# Engineering

— Although a vibration isolator will provide some degree of shock isolation, and vice versa, the principles of isolation are different, and shock and vibration requirements should be analyzed separately. In practical situations, the most potentially troublesome environment, whether it be vibration or shock, generally dictates the design of the isolator. In other applications, where both are potentially troublesome, a compromise solution is possible.

Before a selection of a vibration and/or shock/isolator can be made, the engineer should have a basic understanding of the following definitions, symbols, and terms:

**Vibration**

A magnitude (force,displacement, or acceleration) which oscillates about some specified reference where the magnitude of the force, displacement, or acceleration is alternately smaller and greater than the reference. Vibration is commonly expressed in terms of frequency (cycles per second or Hz) and amplitude, which is the magnitude of the force, displacement, or acceleration. The relationship of these terms is illustrated in Figure 3 below.

**Frequency**

Frequency may be defined as the number of complete cycles of oscillations which occur per unit of time.

**Period**

The time required to complete one cycle of vibration.

**Forcing Frequency**

Defined as the number of oscillations per unit time of an external force or displacement applied to a system.

**Forcing Frequency** **= ****f**_{d}

**Natural Frequency**

Natural frequency may be defined as the number of oscillations that a system will carry out in unit time if displaced from its equilibrium position and allowed to vibrate freely. (See figure 3 above).

Eq.1

Eq. 2

Eq. 3

Natural frequency in terms of static deflection:

Eq. 4

Also, natural frequency for torsional vibration:

Eq. 5

Equations 1 through 5 all neglect the effects of damping. When damping is considered, Equation 2 becomes:

**Amplitude**

The amplitude of a harmonic vibration such as displacement, velocity, or acceleration is the zero to peak value corresponding to the maximum magnitude of a harmonic vibration time-history. (See Figure 3 above).

**Displacement**

Displacement is a vector quantity that specifies the change of the position of a body or particle and is usually measured from the mean position or equilibrium position. In general it can be represented by a translation of rotation vector or both. (See Figure 3).

**Velocity**

Velocity is a vector that specifies the time rate change of displacement with respect to a frame of reference.

**Acceleration**

Acceleration is a vector that specifies the time rate of change of velocity with respect to a frame of reference. The acceleration produced by the force of gravity, which varies with the latitude and elevation of the point of observation is given by g = 980.665 cm/sec^{2} = 386.093 in/sec^{2} = 32.1739 ft/sec^{2}, which has been chosen as a standard acceleration due to gravity.

**Deflection**

Deflection is defined as the distance a body or spring will move when subjected to a static or dynamic force, F.

**Spring Stiffness**

Described as a constant which is the ratio of a force increment to a corresponding deflection increment of the spring.

Eq, 7

Rational spring stiffness:

Eq. 8

**Elastic Center**

The elastic center is defined as a single point at which the stiffness of an isolator or system isolators can be represented by a single stiffness value.

**Damping**

Damping is the phenomenon by which energy is dissipated in a vibratory system. Three types of damping generally encountered are: coulomb, hysteresis and viscous.

**Coulomb Damping**

If the damping force in a vibratory system is constant and independent of the position of velocity of the system, the system is said to have coulomb or dry friction damping.

**Hysteresis (Inherent) Damping**

Damping which results from the molecular structure of a material when that material is subjected to motion is referred to as hysteresis damping. Elastomers are good examples of materials which posses this type of damping.

**Viscous Damping**

If any particle in a vibrating body encounters a force which has a magnitude proportional to the magnitude of the velocity of the particle in a direction opposite to the direction of the velocity of the particle, the particle is said to be viscously damped. This is the easiest type of damping to model mathematically. All of the equations in this text are based on use of a viscous damping coefficient. Although most isolators do not use viscous damping, equivalent viscous damping usually yields excellent results when modeling systems.

**Damping Coefficient**

Damping for a material is expressed by its damping coefficient.

**Critical Damping**

A system is said to be critically damped when it is displaced from its static position and most quickly returns to this initial static position without any over-oscillation. The damping coefficient required for critical damping can be calculated using:

Eq. 9

**Damping Factor**

The non-dimensionless ratio which defines the amount of damping in a system.

**Resonance**

When the forcing frequency coincides with the natural frequency of a suspension system, this condition is known as resonance.

**Transmissibility**

Defined as the ratio of the dynamic output to the dynamic input.

Eq. 10

For negligible damping (C/C_{c}=0), T becomes:

When resonance occurs, and T is at its max and Equation 10 becomes:

**Shock**

Defined as a motion in which there is a sharp, nearly sudden change in velocity. Examples of this are a hammer blow on an anvil or a package falling to the ground. Shock may be expressed mathematically as a motion in which the velocity changes very suddenly.

**Shock Pulse**

Shock Pulse is a primary disturbance characterized by a rise and decay of acceleration from a constant value in a very sort period of time. Shock pulses are normally displayed graphically as acceleration vs. time curves.

**Shock Transmission**

Shock transmitted to the object subjected to the shock. This can be calculated with the following equation:

Eq. 13

In this equation, V represents an instantaneous velocity shock. Most shock inputs can be approximated by an instantaneous velocity shock.

The associated dynamic linear deflection of an isolator under shock can be determined by the use of the following equation:

Eq. 14