Design Curves

 

(a) Transmissibility vs. damping ratio

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 :

 

(7)

 

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 System

 

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. natural frequency

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 :

(8)

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.

 

EXAMPLE 1

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 :