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1.1 Classification of Vibrations........................................................................................ 1
1.2 Elementary Parts of Vibrating Systems ................................................................... 2
1.3 Periodic Motion ........................................................................................................... 3
1.4 Discrete and Continuous Systems ............................................................................ 4
1.5 Vibration Analysis ...................................................................................................... 4
1.5.1 Components of Vibrating Systems ............................................................... 6
1.6 Free Vibration of Single Degree of Freedom Systems............................................. 8
1.6.1 Free Vibration of an Undamped Translational System ............................. 8
1.6.2 Free Vibration of an Undamped Torsional System .................................. 10
1.6.3 Energy Method ............................................................................................ 10
1.6.4 Stability of Undamped Linear Systems..................................................... 11
1.6.5 Free Vibration with Viscous Damping ...................................................... 11
1.6.6 Logarithmic Decrement .............................................................................. 13
1.6.7 Torsional System with Viscous Damping .................................................. 14
1.6.8 Free Vibration with Coulomb Damping .................................................... 14
1.6.9 Free Vibration with Hysteretic Damping.................................................. 15
1.7 Forced Vibration of Single-degree-of-freedom Systems ........................................ 15
1.7.1 Forced Vibrations of Damped System ....................................................... 16
1.7.1.1 Resonance ........................................................................................ 18
1.7.2 Beats ............................................................................................................. 19
1.7.3 Transmissibility ........................................................................................... 19
1.7.4 Quality Factor and Bandwidth .................................................................. 20
1.7.5 Rotating Unbalance..................................................................................... 21
1.7.6 Base Excitation ............................................................................................ 21
1.7.7 Response Under Coulomb Damping .......................................................... 22
1.7.8 Response Under Hysteresis Damping ....................................................... 22
1.7.9 General Forcing Conditions And Response ............................................... 22
1.7.10 Fourier Series and Harmonic Analysis ..................................................... 23
1.8 Harmonic Functions ................................................................................................. 23
1.8.1 Even Functions ............................................................................................ 23
1.8.2 Odd Functions ............................................................................................. 23
1.8.3 Response Under a Periodic Force of Irregular Form ............................... 23
1.8.4 Response Under a General Periodic Force ................................................ 24
1.8.5 Transient Vibration ..................................................................................... 24
1.8.6 Unit Impulse ................................................................................................ 25
1.8.7 Impulsive Response of a System ................................................................ 25
1.8.8 Response to an Arbitrary Input ................................................................. 26
1.8.9 Laplace Transformation Method ................................................................ 26
1.9 Two Degree of Freedom Systems ............................................................................ 26
1.9.1 Equations of Motion .................................................................................... 27
1.9.2 Free Vibration Analysis .............................................................................. 27
1.9.3 Torsional System ......................................................................................... 28
1.9.4 Coordinate Coupling and Principal Coordinates ...................................... 29
1.9.5 Forced Vibrations ........................................................................................ 29
1.9.6 Orthogonality Principle .............................................................................. 29
1.10 Multi-degree-of-freedom Systems ........................................................................... 30
1.10.1 Equations of Motion .................................................................................... 30
1.10.2 Stiffness Influence Coefficients .................................................................. 31
1.10.3 Flexibility Influence Coefficients ............................................................... 31
1.10.4 Matrix Formulation..................................................................................... 31
