 Open Access
 Total Downloads : 3534
 Authors : T. Ravi Teja, S. Krishna Chaitanya
 Paper ID : IJERTV1IS10544
 Volume & Issue : Volume 01, Issue 10 (December 2012)
 Published (First Online): 28122012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Computation Of Natural Frequencies Of Multi Degree Of Freedom System
T. Ravi teja M.Tech (Robotics) S. Krishna chaitanya M.Tech
Assistant professor Associate professor
Narasaraopeta Engineering college St. Marys Group of institutions
Abstract
Wear and tear of machinery manifests into vibrations or changing the pattern of vibrations. Analysis of vibrations for frequency response and time response has become indispensable for major process machinery in trouble shooting. Computation of natural frequencies for an ndegree freedom systems and relative amplitudes of vibrating masses help the designer in choosing the parameters of the system and operating frequencies for the safer operations. Classical mathematical methods are not enough for real time computation of frequency response as they consume more time and effort. In this project we have attempted to analyze a multidegree freedom system for natural frequencies and thereby pure mode shapes. The pure mode shapes can be later superimposed to get the actual displacement pattern of the system. We have developed a multidegree freedom system by developing a program in Mat Lab platform.

Introduction
Vibrations occur in many aspects of our life. For example in human body, there are low frequency oscillations of lungs and the heart, high frequency oscillations of ear, oscillations of the larynx as one speaks, and oscillations induced by rhythmical body Motions such as walking, jumping, and dancing. Many manmade systems also experience or produce vibrations. For example, any unbalance in machines with rotating parts such as fans, ventilators, centrifugal pumps, rotary presses, and turbines can cause vibrations. Building and structures can experience vibrations due to operating machinery, passing vehicular, air and railway traffic; or natural phenomena such as earthquakes and winds. Pedestrian bridges and floors in buildings also experience vibrations due to human moment on them.
From the above examples the definition of vibrations can be drawn as Fluctuations of mechanical or structural systems about an equilibrium position. Vibrations are initiated when an inertia element is displaced from its equilibrium Position due to an energy imparted to the system by an external source.
Fig.1 Simple pendulum

Modelling of vibration systems
All physical systems are inherently non linear. Exact mathematical modelling of a physical system leads to nonlinear differential equations, which often have no analytical solutions. So the mathematical model is only an approximation to the true physical system. The elements that comprise a vibratory system model are:

System elements

Inertia elements

Dissipation elements
External applied forces and moments and external disturbances from prescribed initial displacements and/or initial velocities.
A linear spring obeys a force displacement law of F=K*X
Where K is called the spring stiffness and has dimensions of force/ length


HARMONIC EXCITATION OF ONE DEGREE OF FREEDOM SYSTEMS:
FREE VIBRATIONS:
Free vibrations are oscillations about a systems equilibrium position that occurs in the absence of an external excitation. Free vibrations are a result of a kinetic energy imparted to the system or a displacement from the equilibrium position that leads to a difference in potential energy from the systems equilibrium position.
FORCED VIBRATIONS:
Forced vibrations of a one degree freedom system occur when work is being done on system while the vibrations occur. Examples of forced excitation include the ground motion during an earthquake or the motion caused by an unbalanced reciprocating component. Consider a spring mass system excited by a sinusoidal forcing function F0t as shown in figure. At any instant, when the mass is displaced from the mean position through a distance x in the downward direction, the external forces acting on the system are

kx, in the upward direction.

F0sinwt,in the downward direction
4 .HARMONIC EXCITATION OF MULTI DEGREE OF FREEDOM SYSTEMS
The procedure for analyzing multidegree of freedom system is only an extension of the method used for analyzing single degree of freedom system. However, the use of influence coefficients and writing the equations in matrix form made the things sufficiently simpler. Besides, the use of equations in matrix forms facilitates the application of computer methods for their solution. Two methods are introduced to drive the differential equations governing the motion of multidegree of freedom systems.

The free body diagram method

The equivalent systems method
Fig 3. N DOF system

METHODOLOGY:
NUMERICAL METHODS:
The actual solution of
Fig 2. Spring mass system
m x = kx + F0 sin (t)
This is a linear, nonhomogeneous, second order differential equation. The solution of this quation consists of two parts, complementary function and particular integral. The complementary function is obtained from the differential equation.
the determinants of higher order becomes more and more difficult with increasing number of degrees of freedom as the exact analysis is associate with laborious calculations. Even with highspeed digital computers that can solve equations of many degreeof freedoms that result beyond the first few normal modes are often unreliable and meaningless. So, Numerical methods are used to solve these problems.

