ﻲﻳﺎﻀﻓ ﺖﻟﺎﺣ )

ﻪﺳ يﺪﻌﺑ ( دراد ) ار ﺖﻓﺎﺷ ﻧ مود وﺮﻴﭘ ﺖﻓﺎﺷ و هﺪﻨ

ﻲﻣ ﺪﻨﺷﺎﺑ اﺰﺠﻣ ﻪﺤﻔﺻ ود رد ﺪﻨﻧاﻮﺗ .(

ﻲﺘﺨﺳ ياراد ﺖﻓﺎﺷ ﺮﻫ

ﻲﺸﭽﻴﭘ ki

ﻲﺸﭽﻴﭘ زﻮﻜﺴﻳو كﻼﻬﺘﺳا و ci

ﺖﺳا و ﻚﻳ ﻪﺑ

ﻲﻧارود ﻲﺳﺮﻨﻳا ﺎﺑ ﻚﺴﻳد Ji

ﻞﺼﺘﻣ ﺖﺳار ﺖﻤﺳ يﺎﻬﺘﻧا رد

ﻪﺘﺸﮔ ﺖﺳا . رد رﺎﻴﻃ خﺮﭼ ﻚﻳ ،هﺪﻨﻧار ﺖﻓﺎﺷ ﭗﭼ ﺖﻤﺳ يﺎﻬﺘﻧا

ﺖﺑﺎﺛ ﺖﻋﺮﺳ ﺎﺑ ﺖﻓﺎﺷ ﻦﻳا ﺎﺗ دراد دﻮﺟو

Ω0

ﺪﻨﻛ نارود .

ﻦﻳا

ﻲﻣ لﺪﻣ ﻫد نﺎﺸﻧ ار ﻢﺘﺴﻴﺳ ﻪﻳﺎﭘ يﺎﻫﺪﻣ رﺎﺘﻓر ﺪﻧاﻮﺗ رد و ﺪ

ﻢﺘﺴﻴﺳ ياﺮﺑ ﻂﻘﻓ ﻪﺠﻴﺘﻧ ﻲﻣ ﻲﻳﺎﻫ

دﺮﻴﮕﺑ راﺮﻗ هدﺎﻔﺘﺳا درﻮﻣ ﺪﻧاﻮﺗ

ﺲﻧﺎﻛﺮﻓ ﻪﻛ ﺖﻓﺎﺷ ﻪﻳﺎﭘ ﻲﺸﭽﻴﭘ يﺎﻫ

نآ يﺎﻫ ﻢﻫ ﺎﻫ ﺪﻨﺷﺎﺑ ﻪﺒﺗﺮﻣ .

ﺑ ﺮﻳز ﻂﺑاور ﻖﺒﻃ ﻢﺘﺴﻴﺳ ﻲﺸﭽﻴﭘ تﺎﺷﺎﻌﺗرا ﻪ

ﻲﻣ ﺖﺳد ﺪﻳآ

) ﺖﺳﻮﻴﭘ دﻮﺷ ﻪﻈﺣﻼﻣ ﻒﻟا .(

1 1 1 1 01 2 2 1 1 01 1 2 0

Jθ+cθ −η c θ +kθ −η k θ =

2 2 01 1 01 0 1

2 2 02 3 3 2 2 02 3 3

[ ( )]

0 J

c c k k

θ η θ η θ

θ η θ θ η θ

+ + Ω +

+ − + − =

3 3 02 2 01 1 01 0 1

02 2 01 0 1 3 3 3 3

[ ( ( ( )))