1.10.5 Inertia Influence Coefficients ..................................................................... 32
1.10.6 Normal Mode Solution ................................................................................ 32
1.10.7 Natural Frequencies and Mode Shapes..................................................... 33
1.10.8 Mode Shape Orthogonality ......................................................................... 33
1.10.9 Response of a System to Initial Conditions ............................................... 33
1.11 Free Vibration of Damped Systems ........................................................................ 34
1.12 Proportional Damping.............................................................................................. 34
1.13 General Viscous Damping ....................................................................................... 35
1.14 Harmonic Excitations............................................................................................... 35
1.15 Modal Analysis for Undamped Systems ................................................................. 35
1.16 Lagrange’s Equation ................................................................................................ 36
1.16.1 Generalized Coordinates............................................................................. 36
1.17 Principle of Virtual Work ......................................................................................... 37
1.18 D’Alembert’s Principle ............................................................................................. 37
1.19 Lagrange’s Equations of Motion .............................................................................. 38
1.20 Variational Principles .............................................................................................. 38
1.21 Hamilton’s Principle ................................................................................................. 38
References ................................................................................................................. 38
Glossary of Terms ..................................................................................................... 40
2.1 Introduction .............................................................................................................. 53
2.1.1 Starting and Quitting MATLAB.................................................................. 54
2.1.2 Display Windows .......................................................................................... 54
2.1.3 Entering Commands..................................................................................... 54
2.1.4 MATLAB Expo .............................................................................................. 54
2.1.5 Abort .............................................................................................................. 54
2.1.6 The Semicolon (;) .......................................................................................... 54
2.1.7 Typing % ........................................................................................................ 54
2.1.8 The clc Command ......................................................................................... 54
2.1.9 Help................................................................................................................ 55
2.1.10 Statements and Variables .......................................................................... 55
2.2 Arithmetic Operations ............................................................................................. 55
2.3 Display Formats ....................................................................................................... 55
2.4 Elementary Math Built-in Functions ..................................................................... 56
2.5 Variable Names ........................................................................................................ 58
2.6 Predefined Variables ................................................................................................ 58
2.7 Commands for Managing Variables........................................................................ 59
2.8 General Commands .................................................................................................. 59
2.9 Arrays ........................................................................................................................ 61
2.9.1 Row Vector ................................................................................................... 61
2.9.2 Column Vector ............................................................................................. 61
2.9.3 Matrix ........................................................................................................... 61
2.9.4 Addressing Arrays ....................................................................................... 61
2.9.4.1 Colon for a Vector ............................................................................ 61
2.9.4.2 Colon for a Matrix ........................................................................... 62
2.9.5 Adding Elements to a Vector or a Matrix.................................................. 62
2.9.6 Deleting Elements ....................................................................................... 62
2.9.7 Built-in Functions ....................................................................................... 62
