Mathematical relationships

Simple harmonic motion

The three parameters described above are mathematically connected in the case of a simple harmonic or sinusoidal vibra­tion such as that produced by out-of-balance.

The displacement “e” is proportional to sin cot where cot is an an­gle which goes through 360° in one vibratory cycle. Angular ve-

Locity (or circular frequency) co is equal to 2uf where f is the fre — 27iN

Quency in Hertz, or for balance problems where N

Equals r/min.

The other properties are also sine waves, the velocity “v” hav­ing a 90° phase lead (one quarter or a cycle with respect to time) whilst acceleration “a” is advanced by half a cycle i. e. a 180° phase lead. This is shown in the equations below:

Displacement e = sin rat

Velocity v = coe^ sin^rat +

Acceleration a = o)2epeaksin(ojt + 71)

These three parameters are illustrated in Figure 15.1 whist Ta­ble 15.1 gives their values with respect to epeak.

3 Time Cycle Angle

Figure 15.1 Sinusoidal vibration Point in cycle Dis­ Placement Velocity Acceleration No. Radians Degrees 1 0 0 0 0 2 0.25ft 45 1 71 xepaak °-71 »««W -0.71 x o>2 eosak 3 0.5ji 90 ®Deak 0 -“2 e»,k 4 0.75ji 135 °-71 XЂW -0.71 x -0.71 x <o2 eD, ak 5 Re 180 0 •“«Deak 0 6 1.25ft 225 -°-71xepeak -0.71 x coenp3k 0)2 e™,,k 7 1.5ft 270 "®peak 0 -0.71 x<o2e„mk 8 1.75ft 315 -0.71 x eDeak °-71 xtoeD„k ■O’71 x “2 e„„„k 9 2ti 360 0 0

Table 15.1 Values of parameters expressed as function of peak displacement

Which vibration level to measure

It will be seen that all these quantities vary with time. For analyt­ical purposes it is desirable to reduce them to single figures and those for displacement are shown in Figure 15.2.

Figure 15.2 Relationship between various vibration levels

The peak-to-peak value indicates the total excursion of the wave and is useful in calculating maximum stress values or de­termining mechanical clearances.

The root-mean-square value is probably the most important measure because it takes account of the cycle time and gives an amplitude value which is directly related to the energy con­tent and therefore the destructive capabilities of the vibration.

For sine wave vibrations e. g. out of balanceerms x V2 =6^.

Peak and average values may also be calculated but have a limited value.

Eav = ^;J|e| dt 0

Velocities and accelerations are given in similar terms and the root-mean-square velocity is especially important as it is used in ISO 2954-1975 as the measure of vibration severity in the range 600 to 12000 r/min (10 to 200 Hz).

Again for a sinusoidal vibration:

^mas xV2 — Vpea^.

It must be emphasized that the relationships connecting root-mean-square and peak values only apply to sine waves. Vibrations arising from certain other sources e. g. rough rolling element bearings or air turbulence may not follow this form. Consequently the equivalents in Table 15.1 will not hold and the acceleration values especially may be much higher.

Where sine wave conditions do exist, by taking time-average measurements the effects of phase may be ignored and:

Displacement e = —= f vdt 4ji f 2rtf J

Velocity v = — = f adt

27tf J

Acceleration a = 2nfv

The values of e, v or a may be either root-mean-square or peak as applicable.

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