Motion, vibration and speed sensors

Speed enters every pump performance result. Shaft power is torque times speed, and the affinity laws used to correct test results to rated speed scale flow with speed, head with its square and power with its cube, so a small speed error is magnified. Vibration and shaft motion are recorded in the mechanical run test, and position sensors tell the control system where the throttling valve stands. MEC-380 dealt with what vibration means and how to analyse it; this lesson is about the sensors that measure motion of all kinds: how they work, how good they are and how to install them so they tell the truth.

Measuring rotational speed

Most speed sensors count events per revolution. ST-301 is a magnetic pickup facing a steel wheel with 60 teeth on the coupling hub: each tooth passing the pickup changes the magnetic flux through its coil and induces a voltage pulse. With 60 teeth, the pulse frequency in hertz equals the speed in rpm, a convenient choice. Hall-effect and eddy-current pickups do the same job with output that does not fall at low speed, and optical sensors read a reflective mark or an encoder disc. The electronics then turn the pulse train into speed in one of two ways.

Frequency counting counts the pulses in a fixed gate time. Its resolution is one count in the gate: at 1,760 Hz, a 1 s gate resolves ±1 rpm, 0.06 percent, but a 0.1 s gate only ±10 rpm. Period measurement times the interval between pulses, or over a whole revolution, with a fast reference clock. Timing one revolution of the TP-80 with a 10 MHz clock gives about 341,000 counts and a resolution of 0.005 rpm, far finer than counting. Period measurement is preferred for fast, precise readings; averaging over a full revolution also cancels small errors in the spacing of the teeth, which would otherwise appear as a ripple at running speed. The overall accuracy of a speed measurement then depends on the reference clock, usually a quartz oscillator good to a few parts per million, and is excellent; ST-301's stated ±1 rpm is set by the readout, not the physics.

Owen checks ST-301 once a year with a handheld optical tachometer whose calibration traces to a frequency standard, and at every test against the torque transducer's own speed channel. The pump test standards set tight limits on speed uncertainty because of its effect on the corrected results; at the guarantee point the speed contributes only 3 percent of the efficiency variance in Lesson 3, but after correction to rated speed by the affinity laws its effect on head and power is larger, as Lesson 12 shows.

# Speed, position and vibration sensors on TL-3
import math
# 1. ST-301: 60-tooth wheel and magnetic pickup, so frequency in Hz equals speed in rpm
n_true, teeth = 1760.0, 60
f = n_true*teeth/60
print(f"Pulse frequency at {n_true:.0f} rpm: {f:.0f} Hz")
for gate in (0.1, 1.0, 10.0):                    # counting pulses in a fixed gate time: +/-1 count
    print(f"  counting over {gate:4.1f} s: resolution +/-{60/(teeth*gate):.2f} rpm ({100*60/(teeth*gate)/n_true:.3f} %)")
clk = 10e6                                       # period measurement: time one revolution (60 pulses) with a 10 MHz clock
T_rev = 60/n_true; counts = T_rev*clk
print(f"  timing one revolution with a 10 MHz clock: {counts:.0f} counts, resolution {100/counts:.5f} % = {n_true/counts:.4f} rpm")
# 2. Incremental encoder on the valve actuator test rig: 1,024 lines, quadrature decoding
ppr = 1024; cpr = 4*ppr
print(f"\nEncoder: {cpr} counts per revolution, {360/cpr:.4f} degrees per count")
# 3. Eddy-current proximity probe: 7.87 V/mm (200 mV/mil), gap set near the middle of the linear range
S_prox = -7.87                                    # V per mm, output becomes more negative as the gap grows
for gap in (0.50, 1.00, 1.50, 2.00):
    print(f"  gap {gap:.2f} mm -> about {S_prox*gap:6.2f} V")
x_pp = 0.040                                      # shaft relative vibration, mm peak to peak
print(f"  40 um peak-to-peak shaft motion gives {abs(S_prox)*x_pp*1000:.0f} mV peak to peak")
# 4. Piezoelectric accelerometer, 100 mV/g, and how its mounting limits the usable frequency
mounts = [('stud on a flat machined face', 25000.0), ('adhesive mounting pad', 15000.0), ('flat magnet on a flat face', 7000.0),
          ('two-pole magnet on a curved face', 2000.0), ('hand-held probe tip', 800.0)]
print("\n  mounting (mounted resonance approximate)     resonance   flat within 5 % up to (light damping)")
for name, fr in mounts:
    print(f"  {name:42s} {fr:7.0f} Hz      {0.22*fr:6.0f} Hz")
# Converting the TP-80 outboard bearing reading from acceleration to velocity at running speed
fr1, a_g = 29.33, 0.074
a = a_g*9.80665; v = a/(2*math.pi*fr1); d = v/(2*math.pi*fr1)
print(f"\nAt {fr1} Hz, {a_g} g peak = {a:.3f} m/s2 -> {v*1000:.2f} mm/s peak ({v*1000/math.sqrt(2):.2f} mm/s rms) -> {d*1e6:.1f} um peak")
print(f"Accelerometer output for that vibration: {a_g*100:.1f} mV peak")

Pulse frequency at 1760 rpm: 1760 Hz
  counting over  0.1 s: resolution +/-10.00 rpm (0.568 %)
  counting over  1.0 s: resolution +/-1.00 rpm (0.057 %)
  counting over 10.0 s: resolution +/-0.10 rpm (0.006 %)
  timing one revolution with a 10 MHz clock: 340909 counts, resolution 0.00029 % = 0.0052 rpm