2.10 Operations with Arrays ........................................................................................... 63
2.10.1 Addition and Subtraction of Matrices........................................................ 63
2.10.2 Dot Product .................................................................................................. 64
2.10.3 Array Multiplication ................................................................................... 64
2.10.4 Array Division ............................................................................................. 64
2.10.5 Identity Matrix ............................................................................................ 64
2.10.6 Inverse of a Matrix ...................................................................................... 64
2.10.7 Transpose ...................................................................................................... 64
2.10.8 Determinant ................................................................................................. 65
2.10.9 Array Division ............................................................................................. 65
2.10.10 Left Division ................................................................................................ 65
(xiii)
2.10.11 Right Division .............................................................................................. 65
2.10.12 Eigenvalues and Eigenvectors ................................................................... 65
2.11 Element-by-element Operations ............................................................................. 66
2.11.1 Built-in Functions for Arrays ..................................................................... 67
2.12 Random Numbers Generation ................................................................................. 68
2.12.1 The Random Command .............................................................................. 69
2.13 Polynomials ............................................................................................................... 69
2.14 System of Linear Equations .................................................................................... 71
2.14.1 Matrix Division ............................................................................................ 71
2.14.2 Matrix Inverse ............................................................................................. 71
2.15 Script Files ................................................................................................................ 76
2.15.1 Creating and Saving a Script File .............................................................. 76
2.15.2 Running a Script File .................................................................................. 76
2.15.3 Input to a Script File ................................................................................... 76
2.15.4 Output Commands ...................................................................................... 77
2.16 Programming in Matlab ........................................................................................... 77
2.16.1 Relational and Logical Operators .............................................................. 77
2.16.2 Order of Precedence .................................................................................... 78
2.16.3 Built-in Logical Functions .......................................................................... 78
2.16.4 Conditional Statements .............................................................................. 80
2.16.5 NESTED IF Statements ............................................................................. 80
2.16.6 ELSE and ELSEIF Clauses ........................................................................ 80
2.16.7 MATLAB while Structures ......................................................................... 81
2.17 Graphics .................................................................................................................... 82
2.17.1 Basic 2-D Plots ............................................................................................... 83
2.17.2 Specialized 2-D Plots ..................................................................................... 83
2.17.2.1 Overlay Plots ................................................................................. 84
2.17.3 3-D Plots ......................................................................................................... 84
2.17.4 Saving and Printing Graphs......................................................................... 90
2.18 Input/Output In Matlab ........................................................................................... 91
2.18.1 The FOPEN Statement ................................................................................. 91
2.19 Symbolic Mathematics ............................................................................................. 92
2.19.1 Symbolic Expressions.................................................................................. 92
2.19.2 Solution to Differential Equations ............................................................. 94
2.19.3 Calculus ........................................................................................................ 95
2.20 The Laplace Transforms .......................................................................................... 97
