During a roller coaster ride, the vehicles and their passengers are exposed to multiple times the acceleration of earth's gravity. The corresponding forces must be transmitted by the wheels to the track and the support structure. To validate the load assumptions of static and fatigue analysis, it is necessary to measure these forces. Currently, it is deemed sufficient to measure only the accelerations, providing a limited view of the whole. This article presents a method to measure the actual forces and moments that act on a wheel. Since the diameters of roller coaster wheels are comparatively small (approx. 200 mm) and the loads high (approx. 30 kN), industrially available multi-axis transducers cannot be used. Therefore, a transducer design based on strain gauges was developed and successfully implemented in a roller coaster vehicle. Due to its scalability, its application is not limited to just roller coaster wheels: it can be used for all types of non-driven wheels as well.

Roller coasters are gravity-driven amusement rides that are guided along a
spatial track. This track is designed to expose the passengers and the
vehicle to multiple times the acceleration of earth's gravity, which is the
sensation of such a ride. The movement on this spatial curve and its design
in order to produce the designated accelerations is well researched. The
reader is referred to the following exemplary literature for further studies:

The validation of these models is mostly based on acceleration measurements,
which can be easily conducted according to

A roller coaster vehicle is usually composed of a vehicle body and frame,
which holds the passenger's seat and the restraint system. The wheel bogies
are connected to the frame via a central axle or king pins. The concept with
an axle is shown in Fig.

In contrast to railway vehicles, the wheels are cylindrical or concave and
the track is usually made out of tubular steel pipes

Since all forces and moments resulting from the vehicle's motion are
transferred from the wheel to the track, the measurement of these is of
special interest. On the one hand, they can be used to assess the vehicle
dynamics; on the other hand, the loads for the vehicle and the track can be
determined. Therefore, different measurement methods have been developed and
are currently in use for cars and commercial and rail vehicles. Most of these
methods are based on two fundamental force measurement principles, namely the
use of strain gauges and a deformation element or piezoelectric force
transducers

A commonly used measurement device for the wheel forces of cars is a
dynamometer consisting of a rectangular array comprising four triaxial
piezoelectric force transducers

This arrangement provides 12 force signals which can be used to calculate the
normal force and the two tangential forces at the contact point as well as
the three moments acting on the wheel. Piezoelectric transducers are
characterised by their high linearity and ratio between range and response
threshold as well as low cross-talk and hysteresis. Despite these advantages,
the cost of the sensors is high and the sensor array needs a designated
space, which makes it difficult to fit into small wheels

A different type of piezoelectric measurement wheel is presented in

Other transducers use a deformation element and strain gauges as described in

Rail vehicles usually also employ strain gauges fitted onto the front and
backside of the wheel discs of a wheelset as described in

Apart from this solution, there exist stationary measuring devices as
described in

Roller coaster vehicle.

The calibration provides the relationship between the

The measurement matrix

Number of coefficients for an

It is obvious that the number of coefficients increases when

The highest possible rank of

Matrix

Generally, it is desirable that most of the elements of

Non-rotating dynamometer for a car tire according to

The main focus of the transducer lies on the measurement of the normal force

Transducer specifications.

Measurement wheel with all acting forces and moments at the contact point.

Out of the aforementioned requirements, the most critical constraints for the
transducer are the available space in a roller coaster wheel (diameter
200 mm) in conjunction with the maximum normal force of 30 kN and the
relatively low manufacturing costs. The space constraint makes it nearly
impossible to fit a telemetry system and a power supply for the transducer
rotating with the wheel itself. However, all roller coaster wheels have one
advantage in comparison to most other vehicle wheels: they spin freely and do
not need to transmit longitudinal forces onto the rails. Therefore, the
transducer can be fixed to the wheel suspension and the wheel bandage rotates
around it, as shown in Fig.

The ring-shaped space for the deformation element is determined by the
bearing with an inner diameter of 140 mm. Since the contact point can shift,
especially in the

If the four beams were pin-jointed, no moments would be acting along the
measurement beams

By measuring the eight shear forces

Scheme of the measurement wheel.

The CAD model of the deformation element with the locations of the strain
gauges, each holding two 45

Deformation element with strain gauges.

Each of the corresponding strain gauges indexed with F
(front)/B (back) and O (outside)/I (inside) are connected to a full bridge to
measure eight shear strains. Disturbance forces and moments as well as the
temperature strains are compensated by the connection of the bridges

Finite element analysis of the deformation element with the forces

By changing the dimensions of the web of the I-beam profile, the resulting
strains for the force

Furthermore, the FE analysis shows that the first eigenfrequency of the deformation element with no load is above 1200 Hz.

The full design of a measurement wheel with a diameter of 360 mm is shown in
Fig.

Measurement wheel.

The outer wheel consists of an aluminium rim with a vulcanised polyurethane bandage. It is fixed with a ring onto the middle wheel that holds the bearings, so that the outer wheel can be easily replaced. The bearings in O-arrangement are contactless sealed with a self-made labyrinth seal to minimise friction. The inner ring of the bearing is fitted onto the deformation element with the strain gauges. To prevent its rotation, a feather key is present on the upper, load-free side of the wheel hub. The same deformation element can be used used for a 200 mm diameter wheel with a thin section bearing.

In order to determine the measurement matrix according to
Sect.

