T1-2 1.0    FUNDAMENTALS OF VIBRATION AND SHOCK 1.1    What Is Vibration? Mechanical vibration is a form of oscillatory motion. It occurs in all forms of machinery and equipment. It is what you feel when you put your hand on the hood of a car, the engine of which is running, or on the base of an electric motor when the motor is running. Perhaps the simplest illustration of a mechanical vibration is a vertical spring loaded with weight (W), as shown in Figure 1. In this position, the deflection of the spring from its free state is just sufficient to counterbalance the weight W. This deflection is called the STATIC DEFLECTION of the spring. The position in which the spring is at rest is No. 1. The spring is then slowly extended to position No. 2 and released. The elastic force moves the block W upward, accelerating up to the mean position and then decelerating moving further up. The uppermost position of the weight (position No. 3) is at the same distance from position No. 1 as position No. 2, but in the opposite direction. The subsequent motion of the weight as a function of time, if there is only negligible resistance to the motion, is repetitive and wavy if plotted on a time scale as shown by line 1 in the graph. This simple model exhibits many of the basic characteristics of mechanical vibrations. The maximum displacement from the rest or mean position is called the AMPLITUDE of the vibration. The vibratory motion repeats itself at regular intervals (A₁, A₂, A₃). The interval of time within which the motion sequence repeats itself is called a CYCLE or PERIOD. The number of cycles executed in a unit time (for example, during one second or during one minute), is known as the FREQUENCY. The UNITS OF FREQUENCY are 1 cycle/sec or 1 Hertz (Hz) which is standard. However, "cycles per minute" (cpm) are also used, especially for isolation of objects with rotating components (rotors) which often produce one excitation cycle per revolution which can be conveniently measured in cpm. When, as in Figure 1, the spring-weight system is not driven by an outside source, the vibration is a FREE VIBRATION and the frequency is called the NATURAL FRE- QUENCY of the system, since it is determined only by its parameters (stiffness of the spring and weight of the block). In general, vibratory motion may or may not be repetitive and its outline as a function of time may be simple or complex. Typical vibrations, which are repetitive and continuous, are those of the base or housing of an electric motor, a household fan, a vacuum cleaner, and a sewing machine, for example. Vibrations of short duration and variable intensity are frequently initiated by a sudden impulsive (shock) load; for example, rocket upon takeoff, equipment subject to impact and drop tests, a package falling from a height, or bouncing of a freight car. In many machines, the vibration is not part of its regular or intended operation and function, but rather it cannot be avoided. Vibration isolation is one of the ways to control this un- wanted vibration so that its adverse effects are kept within acceptable limits. 1.1.1    Damping The vibratory motion as a function of time as shown in Figure 1 (line 1) does not change or fade. The elastic (potential) energy of the spring transforms into motion (kinetic) energy of the massive block and back into potential energy of the spring, and so on. In reality, there are always some losses of the energy (usually, into thermal energy) due to friction, imperfections of the spring mate- rial, etc. As a result, the total energy supporting the vibratory motion in the sys- tem is gradually decreasing (dissipated), thus diminishing the intensity (ampli- tude) of the spring excursions, as shown by line 2 in Figure 1 ("decaying vibra- tion"). This phenomenon is called  DAMPING, and energy-dissipating compo- nents are called DAMPERS, Figure 2. The rate of decay of amplitude in a sys- tem with damping is often characterized by LOGARITHMIC (or LOG) DECRE- MENT 3 defined as 3 = log (A_n/A_n-1),                                                                                        (1) W W W x W = Weight Position No. 1; spring at rest (mean position) Weight, W in position No. 2 spring extended Position No. 3 spring contracted Position of weight (x) Amplitude Line 1 Line 2 1 Cycle A₀ A₁ A₂ A_d1 A_d2 A_d3 A₃ Figure 1    Free Vibrations of a Simple Vibratory System Figure 2    Simple Vibratory      System with                   Damping W x c k Object Spring Base Damping Element