Vibrations and Time Response Presentation

Vibrations and Time Response
(Meriam & Kraige,
Chapter 8,p 601 – 633)
Applied Mechanics 2A
Introduction
Rigid body
Linear and angular motion
Oscillates
Responds to applied disturbances
Structures to earthquakes
Vibrations of unbalanced rotating machinery etc.
Simplified model very accurate  some
protions of a continuous system are massive
in comparison with other portions
1-D vibrations for this chapter
Applied Mechanics 2A
Free vibrations of particles
• Spring mounted body
disturbed from equilibrium
position
absence of imposed external
forces = free vibration
• Retarding forces excluded for
example  friction
Applied Mechanics 2A
Equations of motion for un-damped
free vibrations
• Consider a frictionless massspring system
 x  displacement from
equilibrium condition
 -kx  restoring force on mass
• From Newton’s second law:
..
 kx  m x
..
m x  kx  0
Applied Mechanics 2A
Equations of motion for un-damped
free vibrations
• Oscillation of a mass to a
linear restoring force 
simple harmonic motion
• Generally expressed as
follows:
..
x   n2 x  0
n 
k
m
Applied Mechanics 2A
Solution for un-damped free
vibrations
x  A cos  n t  B sin  n t or x  C sin  n t   
A, B constants
 nt    Phase Angle
C Amplitude
 Phase constant
 n Natural frequency
Applied Mechanics 2A
Solution for un-damped free
vibrations
x0  C sin 
.
x 0   n C cos
At time t = 0s
Initial displacement = x0
Initial velocity = dx0/dt
thus


  n x0 
  arctan  . 
 x 
 0 
 
 x0 
2
C  x0   
 n 
 
.
Applied Mechanics 2A
2
Graphical representation of motion
• Motion  graphically  x is seen to be the
projection onto a vertical axis of a rotating vector
of length C
• Vector rotates at constant angular velocity n
• Number of complete cycles per unit time = natural
frequency  fn = n/2
• f  hertz (Hz)
• Time to complete one motion cycle   = Period =
1/ fn = 2/n
Applied Mechanics 2A
Graphical representation of motion
Applied Mechanics 2A
Graphical representation of motion
Significance of the phase angle
A  A cost   
can be expressed as follows
1  cost   
 time required to move t    0; 2n; n  0,1,2..... 
from zero to first maximum

t 

amplitude
m
m
m
m
Applied Mechanics 2A
Graphical representation of motion
If we started the oscillator at its
maximum positive
displacement
x  A cos
Applied Mechanics 2A
t 
Velocity and acceleration of a simple
harmonic motion
x  A cost 
dx


v
  A sin t   A cos t  
dt
2

dv
2
a
  A cost 
dt
Applied Mechanics 2A
Damped Free vibrations
• Every system possesses a
degree of friction 
dissipates mechanical enegy
 viscous damper intentionally
added to a system to limit
vibration
• Constant of proportionality c
 viscous damping
coefficient  N.s/m
• Damping force is directed as
to oppose the velocity
Applied Mechanics 2A
Damped Free vibrations
.
..
..
.
 kx  c x  m x or m x  kx  c x  0
n 
k
c
; 
 viscous damping factor(zet a)
m
2 m n
..
.
x  2n x   n2 x  0
Applied Mechanics 2A
Solution for damped free vibrations
Assume the solution has the form x = Aet
Substitute in equation and solve quadratic equation
..
.
x  2n x   n2 x  0;

1, 2   n     2  1

Property of superposit ion of linear system
x  A1e 1t  A2 e 2t
Applied Mechanics 2A
Categories of damped motion
• Because  lies between 0 and  and the radicand
(2 -1) may be positive, negative or zero  3
categories of damped motion
–  > 1  over damped  Roots 1 & 2  distinct,
real, negative numbers
– Motion decays  x approaches zero for large values
of time
Applied Mechanics 2A
Categories of damped motion
• Because  lies between 0 and  and the radicand
(2 -1) may be positive, negative or zero  3
categories of damped motion
–  = 1  critically damped  Roots 1 & 2  equal,
real, negative numbers = -n
x   A1  A2 e
Applied Mechanics 2A
nt
Categories of damped motion
• Because  lies between 0 and  and the radicand
(2 -1) may be positive, negative or zero  3
categories of damped motion
–  < 1  radicand (2 -1) < 0


x  A1e i 1  n t  A2 e i 1  n t e  n t
2
2
 d   n 1   2 (damped natural frequency)
e  ix  cos x  i sin x
x   A3 cos  d t  A4 sin  d t e  n t
A3  A1  A2 ; A4  A1  A2


x  Ce  nt sin  d t   
2
 d Applied

(damped2Aperiod)
Mechanics
d
Forced vibrations of particles
• Motion excited by disturbing force
• Force  internally or externally generated
• May be caused by motion of the foundation of
the system
Applied Mechanics 2A
Harmonic Excitation
• Consider a body excited by an external force 
F = F0 sin t; F0 = force amplitude &  = driving
frequency
• Don’t mix up the two frequencies
..
 kx  cx  F0 sin t  m x
F0 sin t
x  2n x   x 
m
..
.
2
n
Applied Mechanics 2A
Un-damped Forced Vibration
Damping  negligible  c=0
F0 sin t
x  x 
m
..
2
n
Solution = sum of complementary and particular
solution
xc  A cos  n t  B sin  n t  C sin  n t   
x p  X sin t
Applied Mechanics 2A
Un-damped Forced Vibration
Substituting xp & derivatives
F0
k
X 
2
 
1   
 n 
• Complementary (transient) Solution  dies out
with time & is of no interest
• Particular (steady state) solution  continuos
motion with period   = 2/
Applied Mechanics 2A
Un-damped Forced Vibration
• Amplitude of importance
• Assume a static force F0 cause static
spring deflection st  F0 = kst
• Amplitude ratio or magnification
factor M  measure severity of
vibration
• Define M  ratio of forcing
amplitude to static deflection
• Approaches infinity if  approaches
n
Applied Mechanics 2A
M 
X
 st

1
 
1   
 n 
2
Un-damped Forced Vibration
• Value of n  resonant or critical
frequency of system
• If  is close to n  resulting in
large displacement amplitude 
resonance  Tacoma bridge
• If <n  M positive & vibration
in phase with force F
• If >n  M negative & vibration
180 degrees out of phase with force
F
Applied Mechanics 2A
Damped Forced Vibration
F0 sin t
x  2n x   x 
m
..
.
2
n
• Solution = sum of complementary and particular
solution
• Complementary solution presented for over
damped, under damped and critically damped
• Particular solution = any solution to complete
equation
x p  X 1cos t  X 2 sin t  X sin t   
Applied Mechanics 2A
Damped Forced Vibration
Substituting xp & derivatives
X 
F0
k
2
      2  2
1      




  n    n 


2
 2 




n
  arctan 
2 
   
 1     
  2A n  
Applied Mechanics
Damped Forced Vibration
Complete solution for a damped system
x  Ce
 n t
sin d t    X sin t   
• First term to right side diminishes with time 
transient solution
• Particular solution of interest  steady state
solution
Applied Mechanics 2A