Slovak University of Technology
Faculty of Material Science and Technology in Trnava
APPLIED MECHANICS
Lecture 09
ANALYSIS OF CONTINUOUS SYSTEMS
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Vibrations of strings, rods and shafts are described by the same
mathematical model - analysis can be discussed simultaneously.
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Free vibration of strings, rods and shafts - governing equation for
free vibrations:
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Boundary conditions - natural frequencies and natural modes
Predict the particular solution - form of the product of two functions.
One of them is a function of the position z and the other one is a
harmonic function of time t
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ANALYSIS OF CONTINUOUS SYSTEMS
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Introduction of the predicted solution into equation of motion yields
the following ordinary differential equation
or
where
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The general solution
ANALYSIS OF CONTINUOUS SYSTEMS
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Parameter bn is expressed:

strings:

rods:
.

shafts:
ANALYSIS OF CONTINUOUS SYSTEMS
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The parameter βn as well as the constants Sn and Cn should be
chosen to fulfill the boundary conditions.
To demonstrate the way of the determination of the natural
frequencies and the corresponding natural modes, let us consider
the fixed on both ends shaft:
 the boundary conditions:
Introduction of this boundary conditions - set of two homogeneous
algebraic equations linear . with respect to the constants Sn and Cn
ANALYSIS OF CONTINUOUS SYSTEMS
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The set of equations has non-zero solutions if its characteristic
determinant is equal to zero:
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In this particular case
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This equation - characteristic equation, has infinite number of solution.
βn, l - always positive - only positive roots of the equation have the
physical meaning
.
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ANALYSIS OF CONTINUOUS SYSTEMS
.
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The natural frequencies
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For each of these natural frequencies the set of homogeneous
algebraic equations becomes linearly dependant - one of the
constants can be chosen arbitrarily.
The predicted solution for this case
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.
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The functions Yn(z) are called eigenfunctions or natural modes and
the corresponding roots ωn are called eigenvalues or natural
angular frequencies.
ANALYSIS OF CONTINUOUS SYSTEMS
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
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The above analysis allows to conclude that a continuous system
possesses infinite number of the natural frequencies and infinite
number of the corresponding natural modes.
The first mode is called fundamental mode and the corresponding
frequency is called fundamental natural frequency.
In the case of free vibrations of the shaft, the natural modes
determine the angular positions of the cross-section of the shaft
j(z). A few first of them are shown infollowing figure:
.
.
ANALYSIS OF CONTINUOUS SYSTEMS
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Some of the boundary conditions for strings, rods and shafts:
.
.
ANALYSIS OF CONTINUOUS SYSTEMS
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Free vibration of beams - governing equation for free vibrations:
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The solution of the above equation - product of two functions:

function of the position z,
 harmonic function of time t
.
.

It yields the following ordinary differential equation
.
where
ANALYSIS OF CONTINUOUS SYSTEMS
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The solution:
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The characteristic equation
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The roots of the characteristic equation
.
.
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The linearly independent particular solution
.
or alternatively
ANALYSIS OF CONTINUOUS SYSTEMS
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The general solution - linear combination of particular solutions is:
Yn ( z )  An sinh( b n z )  Bn cosh( b n z )  Cn sin( b n z )  Dn cos(b n z )
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Values for the parameter βn and for the constants An, Bn, Cn and Dn
should be chosen to fulfill boundary conditions.
The equation of motion of fourth order - produce four boundary
conditions reflecting the conditions at both ends of the beam.
The first three derivatives with respect to z:
.
YnI ( z )  b n [ An cosh( b n z )  Bn sinh( b n z )  Cn cos(b n z )  Dn sin( b n z )]
YnII ( z )  b 2n [ An sinh( b n z )  Bn. cosh( b n z )  Cn sin( b n z )  Dn cos(b n z )]
YnIII ( z )  b3n [ An cosh( b n z )  Bn sinh( b n z )  Cn cos(b n z )  Dn sin( b n z )]
ANALYSIS OF CONTINUOUS SYSTEMS
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The boundary conditions for the free-free beam:
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The following set of algebraic equations - linear with respect to the
constants An, Bn, Cn and Dn
.
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Non-zero solution - characteristic determinant is equal to zero.
ANALYSIS OF CONTINUOUS SYSTEMS
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The characteristic equation
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Characteristic equation is transcendental - infinite number of roots.
Solution of this equation - within a limited range of the parameter bnl
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The characteristic equation has double root of zero magnitude
.
ANALYSIS OF CONTINUOUS SYSTEMS
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Beam considered is free-free in space. The zero root is associated
with the possible translation and rotation of the beam as a rigid body.
The two modes, corresponding to the zero root are shown in figure.
Modes corresponding to the non-zero roots are determined according
to the following procedure:

For any root of the characteristic equation the set of algebraic equations,
since its characteristic determinant is zero, becomes linearly dependant.
 Therefore - choose arbitrarily one of the constants (for example An) and
the other can be obtained from equations:
.
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For the first non-zero root β1l = 4.73 and An = −1
ANALYSIS OF CONTINUOUS SYSTEMS
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Corresponding mode
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In the same manner - modes for all the other characteristic roots are
determined.
Modes for β1l = 4.73 (c), β2l = 7.85 (d), β3l = 11 (e)
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.
ANALYSIS OF CONTINUOUS SYSTEMS
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Formula for the natural frequencies
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The particular solution
.
where Yn(z) and ωn are uniquely determined,
Sn is an arbitrarily chosen constant
ANALYSIS OF CONTINUOUS SYSTEMS
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The boundary conditions for some cases of beams:
.
.