( ( ))] 0

J

c k

θ η θ η θ η θ

η θ η θ θ θ

+ + + Ω +

+ + Ω + + + =

) 1 (

) 2 (

) 3 (

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

ﻚﻳ يراﺪﻳﺎﭘ كﻮﻫ لﺎﺼﺗا ﺎﺑ رﻮﺤﻣ ﻪﺳ زا ﻞﻜﺸﺘﻣ ﻢﺘﺴﻴﺳ

نﺎﻄﻠﺳ دﻮﻌﺴﻣ ﻲﻳﺎﺿر

رﺎﻜﻤﻫ و نا

72

سرﺪﻣ ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ

ةرود 12 ةرﺎﻤﺷ 6 ، ﺪﻨﻔﺳا 1391

ﻂﺑاور ﻦﻳا رد θ1

2 ، و θ θ3

ﻲﺸﭽﻴﭘ تﺎﺼﺘﺨﻣ و ﺪﻧا

θ1

،

θ2 3 ، θ

1 و θ

2 ، θ

3 ، θ ﻪﺑ نﺎﺸﻧ ﺐﻴﺗﺮﺗ و لوا تﺎﻘﺘﺸﻣ هﺪﻨﻫد

ﺖﺳا نﺎﻣز ﻪﺑ ﺖﺒﺴﻧ مود

01 . و η η02

رد ﺖﻋﺮﺳ لﺎﻘﺘﻧا ﺖﺒﺴﻧ

ﻲﻣ كﻮﻫ لﺎﺼﺗا ﻦﻴﻣود و ﻦﻴﻟوا ﺪﺷﺎﺑ

] 19 [

1 1

1 2 2

1 1 1 1

cos

( ) 1 sin sin

out in

η β

ϕ β ϕ

= Ω =

Ω = −

2 2

2 2 2

2 2 2 2

cos

( ) 1 sin sin

out in

η β

ϕ β ϕ

= Ω =

Ω = −

ﻂﺑاور ﻦﻳا رد ﻪﻛ

1 0t 1

ϕ = Ω +θ

2 out1t 2

ϕ = Ω +θ

ﻲﻄﺧﺮﻴﻏ ﻢﺘﺴﻴﺳ ﺖﻛﺮﺣ تﻻدﺎﻌﻣ ﻲﻣ

ﺪﻨﺷﺎﺑ . نآ ﻂﺴﺑ ﺎﺑ ﺎﻫ

ﻪﺑ ﻚﻣ يﺮﺳ ﻪﻠﻴﺳو نارﻮﻟ

θ1 2 ، θ

3 ، θ

1 ، θ

2 ، θ

3 و θ و

فﺮﺻ ﻚﭼﻮﻛ ضﺮﻓ ﺎﺑ ،ﻲﻄﺧﺮﻴﻏ تﻼﻤﺟ زا ﺮﻈﻧ ﺮﻳدﺎﻘﻣ ندﻮﺑ

ﺲﻧﺎﻛﺮﻓ دﻮﺟو مﺪﻋ و ﻲﺷﺎﻌﺗرا تﻻدﺎﻌﻣ ،ﻻﺎﺑ ﻪﺒﺗﺮﻣ يﺎﻫ

) 1 ( ﺎﺗ

) 3 ( ﻪﺑ رد ﻲﻄﺧ ترﻮﺻ ﻲﻣ

ﺪﻨﻳآ ) ﺖﺳﻮﻴﭘ دﻮﺷ ﻪﻈﺣﻼﻣ ب .(

ﻲﺑ يﺎﻫﺮﺘﻣارﺎﭘ ﻒﻳﺮﻌﺗ ﺎﺑ ﺪﻌﺑ

0 1

0

1 1 1 1

, , c

t k J k J

τ = Ω Ω = ^Ω ζ =

ﻪﺑ ﺖﻛﺮﺣ تﻻدﺎﻌﻣ ﻲﺑ ترﻮﺻ

رد ﺪﻌﺑ رد و هﺪﻣآ يرادﺮﺑ مﺮﻓ

-

ﻪﺑ ﻲﺴﻳﺮﺗﺎﻣ ﻲﻣ ﻪﺘﺷﻮﻧ ﺮﻳز ترﻮﺻ

ﺪﻧﻮﺷ .

