CMVA – Vibration Math

This formula sheet will be provided with the CMVA exam for Cat 2 and 3 but there is no guarantee that the exam will have related questions. Instead, make sure you master the stated performance objectives for the category you are writing, as well as for the lower categories. English units. acceleration: inches per second per second, written General Note: All the math formulae are written in 2 the standard mathematical way which is ins/sec/sec or ins/sec . independent of units as long as consistency is velocity: inches per second, written ins/sec or IPS. maintained. All amplitudes are therefore zero to displacement: inches, written ins. peak, including displacement. When applying these formulae with English units, use inches for length, Metric units. acceleration: millimetres per second per second, pounds-force (lbf) for force and seconds (sec) for 2 time, with “g” in ins/sec/sec. (ins/sec2). written mm/s/s or mm/s . For metric units, although S.I. strictly applied uses velocity: millimetres per second, written mm/s. metres (m) for length, for this subject millimetres displacement: millimetres, written mm. (mm) are the convention and are used here in most places. Newtons (N) are the metric units for force, Acceleration from Displacement: 2 kilograms (kg) for mass and seconds (s) for time A = [ 2π f] D 2 with “g” in mm/s/s (mm/s ). where D is peak displacement . Mils, feet, microns (µm), and peak to peak unit English units for displacement: inches (ins) . conventions all have to be converted to the standard Metric units for displacement: millimetres (mm). units and conventions before doing any parameter conversions (e.g. displacement to velocity) and the Convert acceleration from standard units to reverse is true when you want your answer in these multiples of the acceleration due to gravity (g-s): units. See Page 3. A in dimensional units Frequency of vibration, f (Hz). A(units of g) = g 1 f = Hz English units: T A is in ins/sec2, where T = period of vibration in seconds. g = 386.1 ins/sec2 . Hz = Hertz = cycles per second = CPS = cps. Metric units: A is in mm/s2, Motion - Basic conversion formulae g = 9810 mm/s2. Velocity from Displacement: Conversion of displacement, velocity and V = 2π f D acceleration by manipulating formulae where V is peak velocity, D is peak displacement, V f is frequency in Hz. V = 2π f D D = 2fπ

Acceleration from Velocity: A A = 2π f V V = A = 2π f V 2fπ where A is peak acceleration , 2 A V is peak velocity and A = [] 2π f D D = f is frequency in Hz. []2fπ 2

Page 1 of 4 CMVA_Formula_Page_Ver_15.4.doc Prepared by Tony Taylor and Brian Howes, with input by Ron Eshleman, Bob Rogers and Bill Eckert. July 4th, 2005.

Spring Force

Fxx = Kx Lowest Resolvable Frequency LRF (also called Band Width but not recommended). where Fx is the force in a spring in the x direction produced by deflection x (from undeformed F condition) and Kx is the stiffness of the spring in the LRF = max x direction. N

English units Note the term Band-Width has been widely adopted Force: pounds force (lbf). by other disciplines with a quite different meaning. Spring stiffness: pounds force / in (lbf / in). Resolution Metric units In order to resolve two adjacent frequency Force: Newtons (N). components in most instances with Hanning, Spring stiffness: Newtons / mm (N/mm) or uniform (rectangular) or flat-top windows, the Newtons / m (N/m). following formula will give the measurement resolution for most situations: Damping Force F= Cv F dr Effective Resolution (Hz)= 2 ××max WF C = Damping coefficient (force per unit velocity): N vr = relative velocity. English units. where WF = Window Factor. Force: pounds force (lbf). WF = 1.5 for the Hanning window, 3.8 for the flat C is in pounds force - seconds per inch (lbf sec/in). top window and 1.0 for the Uniform (rectangular) vr is in ins/sec. window (Refs. 1, 2). Metric units. Force: Newtons (N). Amplitude in deciBels (dB). C is in Newton - seconds per millimetre (Ns/mm) . vr is in mm/s.  V  Amp (dB) = 20 Log   V Data Acquisition Time T for a single continuous  r  sample (secs). V = voltage (or other measure) N V r = reference value of voltage or other measure T = having same units as numerator. F max Relative Amplitude in deciBels V2 , V1. where N is number of spectral lines and Fmax = VV21 V 2 frequency span in Hz. 20Log  − 20Log = 20Log dB VV V  rr 1 Total Data Acquisition Time Ttotal for multiple sample average with overlap. Dynamic Range Dynamic range is the difference between the NP% highest (VH) and lowest (VL) level signals Ttotal = 1+ ()n s -1 × 1- measurable in terms of the voltage ratio, usually F 100 max  expressed in deciBels (dB). where ns = number of samples in average, VH P% is percentage overlap. Dynamic Range = 20 Log  dB VL

Page 2 of 4 CMVA_Formula_Page_Ver_15.4.doc Prepared by Tony Taylor and Brian Howes, with input by Ron Eshleman, Bob Rogers and Bill Eckert. July 4th, 2005.