RAYLEIGH METHOD

RAYLEIGHRITZ METHOD

HOLZERS METHOD

METHOD OF MATRIX ITERATION
Fig 4. Different masses in nDOF spring system
By using differential equations of motions for the three masses in terms
x = m x + m x + m x
x =mx+mx+mx3
x = m x + m x +m x Replacing x by Â²x we have
x = m Â² x + m Â² x + m Â² x x = m Â² x + m Â² x +m Â² x x = m Â² x + m Â² x +m Â² x
The modes have been repeated with sufficient accuracy in the 4th iterative
Â²m/3k x 14.28 = 1 = 0.458 (k/m)
Natural frequencies are = 1, 3.16, 4


MAT LAB
MATLAB (short for Matrix Laboratory) is a special purpose computer program optimized to perform engineering and scientific calculations. It is a high performance language for technical computing. It integrates computation, visualization, and programming in an easytouse environment where problems and solutions are expressed in familiar mathematical notation. Typical uses include:

Math and computation

Algorithm development

Modelling, simulation and prototyping

Data analysis, exploration and visualization

Scientific and engineering graphics

Application development, including Graphical User Interface (GUI) building


Test Cases:
% DOF = input ('enter the number of degrees of freedom: ');
% MASS = input ('enter masses');
% STFNS_VALS = input ('enter stiffness values of springs: ');
% AMPLITUDE = input ('Enter Random values for amplitude: ');
%creation of global stiffness matrix DOF = 3
STFNS_VALS = [3 1 1 0]
MASS = [4 2 1]
AMPLITUDE = [1; 2; 3]
STFNS_MTRX = zeros (DOF, DOF); MASS_MTRX = zeros (DOF, DOF); AMPLITUDE_MTRX = zeros (DOF, 1);
fprintf (the global stiffness matrix is:\n'); for i = 1: DOF
for j = 1:DOF if i == j
STFNS_MTRX(i,j) = (STFNS_VALS(i) + STFNS_VALS(i+1));
elseif j==i+1
STFNS_MTRX(i,j) = STFNS_VALS(j);
elseif i==j+1
STFNS_MTRX (i,j) = STFNS_VALS(i);
else STFNS_MTRX(i,j)=0; end
fprintf ('%d ',STFNS_MTRX(i,j)); end
fprintf ('\n'); end
SPRSE_STFNS_MTRX = sparse (STFNS_MTRX); MASS_MTRX = diag(MASS) SPRSE_MASS_MTRX = sparse (MASS_MTRX); INV_MASS_MTRX = inv (SPRSE_MASS_MTRX); MASS_X_STFNS = (INV_MASS_MTRX * STFNS_MTRX);
cond = 1;
while (cond ~= 0)
AMPLITUDE_MTRX = (MASS_X_STFNS * AMPLITUDE);
AMPLITUDE_MTRX = (AMPLITUDE_MTRX/AMPLITUDE_MTRX(1));
AMPLITUDE_MTRX = num2str (AMPLITUDE_MTRX,3);
AMPLITUDE = num2str (AMPLITUDE,3);
AMPLITUDE_MTRX = str2num (AMPLITUDE_MTRX);
AMPLITUDE = str2num (AMPLITUDE); cond = AMPLITUDE –
AMPLITUDE_MTRX;
cond = sum (cond(:)); AMPLITUDE = AMPLITUDE_MTRX;
end
fprintf ('the final amplitude of the system are:\n'); fprintf ('%g \n',AMPLITUDE_MTRX);
LAMDA = eig(MASS_X_STFNS);
fprintf ('the natural frequencies of the system in rad/sec are:');
OMEGA = sqrt(LAMDA)
for i = 1:DOF
fprintf('the displacement when vibration with w %d natural frequency is \n',i );
TIME = linspace (0, 20, 123); WT = (OMEGA (i, 1)* TIME);
DISP = AMPLITUDE_MTRX (i,1)*sin(WT);
fprintf ('%g \n', DISP);
figure, plot (TIME, DISP ); title('Displacement Vs Time') xlabel ('Time (sec)')
set (gca, 'XGrid', 'on', 'YGrid', 'on') ylabel ('Displacement (mts)')
end

Results:
The final amplitude of the system is: 1
3.15
3.98
The natural frequencies of the system in rad/sec are: OMEGA = 0.4576
1.0000
1.3381
Graph 1.The Natural frequency 1 graph
Graph 2.The Natural frequency 2 graph
Graph 3.The Natural frequency 3 graph

Conclusion
Computer of natural frequencies and relative amplitudes has become indispensable in the design of multi degree freedom vibration systems. The code we have developed in Matlab helps a perspective designer in choosing the operating speed of the system. The program also plots the amplitudes of the vibrating bodies, with the help of which the designer can tryout different system parameters in real time to arrive at optimum specifications of the system. Thus optimization of the parameters can be achieved without the application of tools like artificial neural networks

References

Theory of vibration with applications by william t. thomson and marie dillon dahleh.

Vibrations problem solving companion by rao v. dukkipati and j.srinivas

Mechanical vibrations by g.k. grover

R.J. Astley, G. Gabard, Computational Aero Acoustics (CAA) for Aircraft Noise Prediction, Elsevier Ltd Pages 40814082