Measuring Drop Heights Using a Force Plate
Introduction. The force plate consists of a rectangular metal frame and a plate supported by four force transducers, one at each corner of the plate, which measure the vertical force. The force plate thus works as a weighing device. If one stands on the plate and makes a jump we can in principle calculate the jumping height from the force measured by the transducers and recorded by a computer. By integrating the reaction force over time up to the take-off we obtain the momentum and thus the vertical take-off velocity from which we can finally calculate the maximum height of the center of mass during the jump. In order to assess the accuracy of this method we dropped a sandbag from known heights onto the force plate and compared the given drop heights with the one calculated by integrating the impact force. Our preliminary tests indicate that using the force integration method one can reach an accuracy around 2 - 3 mm or better on the average. It is found that the dominant source of the error derives from the discretization and the numerical integration, and these errors overshadow the errors due to the inaccuracies of the force transducers.
h sand bag
The momentum method. If an impact occurs at t = 0, and the impact force is F(t), and the impact momentum is Mv, then we should have (integrating the second law of Newton; x is here the vertical coordinate of the center of mass)
(1) Mv=
∫
0 T
M x(t) dt=
∫
0 T
(F(t)Mg)dt
where T is a time when the system has settled down ( F(t) = Mg for t > T). M is the mass of the impacting object (a 27 kg sandbag in our case). A similar procedure enables one to calculate the take-off velocity v for jumps. Knowing the impact velocity (or take-off velocity) v we obtain the drop height (jump height) from
(2) h= v2 2 g
where g is the gravitational acceleration (whose standard value1 is 9.807 m/s2). Besides the usual measurement errors there is a principal source of error in the method based on (1) due to the internal friction of the force transducers which means the they do not measure exactly the force acting on them. A simple model will illustrate the problem and enables us to assess the size of this error.
A simple model of the transducers plus the force plate. We suppose the transducers behave as harmonic oscillators with a damping (friction) term. Thus, the deflection x of a transducer acted on by a force F is supposed to satisfy an equation of the form
(3) µ xγ xk x=F (t)
Here k is the spring constant of the transducer, and Fs = kx is the force actually measured by the transducer which is different from F. If we integrate (3) over time we obtain
(4)
∫
0 T
F(t)dt=γ(x(T ) – x(0))
∫
0 T
k x(t) dt
since
x(0)= x(T )=0
Thus the momentum change calculated on the basis of the force Fs measured by the transducers will differ from the true momentum change by an amount of
(5) γ(x(T ) – x(0))=γ k M g
The corresponding error in the take-off velocity and drop height estimates will then be
(6) ∆ v=γ
k g
1 g varies from ca 9.780 m/s2 at the equator to about 9.832 m/s2 at the pole; i.e. a circa ± 0.27 % variation around the standard value.
and
(7) ∆ h=γ
k 2 g h
From the data supplied by the transducer manufacturer (Raute Precision) we have estimated the spring constant to be (per transducer)
(6) k = 5 × 106 kg/s2
The general solution of (1) for unforced vibrations is of the form
(7) x(t)=a e
γ 2µt
cos(ω tb)
with
(8) ω= k
µ
γ2 4µ2
The characteristic damping time τ (for a halving of the amplitude) is therefore
(9) τ=2µ γ ln(2)
According to a rough estimate we have τ ∼ 30 ms whence
(10) γ∼46µ⋅sec1
which means that
(11) ∆ h∼46 µ
5×106 2 g h
If we use the values µ = 20/4 kg (the force plate weighs about 20 kg), h = 0.5 m, (11) gives
∆ h∼0.2 mm
i.e. an error of the order of 0.1 %.