For the theoretical calibration, the geometry of the deformation element without the other parts of the wheel is analysed with a finite element analysis using different load cases to determine the average strains acting on the area of the strain gauges. The main advantage of this method is that no actual manufactured parts as well as a calibration test rig are required. Also, the geometry and the load application are ideal. Thus this method describes the best possible conditions.

As described in Sect.

This formula uses the symmetries of the deformation element and for the
calibration only the measurable components

The measurement matrix

The absolute values of only six coefficients are significantly larger than
zero. Again, the mechanical isolation of the measuring points

Calibration rig.

The mechanical properties of the deformation element, its manufacturing, and
the positioning of the strain gauges are always subject to certain
tolerances. Therefore, an experimental calibration is necessary. This is
carried out with a specially designed calibration rig as shown in
Fig.

For each component, apart from the force

The overall agreement between the finite element calibration and the
experiment is good. As expected, the mechanical and electrical isolation of
the measuring points is less distinct. The corresponding coefficients are
still close to zero but several times larger than in
Eq. (

To assess the linearity deviation, the actual values of the force and moment
components, calculated by multiplying the measurement matrix by signals of
the strain gauges, are compared to the target value of the reference
transducers. If these are plotted together, they should ideally show a
straight line through the origin with a slope of 1. The deviations from the
ideal measurement are presented in Fig.

Linearity deviations.

The maximum deviation of the actual to the target values divided by the
nominal value is defined as the linearity deviation

To get a more global view of the deviations and to incorporate effects of the
cross-talking, the deviations can also be treated as normally distributed

Histogram of the deviations.

It can be observed that they are approximately normally distributed; thus, an
interval with the positive and negative limits twice the standard deviation
holds 95.5 % of all deviations

The four manufactured measurement wheels are installed in a wheel bogie of an
existing roller coaster train as running and side wheels (Fig.

To determine the random deviation, the concepts of the

Standard deviation for five unfiltered measurements from a fictive mean value measurement:

The standard deviations are similar to the previously discussed normally
distributed deviations (Sect.

As previously stated, the deviations of the moments are a result of inner
stresses that shift the zero point if the wheel is turned. Thus, they can be
reduced if the data are low-pass filtered, which is achieved here with a
moving average filter with a width of 200 ms. The random deviations decrease
significantly, as shown by their standard deviation in relation to the
nominal value – Fig.

Standard deviation for five filtered measurements from a fictive mean value measurement:

To evaluate the measurements, they are compared to the results of a multibody
simulation (MBS) of the roller coaster train on the same track. The MBS was
carried out with SIMPACK employing the FASTIM algorithm

Tangential force

Normal force

The overall agreement is good and it is especially remarkable that the
FASTSIM algorithm predicts the tangential forces due to slip very well even
for viscoelastic wheels that are treated herein as purely elastic. However,
it must be noted that according to

The normal forces on the wheel are also mostly generated by the track
geometry itself. They represent the main loads on the wheel. As can be
observed in Fig.

By comparing the normal force

The article proposes a simple and low-priced design for a force and torque transducer incorporated into an non-driven wheel with an outer diameter as small as 200 mm while still being able to carry and measure a maximum normal force of 30 kN. Also, it is possible to measure tangential forces in the axial direction, which are only one-tenth of the normal force. Furthermore, the application point of the forces does not need to be specified exactly. The design is easily scalable to measure different forces and can be used for all kinds of non-driven wheels.

By performing a theoretical calibration, the best possible results of the developed transducer under ideal conditions were shown. The experimental calibration showed higher deviation to the reference, but for measuring the forces these were still comparatively low (less than 2 % of the nominal value). Due to manufacturing tolerances of the transducer, the measurement of the moments acting on the wheel has only an orientating character. It can be dramatically improved by low-pass filtering the data; however, this reduces the frequency range of the transducer. Therefore, further studies should focus on improving the manufacturing quality and eliminating the zero shift that results from turning the wheel. Especially improving the circularity or changing the fit of the bearing seat could reduce these deviations significantly. One should investigate whether it is possible to measure the moments with acceptable deviations.

By employing the measurement wheel in a roller coaster vehicle, the real-life
application of the transducer design was successfully demonstrated. The
comparison to a multibody simulation under similar conditions showed a good
agreement. Therefore, the proposed design can be used to assess the loads
acting on a roller coaster wheel and thus also acting on the rail, its
support structure and the adjacent vehicle frame. Furthermore, it could be
possible to use the designed wheel not only for the purpose of validating the
load assumptions, but also to integrate it permanently into the roller
coaster vehicle. Then it would be possible to monitor the condition of the
vehicle and track during operation. The wheel could communicate with the
control system and the system could react as described in

The measured data are not publicly available. They were only used as an example to show the capability of the developed transducer. The data are not necessary to develop or reproduce a similar transducer, which is the main focus of the article.

AS developed the transducer, performed the calibration as well as the experiments and wrote the article; CS reviewed and provided corrections for the article.

Andreas Simonis was previously employed by Gerstlauer Amusement Rides GmbH and plans to work henceforth. However, he would like to state that his work was not influenced by this employment.

The authors would like to thank Gerstlauer Amusement Rides GmbH for their support. Edited by: Bernhard Jakoby Reviewed by: three anonymous referees