Vibrations in linear 1-degree of freedom systems;
I. undamped systems
(last updated 2011-08-26)
Kjell Simonsson
1
Aim
The aim of this presentation is to give a short review of basic vibration
analysis in undamped linear 1 degree-of-freedom (1-dof) systems.
For a more comprehensive treatment of the subject, see any book in
basic Solid Mechanics or vibration analysis.
Kjell Simonsson
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Linear 1-dof systems
A 1-dof system is a simplification of reality, where it has been assumed that the
mass of the body/structure
• can be associated with one specific small part of the body
• is restricted to one "major vibration mode"
A typical example of a situation for which a 1-dof assumption is applicable is
a slender structure on which a single heavy detail is attached, where the latter
mainly moves in one (translational or angular) direction, see below where the
movement is depicted in blue.
EIL
The term linear 1dof-systems,
implies that all the governing
relations are linear, which
e.g. is the case of a linear elastic
structure (material linearity), subjected
to so small deformations that the
ordinary stress and strain measures
may be used (geometric linearity).
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m
lateral motion
E AL
m
horizontal motion
LG K
rotational motion
J
A simple example
Let us study some basic phenomena of linear 1-dof vibration analysis
(of un-damped) systems, by considering one of the simple examples
shown above!
m
EIL
F  F0 sin 0t
By making a free body diagram of the
(point) mass, we get
EIL
S
S
x
m
F
where the coordinate x describes the position of the mass (equal to the
deflection of the rod), and where the force S represents the interaction
between the rod and the point mass. Elementary solid mechanics now gives
 : mx   S  F 
3EI

3
  mx  3 x  F
SL
x
L


3EI
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A simple example; cont.
By introducing the lateral/bending stiffness k, we get
mx  kx  F
, k
3EI
L3
or, by dividing with m and introducing the complete expression for F
x 
F
k
x  0 sin 0t
m
m
Thus, the lateral motion of the mass as a function of time is governed by a
linear 2'nd order ordinary differential equation with constant coefficients!
The solution is given by the sum of the homogeneous solution and the
particular solution, x  xhom  x part , where the former corresponds to the
self vibration of the mass caused by the initial conditions (placement and
velocity), while the latter is caused by the applied force.
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A simple example; cont.
Homogeneous solution
For the homogeneous solution we have
k
k
 0  r  i
 xhom  C1ei k / m t  C2e i
m
m
k
k 
k
k 


 C1  cos
t  i sin
t   C2  cos
t  i sin
t 
m
m 
m
m 


k
k
 C1  C2  cos
t  iC1  C2 sin
t








m
m
r2 
A
k /m t

B
As can be seen, for free vibrations with F=0, the point mass can only vibrate
with the frequency k / m , which we refer to as the natural frequency or
eigenfrequency of the structure. By denoting it  e , we get
xhom  A cos et  B sin et , e  k / m
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A simple example; cont.
Homogeneous solution; cont.
Alternatively we may express this as
xhom
xhom


 A

B
 A cos et  B sin et  X  cos et 
sin et  
X
X



cos
 sin 

A & B or X & ϕ are found by
1  A 
 X sin et    ,   tan   using the initial conditions
B
Since we in all physical contexts have some kind of damping, i.e. processes
that transform the mechanical energy into e.g heat, it follows that the
self vibration caused by some initial conditions sooner or later will vanish.
Thus, at such a stationary state ("fortvarighet" in Swedish), only the particular
solution will remain.
It can finally be noted that the circular eigenfrequency fe and the eigenperiod
Te are defined as f e  e / 2 , Te  1 / f e
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Resonance
Homogeneous solution; cont.
We may also analyze the eigenfrequency of a structure/body by the
Finite Element method (FEM), see below where an animation of the
vibration mode (eigenmode) can be found for a similar type of structure.
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A simple example; cont.
Particular solution
For the particular solution we let
x part  C3 sin 0t 
F0
sin 0t 
m
F
1
 0 2
sin 0t
2
m e  0
 02C3 sin 0t  e2C3 sin 0t 
C3 
F0
1
m e2  02
 x part
As can be seen, for forced vibrations the displacement amplitude will
depend on how close to the eigenfrequency the applied frequence is.
Theoretically, we will get infinite amplitudes for the case 0  e.
We refer to this as resonance!
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A simple example; cont.
Particular solution; cont.
It may be noted that the particular solution can be recast in another form,
as illustrated below
x part 
F0
F0 1
1
sin

t

0
m e2  02
m e2

1
2
 
1  0 
 e 
where the factor
1
 
1  0 
 e 
xstat sin0 t
1
 
1  0 
 e 
F
, xstat  0
k
2
is the so called dynamic impact factor
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2
sin0 t 
Resonance
As we saw above, the amplitude of the particular solution goes to infinity when
the applied frequency approaches the eigenfrequency of the system. Since the
obtained particular solution is not valid for this case (we are not allowed to
divide by zero), we instead have to proceed as follows
F0

 sin et 
m
 
 C4t cos et

xpart  e2 x part
x part
F0
d
2
C4 cos et  eC4t sin et   e C4t cos et  sin et 
dt
m
 eC4 sin et  eC4 sin et  e2C4t cos et
 e2C4t cos et
F0
F0
 eC4 sin et  eC4 sin et  sin et  C4  
m
2me
x part
F0

t cos et
2me
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F0
 sin et 
m

Resonance
On the previous slide we found that the amplitude of the forced vibration at
resonance grows linearly with time
x
part
x part
F0

t cos et
2me
i.e. we get the type of behavior
illustrated to the right
t
Even though (as will be seen later on) damping will make the theoretical
vibration amplitude finite, it may still be very large near the "resonance"
frequency. The same is true for multi-dof systems and continuous systems.
As a consequence, all engineering designs must be designed such that they
are not operated in a resonance region, since that inevitably will cause
premature failure unless additional damping devices are used.
Kjell Simonsson
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