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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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