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# How to measure torsional vibrations

Various measurement techniques are available for torsional vibration. The best sensor can be selected for each individual case based on the quantity to be measured, the type of analysis, the accessibility of the shaft, the ease of instrumentation and the required accuracy or level of detail.

# Content

1. Direct measurements
2. Coder-based techniques
3. Pulse signal conditioning
4. RPM and angle reconstruction methods
5. Number of pulse per rotation

# a) Direct measurements

Linear accelerometers
Two linear accelerometers are fixed in parallel on the rotating shaft. The two accelerometers will measure the tangential acceleration. As they have opposite direction in the fixed system of the rotation axis, any translational acceleration of the shaft is cancelled by taking the average of both accelerometers. The angular displacement, velocity and acceleration can be computed according to equations.

Linear accelerometers configuration

Classically for NVH analysis we will prefer the rpm quantity.

 RPM = 60 ∫ a1 + a2 dt 2π 2r

• High dynamic range directly determined by the dynamic range of the used accelerometers.
• Low sensitivity to the shaft translational vibrations when the two accelerometers are properly aligned.

• Expensive telemetry system or sensitive slip rings are needed to transfer the acceleration signals from the rotating shaft to the measurement hardware.
• Mass loading for relatively small shafts influencing the structural behavior of the shaft, e.g. causing torsional resonances to shift in frequency or shaft unbalance.
• Bigger shafts at relatively high rpm cause centrifugal force that may lead to dangerous loss of accelerometers and measurement equipment when not sufficiently well glued.
• Since acceleration is measured, and angle and speed can only be derived by integration, no absolute angular position is available. Angle domain processing for example will not be possible.

Angular accelerometers
Sensor manufacturers propose fully integrated angular accelerometers to be fixed on shaft extremities including the sensors and the slip ring. Some are based on linear accelerometers applying the method described in previous paragraph. Other techniques like torsional springs can also be used.

Strain gauges
Strain gauges are glued on the shaft to measure the torsional elongation or stress (shear stress). As the stress is directly measured, this method is often used in durability tests to estimate the torsional fatigue.

Torsional vibration is mostly measured in half-bridge configuration, with two strain gauges positioned at a 45 degree angle on the shaft. For a full-bridge configuration, four strain gauges are placed, two on the shaft’s front side and two are on the shaft’s back side. The symmetric configuration compensates for unwanted measured quantities like bending stress.

Half bridge and full bridge configuration applied to torsional measurement

• Direct measurement of the torsional elongation of shear stress.
• Low sensitivities to other deformation than torsional.

• Expensive telemetry system or sensitive slip rings are needed to transfer the acceleration signals from the rotating shaft to the measurement hardware. These systems may influence the structural behavior of the shaft due to mass loading.
• Bigger shafts at relatively high rpm cause centrifugal force that may lead to dangerous loss of accelerometers and measurement equipment when not sufficiently well glued.
• Exact angular speed and position are not known. Since acceleration is measured, and angle and speed can only be derived by integration, no absolute position is available. Angle domain processing for example will not be possible.

Dual-beam laser interferometers
Laser interferometers can be used as well to measure torsional vibration. Laser manufacturers typically proposed specific systems for rotating measurement based on dual beam techniques to cancel the effect of translational movement of the shaft.

The angular velocity is computed from the velocity measured in the direction of the laser beams on the two pointed areas.

Torsional laser principle based on Doppler Effect

• Contactless measurement
• Low sensitivity to shaft translation vibration
• Low sensitivity to the shape of the shaft
• Easy instrumentation

• Expensive device. Since it is often required to measure torsional vibrations at different shaft locations simultaneously, this is often a very large drawback.
• Exact angular speed and position are not known. Since velocity is measured, angle can only be derived by integration, no absolute position is available. Angle domain processing for example will not be possible.
• The size of the device does not allow using it in confined environment. Its use in real-life mobile conditions is very difficult or nearly impossible.

Torsional laser setup

# b) Coder-based techniques

Coder-based techniques make use of equidistantly spaced markers on the shaft or rotating part. The sensor measures every time a marker passes in front of a sensor and the time difference between two markers is used to estimate the angular velocity.

Coder-based measurement principle

The coder-based techniques have the advantage to deliver rpm and discrete angle position. The data resolution is determined by a number of markers: the more markers, the more accurate information although for low speeds, one will need more markers than for high speeds.

Different types of set-ups to provide markers (coder) are used, for example, stripes drawn on the shaft or the teeth of the gears. Also different sensors are available to detect the markers, such as electro-magnetic pick-up or optical sensors. Incremental encoders are devices combining the coder and the sensor in a single hardware.

Magnetic pick-ups
Magnetic pick-ups detect changes in the magnetic field or magnetic flux, typically resulting from metallic teeth passing the sensor. They are often used in industrial applications because of their robustness and low sensitivity to ambient dust. Set-ups for this are often very practical as well, since existing gear sets on the machine can be used as coder, e.g. gear-teeth on flywheels of transmissions. Resulting from that, magnetic pick-ups are very popular to measure torsional vibration, as they are easy to set up and, as they work very well with existing gear teeth, and as they are very robust. Most combustion engines today by default already, are equipped with them to transfer the different shaft positions to the engine or gearbox controllers.