[ ] [ ] ( )

( ) ( )

1 1 1

2 2 2 01

3 3 3 01 02

0

E F

θ θ θ

θ θ θ η τ

θ θ θ η τ η τ

′′ ′  

     

       

′′ + ′ + = − ′

       

 ′′  ′   − ′ ′ 

       

ﻪﻄﺑار ﻦﻳا رد θ1′

2 ، θ′

3 ، θ′

1 و θ′′

2 ، θ′′

3 ، θ′′

ﻪﺑ ﺐﻴﺗﺮﺗ

نﺎﺸﻧ هﺪﻨﻫد ﻪﺑ ﺖﺒﺴﻧ مود و لوا تﺎﻘﺘﺸﻣ ﻲﻣ τ

ﺷﺎﺑ ﻨ و ﺪ و E

ﺲﻳﺮﺗﺎﻣF يﺎﻫ 3

× 3 ﻪﻳارد ﻪﻛ ﺪﻨﺘﺴﻫ ﺖﺳا ﻦﻴﻨﭼ نﺎﺸﻳﺎﻫ

:

11 2 E = ζ Ω

12 2 01

( )

1

E = − η τ α ζ Ω

13 0 E =

( ) ( )

21 01 2 01

E =η′ τ − η τ ζ Ω

(

²

( ) )

22 2 ( 1 ) 01 1 1

E = α ζ Ω η τ + γ

23 (2 2 1) 02

( )

E = − α ζ Ωγ η τ

( ) ( )

31 01 02

E =η τ η′ ′ τ

( ) ( )

32 02 (2 1 1) 02

E =η^′ τ − α ζ Ωγ η τ

(

²

( ) )

33 2( 2 ) 02 1 1

E = α ζ Ω η τ γ +

2 11 1 F = Ω

( )

²

12 01 1

F = −η τ µ Ω

13 0 F =

( )

²

21 01

F =η τ Ω

( ( ) )

2 2

22 ( 1 ) 01 1 1

F = µ Ω η τ + γ

( )

2

23 ( 2 1) 02

F = − µ Ωγ η τ

( ) ( )

31 01 02

F =η τ η′′ ′ τ

( )

2

32 ( 1 1) 02

F = µ Ωγ η′′ τ

( ( ) )

2 2

33 ( 2 ) 02 1 1 2

F = µ Ω η τ γ + γ

ﻪﻟدﺎﻌﻣ ) 9 ( ﻲﻄﺧ ﻪﻄﺑار ، ﻲﺸﭽﻴﭘ تﺎﺷﺎﻌﺗرا ياﺮﺑ هﺪﺷ يزﺎﺳ

نآ بوﺎﻨﺗ هرود ﻪﻛ ﺖﺳا ﺮﻈﻧ درﻮﻣ ﻢﺘﺴﻴﺳ نﺎﻳدار π

ﻲﻣ ﺪﺷﺎﺑ .

1

01 01 2 2

1

( ) ( ) cos

1 sin sin η τ η τ π β

β τ

= + =

−

2

02 02 2 2

2

( ) ( ) cos

1 sin sin η τ η τ π β

β τ

= + =

−

4 - ﻲﻜﻴﻣﺎﻨﻳد يراﺪﻳﺎﭘ ﻞﻴﻠﺤﺗ

ﻪﻟدﺎﻌﻣ ) 9 ( تﻻدﺎﻌﻣ هﺎﮕﺘﺳد ﻚﻳ د

ﺎﺑ ﻲﺒﻳاﺮﺿ ﺎﺑ ﻲﻄﺧ ﻞﻴﺴﻧاﺮﻔﻳ

بوﺎﻨﺗ هرود )π

ﻮﻴﺘﻣ تﻻدﺎﻌﻣ هﺎﮕﺘﺳد ﻚﻳ -

1ﻞﻴﻫ ( ﻲﻣ ﺪﺷﺎﺑ .

