Forced vibration of single degree of freedom system Theory

Forced vibration of single degree of freedom system
Forced vibration of single degree of freedom system
Theory
Learning objectives
After completing this simulation experiment one should be able to
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Model a given real system to an equivalent simplified model of single degree freedom
system with suitable assumptions / idealisations.
Calculate the natural frequency of undamped oscillation, critical damping constant
and damping ratio of the proposed single degree freedom system.
Study the response of the system for different excitation frequency and amplitude.
Calculate the magnification factor and phase angle as a function of forcing frequency.
Forced vibration of single degree of freedom system
Introduction
Forced vibrations in a system occur (in a system occur when any force persistently disturbs it
from its equilibrium) work is being done on it. In engineering practice, we are almost
invariably interested in predicting the response of a structure or mechanical system to
external forcing. For example, we may need to predict the response of a bridge or tall
building to wind loading, earthquakes, ground vibrations due to traffic, tool chattering during
machining operation, ground vibrations transmitted through the suspension to the rider, an
offshore structure subjected to wave loading etc.
(a) Offshore structure subjected to wave loading
(c) Tacoma Narrows bridge during wind-induced vibration.
(b) Coconut trees under high wind loading
(d) Vibration of vehicle due to bumpy road
Fig 1: Structures subjected to forced vibration.
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Forced vibration of single degree of freedom system
To study the forced response of a given real system, we will use a single degree spring-mass
system as an equivalent model as shown in fig 2 and it can be thought of as representing a
single mode of vibration in a real system, whose natural frequency and damping coefficient
coincide with that of proposed spring-mass system. The system is excited by a harmonic
forcing function F0 sin ωt to simulate forced vibration condition as in real system.
Fig 2: a single degree spring-mass system
Differential equation of motion of the system is given by,
mx  cx  kx  F0 sin t
... (1)
The general solution of this mathematical model consists of two parts, the complementary
function which is the solution of the homogenous equation, and the particular integral. The
particular solution to the preceding equation is a steady-state oscillation of same
frequency ω as that of excitation. We can assume particular solution to be of the form
x  X sin(t   )
... (2)
Where X is the amplitude of oscillation and  is the phase lag of the displacement with
respect to the exciting force.
By substituting the equation 2 in equation 1, amplitude X and phase  is given as,
X
F0
(k  m 2 )2  (c )2
... (3)
and
  tan 1
c
(k  m 2 )
... (4)
Where,
ω is the excitation frequency
k is the equivalent spring stiffness
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m is the equivalent mass of the system
c is the equivalent damping coefficient
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Forced vibration of single degree of freedom system
The non-dimensional form of expression 3 and 4 are given by,
X
1
1


2
2
2
X st
2
   2  
1   2   2 

1       2

  n    n 



 2    
  

 n  
1
  tan 
2
     
 1     
   n   

... (5)
... (6)
Where,
The above equations may be represented in terms of following quantities:
ωn = √(k/m) = natural frequency of un-damped oscillation
cc = 2mωnf = critical damping constant
ζ = c/cc = damping ratio
Xst= F0/k
η is the frequency ratio
The quantity X/Xst is called the magnification factor, amplification factor, or amplitude ratio.
The variations X/Xst and  of and with the frequency ratio η and the damping ratio ξ are
shown in fig 3.
Fig 3: Variation of X/Xst and
 with frequency ratio η.
Let’s try to understand these equations by doing a few simple simulations, go to next tab
procedure to find out how to run the simulation to EXPLORE (expR) and to EXPERIMENT
(expT). A talking tutorial or a self-running demo with narration can be seen at EXPLAIN
(expN)
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