High Strain-Rate Application

Measuring forces in falls on very stiff ropes or slings such as Dyneema can be challenging. The stiff nature of the rope means that the natural frequency of oscillation is relatively high, typically on the order of 10 to 40 Hz, depending on the stiffness, length of the rope and size of the mass, with a short and stiff rope with a small mass resulting in the highest frequency oscillations. In such tests we experience fast peaks separated by the rebound of the mass. The motion of the mass (to a good approximation) is governed by the damped harmonic oscillation equation


where c is the damping constant, k is the spring constant and m is the mass, and the general equations of motion in gravity during free fall, most importantly
        
          

From this we know that the force profile will be sinusoidal (based on the under-damped solution of the harmonic equation) and from that we can estimate the worst case error caused by the sampling process of the ADC.

For an arbitrary sine wave of frequency fin (period Tin), sampled at rate fs (period Ts) we get the worst case amplitude error when two samples land equally spaced from the peak of the sine wave, as shown in the figure below.


The error is fully determined by the sampling period and the sine wave that is being sampled as


When sampling at the Nyquist rate, twice the signal frequency we have


This makes intuitive sense as when sampling at the Nyquist rate there are two samples per period and in the worst case they lie on the x-axis resulting in zero amplitude and thus 100% error.

Before moving on, it is important to note that there are other sources of error. Other amplitude errors include the quantization error of the ADC, thermal and flicker noise of the electronic circuits, noise from the strain gauges in the load-cell, noise coupled to cables, wires, PCB traces ect. Additionally there is also some small amount of timing error, although this is usually very small. The figure below shows how each sample has some margin of error.


This means that higher sampling rates will generally result in smaller amplitude sampling errors (here we don't consider that faster sampling rates generally reduce the accuracy of the ADC), which was quite obvious from the start. However, we now can calculate both the maximum amplitude error due to sampling for a specific sample rate and frequency of the signal we want to measure and it should be noted that it does not have to be periodic, only reasonably close to a sinusoidal waveform. For a typical 30 Hz signal (such as a short Dyneema sling with small 5 kg mass), sampled at 500 Hz, we see that the maximum amplitude error should be

 



We can also use the equation for the error to plot a graph with the error on the x-axis and the corresponding ratio of sample rate to signal frequency on the y-axis.




This shows that for ratios above 15 the error will be less than 2%, thus you can sample signals of up to 33Hz with a 500 S/s DAQ which should cover most tests for rigging scenarios, even small mass drops on Dyneema slings. We see that with some careful consideration and correct test set-up we can measure relatively high frequency spikes with moderate sampling rates.