ﻲﻣ مﺎﺠﻧا ﺲﻳﺮﺗﺎﻣ ﻲﻣرﺪﻨﻣ شور ﻪﺑ هﺎﮕﺘﺳد ﻦﻳا يراﺪﻳﺎﭘ دﻮﺷ

.

ﺲﻳﺮﺗﺎﻣ ﻲﻣرﺪﻨﻣ شور ]

20 [ ﺖﺳا نﺎﻨﻴﻤﻃا ﻞﺑﺎﻗ و هدﺎﺳ ﻲﻜﻴﻨﻜﺗ

ياﺮﺑ و ﻞﻴﻠﺤﺗ ﻢﺘﺴﻴﺳ رد يراﺪﻳﺎﭘ ﻚﻳﺮﺘﻣارﺎﭘ ﻚﻳﺮﺤﺗ ﻪﻛ ﻲﻳﺎﻫ

درﻮﻣ ﺖﺳا نﺎﻣز ﻪﺑ ﻪﺘﺴﺑاو و ﻚﻴﻧﻮﻣرﺎﻫ ،ﺐﻳاﺮﺿ راﺪﻘﻣ و ﺪﻧراد ﻲﻣ راﺮﻗ هدﺎﻔﺘﺳا دﺮﻴﮔ

. شور ﻦﻳا رد

، ﻚﻳ رد طﺎﻘﻧ مﺎﻤﺗ يراﺪﻳﺎﭘ

هدوﺪﺤﻣ ﺮﺑ ،هﺪﺷ ﻦﻴﻴﻌﺗ

ﻲﺳرﺮﺑ ﻢﺘﺴﻴﺳ يﺎﻫﺮﺘﻣارﺎﭘ سﺎﺳا

ﻲﻣ دﻮﺷ ﻲﻣ و دراد ﻲﻳﻻﺎﺑ ﺖﻗد ور ﻦﻳا زا و طﺎﻘﻧ مﺎﻤﺗ ﺪﻧاﻮﺗ

ﺎﺑ ار رﺎﻛ ﻦﻳا ﻪﺘﺒﻟا ؛ﺪﻳﺎﻤﻧ ﻦﻴﻌﻣ ار هدوﺪﺤﻣ رد دﻮﺟﻮﻣ راﺪﻳﺎﭘﺎﻧ ﻲﻣ مﺎﺠﻧا ﻲﺿﺎﻳر تﺎﺒﺳﺎﺤﻣ زا يدﺎﻳز ﻢﺠﺣ مﺎﺠﻧا ﻦﻴﻤﻫ ﻪﺑ و ﺪﻫد

ﺦﺳﺎﭘ ياﺮﺑ يدﺎﻳز ًﺎﺘﺒﺴﻧ نﺎﻣز ﻞﻴﻟد ﺖﻗد ﻪﭼﺮﻫ و دراد زﺎﻴﻧ نداد

ﻘﻧ داﺪﻌﺗ ،ﺪﺷﺎﺑ ﺮﻈﻧ درﻮﻣ يﺮﺗﻻﺎﺑ رد ،هدﺮﻛ ﻲﺳرﺮﺑ ار يﺮﺘﺸﻴﺑ طﺎ

ﻲﻣ فﺮﺻ يﺮﺘﺸﻴﺑ نﺎﻣز ﻪﺠﻴﺘﻧ ﺪﻨﻛ

. رد ﻦﻳا شور

، ﻪﺑ ﺞﻳﺎﺘﻧ

ﺮﺑ يراﺪﻳﺎﭘ لواﺪﺟ ترﻮﺻ ﺐﺴﺣ

ﻳ يﺎﻫﺮﺘﻣارﺎﭘ زا ﺖﻔﺟ ﻚ

1. A set of Mathieu-Hill equations

) 4 (

) 5 (

) 6 (

) 7 (

) 8 ( )

10 (

) 11 (

) 9 (

[ DOR: 20.1001.1.10275940.1391.12.6.2.7 ] [ Downloaded from mme.modares.ac.ir on 2022-10-31 ]