Passive sensors like the Magneto-resistive or the Magneto-inductive are the most popular and most cost effective. The measured voltage from the sensor is generated by the changing flux, given by Faraday’s law:

 V = –N. dΦ dt

With
V generated magnetic voltage
N number of wires in coil of the sensor
Φ magnetic flux

As the change of the magnetic flux comes from the rotation of the shaft, the sensor does not need to be powered. The amplitude and the shape of the delivered signal however varies with the speed of the shaft (as this affects the differential of the magnetic flux) and may affect the accuracy of the teeth detection, mostly at low rotational speeds.

Influence of the rotational speed on the pulse level

Other magnetic sensors are based on the Hall-effect. Often those sensors are equipped with miniaturized electronic circuit to condition the output to deliver a TTL type of output signal. They will also need to be powered.

• Price. Mass production of magnetic pick-up for automotive or industrial application has a very positive influence on their end-user price.
• Simplicity of instrumentation. The sensor is typically fixed on non-rotating components, which removes the need, e.g. telemetry. Coders are mostly generated by existing gear sets.

• The number of teeth on the gear set limits the number of pulses per rotation which could be insufficient to capture all torsional content.
• Accuracy of the measurement will be very much dependent on the machining accuracy and the deformation of the gear teeth.
• The sensor must be fixed very close to the rotating shaft (less than 0.5 cm) which is sometimes difficult, e.g. when the shaft has an important translational movement.
• Relative displacement between the magnetic pick-up and the shaft, due to shaft bending, or due to displacement of the sensor, attached on a too soft mounting, will influence the quality of the measured pulses and generate a fictive torsional vibration.

Missing pulse correction
Coder wheels sometimes come with one or two consecutive missing teeth to generate an absolute angular reference position per rotation. Engine manufacturers use this possibility to synchronize the equipment linked to the coder with the cylinders’ top death center (TDC).

However, when such setup is used to provide coder pulses for torsional vibration analysis, missing pulse correction algorithms are needed. If not, the missing coder pulse would generate a spike into the rpm or angle estimation.

Although it’s clear that such coder will never be as accurate for torsional vibration measurements as the ones without missing pulses, they are often used when doing measurements on engines because of their advantage of providing the TDC reference and of being present on most engines.

Missing pulses correction algorithm. An interpolation method is used to re-generate the missing pulse. The next pulse directly after the missing one is used as absolute angle reference.

This configuration requires special care for the magnetic pick-up sensor. The hysteresis of the sensor does not always allow a good and accurate detection of the pulse located directly after the missing one.

Optical sensors
Many types of optical sensors can be found on the market, however most of them are designed for object detection. To measure torsional vibrations, the sensor not only needs to be able to detect a high rate of events per second, also the timing accuracy of that detection is very important and this accuracy is often insufficient.

Optical sensors generate an electric signal proportional to the received versus light intensity. Optical fibers are used to conduct the light from the emitter to the sensor head, and back from the sensor head to the receptor. They can be configured in reflection or transmission configuration.

Optical sensors can be used with much different type of coders as soon as the visible contrast between the stripes can be sufficient. Most optical sensors deliver TTL output signal.

• The instrumentation can be extremely simple as the sensor is typically fixed on non-rotating components. Only the coder needs to rotate.
• Optical sensors can be directly instrumented on gears as we would do with magnetic pick-ups, under condition that the reflection of the material gear surface is sufficient
• Coders can easily be implemented on the shaft with contrasted paint or ‘Zebra’ tapes.
• Quick response and good phase accuracy of good quality optical sensor allow the measurement of very high pulse rate.

• The sensitivity to ambient the light and/or the quality of the material reflection complicate the direct instrumentation of the gears in gearboxes.
• The sensor must be fixed very closely to the rotating shaft (less than 0.5 cm) which is sometimes difficult when the access is limited or when the shaft has some important translational movement.
• Relative displacement between the magnetic pick up and the shaft, due to shaft bending, or due to displacement of the sensor, attached on a too soft mounting, will influence the quality of the measured pulses and generate a fictive torsional vibration.

Zebra tapes
Black and white tapes are more and more used to quickly implement a coder on a shaft. It can be used to create a coder where no gear wheel is available or when a higher number of pulses per rotation are needed. There are two families of tape depending if it must be glued around the shaft (zebra tape) or on the extremity (zebra disc). Zebra tapes and discs exist in multiple stripe width to adapt the number of pulses per revolution in function of the shaft diameter.

Although zebra tape is very easy to instrument, an error will be introduced onto the measurement at the location where two zebra tape endings come together. When this point passes the optical sensor it will introduce a discontinuity in the rpm signal. The automatic correction method detailed in the paper ‘Zebra Tape Butt Joint Algorithm for Torsional Vibrations’ must be applied before analysis of the measured signal. Guidelines to obtain optimal results with zebra tapes are developed in annexes.