Noise and quantization. In the measurement setup we used a 16 bit AD-card. The transducer has a voltage range of 2.5 V and a nominal load of 200 kg which gives a resolution of about 800 N/V. A typical noise rms value is around 1 mV which implies a fluctuation∆F in the force of the order of 0.8 N. If we suppose the errors are uncorrelated the total standard error∆F summing the errors from the four force transducers will be around√4×0.8 N = 1.6 N. Since the quantization error (∼0.15 mV) is considerably less than 1 mV it can be neglected. Integrating the force the fluctuations will make the integral fluctuate with an rms error of the order of
(12) ∆ F⋅ T f
where f is the sampling rate. If ∆F ∼2 N, T = 0.2 s, and f = 1000 S/s, then (12) becomes about 0.03 N s. This corresponds to only a 0.035 % error in the impact velocity estimate, or a 0.07 % error in the drop height estimate, for the sandbag dropped from h = 0.5 m.
Discretization. Even if we suppose that the measured force values are exact we get an error from the discretization (in time) of a continuous force curve and its numerical integration. The result is especially sensitive to the choice of the starting point, since the touch down phase, where the impact force rises from 0 to Mg, is only about 3 ms for the sandbag. For discrete data F(ti) the integral (4) is replaced by the numerical estimate (time step ∆t = 1/f = 1/1000 s)
(13) F(t0)
2 ∆ tF (t1)∆ t F (tn1)∆ tF(tn) 2 ∆ t
A displacement of the starting point by one step may cause an error of the order of
(14) ∆ v= g 2 f
in the impact velocity estimate. For f = 1000 S/s and impact velocity around 3 m/s (14) implies an
error about 0.15 % in the velocity estimate and 0.3 % error in the drop height estimate. We used the following rule for choosing the starting point of integration: we selected the first time point ti such that F(ti) > base level + 10 N, and such that the next point satisfied F(ti+1) > F(ti) +10 N.
The force peak may be quite large at the impact. Dropping the bag from 52 cm caused an impact force peak 22.5 times the weight of the sandbag (corresponds to circa 605 kg). The force peak in our case was covered by about 20 - 30 data points when using the sampling rate of 1000 S/s, and the force increase per step could be as high as 700 N. If too few points are used this will be the single most important source of error. The range of this error could e.g. be determined by simulating the discretization for some model force curves. Suppose the exact value of an integral is S(0) and that a numerical integration using the step∆t yields the result S(∆t), then, according to the Richardson's method, we may assume that S(∆t) deviates from the exact value, to the second order in ∆t, as
(15) S(∆ t)∼S (0)a∆ t2
Using (15) also for the step 2∆t one can solve for the parameter a and insert it into (15) which gives an estimate for the error δ,
(16) δ=S (∆ t)S (0)∼1
3
(
S(2∆ t)S (∆ t))
Using the method (16) to the force integration we obtain for the error estimate δ = 0.043 ± 0.002 N s in the resulting impulse, which implies only a circa 0.1 % error in the drop height estimate for the h
= 0.5 m case.
Another rougher method is based on a special case of the Euler-Maclaurin series,
(17)
f(t0) f (tn)= 1
∆ t
∫
t0 tn
f(t) dt1
2
(
f(t0) f (tn))
∆ t
12
(
f(1)(tn) f(1)(t0))
n720∆ t4 f(4)(u)(t0<u<tn, ti=t0i∆ t)
From this we see that the critical term for the error between the exact integral and the numerical integration is given by (in our case the first order derivatives at the end points will be close to zero)
(18) T ∆ t4
720 f(4)(u) (T =n∆ t)
In our case we have T = 0.2 s, ∆t = 1/1000 s, and the magnitude of the 4th derivative of the force may be estimated according to the model (1), (7), to be of the order ω4 Fpeak, or less, where Fpeak
denotes the peak force. Using Fpeak = 6000 N and
ω∼2π 20
2027 4
⋅80∼656
we get for the maximum magnitude of (18) about 0.3 N s. Here we have rescaled the natural frequency of 80 Hz (for 20 kg per force transducer) due to the added mass of the sandbag lying on the force plate2. Anyway this rough estimate suggests a upper limit for the discretization error to be around 0.8 % in the drop height.
