Basic Principles
When measuring vibrations in general, we can distinguish between two fundamentally different types, namely relative and absolute vibration measurement.
Relative vibration would be the measurement of a variable distance, for example the gap between a rotating shaft and a bearing. It can be measured by a proximity sensor installed on the bearing which is pointing at the shaft. This provides the motion of the shaft relative to the bearing.
An accelerometer (piezoelectric or other) is an inertial sensor and measures absolute vibration. The accelerometer measures vibration in space and not with respect to some other system. The inertial mass provides this characteristic because it is "suspended" by the sensing element in the sensor. The acceleration acts on the inertial mass and generates a force or movement of this mass which is measured by the sensing element. For absolute or seismic measurements either a velocity or acceleration sensor can be used but there are also absolute vibration sensors which measure displacement. In the case of a piezoelectric accelerometer we measure the force of the accelerated inertial mass. However the piezo sensing element is not completely rigid and therefore it allows for a tiny movement of the inertial mass. We will see in this chapter that this movement of the inertial mass is strongly dependent from the vibration frequency and leads to a most important frequencyrelated dynamic behavior.
The Natural Frequency of a Seismic Transducer
We retain that

The measuring element of an absolute vibration transducer can be seen as a spring. Its dynamic behaviour is therefore, in a first approximation, the same as that of a springmass oscillator and it is characterised by its natural frequency.

Increasing the inertial mass m leads to a lower natural frequency while increasing the stiffness k of the suspension (spring) will increase it.

Expressed in an equation the natural frequency a piezoelectric accelerometer which is an absolute vibration transducer becomes
Using the angular frequency the natural frequency becomes:
But beware: Strictly speaking this simple formula applies only to the natural frequency of an undamped oscillator however it is a good enough approximation for the natural frequency of a piezoelectric accelerometer.
SpringMass Oscillator without Damping
We look at our springmass oscillator model standing on a fixed base. The mass is m [kg] and the spring constant is k [N/m].
According to Sir Newton, the force acting on the mass due to the acceleration is m· ẍ. An additional force is due to the spring and is k·x. The gravitational force m·g is constant. We can therefore omit it and write:
or
The solution to this differential equation becomes
This is a continuous, harmonic oscillation with the frequency ω₀.
is called the natural frequency of the system.
Undamped springmass oscillator
SpringMass Oscillator with Damping
The natural frequency of a damped oscillator is slightly different from that of the same oscillator without damping.A truly undamped oscillator does not exist in the real world but the damping of a piezoelectric accelerometer is very small and can practically be neglected. Other absolute vibration sensors (e.g. MEMS sensors or velocity sensors), show significant damping which must be taken into account.
For this reason and for the sake of completeness, both damped and undamped versions are mentioned here.
For more details please refer to the
For the calculation it is normally assumed that the frictional force is proportional to the velocity. In the illustration the damping is presented by the added "dashpot" with the damping coefficient c [N/m/s].
The damping force then becomes c · ẋ and the summation of all forces leads to
Here it is useful to introduce the relative damping ζ which is defined as
Springmass oscillator with damping
Again with , the differential equation can be written in the form
Depending on the initial conditions, with x(t=0)=0 for example, the solution is
This is a sinusoidal oscillation with the frequency ωD and the initial amplitude A. The oscillation is decaying exponentially with the exponent  ζ· ω0 (see figure ).
The oscillation frequency ωD is the damped natural frequency.
With
it is slightly different from ω0 of the undamped oscillation depending on ζ .
For small ζ however (as we find in a piezoelectric accelerometer) we can safely take ω0 instead of ωD.
Decaying sinusoidal oscillation
Frequency Characteristics of a Seismic Transducer
In this section we deal with inertial vibration sensors or seismic transducers in general. For piezoelectric accelerometers, we can use the mathematical model without damping, except for the region around the resonance. Most other inertial sensors, be it velocity or acceleration sensors, usually have significant damping that must be taken into account.
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Frequency Response
We retain that
Inertial Sensor Frequency Response
As a model, we use the same damped oscillator. This time however, it does not stand still but moves with respect to the stationary space with the function u=g(t). Thus we force the oscillator to move and we speak of forced vibration.
A sensor that is built like this can measure vibration displacement, velocity or acceleration, depending on which part of the frequency range of the device is used. The division is made by the natural frequency. Below the natural frequency the vibration is measured in acceleration terms and in the upper frequency range as displacement, while around the natural frequency the sensor signal corresponds more to the velocity of the movement.
In the case of forced vibration we have the excitation u of the base and the relative motion x of the mass. The position of the mass m is given by u+x.
The acceleration force therefore becomes
The forces of spring and damping between base and mass are relative and remain the same as for the fixed oscillator. So we get for the equation of the equilibrium of the forces
or
Springmass oscillator under forced vibration
On the righthand side is the second derivative of the driving function, which means the acceleration of the base. We therefore choose
i.e. a harmonic oscillation with a constant acceleration amplitude UA.