Undamped Natural Frequency

Ball spin frequency: 1K fn = 2Mπ 2 P RPM  B f = undamped natural frequency in Hz of a single BSF = 1cos−Φ2 n   degree of freedom system or of a principal mode of 260BP  a system. English units: where: K = stiffness, lbf/in, RPM = machine speed revolutions per minute M = mass = w /g ; w = weight in lbf. Ф = contact angle P = pitch diameter g = gravitational constant 386.1 in/sec2. n = number of rolling elements Metric units B = ball or roller diameter k = stiffness in Newtons/metre (N/m). M = mass in kg. Roll rotational speed

The speed (frequency of rotation) fr of a roll:

Machine Calculations V Magnetic frequency (F ) = 2 times line frequency fr = Hz m π D divided by number of poles (or line frequency divided by number of pairs of poles). where V = web velocity. D = roll diameter. Note units must be consistent.

Induction Motor Slip frequency (Fs) = Fm - Fr Pulley Speed where Fr = rotational speed of rotor. Df11 = Df 2 2 Gearboxes (simple speed increaser or reducer). where: Gearmesh frequency (GM) is given by: D 1 = diameter of drive pulley. f 1 = speed of drive pulley, rpm or Hz

GM = F11 N= F 2 N 2 D 2 = diameter of driven pulley where: f 2 = speed of driven pulley, rpm or Hz Note both speeds have to be in the same units. Input speed is F1, input number of teeth is N1.

Output speed is F2, output number of teeth is N2. Belt Frequency for a 2 pulley system. N1 F21=× F N2 2RRPM2RRPM×π ××11 ×××π 2 2 Bearing Calculations (Hz) fb = = Hz 60××ll 60 Fundamental train frequency (cage frequency):

where: f is the rotational speed of a drive belt and b 1cosRPM  B Φ  in the case of a defect on the belt is the frequency of FTF =− 1  260 P  impacts of the defect on each pulley,

Inner race ball pass frequency: R1 = radius of pulley 1, R = radius of pulley 2, nRPMB cos Φ  2 BPFI =+ 1  RPM1 = speed of pulley 1 RPM, 260 P    RPM2 = speed of pulley 2 RPM, Outer race ball pass frequency: l is the belt length.

n RPM  Bcos Φ  BPFO =− 1  260 P  Page 3 of 4 CMVA_Formula_Page_Ver_15.4.doc Prepared by Tony Taylor and Brian Howes, with input by Ron Eshleman, Bob Rogers and Bill Eckert. July 4th, 2005.

Force due to Unbalance The rotating force F due to unbalance is given by: F = Meω 2 where: 2Nπ ω = angular velocity (omega) = rads/sec 60 In English units: M = rotor mass = W/g. W = weight of rotor in lbf. e = rotor eccentricity in inches. g = gravitational constant, 386.1 in/sec2. N = machine speed in rpm. F is unbalance force in pounds force (lbf). In Metric units: M is rotor mass in kg. e = rotor eccentricity in metres. N = machine speed in rpm. F is unbalance force in Newtons.

Conversions To convert mm into inches divide mm by 25.4.

To convert microns (micrometers µm) to mils divide by 25.4.

To convert inches into mm multiply inches by 25.4.

To convert mils to microns (micrometers µm) multiply by 25.4.

Hz = CPM divided by 60 = CPS = cycles/second.

CPM = Hz times 60. where CPM = cycles / minute.

For a pure sine wave, Peak = RMS times the square root of two, RMS = Peak divided by the square root of two, Peak-to-peak = 2 times peak.

References: 1. J. Frarey An Examination of Signal Resolution in an FFT Analyzer, Vibrations Vol. 13 Number 2 June 1997. 2. CMVA Note by Tony Taylor Effective Resolution vs Lowest Resolvable Frequency.

Page 4 of 4 CMVA_Formula_Page_Ver_15.4.doc Prepared by Tony Taylor and Brian Howes, with input by Ron Eshleman, Bob Rogers and Bill Eckert. July 4th, 2005.