Zebra tape instrumentation and LMS automatic correction

Zebra discs do not suffer for the inherent butt joint issue, however proper care needs to be taken to properly centering the disc. Since perfect centering is never possible, harmonic or order 1 is typically not reliable when using this coder set up.

Incremental encoders
Incremental encoders are devices typically used in automat or robotic applications for accurate detection of shaft positions. Their high accuracy makes them very attractive for torsional vibration analysis applications as well. Often based on optical technology, incremental encoders combine the coder and the sensor in one single device. This means in practice it consists of both a rotating (rotor) and a static (stator) component and the full sensor needs to be mounted on the set up. Incremental encoders come in many different shapes and sizes, to cover all required applications.

The incremental encoder makes use of three embedded coders: one detecting one single pulse/revolution, called index, as absolute angle reference and two more high resolution encoders called A and B. The A and B signals have the exact same number of pulses but the B signal is phase shifted with a quarter of a pulse cycle (90 degrees) compared to A. Combination of these two coder signals allows detection of the sense of rotation of the coder.

• The fully integrated approach allows developing accurate coders with potentially very high pulse rate. Incremental encoders can be delivered with the appropriate number of pulses, depending on the application and desired accuracy (typically 50 to 500).
• The sense of rotation can be a great advantage i.e. for the investigation of start/stop behavior on engines.
• The integrated index signal by definition allows duty cycle related analysis with accurate TDC identification (e.g. engine combustion analysis).

• The relative complex instrumentation limits their usage for in-vehicle or mobile measurements. Incremental encoders are mainly used when working on test benches where the instrumentation makes part of the test bench equipment.

# c) Pulse signal conditioning

Magnetic pick-ups, optical sensors and incremental encoders all output periodic signals to be processed into angular velocity and/or position by acquisition hardware. Some sensors have dedicated circuit to pre-process or pre-condition the signal into a standardized type of output signal like TTL or RS422/485.

The torsional vibration measurement system must detect accurately the time stamps, named tacho moments, at which a predefined level is crossed by the periodic sensor signal assuming a fixed angle increment between pulses. The required hardware signal conditioning, depends of the type of sensor being used and its’ corresponding signal type.

Analog Tacho
The output signal is delivered as measured by the sensor, which means in practice that it can have any shape. A user defined trigger level is used to identify the tacho moments.

Digital Tacho – TTL
Most optical sensors and some magnetic pick-ups are equipped with dedicated level detection circuit delivering a standardized digital signal (TTL), which allows clear and accurate detection of the tacho moments at higher pulse rates than the classical analog tacho.

Digital Tacho – Differential TTL
Single ended signals are more sensitive to electrical noise than differential signals, especially when longer cable lengths are needed. That is why the differential standards RS422/485 are recommended for incremental encoders, where the distance between the emitter and the sensor head cannot be changed.

# d) Rpm and angle reconstruction methods

When using coder based sensors, each pulse detection moment corresponds to a known increase in angular position, i.e. at each tacho moment the angular position with respect to the first tacho moment is known.

To process the data in time or frequency domain, interpolation techniques must be used to estimate the position or the speed between two detected tacho moments and to generate a time-equidistant rpm or angle trace at the desired sampling frequency.

The interpolation method used can have an important impact on the quality of the analysis. Best techniques use digital reconstruction filters to avoid aliasing.

# e) Number of pulse per rotation

The quality of coder based torsional measurements is heavily impacted by a correct selection of minimum number of pulses per rotation. In case too few markers per revolution are available, this will add an error, known as angle domain aliasing to the measured data. The minimum coder resolution can be identified following the three following principles, summarized in the picture below.

Nyquist-Shannon sampling theorem
In principal, the number of pulses per revolution required is determined by the Nyquist theorem applied to an angle domain acquisition:
If a angular function x(α) contains no order higher than Omax, it is completely determined by giving its ordinates at a series of points spaced 2π/(2Omax) radians apart.
In other words, the number of pulses per rotation must be at least two time higher than the maximum order.
ppr > 2 Omax

Response bandwidth of mechanical systems
Mechanical system responses to a certain excitation are band limited in frequency. The relevant frequency range for torsional vibrations is limited to a maximum and this maximum depends of the structural dynamics of the mechanical system (shaft, or component being analyzed). It also means that once for a specific overall RPM the order exceeds the bandwidth, this order is no longer of interest.

Relation order bandwidth for rotating machinery
The maximum order observed is function of the bandwidth of the system and the rotational speed. For varying speeds (run-ups, run downs…), the minimum rpm determines this maximum observed order.
Omax = Bandwidth . 60 / Rpmmin

Optimal number of pulses per rotation
Based on those three principals, the optimal coder resolution can be estimated as follow:
ppr = 2. Omax = 120 . Bandwidth / Rpmmin
As the bandwidth of the system cannot always easily be estimated, a safety factor of 2 or 4 is often applied.
In many cases, test engineers do not have the luxury of selecting the optimal coder for their test. When using coders available as part of the structure, like for example teeth in gearboxes, it’s always important to estimate the error made with too few pulses. The reversed equation can be used to estimate the bandwidth of the coder for specific rpm conditions.
Bandwidth = ppr . Rpmmin / 120

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