Vibration absorption
EXPERIMENTAL METHODOLOGY
Vibration Absorber
1.1 Vibration absorber undamped
Vibration absorber is a 2DOF system, intelligently designed to absorb the energy of main system to a secondary system. It has some major application in reducing the vibration response at machine foundation. To reduce the vibration of foundation some tuned system is generally attached to the machine to absorb most of the energy.
The equation of motion of the undamped MDOF system is
Now in order to solve the eq-(1.1) assume
So, after taking the differentiation with respect to time eq-(1.2) yields
So, Eq-(1.1) can be rewritten as
Last row of eq-(1.5) gives
Where
First row of eq-(1.5)
So, kinetic energy in mass-1
So, kinetic energy in mass-2
So, energy distribution ratio RD
The energy distribution for mass-1 and mass-2 is shown in Figure 1and Figure 2 respectively.
According to the Figure 1 if the forcing frequency matched with the frequency of the inner system then the main system will not feel any energy. In other word it can be stated that system1 will be insulated or isolated from the force field. Actually the energy is attracted by the system2 for this frequency band as shown in Figure 2.
According to the Figure 1 insulated frequency band is proportional with the mass ratio. If the mass of the insulator (system-2) is high then it can attract more energy to it.
1.2 Vibration absorber damped
Vibration absorber is a 2DOF system, intelligently designed to absorb the energy of main system to a secondary system. It has some major application in reducing the vibration response at machine foundation. To reduce the vibration of foundation some tuned system is generally attached to the machine to absorb most of the energy.
The equation of motion of the damped MDOF system is
Now in order to solve the eq- Error! Reference source not found. assume
So, after taking the differentiation with respect to time eq- Error! Reference source not found. yields
So, Eq- Error! Reference source not found. can be rewritten as
Last row of eq- Error! Reference source not found. gives
where
First row of eq- Error! Reference source not found.
So, the dynamic amplification factor
So, kinetic energy in mass-1
So, kinetic energy in mass-2
Because,
So, energy distribution ratio RD
Example
Determine the frequency, modal mass, modal participation factor and effective mass of first 2
mode of the following system.
The frequency of the system is det ( K - ω2M) = 0
Assume the frequency of each spring mass system is and ratio of two springs is
So,