Using ω0 and ζ again as defined before, we obtain a similar differential equation as for the free oscillation, but it is this time inhomogeneous due to the term on the right:
The solution of this equation consists of a first decaying part (as before in free oscillation) and a second stationary part, which follows the excitation u=g(t).
After a while the first part disappears. It remains the stationary part of the solution, the function of the displacement x(t) of the mass with respect to the acceleration of the base:
Here it is useful to introduce the dimensionless relative frequency ωR i.e. the excitation frequency in relation to the natural frequency
It is also common to use the frequency response representation (split into amplitude response and phase response), i.e. the ratio of the output amplitude to the input amplitude as a function of the excitation frequency.
The input U represents an acceleration while the output X in our model is a displacement.
The frequency response is therefore denoted by acceleration amplitude response Φa
The dimension of the function is [s²] because it is displacement / acceleration, X0 / UA.
The double logarithmic diagram shows the function for different values of the relative damping ζ .
Here we see the phase response. It turns out to be symmetric about the resonance when plotted against a logarithmic frequency scale
The phase of the output is lagging behind the input
Acceleration amplitude response
Phase angle frequency response
Resonance Frequency
Resonance (Latin resonare "to echo") is the amplified oscillation of a vibratory system under an external, cyclic excitation close to the natural frequency of the system.
With each cycle the system absorbs more energy, which continuously increases the amplitude of the system until the energy absorption is in balance with the energy loss due to damping.
In sensor technology, resonance is simply defined as the maximum of the amplitude response to an input signal with constant amplitude but varying frequency.
In fact we get, for greater ζvalues, slightly different resonant frequencies depending wether the acceleration, velocity or the displacement of the input signal is kept at constant, while the frequency is swept.
(With ζ = 0.1 the difference makes about 1%)
Acceleration resonance frequency:
Velocity resonance frequency:
Displacement resonance frequency:
But how can it be that you get three different solutions for the same physical phenomenon?
If you want to learn more about (incl. the derivation of the formulas) visit the
Quality Factor Q and Bandwidth
The frequnecy response or transfer function of a singlemass oscillator can also be regarded as a bandpass filter. The amplification at resonance is then called quality factor Q and the width of the filter is called bandwidth B. Both Q and B depend on the damping. The damping of the oscillator can therefore be determined by recording the maximum response amplitude Фv.
The following relationship applies:
The damping in a system is also characterised by the sharpness or width of the resonance peak.
The frequency difference Δω measured at 3dB from (Фv)max is called the bandwidth ( 3dB corresponds to Q/√2 ).
For the damping of the system we find for small values of ζ
Quality factor and bandwidth
Accelerometer Natural Frequency / Resonance
Very often, especially in connection with piezoelectric accelerometers, the natural frequency is referred to as resonance, but strictly speaking resonance is something different. The two things are closely related but not identical.
The frequency at which a system would oscillate after an excitation, if there were no driving force and no damping force is called natural frequency.
While the natural frequency is an internal property of the system, resonance is the interaction with an external excitation at a frequency equal or close to its natural frequency. For systems with a considerable damping the resonance frequency is in fact slightly different from the natural frequency, whereas for a piezoelectric accelerometer the two values can be regarded as the same.
Contributing design elements
We have seen that the natural frequency depends on the inertial mass and the spring constant. In a piezoelectric accelerometer, the "spring" is made up by the stack of piezoelectric elements.
The spring constant ki of each part of the sensing element is given by the surface area Ai divided by the thickness ti times the Young’s modulus Ei
With parts arranged in series and the total spring constant k becomes
Thus, for high stiffness, we prefer the smallest number of elements with a large surface area and small thickness. Besides the number of parts there are also the interfaces between them that reduce the stiffness. Therefore, attention must be paid to the flatness and surface quality of all parts.
Mounted vs. NonMounted Natural Frequency or Resonance
When defining and calculating the natural frequency of the sensor (or oscillator), we assumed that the base is fixed in space. In the real world this is approximately the case when the sensor is mounted on a heavy and rigid body. In this case the vibration pattern has a node at the base and an antinode at the inertial mass. When the sensor is not mounted, the seismic mass and the base oscillate in counter action around a node located somewhere inside the sensing element. The natural frequency of the free, nonmounted sensor is therefore considerably higher than when it is mounted.
In order to say something about the high frequency capability of an accelerometer we need to look at the mounted condition. The nonmounted value can be used to check the conformity of a set of sensors, e.g. in a quality control.
Measurement of the Natural Frequency / Resonance
The natural frequency f0 of a piezoelectric accelerometer can be excited by a short mechanical pulse. To measure f0 in the mounted state, we need a block with minimum ten times the weight of the sensor and a high Young's modulus. It is recommended to use a cube made of tungsten.
The following methods can be used to measure the mounted or nonmounted natural frequency.