كﻮﻫ لﺎﺼﺗا ﺎﺑ رﻮﺤﻣ ﻪﺳ زا ﻞﻜﺸﺘﻣ ﻢﺘﺴﻴﺳ ﻚﻳ يراﺪﻳﺎﭘ نﺎﻄﻠﺳ دﻮﻌﺴﻣ

ﻲﻳﺎﺿر رﺎﻜﻤﻫ و نا

سرﺪﻣ ﻚﻴﻧﺎﻜﻣ ﻲﺳﺪﻨﻬﻣ

ةرود 12 ةرﺎﻤﺷ 6 ، ﺪﻨﻔﺳا

1391

73

) 13 (

) 16 (

ﻲﻣ نﺎﻴﺑ ﻢﺘﺴﻴﺳ دﻮﺷ

. رد شور ﻦﻳا ﻪﺑ ﻪﻣادا

هدروآ ﺮﺼﺘﺨﻣ رﻮﻃ

ﺖﺳا هﺪﺷ .

ﻲﻣ ترﻮﺻ ﻦﻳﺪﺑ ﻢﺘﺴﻴﺳ زا ﺖﻟﺎﺣ يﺎﻀﻓ ﻚﻳ ﺪﺷﺎﺑ

:

1 2 1 2 1 2 1 2

( , , , , , , , , ).

u′ =H α α β β γ γ µ µ τ u

{ 1, 2, 3, 1, 2, 3}^T

u = θ θ θ θ θ θ′ ′ ′

و ﺲﻳﺮﺗﺎﻣ ﻚﻳ H 6

× 6 هرود ﺎﺑ

بوﺎﻨﺗ ﻪﺑ π

ﺖﺳا ﺮﻳز ترﻮﺻ .

1 2 1 2 1 2 1 2

( , , , , , , , , ) O I

H α α β β γ γ µ µ τ =^ F E^

− −

 

ﻚﻳ I ﺪﺣاو ﺲﻳﺮﺗﺎﻣ 3

× 3 و ﺮﻔﺻ ﺲﻳﺮﺗﺎﻣ ﻚﻳ O 3

× 3

ﻲﻣ ﺪﺷﺎﺑ . ﻪﻛﻮﻠﻓ يرﻮﺌﺗ ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ ]

18 [ ﻞﺣ ، ﻦﻳﺪﺑ هﺎﮕﺘﺳد ﻦﻳا

ﺖﺳا ترﻮﺻ :

( )τ Q( )τ e^Rτ

Φ =

بوﺎﻨﺗ هرود ﺎﺑ ﺲﻳﺮﺗﺎﻣ ﻚﻳQ و ﺖﺳا π

ﺲﻳﺮﺗﺎﻣ ﻚﻳ R

ﻲﻨﻌﻳ ﺮﮕﻳد ﺖﺑﺎﺛ ﺲﻳﺮﺗﺎﻣ ﻪﺑ ﻪﻛ ﺖﺑﺎﺛ )S

ﻲﻣرﺪﻨﻣ ﺲﻳﺮﺗﺎﻣ (

ﻪﺑ

ترﻮﺻ

(

¹

)

^log

R = π S

ﺖﺳا ﻪﺘﺴﺑاو .

ﻲﻄﺧ تﻻدﺎﻌﻣ ﺮﮔا

هﺪﺷ ﺪﻨﺷﺎﺑ

، (0) I Φ = و

( ) S = Φπ ﻲﻣ

ﺪﺷﺎﺑ .