Summing all the errors we may thus expect the standard error in the drop height estimate to be of an order around 1 % or less, and a major variation coming from the method of selecting the touch down point from which to start the integration.
Method of measurement. A sandbag of mass M = 27 kg was suspended by a string from a beam right above the force plate (made by HUR Co). The height h of the sandbag (measured from the lowest bottom point of the bag) above the surface of force plate was measured (the error something around±2 mm). The sandbag was tightly compressed so as to minimize its change of form when it hit the force plate. The sandbag was released by cutting the string with a pair of scissor. The force plate, supported by the transducers, has a mass of 19.6 kg. Force transducers of the type B5A (Raute Precision) with a nominal load of 200 kg are rigidly attached to each corner of the force plate with bolts. The transducers have plastic "feet" in order to protect the floor (in our case a concrete floor covered with a plastic carpet). The nominal sensitivity of the transducer as given by the technical specifications is 2 mV/V, and the combined error is reported to be less than 0.02 % of the nominal load (i.e. less than 0.4 N). The natural frequency of the force transducer with a 20 kg load is 80 Hz, and for an unloaded transducer the manufacturer reports a natural frequency of 900 Hz. (We could confirm this by an experiment were we attached an accelerometer to a force transducer and deter- mined its vibration spectrum.) The force transducers were individually calibrated using a 19.15 kg calibration weight. An accelerometer (SCA 600, VTI Hamlin) was also glued to the center of one of the edges of the force plate in order to measure the vibrations of the plate. The transducers were interfaced, via an amplifier and a 16-bit AD-card (National Instruments), with a standard PC. Data were sampled at a rate of 1000 S/s and each measurement covered a time of 4 seconds (4000 points of data) in order to be sure of catching the event. The data was imported into the Mathcad program (MathSoft) for further processing. No smoothing of the data was applied since we integrate the data whence the fluctuations will tend to cancel anyway. The voltage data was converted to force units (Newton). The force integration was started on the first point which exhibited a significant force rise (10 N) above the base level signifying the moment of the impact.
2 This assumes that the sandbag partakes in the oscillating motion.
The integrated force, or momentum.
Results. The results of the six measurements are presented in the following table.
Measured drop height (mm) Calculated drop height (mm)
246 228
425 408
425 407
530 507
500 479
320 303
Measured force (red, units Newton) of the impact. Blue graph shows the accelerometer output (cut-off due to restricted voltage range).
The difference between measured drop height and calculated drop height is 18.83±2.33 mm and is due to the difference in the shape of the free-hanging sandbag and the floor-resting sandbag. If we add this correction (rounded to 19 mm) to the calculated values we obtain the table
Measured drop height (mm) Corrected calculated drop height (mm)
Difference Relative error (%)
245 247 -2 -0.7
425 427 -2 -0.4
425 426 -1 -0.2
530 526 4 0.8
500 498 2 0.4
320 322 2 0.6
The difference becomes -0.003 ± 2.34 mm (with unrounded values). Thus, we may expect the accuracy of the impulse method for calculating drop heights to be around ± 2.3 mm which is less than 1% of the typical jump heights. Indeed, the average of the magnitudes of the relative errors becomes circa 0.5 %. The mean drop height in our experiment was 407.5 mm. To be observed is that the standard error ±2.3 mm also contains the error from the measuring of the heights with the tape measure.
Discussion. Our experiments suggest that a force plate with the characteristics we have described above is suited for measuring drop heights (and jumping heights) using the force integration method. The main error may come from the discretization whence a sampling rate of 1000 S/s or higher is recommended and with no smoothing of the data. Typically one could by this method reduce the standard error to less than 1% of the drop height. Jumps are of course more complicated phenomena than an impacting sandbag but the physics is basically the same. The biggest difference with regards to the computational accuracy may be due to the oscillations when the person stands on the force plate before the take-off. Also the choice of the take-off end point for the integration will affect the result.