A mechanical shock can be applied, for example, by a small metal ball attached to a thread. When striking with the suspended ball, make sure that it does not hit twice.
Steel ball impact
Another possibility is the HsuNielsen test method. This is a simple method in which an excitation signal with a wide frequency range is generated mechanically by breaking a pencil lead. Originally developed for testing acoustic emission sensors, this method is also excellent for testing the natural frequency of an accelerometer.
Hsu Nielsen source test method
After such an impact the inertial mass will be «ringing» at f0 and with a signal analizer or transient recording device we can read out f0. Basically, the damping rate can also be determined, although the signals need to be clean enough to do so.
We have to take into account that we are not only exciting the natural frequency of the sensing element, but also many other resonance frequencies within the sensor. This means that we usually need a critical interpretation of the measurement result.
There is also a completely different method that uses the inverse piezoelectric effect to excite the natural / resonant frequency, by injecting an electrical signal.
As injection signal we choose a single pulse or a alternatively a white noise.
By calculating the FFT transfer function of the signal fed in via the sensor compared to the injection signal, the natural / resonant frequency can be easily determined.
Signal injection test method
Accelerometer Frequency Response
Frequency response in the upper range
One of the principal features of a piezoelectric accelerometer is its frequency response. As we have seen above the frequency response function is depending on the resonance of the sensor.
The resonant frequency as such depends strongly on the construction of the sensor, but the evolution of the response curve is usually very close to the theoretically expected curve.
The figure shows the typical frequency response of a piezoelectric accelerometer with a strong amplification at the resonance. We find Qfactors of about 50 or even 100, which means ζ values in the order of 0.01 to 0.005.
For the frequency we are using the relation
As a rule of thumb, the response curve stays within 5% up to about 1/5 of the resonant frequency and the +3dB point is at about half the resonance frequency.
The phase shift is negligible below the resonance. At the resonance frequency however, it changes almost instantly to 180°.
Typical accelerometer frequency response
Frequency response in the lower range
Towards the lower frequencies, the correct reproduction of the acceleration signal is limited by the fact that the charge built up in the piezo element decays according to the time constant RC. The time constant itself is given by the capacitance C and the internal resistance R of the piezo element.
In the frequency domain, the RC element results in a highpass filter with the cutoff frequency
By using a charge amplifier, the time constant can be increased considerably and the cutoff frequency of the resulting highpass filter can be reduced enormously. The time constant is then imposed by the charge converter, supposing the internal resistance of the piezo element is sufficient.
The detailed relations are shown in the chapter "Signal Conditioners".
Typical single pole highpass filter frequency response
The figure shows a typical characteristic for such a single pole highpass filter. At the cutoff frequency fc we find an attenuation of 3 dB. If we search the 5% point we have to go to the region of about 10 times fc.
Design and other Contributing Elements
Similar to the dynamic range of an accelerometer we find for the frequency response that the high end is rather defined trough the mechanical sensor design while the low end is rather given by the following electronics.
The frequency response at higher frequencies is determined by the resonance, therefore we have to consider the same points mentioned for the natural frequency. Furthermore, it must be taken into account that the mass of the accelerometer can influence the frequency behaviour of the measured object or create an additional resonance in the mechanical measuring path. This means the sensor with the lowest mass will normally be the best to measure high frequencies.
In addition to the sensor mass, the mounting surface of the accelerometer and the contact surface to the measured object are of particular importance.
Measurement of the Frequency Response and Resonance
For measuring the frequency response, we basically use the same setup as for the calibration. I.e. a backtoback mounting with a central block on which we mount the unit under test on one side and a reference transducer on the other side. Of course, the frequency response of the reference transducer also plays a role. Ideally, it should have a resonance that is about 10 times higher than the highest frequency to be measured. If this is not the case, the excitation amplitude must be adjusted accordingly. With a Laser reference this problem does not exist.
In addition, utmost attention must be paid to the surfaces of the central block. The flatness and roughness of these surfaces have a great influence on the resonance frequency. For very high frequencies, an oil film should be applied.
Unit under test (UUT)
Reference
Accelerometer
Setup for the frequency response measurement
To carry out a frequency response test we either set the drive signal to different fixed frequencies and read the signal of the UUT or we sweep the frequency of the drive signal slowly from the lowest to the highest frequency of interest.
By increasing the frequency until we reach the maximum output signal of the UUT, we measure the resonance. If we are interested in the Qfactor, care must be taken that the resonance peak is really hit by the excitation and remains long enough for the signal to rise to the actual maximum
Influence of the surface / coupling condition
The graph shows a series of real measurements of the resonance frequency of the same sensor mounted on a block with different surface finishes from ISO N3 (lapped) to N6 (ground). The resonance varies from 37.7 kHz down to 27.2 kHz for the dry mounted condition, which corresponds to a loss of 27 %!
With an oil film applied, the result is much better, but the influence of the surface is still visible.
Measured values of the accelerometer resonance with different surface qualities and coupling conditions