ﻲﻣرﺪﻨﻣ ﺲﻳﺮﺗﺎﻣ هﮋﻳو ﺮﻳدﺎﻘﻣ

; 1, 2,3, 4,5,6

k k

σ =

ياﺮﺑ

ﺑ ﻢﺘﺴﻴﺳ يراﺪﻳﺎﭘ ﻞﻴﻠﺤﺗ ﻪ

ﻲﻣ رﺎﻛ ﺪﻧور . ﻴﺳ ﺖﺳا راﺪﻳﺎﭘ ﻢﺘﺴ

، ﺮﮔا

ﺑ ﺮﮔا ﺎﻬﻨﺗ و ﻪ

مﺎﻤﺗ يازا ﺎﻫk

mod(σ_k) 1≤ ﺎﻳ و ﺪﺷﺎﺑ

Re(σ_k)≤0 .

ﻪﺑ ﻲﻣرﺪﻨﻣ ﺲﻳﺮﺗﺎﻣ ﻪﻟﺎﻘﻣ ﻦﻳا رد لاﺮﮕﺘﻧا ﻪﻠﻴﺳو

يﺮﻴﮔ

ﻪﻟدﺎﻌﻣ زا يدﺪﻋ )

12 ( ﺑ ﻪ نآ هﮋﻳو ﺮﻳدﺎﻘﻣ و ﺖﺳا هﺪﻣآ ﺖﺳد

هﺪﺷ ﻪﺒﺳﺎﺤﻣ ﻢﺘﺴﻴﺳ يراﺪﻳﺎﭘ ﺎﺗ ﺪﻧا

ﻲﺑﺎﻳزرا دﻮﺷ . شور

ﺞﻧار

1ﺎﺗﻮﻛ ) 45 (RK لاﺮﮕﺘﻧا ياﺮﺑ هدﺎﻔﺘﺳا يدﺪﻋ يﺮﻴﮔ

راﺪﻘﻣ و

ﺎﻄﺧ لﺮﺘﻨﻛ ﺲﻧاﺮﻠﺗ

، 2

ﻪﺘﺷﺬﮔ ﻊﺟاﺮﻣ ﻪﺑﺎﺸﻣ

8 ، 10-

رد ﺮﻈﻧ ﻪﺘﻓﺮﮔ

ﺖﺳا هﺪﺷ ]

9 [ .

5 - ﻪﺘﻓﺮﮔ ترﻮﺻ ﻞﻴﻠﺤﺗ و لﺪﻣ ﺖﺤﺻ ﻲﺳرﺮﺑ

ﻪﺑ دﻮﺟﻮﻣ ﺮﻈﻧ درﻮﻣ ﻪﻠﺌﺴﻣ ﻞﺣ ﻒﻠﺘﺨﻣ ﻊﺟاﺮﻣ رد ﻪﻜﻨﻳا ﻞﻴﻟد

ﻗد ﻲﺳرﺮﺑ رﻮﻈﻨﻣ ﻪﺑ ،هدﻮﺒﻧ ﺑ ﻞﺣ شور ﺖ

ﻪ رد ،هﺪﺷ ﻪﺘﻓﺮﮔ رﺎﻛ

ﻪﻠﺌﺴﻣ اﺪﺘﺑا نوﺪﺑ

مود وﺮﻴﭘ ﺖﻓﺎﺷ ﻪﺠﻴﺘﻧ رد و

مود كﻮﻫ لﺎﺼﺗا

،

ﻊﺟاﺮﻣ رد دﻮﺟﻮﻣ ﺞﻳﺎﺘﻧ ﺎﺑ و ﻞﺣ ﻪﺴﻳﺎﻘﻣ

ﺖﺳا هﺪﺷ .

ﻦﻴﻨﭼ

ﻪﺤﻔﺻ ﺖﻟﺎﺣ لﺪﻣ ﻪﻛ ﺖﺳا نآ يﺎﻨﻌﻣ ﻪﺑ ﻲﺿﺮﻓ رد و ﻪﺘﺷاد يا

1. Runge Kutta method

2. The value of error control tolerance

دراد ﺖﻛﺮﺣ ﻲﻳﺎﻧاﻮﺗ ﺪﻌﺑ ود .