Encoder: 4096 counts per revolution, 0.0879 degrees per count
  gap 0.50 mm -> about  -3.94 V
  gap 1.00 mm -> about  -7.87 V
  gap 1.50 mm -> about -11.80 V
  gap 2.00 mm -> about -15.74 V
  40 um peak-to-peak shaft motion gives 315 mV peak to peak

  mounting (mounted resonance approximate)     resonance   flat within 5 % up to (light damping)
  stud on a flat machined face                 25000 Hz        5500 Hz
  adhesive mounting pad                        15000 Hz        3300 Hz
  flat magnet on a flat face                    7000 Hz        1540 Hz
  two-pole magnet on a curved face              2000 Hz         440 Hz
  hand-held probe tip                            800 Hz         176 Hz

At 29.33 Hz, 0.074 g peak = 0.726 m/s2 -> 3.94 mm/s peak (2.78 mm/s rms) -> 21.4 um peak
Accelerometer output for that vibration: 7.4 mV peak

Displacement and position sensors

Linear and angular position are measured by several families of sensor. A potentiometer is a resistive track with a sliding contact, cheap and simple but subject to wear and electrical noise from the wiper. A linear variable differential transformer, or LVDT, has a primary coil and two secondaries around a movable iron core; displacing the core changes the coupling to the two secondaries, and their difference gives a voltage proportional to position, with no contact, no wear and essentially unlimited resolution. LVDTs are standard for laboratory displacement measurement and for the stem position of control valves. Magnetostrictive sensors, which time a torsional pulse along a waveguide to a moving magnet, measure long strokes precisely, and the FV-301 valve actuator from MEC-390 uses one inside its cylinder.

Rotary and linear encoders are digital position sensors. An incremental encoder produces two square waves, A and B, a quarter cycle apart; counting their edges gives position, and which one leads gives direction. With quadrature decoding, a 1,024-line encoder resolves 4,096 counts per revolution, under 0.09 degrees. Incremental encoders lose their count if power fails, so they need a reference mark and a homing routine at start-up. Absolute encoders encode the full position in a pattern on the disc and know where they are as soon as they are switched on.

Eddy-current proximity probes measure the gap between the probe tip and a conductive target without contact. A high-frequency field from a coil in the tip induces eddy currents in the target, and the resulting loss in the coil, detected by a driver, gives an output that varies linearly with gap over a range of about two millimetres. Machinery-protection systems for large machines with fluid-film bearings commonly use a standardized sensitivity of 7.87 V/mm, or 200 mV/mil, and an installed pair of probes at 90° to each other measures the shaft's orbit within its bearing. Proximity probes measure shaft motion relative to the bearing, which is what matters for fluid-film bearings, whereas casing sensors measure the absolute motion of the housing. Their readings depend on the target material and its surface: scratches, electrical or magnetic variations in the shaft show up as false vibration, called runout, which must be checked by turning the shaft slowly.

Vibration sensors

Housing vibration is usually measured with one of two seismic sensors. A velocity pickup, the traditional choice, is a coil and magnet on a soft spring; above the spring's natural frequency of about 10 Hz the magnet stays nearly still while the case moves, and the coil generates a voltage proportional to relative velocity. Velocity pickups need no power, but they have moving parts and a limited frequency range. The piezoelectric accelerometer has largely replaced them. A small mass sits on a piezoelectric crystal; when the base accelerates, the force needed to accelerate the mass compresses the crystal, which generates charge proportional to acceleration.

An accelerometer is a second-order system with a very high natural frequency, often 30 kHz or more for a general-purpose sensor, so below a fraction of that frequency its output is proportional to acceleration, exactly as Lesson 4 described. Most industrial accelerometers now contain a small built-in charge amplifier and run on a constant-current supply over the same two wires that carry the signal, an arrangement often called IEPE; their sensitivity is typically 100 mV/g. Because they measure acceleration, they emphasize high frequencies, which is ideal for detecting bearing defects and gear mesh, and their signal is integrated electronically to give velocity, the quantity used for most vibration severity limits.

The program converts the TP-80's outboard bearing reading at running speed, 0.074 g peak at 29.33 Hz, to 3.94 mm/s peak, 2.78 mm/s rms and 21.4 µm peak, agreeing with the figures used in MEC-380. The accelerometer's output for that vibration is only 7.4 mV, a reminder that the conditioning and cabling practices of Lesson 5 apply to vibration channels too.

Mounting accelerometers

An accelerometer's own natural frequency is rarely the limit in practice; the way it is attached is. The mounting acts as a spring between the sensor and the machine, and a softer mounting lowers the natural frequency of the combination. A sensor screwed onto a stud in a flat, machined, lightly greased surface keeps a mounted resonance near the sensor's own, perhaps 25 kHz. An adhesive pad lowers it somewhat, a flat magnet on a flat clean face further, a two-pole magnet on a curved surface to a few kilohertz, and a probe held in the hand to well under a kilohertz. The approximate figures in the program show the consequence: with light damping, the hand-held probe is accurate to 5 percent only up to about 180 Hz, which does not even reach the TP-80's vane-pass frequency, while a stud mounting is flat to over 5 kHz.

For the mechanical run test, Gilles mounts VT-401 and VT-402 on studs screwed into spot-faced pads on the TP-80's bearing housings, in the horizontal and vertical directions, as the customer's specification and the pump vibration standards ask. For quick surveys of pumps in the field, magnets are used, and the survey's report states the mounting so that readings above a kilohertz or two are treated with suspicion. The accelerometer's cable is clamped close to the sensor, because a cable that whips around generates charge of its own in some designs, and kept away from motor cables. A sensor's mass also matters on light structures: an accelerometer of 100 grams on a thin cover plate changes the plate's own vibration.