Many drive system problems can be traced to mechanical vibrations. But, a basic understanding of vibration principles and the benefits of elastomeric couplings can help you avoid many of the unwanted effects of vibration.

### Noise and vibration sources

Mechanical vibrations occur in both rotating and reciprocating equipment. In rotating equipment, torsional pulsations initiated by either the driving or driven equipment are the predominant cause of vibration.

Among types of driving equipment, four-stroke engines generate pulsations equal to the number of cylinders times half the engine speed, and two-stroke engines generate pulsations equal to the number of cylinders times the engine speed. Even seemingly smooth-running electric motors generate torsional pulsations that are a function of the number of stator poles and the motor operating speed.

Driven equipment can also generate torsional vibrations, as in the case of a horizontal mixer where the paddle loading varies as it rotates through the mixing medium. Here, the vibration frequency equals the number of blades times the rotational speed.

Abrupt starts and stops can create vibration in the form of damaging shock waves. Shock is a high-intensity noncyclic vibration.

Where vibrations generate audible signals, the result is called structure-born noise. This noise usually originates from the relative motion of structural components or from sound waves generated by high-frequency vibration in structural members. And, it occurs most often in flat plates and equipment enclosures.

If mechanical vibrations areunchecked, they can cause human discomfort and premature wear. In the mixer example, if the cyclic loading is transmitted back into the driving gearbox, resonances may occur within the gearbox, which can reduce the service life of drive components. Vibrations can also be amplified by the surrounding structure, causing fatigue failure of support members.

### Control methods

There are two basic ways to control noise and vibration:

**Minimize them.** The primary method of reducing noise and vibration is to minimize their generation. This can be accomplished by:

• Balancing rotating components.

• Reducing the mass of reciprocating members.

• Loading the driven device as uniformly as possible.

• Incorporating a soft-start device to minimize start-up shock.

**Accommodate them.** Once vibrations are minimized at the source, the next step is to reduce their harmful effects. Here, the key issues are system design and drive coupling selection. Vibrations can be either amplified or isolated depending on the mass and stiffness characteristics of the system. The main parameter to control is the relationship between natural and disturbing frequencies of the system.

*Natural frequency* is the rate at which a spring-mass system vibrates in an unrestrained state. All systems exhibit a natural frequency depending on their stiffness and inertia characteristics. Vibration frequency is usually expressed in cycles per second or Hertz (1 cps = 1 Hz). For a rotational system the natural frequency is:

*Fn* = (*1/2π*)(*K/J*)^{1/2}

where:*J* = Mass moment of inertia of driven equipment, in.-lb-sec^{2}/rad * K* = Torsional stiffness of the system, in.- lb/rad

*Disturbing frequency* is the rate at which torsional pulsations are applied to the system. They can be either steady state, such as firing pulses created by a diesel engine, or transient, as created by an abrupt change in shaft speed.

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A spring-mass system, *Figure 1*, illustrates the desired relationship between the natural and disturbing frequencies.

With an accelerometer attached to the mass and another to the base, the ratio of the mass-to-base acceleration readings (referred to as the transmissibility) at different frequencies appears as shown in *Figure 2*.

This curve shows that if the system operates at or near its natural frequency, *Fn*, the amount of transmitted vibration increases dramatically. If a machine is operated in this region, catastrophic failure is likely to occur even if the exposure is only for a short time. As the disturbing frequency increases above the natural frequency, vibration decreases. Eventually the system enters a region where the transmitted vibrations become less than the input vibrations. This is called the isolation region. All systems should be designed to operate in the isolation region. This is true for both linear systems, *Figure 1*, and rotational systems.

### Role of elastomeric couplings

Drive shaft couplings with flexible elastomeric elements are commonly used to reduce the transmission of noise and vibration as well as to accommodate shaft misalignments.

These elastomeric couplings allow relative motion between the driving and driven shafts. When a torsional pulse is transmitted to the coupling, it is absorbed and momentarily stored as potential energy. Then, the energy is released to the driven shaft over a period of time corresponding to the system’s natural frequency. In a well-designed system, the natural frequency is lower than the disturbing frequency by at least 30%. Under these conditions, it takes more time to release the energy than to absorb it. Because the same amount of energy is transmitted over a longer period of time, the shock pulse is reduced, producing a smoother-running system.

Several types of elastomeric couplings are available, each suited to different operating conditions. For example, the tire type coupling, *Figure 3*, is commonly used for vibration control because of its high deflection capability and low torsional stiffness.

### Selecting a coupling to reduce vibration

Here’s a simplified method for selecting an elastomeric coupling to minimize the transmission of noise and vibration.

**1. Determine the disturbing frequencies.** For a gasoline or diesel engine, the disturbing frequency is obtained from:

Four-stroke engine: *Fd* = (*N* x *S*)/ 2 x 60 Two-stroke engine: *Fd* = (*N* x *S*)/60

where: * Fd* = Disturbing frequency, Hz

*N*= Number of cylinders.

*S*= Speed, rpm. For electric motors, the disturbing frequency is:

*Fd* = (*N* x *S*)/60

where:*N* = Number of stator poles.*S* = Speed, rpm.

To protect against start-up shock, measure the time (in seconds) required for the system to come up to speed. This is the shock pulse duration, T. Convert this into a disturbing frequency by using the formula *Fd* = 1/2 *T*.

For torsional disturbances created by the driven device, determine the disturbing frequency by estimating the number of load applications per revolution and multiplying it by the shaft speed.

**2. Determine the system inertia.** To calculate inertia, divide the driven mass into a series of simplified geometric sections. Then, use standard formulas (from *PTD Guide to PT Products* or standard engineering textbooks) to estimate the inertia for each section. The total inertia equals the sum of the individual section inertias. For circular discs or cylinders, the mass moment of inertia is:

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*J*= (*π* x *d*^{4} x *L* x *D*)/32g

where: * J* = Mass monent of inertia, in.-lb-sec

^{2}/rad

*= Outer diameter of cylinder, in.*

d

d

*= Length of cylinder, in.*

L

L

*= Material density, lb/in.*

D

D

^{3}

*g*= Acceleration due to gravity, 386.4 in./sec

^{2}

**3. Determine the maximum allowable natural frequency, Fn.** Use the following formula to determine the maximum allowable system natural frequency. Keeping the natural frequency below the disturbing frequency in accordance with this formula prevents the amplification of torsional vibrations and reduces the transmission of both vibration and noise.

*Fn* = *Fd*/1.414

**4. Determine the required coupling spring rate.** From the following formula, determine the required coupling spring rate (stiffness) from:

*K* = *J* x (2 π x *Fn*)^{2} in.-lb/rad

where *K* can be converted to in.-lb/deg by dividing it by 57.3.

**5. Select the appropriate coupling.** Once the required spring rate has been determined, select a coupling with the required torque, speed, and misalignment capability. A properly selected elastomeric coupling can reduce transmitted torsional vibrations by up to 90%.

A concept that is often misunderstood, even among experienced engineers, is the relationship between damping and isolation. • Damping is the dissipation of vibrational energy through the conversion of mechanical energy into thermal energy. The figure shows the effect of adding damping to a spring mass system. In many cases, it is impossible to add enough damping to reduce peak transmitted vibrations to acceptable levels. Therefore, the system engineer must ensure that the natural frequency of the system is sufficiently below the disturbing frequency so the transmitted vibrations are minimized. |