ﺎﺑ يراﺪﻳﺎﭘ ﻞﻴﻠﺤﺗ ﻦﺘﻓﺮﮔ ﺮﻈﻧرد

ﻦﻳا

ﻲﺑ يﺎﻫﺮﺘﻣارﺎﭘ ياﺮﺑ ﺮﻳدﺎﻘﻣ ﺖﺳا هﺪﺷ مﺎﺠﻧا ﻪﻠﺌﺴﻣ ﺪﻌﺑ

:

1 1 1

0.001, 1, 1, 10

ζ = α = µ = γ =

ﺞﻳﺎﺘﻧ ﻪﺘﺷﺬﮔ يﺎﻫرﺎﻛ ]

16 [ ﻪﺑ فاﺮﮔ ترﻮﺻ ﻲﻳﺎﻫ

ﻞﻜﺷ رد

2 ﻪﻛ هﺪﺷ ﻪﺋارا ﻢﻫﺎﻧ ﻪﻳواز ﺮﺛا

ﺖﻋﺮﺳ ﺖﺒﺴﻧ رد ار ﻲﻳﺎﺘﺳار يﺎﻫ

ﻲﻜﻴﻣﺎﻨﻳد يراﺪﻳﺎﭘ ﺮﺑ ﻒﻠﺘﺨﻣ ﺎﻫرﻮﺤﻣ

ﻲﻣ نﺎﺸﻧ ﺪﻨﻫد .

ﺪﻌﺑ ﻪﻠﺣﺮﻣ رد

، ﻦﻴﻤﻫ ﻦﺘﻓﺮﮔ ﺮﻈﻧرد ﺎﺑ ﺮﻳدﺎﻘﻣ

ﻞﻴﻠﺤﺗ ﻪﺑ ،

ﻪﺋارا لﺪﻣ يراﺪﻳﺎﭘ ﻲﻣ ﻪﺘﺧادﺮﭘ ﻪﻟﺎﻘﻣ ﻦﻳا رد هﺪﺷ

ﺖﺤﺻ ﺎﺗ دﻮﺷ

دﺮﻴﮔ راﺮﻗ ﻲﺳرﺮﺑ درﻮﻣ نآ .

ﻞﻜﺷ رد ﺞﻳﺎﺘﻧ 2

هداد شراﺰﮔ

ﺖﺳا هﺪﺷ .

نﺎﻨﭼ زا ﻪﻛ ﻞﻜﺷ 2 ﺞﻳﺎﺘﻧ ،ﺖﺳا ﺺﺨﺸﻣ يراﺪﻳﺎﭘ ﻞﻴﻠﺤﺗ

زا

ﻪﻠﻗ نﺎﻜﻣ و يﺮﻫﺎﻇ ﻞﻜﺷ ﺮﻈﻧ توﺎﻔﺗ و ﻪﺘﺷاد ﻲﺑﻮﺧ قﺎﺒﻄﻧا ﺎﻫ

دراﺪﻧ دﻮﺟو ﻲﻬﺟﻮﺗ ﻞﺑﺎﻗ .

ﻦﻳاﺮﺑﺎﻨﺑ

، هدوﺪﺤﻣ ﻲﻳﺎﻫ ﻂﺳﻮﺗ ﻪﻛ

ﻪﺋارا لﺪﻣ ،هﺪﺷ

ﺶﻴﭘ راﺪﻳﺎﭘﺎﻧ هﺪﺷ ﻲﻨﻴﺑ

ﻊﺟاﺮﻣ ﺪﻴﻳﺄﺗ درﻮﻣ ﺪﻧا

ﻲﻣ ﺰﻴﻧ ﺪﻨﺷﺎﺑ . ﺐﻟﺎﻗ رد يراﺪﻳﺎﭘ ﻞﻴﻠﺤﺗ ﻪﺋارا زا ﻞﺒﻗ دراﻮﻣ

ﻲﺜﺤﺑ ،يدﺪﻋ

ﺖﺳا حﺮﻃ ﻪﺑ مزﻻ ،رﺎﻈﺘﻧا درﻮﻣ راﺪﻳﺎﭘﺎﻧ ﻲﺣاﻮﻧ نﻮﻣاﺮﻴﭘ .

