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External disturbance can be force or motion, and depending on that, force isolation (related to force transmissibility) or motion isolation (related to motion transmissibility) would be applicable in the design of the isolator. Luckily, the design is quite similar for the two situations.

In force isolation, vibration forces that would be ordinarily transmitted directly from a source to a supporting structure (isolated system) are filtered out by an isolator through its flexibility (spring) and dissipation (damping) so that part of the force is routed through an inertial path. In motion isolation, vibration motions that are applied to a system (e.g., vehicle) by a moving platform are absorbed by an isolator through its flexibility and dissipation so that the motion that is transmitted to the system of interest is weakened. The design problem in both cases is to select applicable parameters for the isolator so that the vibrations entering the system of interest are below the specified values within a fre-quency band of interest (the operating frefre-quency range). This design problem is essentially a situation of mechanical impedance matching because impedance parameters (mechanical) of the isolator are chosen depending on the impedance parameters of the isolator.

Note: As indicated before, generalized impedance (across variable/through variable) corresponds to mechanical mobility, which is the inverse of what is traditionally called mechanical impedance.

Force isolation and motion isolation are represented in Figure 2.8. In Figure 2.8a, vibration force at the source is f(t). In view of the isolator, the source (with mechanical impedance Z_m) is made to move at the same speed as the isolator (with mechanical impedance Z_s). This is a parallel connection of imped-ances. Hence, the force f(t) is split so that part of it is taken up by the inertial path (broken line) of Z_m and only the remainder (f_s) is transmitted through Z_s to the supporting structure, which is the isolated system. Force transmissibility is given by

T f

In Figure 2.8b, vibration motion v(t) of the source is applied through an isolator (with impedance Zs and mobility Ms) to the isolated system (with mechanical impedance Zm and mobility Mm). The resulting force is assumed to transmit directly from the isolator to the isolated system and hence, these two units are connected in series. Consequently, we have the motion transmissibility.

T v

It is noticed that, according to these two models, we have

Tf =Tm= (2.15)T

As a result, usually both types of isolators can be designed in the same manner using a common transmissibility function T.

Simple examples of force isolation and motion isolation are shown in Figure 2.8c and d. First we obtain the transmissibility (force transmissibility) function for system in Figure 2.8c. Then, in view of Equation 2.15, the motion transmissibility of the system in Figure 2.8d is equal to the same expression.

First consider the force transmissibility problem of Figure 2.8c, which is shown again in Figure 2.8e.

This may represent a simplified model of a machine tool and its supporting structure.

Clearly, the elements m, b, and f are in parallel, since they have a common velocity v across them.

Hence, its mechanical-impedance circuit is shown in Figure 2.8f. In this circuit, the impedances of the basic elements are Zm = mjω, Zb = b, and Zk = k/(jω) for mass (m), spring (k), and viscous damper (b), respectively (see Table 2.1). Substitute the element impedances into the force transmissibility expression (which is obtained by using the laws of element interconnection shown in Table 2.2 and the circuit in Figure 2.8f):

FIGURE 2.8 (a) Force isolation, (b) motion isolation, (c) force isolation example, (d) motion isolation example, (e) a simplified model of a machine tool and its supporting structure, and (f) mechanical impedance circuit of the force isolation problem.

The last expression is obtained by dividing the numerator and the denominator by m. Now use the fact that k m/ = wⁿ² and b/m = 2ζωn (or, wⁿ= k m/ = undamped natural frequency of the system; z =b/(2 km)= damping ratio of the system) and divide (2.17) throughout by wn2. We get,

T j

where the nondimensional excitation frequency is defined as r = ω/ωn.

The transmissibility function has a phase angle as well as magnitude. In practical applications of vibration isolation, it is the level of attenuation of the vibration excitation that is of primary importance, rather than the phase difference between the vibration excitation and the response. Accordingly, the transmissibility magnitude

To determine the peak point of |T_f| differentiate the expression within the square root sign in (2.20) and equate to zero:

The root r = 0 corresponds to the initial stationary point at zero frequency. That does not represent a peak. Taking only the positive root for r² and then its positive square root, the peak point of the trans-missibility magnitude is given by

r = TABLE 2.1 Mechanical Impedance and Mobility of Basic Mechanical Elements

Element Time-Domain Model Impedance Mobility (Generalized Impedance)

Mass, m mdv

TABLE 2.2 Interconnection Laws for Mechanical Impedance (Z) and Mobility (M)

