1
Lecture 17: Vibration Absorbers
Note: This lecture note is a condensed version of section 5.3 of the prescribed text
“Engineering Vibration” by Inman and resources from the internet.
Vibration Absorbers:
For a starter, please see the following youtube video on an industrial vibration absorber –
https://www.youtube.com/watch?v=uDu1sdGX2vw.
As shown in the adjoining figure,
another approach in protecting a device
from
steady-state
harmonic
disturbance at a constant frequency is a
vibration absorber.
Unlike a vibration isolator, an absorber
consists of a second spring-mass combination added to the primary device to protect it
from vibrating. The major effect of adding the second spring-mass system is to change
from a single degree of freedom to a two-degree-of-freedom system. The new system has
two natural frequencies. The added spring-mass system is called the absorber. The values
of the absorber mass and stiffness are chosen such that the motion of the original mass is
a minimum. This is accompanied by substantial motion of the added absorber system.
For the given system above, the equations of motion are given as –
(
𝑚
0
𝑘 + 𝑘𝑎
0
𝑥̈
)( ) + (
𝑚𝑎 𝑥̈ 𝑎
−𝑘𝑎
−𝑘𝑎 𝑥
𝐹 sin(ωt)
) (𝑥 ) = ( 0
)---------------------------------(1),
𝑘𝑎
𝑎
0
where x = x(t) is the displacement of the table modelled as having mass m and stiffness k,
xa is the displacement of the absorber mass (of mass ma and stiffness ka), and the harmonic
force F0 sin(ωt) is the disturbance applied to the table mass.
It is desired to design the absorber (ie., choose ma and ka) such that the displacement of
the primary system is small as possible in steady state. Here it desired to reduce the
vibration of the table, which is the primary mass.
2
Fine-tuning the absorber spring-mass system:
At the steady state if the displacements are given as 𝑥(𝑡) = 𝑋 sin 𝜔𝑡 and 𝑥𝑎 (𝑡) =
𝑋𝑎 sin 𝜔𝑡, the from (1), we have –
𝑋 = (𝑘+𝑘
(𝑘𝑎 −𝑚𝑎 𝜔2 )𝐹0
2
2
2
𝑎 −𝑚𝜔 )(𝑘𝑎 −𝑚𝑎 𝜔 )−𝑘𝑎
-------------------------------------------------------------(2) and
𝑘𝑎 𝐹0
2
2
2
𝑎 −𝑚𝜔 )(𝑘𝑎 −𝑚𝜔 )−𝑘𝑎
and 𝑋𝑎 = (𝑘+𝑘
From (2) if we choose 𝜔2 =
------------------------------------------------------------(3).
𝑘𝑎
𝑚𝑎
, X is exactly zero. In other words, if the absorber
parameters are chosen to satisfy the tuning conditions, the steady-state motion of the
primary mass is zero (ie., X = 0). In this case the steady state motion of the absorber mass
𝐹
is given as 𝑥𝑎 (𝑡) = − 0 sin 𝜔𝑡.
𝑘𝑎
Fine-tuning the force on the primary mass:
Little deviation from the driving frequency can disturb the fine-tuning of the absorber
system and if near resonance frequency, the force on the primary mass can increase.
Define the mass ratio 𝜇 =
𝑚𝑎
𝑚
, the original natural frequency of the primary system
without the absorber attached as 𝜔𝑝 = √
𝑘
𝑚
, and the natural frequency of the absorber
𝑘
system before it is attached to the primary system as 𝜔𝑎 = √ 𝑎 . With these definitions
𝑚𝑎
𝑘
we can write, 𝑎 = 𝜇
𝑘
2
𝜔𝑎
2
𝜔𝑝
= 𝜇𝛽 2 , where 𝛽 =
𝜔𝑎
𝜔𝑝
is the frequency ratio.
The normalized amplitude from (2) becomes –
𝜔2
𝑋𝑘
𝐹0
=
1− 2
𝜔
𝑎
2
𝜔𝑎
𝜔 2
𝜔 2
𝜔
(1+𝜇( ) −( ) )(1−( ) )−𝜇( 𝑎 )2
𝜔𝑝
𝜔𝑎
𝜔𝑎
𝜔𝑝
----------------------------------------------------(4).
3
As shown in the adjoining figure the two
natural frequencies of vibration are at 0.781
and 1.281. The amplitude is very high near
resonance.
To minimize the amplitude with force the
ideal range would be between 0.908 and
1.118.