(a) Transmissibility vs.
The curve in Figure 3 demonstrates the
effectiveness of an isolator to reduce vibration. Figure 3 also indicates a number of
important concepts: (i) isolators should be chosen so as not to excite the natural
frequencies of the system; (ii) damping is important in the range of resonance whether the
dynamic system is operating near resonance or must pass through resonance during start-up;
(iii) in the isolation region, the larger the ratio (i.e., the smaller the value of ), the smaller the transmissibility will be.
(b) Isolation efficiency vs. w and
Another graphical method of
illustrating the regions of isolation and amplitude as a function of the disturbing
frequency and the natural frequency of the system is shown in Figure 4. In using this
figure we must note that percent isolation is defined by the expression :
Figure 3 Design Curves
for the Transmissibility vs. the Frequency ratio as a Function of the Damping Ratio z for a Linear Single-Degree-of-Freedom
The forcing frequency on the ordinate
and the percent isolation lines in the graph locate a point, the abscissa of which is the
natural frequency of the system necessary to achieve the required isolation. The system
parameters may then be selected or adjusted to obtain this desired natural frequency.
(c) Static deflection vs.
The static deflection is the
deflection of an isolator that occurs due to the dead weight load of the mounted
equipment. Since the static deflection is given by the expression , and since the undamped natural
frequency of a single-degree-of-freedom system is determined by the equation :
where is in centimeters.
Figure 4 Design Curves for Isolation
Efficiency vs. Frequency
(Damping Ratio, z = 0)
The graphical presentation of this
equation is given in Figure 5. Thus, we can determine the natural frequency of a system by
measuring the static deflection. This statement is correct provided that the spring is
linear and that the isolator material possesses the same type of elasticity under both
static and dynamic conditions. As mentioned previously, however, we are assuming a
single-degree-of-freedom linear system throughout our analysis, and thus all the design
curves presented above are applicable.
Figure 5 Design Curves for the
Static Deflection vs. Natural Frequency for a Linear Single-Degree-of-Freedom System
The examples that follow
demonstrate the use of these design curves.
A pump in an industrial plant is
mounted rigidly to a massive base plate. The base plate rests on four springs, one at each
corner. If the static deflection of each spring is two centimeters, then the natural
frequency of the system is given by :