1 1 1

1 2

Z =Z +Z 1 1 1

1 2

M=M +M

For small ζ, Taylor series expansion gives 1 8+ z² » +1 (1 2/)´8z²= +1 4z². With this approximation, (2.20) evaluates to 1. Hence, for small damping, the transmissibility magnitude will have a peak at r = 1 and, from Equation 2.19, its value is

Tf » 1 4+ » + ´ exact expression (2.19), and can be generated using the following MATLAB^® program:

clear;

From the transmissibility curves in Figure 2.9a we observe the following:

1. There is always a nonzero frequency value at which the transmissibility magnitude will peak. This is the resonance.

2. For small ζ the peak transmissibility magnitude is obtained at approximately r = 1. As ζ increases, this peak point shifts to the left (i.e., a lower value for peak frequency).

3. The peak magnitude decreases as ζ increases.

4. All the transmissibility curves pass through the magnitude value 1.0 at the same frequency r = 2.

5. The isolation (i.e., |Tf| < 1) is given by r > 2. In this region, |Tf| increases with ζ.

6. In the isolation region, the transmissibility magnitude decreases as r increases.

As two particular situations, from the transmissibility curves we observe the following:

For |Tf| < 1.05; r > 2 for all ζ

For |Tf| < 0.5; r > [1.73, 1.964, 2.871, 3.77, 7.075] for ζ = [0.0, 0.3, 0.7, 1.0, 2.0], respectively

Next, suppose that the device in Figure 2.8e has a primary, undamped natural frequency of 6 Hz and a damping ratio of 0.2. Suppose, it is required that for proper operation, the system achieves a force transmissibility magnitude of less than 0.5 for operating frequency values greater than 12 Hz. We need

1 4 If the requirement was not met (say, if ζ = 0.4), an option would be to reduce damping.

In design problems of vibration isolator, what is normally specified is the percentage isolation, as given by,

I = [1 − |T|] × 100% (2.22)

For the result in Equations 2.19 and 2.3.1 this corresponds to

I r

The isolation curves given by Equation 2.23 are plotted in Figure 2.9b. These curves are useful in the design of vibration isolators.

Normalized frequency r = ω/ω_n (a)

Transmissibility magnitude |Tf| ζ = 0.0

ζ = 0.3

FIGURE 2.9 (a) Transmissibility curves for a simple oscillator model and (b) curves of vibration isolation.

Note: Model in Figure 2.8 is not restricted to sinusoidal vibrations. Any general vibration excitation may be represented by a Fourier spectrum, which is a function of frequency ω. Then, the response vibration spectrum is obtained by multiplying the excitation spectrum by the transmissibility function T. The associated design problem is to select the isolator impedance parameters k and b to meet the specifica-tions of isolation.

Example 2.3

A machine tool, sketched in Figure 2.10a, weighs 1000 kg and normally operates in the speed range of 300–1200 rpm. A set of spring mounts has to be placed beneath the base of the machine so as to achieve a vibration isolation level of at least 70%. A commercially available spring mount has the load–deflection characteristic shown in Figure 2.10b. It is recommended that an appro-priate number of these mounts be used, along with an inertia block, if necessary. The damping constant of each mount is 1.56 × 10³ N/ms. Design a vibration isolation system for the machine.

Specifically, decide upon the number of spring mounts that are needed and the mass of the iner-tia block that should be added.

Solution

First we assume zero damping (since, in practice, the level of damping in a system of this type is small), and design an isolator (spring mount and inertia block) for a level of isolation greater than the required 70%. Then we will check for the case of damped isolator to see whether the required 70% level isolation is achieved.

For the undamped case, Equation 2.19 becomes

T =r -1

2 1 (2.3.1)

Note: We have used the case r > 1 since the isolation region corresponds to r > 2.

Assume the conservative value I = 80% ⇒ |T| = 0.2.

Using Equation 2.3.1, we have

r T have to be placed here An inertia block

FIGURE 2.10 (a) A machine tool and (b) load–deflection characteristic of a spring mount.

The lowest operating speed (frequency) is the most significant one (because it corresponds to the lowest isolation, as clear from Figure 2.19a). Hence,

w=300´ p= p = p

60 2 10 rad/s, rad/s 10 rad/s From the load–deflection curve of a spring mount (Figure 2.10b),

Mount stiffness= N/m

Note that an inertia block of mass 216 kg has to be added.

Now we must check whether the required level of vibration would be achieved in the damped case.

Substitute in the damped isolator Equation 2.19:

T r

This corresponds to an isolation level of 73%, which is better than the required 70%.