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Slovak University of Technology
Faculty of Material Science and Technology in Trnava
APPLIED MECHANICS
Lecture 04
SINGLE-DOF SYSTEM
FREE VIBRATION, WITH DAMPING

Model of single-DOF system with a viscous damper
. ..
x, x, x
b
k
m
b - coefficient of viscous damping,
k - stiffness of spring,
m - mass.
SINGLE-DOF SYSTEM
FREE VIBRATION, WITH DAMPING

Equation of motion
mx  bx  kx  0
b

2m
x 
b
k
x  x  0
m
m
x  2x  02 x  0
- damping ratio,
k
0 
- natural angular frequency of undamped system
m
.

The characteristic equation
r 2  2r  02  0
with the solutions
r1,2     2  02
SINGLE-DOF SYSTEM
FREE VIBRATION, WITH DAMPING
r1,2     2  02

Case 1
 2  02  0  the roots r1 and r2 are complex conjugate
(r1, r2  C, where C - the set of complex numbers )

Case 2
 2  02  0  the roots r1 and r2 real and distinct
(r1, r2  R , where R - the set of real numbers)

Case 3
 2  02  0  the roots r1 = r2 are real and identical
2
(r1, r2  R , where R - rthe
set of real numbers)
SINGLE-DOF SYSTEM
FREE VIBRATION, WITH DAMPING

For case 3
bcr
k
  0 

2m
m
b = bcr is called the critical damping coefficient
bcr  2 km  2m0

One can classify the vibrations with respect to critical
damping coefficient as follows:



Case 1: b  bcr - complex conjugate roots (low damping and
oscillatory motion).
Case 2: b  bcr - real and distinct roots (great damping and
aperiodic motion).
Case 3: b  bcr - real and identical roots (critical damping and
aperiodic motion).
SINGLE-DOF SYSTEM
FREE VIBRATION, WITH DAMPING
Case 1: Complex conjugate roots
2
2
The term d  0  
 2  02  0
is the quasicircular frequency.
The roots
r1,2     2  02    i 02   2    id
The solution
x  e t ( Ad sin( d t )  Bd cos(d t ))  Cd e t sin( d t   d )
where Ad, Bd and Cd, d are integration constants.
SINGLE-DOF SYSTEM
FREE VIBRATION, WITH DAMPING
Using the initial condition
 x  x0
t 0
 x  v0
and the derivative of equation x with respect to time
x  et  ( Ad cos(d t )  Bd sin( d t ))  d ( Bd cos(d t )  Ad sin( d t ))
the integration constants are
v  x0
Ad  0

Bd  x0
2
 v  x0 
C d  Ad2  Bd2  x02   0
 ,



B
x0
tan  d  d 
.
Ad v0  x0
SINGLE-DOF SYSTEM
FREE VIBRATION, WITH DAMPING
The solution is
xe
 t  v0


 x0

sin( d t )  x0 cos(d t ) 


or in amplitude form
x
x02
2
x0 
 v0  x0   t 


 e sin  d t  arctan

v0  x0 



2
 b 

d  0 1  
 bcr 
2
2
Td 

d
2   2
0
- the quasicircular (or quasiangular) frequency
- the quasiperiod of motion
SINGLE-DOF SYSTEM
FREE VIBRATION, WITH DAMPING

The displacement, velocity and acceleration for parameters:
x0  0 m
v0  0,2 m/s
0  0,8 rad/s
SINGLE-DOF SYSTEM
FREE VIBRATION, WITH DAMPING

The rate of decay of oscillation:
Cd e t sin( d t  d )
x(t )
1
Td



e
x(t  Td ) Cd e (t Td ) sin[ d (t  Td )  d ] e Td

The logarithmic decrement L is defined:
L  ln
x(t )
2
b
 ln e Td  Td 

x(t  Td )
02   2 md
SINGLE-DOF SYSTEM
FREE VIBRATION, WITH DAMPING

Case 2: Real and Distinct Roots
 2  02  0
The roots of the characteristic equation
r1     2  02  1 ,
r2     2  02   2 ,
The solution
x  C1e 1t  C2e 2t
Aperiodic motion - tends asymptotically to the rest position t  0, x  0
SINGLE-DOF SYSTEM
FREE VIBRATION, WITH DAMPING

Case 3: Real identical roots
2  02  0
The roots of the characteristic equation
r1  r2  
The solution
xe
 t
(C1t  C2 ) 
C1t  C2
e t
The diagrams of motion are similar to the previous case, with the
same initial conditions. Critical damping represents the limit of
periodic motion; hence, the displaced body is restored to equilibrium
in the shortest possible time, and without oscillation. Many devices,
particularly electrical instruments, are critically damped to take
advantage of this property.