2.20.1 Finding Zeros and Poles of B(s)/A(s) .......................................................... 98
2.21 Control Systems........................................................................................................ 98
2.21.1 Transfer Functions ...................................................................................... 98
2.21.2 Model Conversion ........................................................................................ 98
(xiv)
2.22 The Laplace Transforms ........................................................................................ 101
10.11.1 Finding Zeros and Poles of B(s)/A(s) ....................................................... 102
Model Problems and Solutions ......................................................................................... 102
2.23 Summary ................................................................................................................. 137
References .......................................................................................................................... 138
Problems............................................................................................................................ 138
3.1 Introduction ............................................................................................................ 150
3.2 Example Problems and Solutions ......................................................................... 150
3.3 Summary ................................................................................................................. 197
Problems............................................................................................................................ 198
Example 3.12. Determine the free-vibration response of a two degree of freedom system
shown in Fig. E3.12 with the initial conditions x1(0) =0, x2(0) = 0.005 m, x1 (0) = 0, x2 (0) = 0.
The parameters of the system are given as m = 30 kg, k = 20,000 N/m, and c = 150 N.s/m.
x1
m
2k c
2m
k
x2
Fig. E3.12 Two degree of freedom system.
The differential equations governing the motion of the system are
m
m
x
x c
x
x
k k
k k
x
x
0
0 2
0 0
0
3 2
2 2
00
1
2
1
2
1
2
LNM
OQP
LN M
OQ P
+LNM
OQP
LN M
OQ P
+ −
−
LNM
OQP
LN M
OQ P
=LNM
OQP
or M y + Ky = 0
where M =
0 M
M c
LNM
OQP
; K =
− LNM
OQP
M
K 0
0 ; Y =
xx
LNM
OQP
174 SOLVING VIBRATION ANALYSIS PROBLEMS USING MATLAB
The solution is assumed as
y = φe– γt
where γ are the eigenvalues of M–1K and φ are the eigenvectors. The general solution is a linear
combination over all solutions, that is,
y =
j =
Σ
1
4
c e j j
φ −γ jt
and application of initial conditions gives
y0 =
j
c j
= Σ
1
4
φj = VC
and C = V–1 y0
Solution:
The MATLAB program is given as follows:
m=30; % Mass
k=20000; % Stiffness
c=150; % Damping
% 4 x 4 matrices
disp('4 x 4 Mass matrix');
mt=[0,0,m,0;0,0,0,2*m;m,0,0,0;0,2*m,0,c];
disp('4 x 4 stiffness matrix');
kt=[-m,0,0,0;0,-2*m,0,0;0,0,3*k,-2*k;0,0,-2*k,2*k];
Z=inv(mt)*kt;
[V,D]=eig(Z);
disp('Eigenvalues');
V
disp('Initial conditions');
x0=[0;0;0.005;0]
disp('Integration constants');
S=inv(V)*x0
tk=linspace(0,2,101);
% Evaluation of time dependent response
% Recall that x1=y3 and x2=y4
for k=1:101
t=tk(k);
for i=3:4
x(k,i-2)=0;
for j=1:4
x(k,i-2)=x(k,i-2)+(real(S(j))*real(V(i,j))-imag(S(j))*imag(V(i,j)))
*cos(imag(D(j,j))*t);
x(k,i-2)=x(k,i-2)+(imag(S(j))*real(V(i,j))-real(S(j))
*imag(V(i,j)))*sin(imag(V(i,j))*t);
x(k,i-2)=x(k,i-2)*exp(-real(D(j,j))*t);
end
MATLAB TUTORIAL 175
end
end
plot(tk,x(:,1),'-',tk,x(:,2),':')
title('Solution of problem E3.12')
xlabel('t[sec]')
ylabel('x(m)')
legend('x1(t)','x2(t)')
The output of this program is given below. See also Fig.E3.12(a).
V =
-0.9390 -0.9390 0.5886 - 0.0085i 0.5886 + 0.0085i
0.3428 - 0.0185i 0.3428 + 0.0185i 0.8050 0.8050
0.0001 - 0.0188i 0.0001 + 0.0188i -0.0026 + 0.0440i -0.0026 - 0.0440i
0.0003 + 0.0069i 0.0003 - 0.0069i -0.0044 + 0.0601i -0.0044 - 0.0601i
Initial conditions
x0 =
0
0
0.0050
0
Integration constants
S =
-0.0013 + 0.1048i
-0.0013 - 0.1048i
-0.0019 - 0.0119i
-0.0019 + 0.0119i
Solution of problem E3.12
5
4
3
2
1
0
– 1
– 2
– 3
x 10 –3
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x1 (t)
x2 (t)
x(m)
t[sec]
Fig. E3.12(
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تعریف حسابداری رفتاری
ضرورت حسابداری رفتاری
هدف علم رفتاری و اخلاقیات در تجارت
بررسی پدیده های رفتاری در قلمرو حرفه ی حسابداری
قلمروحسابداری رفتاری
کاربردهای حسابداری رفتاری
سیرتدریجی تکامل حسابداری رفتاری
تحقیقات حسابداری رفتاری
تقسیم بندی حسابداری رفتاری
حسابداری مالی رفتاری ( Behavioral Finance Accounting )
حسابداری مالیات رفتاری ( Behavioral Tax Accounting)
حسابداری مدیریت رفتاری (Behavioral Management Accounting )
اهمیت و ارزش ارائه اطلاعات رفتاری شرکتها توسط حسابداری رفتاری
تاثیر رویه های مختلف حسابداری بر تصمیم
مطالعات انجام شده درباره تاثیرات زبان شناختی تکنیکها و داده های حسابداری
مطالعات انجام شده درباره القای اطلاعات
نقش تحقیقات رفتاری در فرآیند تدوین استانداردهای حسابداری
رهنمودهایی برای پیشگیری از مشکلات رفتاری
پژوهش های انجام شده در حسابداری رفتاری
استفاده از اطلاعات حسابداری در شرکت
ابعاد تحقیق تجربی در حسابداری مدیریت
نحوه قضاوت سرمایه گذاران درباره ریسک اقلام مالی
پژوهش های انجام شده در حسابرسی رفتاری
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مفروضات مربوط به بازارها - مالی کلاسیک
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«تئوری چشم انداز» (Prospect Theory)
تحقیق تهیه شده بسیار کامل و قابل ویرایش بوده و در تهیه آن کلیه اصول نگارشی، املایی و چیدمان و جمله بندی رعایت گردیده است.
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