نﺎﻨﭼ

ﺖﺳا ﺺﺨﺸﻣ ﻪﻛ

، هزﻮﺣ ﺲﻧﺎﻛﺮﻓ ﻪﺑ ﻲﮕﺘﺴﺑ يراﺪﻳﺎﭘﺎﻧ يﺎﻫ يﺎﻫ

ﺪﻧراد ﻢﺘﺴﻴﺳ ﻲﻌﻴﺒﻃ .

ﻲﻣ ﻦﻳاﺮﺑﺎﻨﺑ ﻞﻴﻠﺤﺗ و ﻪﺒﺳﺎﺤﻣ ﺎﺑ ناﻮﺗ

نآ ﻞﺑﺎﻘﺘﻣ تاﺮﺛا ﺑ راﺪﻳﺎﭘﺎﻧ ﻲﺣاﻮﻧ ﺎﺑ ﻪﺴﻳﺎﻘﻣ و ﺎﻫ

ﻪ هﺪﻣآ ﺖﺳد

ﻪﺋارا لﺪﻣ ﻂﺳﻮﺗ ﻲﺳرﺮﺑ درﻮﻣ ار لﺪﻣ و ﻞﻴﻠﺤﺗ ﺖﺤﺻ ،هﺪﺷ

داد راﺮﻗ . ﺲﻧﺎﻛﺮﻓ تاﺮﺛا ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ ،ﺖﻓﺎﺷ ﻢﺘﺴﻴﺳ ﻲﻌﻴﺒﻃ يﺎﻫ

زﺎﺴﻤﻫ ﻲﺒﻴﻛﺮﺗ ﺰﻴﻧ و 3

ﺮﻳز و زﺎﺴﻤﻫ راﺪﻳﺎﭘﺎﻧ ﻲﺣاﻮﻧ ،ﺪﻧراد ﻪﻛ 4

زﺎﺴﻤﻫ ﺑ ﻲﻘﻳﺮﻔﺗ و ﻲﻌﻴﻤﺠﺗ ﻲﺒﻴﻛﺮﺗ ﺰﻴﻧ و5

ﻪ ﻲﻣ دﻮﺟو ﺪﻧروآ .

زﺎﺴﻤﻫ ﻚﻳﺮﺘﻣارﺎﭘ ﺪﻳﺪﺸﺗ ﻲﺣاﻮﻧ ΩH

زﺎﺴﻤﻫ ﺮﻳز و ΩS

ﻪﺒﺗﺮﻣ ﺑ ﺮﻳز ﻂﺑاور زا ماk ﻪ

ﻲﻣ ﺖﺳد ﺪﻨﻳآ ] 15 [.

, ; 1, 2,3; 1, 2,3,...

2 2 1

H i S i

ik ik i k

k k

ω ω

Ω = Ω = = =

− (15)

ﻦﻴﻨﭽﻤﻫ

، ﻲﻌﻴﻤﺠﺗ ﻲﺒﻴﻛﺮﺗ ﻚﻳﺮﺘﻣارﺎﭘ ﺪﻳﺪﺸﺗ ﻲﺣاﻮﻧ

C+

ﻲﻘﻳﺮﻔﺗ ﻲﺒﻴﻛﺮﺗ و Ω

C−

ﻪﺒﺗﺮﻣ Ω ماk ﺖﺳا ﺮﻳز ترﻮﺻ ﻪﺑ .

3 1 3 2

2 1

12 , 13 , 23

2 2 2

1, 2,3; 1, 2,3,...

C C C

k k k

k k k

i k

ω ω ω ω

ω ω

± ± ± ± ± ±

Ω = Ω = Ω =

= =

3. Harmonic 4. Combination 5. Sub-hormonic

) 12 (

) 14 (

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