Since moving from the UK to the USA in 2018, I’ve noticed various little differences beyond the obvious between the two countries. Some are positive differences, like actual sunshine, decently sized parking spaces, drive-thru post boxes and cheap petrol. Other differences however are less than ideal, such as polar vortexes or in particular, shopping trolleys (or carts, for the Claytex USA guys).
As a rule, I’m not too keen on grocery shopping. Whichever side of the Atlantic I am on, I usually do the weekly shop on a Friday evening to avoid most other shoppers. One past Friday I found myself battling an ill-handling shopping cart.
Earlier in the day, I had read an academic paper from the latest International Modelica conference. As part of other work, it discussed the multibody modelling of a castor wheel in Dymola. This sparked an idea on the drive home.
I seemed to recall that British trolleys were more nimble and agile than their American counterparts. My hunch was that this was due to British trolleys having 4 swivel castor wheels, apposed to the common American setup of 2 front swivel castors and two fixed castor wheels at the rear. So to test my hunch I decided to build a couple of multibody Dymola models.
Building the trolley/cart model
Considering a shopping trolley conceptually, I determined it is made up of three main elements. A basket, to hold your shopping; a bed, to place heavy/large flat goods and 4 individual wheels of varying rotational freedom. The base of the trolley can be considered rigidly connected to the basket, as are the wheels to the base. As the only thing to vary between the UK specification and a USA specification trolley (besides the presence/omission of cup holders) are the wheels, I began by building a trolley chassis model as a template.
The TrolleyFrame was considered the origin of the trolley. Two frames were added, TrolleyLeftHandle and TrolleyRightHandle to enable two separate forces to be applied to the trolley, to mimic how people push the trolley itself.
To simplify the modelling, the basket model features a lumped mass approach. Handle frames were offset the required distance from the basketFrame, where the basket connects to the base, by fixed translation each.
A similar approach was taken with the trolley base model. Primarily establishing the base frame with requisite frames for the connection of the castor wheels.
Both the basket and base models featured estimated masses and inertia. Overall dimensions were based on this commercially available shopping cart.
Modelling the wheels
In order to recreate the full shopping trolley experience, correct wheels are required. Casters feature an offset vertical mounting (fork), either free to rotate (swivel) or fixed to the body it is attached to. Typically they have a hard rubber or plastic wheel mounted on a single axle, held by the fork.
From a multibody perspective, we can begin to model by reducing the castor down to its key motions and components. There is a wheel, with mass and inertia, which rotates around an axis parallel to the ground. A fork, intersecting the rotational axis of the wheel, rotates around an axis perpendicular with the ground. Connecting the wheel to the body the castor itself is attached to, the fork, also having a mass and inertia, provides the offset between the wheel contact patch perpendicular to the ground and the fork rotation axis. Finally, there is friction to consider; in the fork swivel, the wheel axis and between the wheel and the ground which acts laterally on the wheel. For these purposes the friction encountered by the rolling of wheel over the ground (rolling resistance) is considered negligible.
Due to there being two rotations to consider, we can determine that for modelling the swivel castor two revolute joints are required, with the required friction. For the fixed castor, one of these revolute joints would be omitted. We also need a wheel model which will rotate freely along the direction of travel, yet restricts lateral sliding to mimic the friction mentioned above. Therefore, I created the following base castor model:
Of specific interest is the wheel, comprised of the wheelSpinAxis revolute joint, rollingContraintVerticalWheel, castorWheelMass and wheelVisualiser. In effect, the rolling dynamics of the wheel are provided by the rolling constraint, Modelica.Mechanics.MultiBody.Joints.Internal.RollingConstraintVerticalWheel, which features a lateral constraint to prevent the wheel slipping laterally, with the revolute joint enabling the mass and the wheel dynamics to roll along the desired axis.
For the swivel castor, extended from the base class, there is an added revolute joint, the forkSwivel, complete with forkSwivelFriction. The fixed castor just features a connection between the castorMountFrame and castorForkMass/forkTranslation. It’s important to note however, that only one of the fixed castors on the rear of the USA specification trolley features the lateral constraint, as they are rigidly connected through the trolley itself.
Testing the trolley models
In order to settle the debate over who builds the superior shopping trolleys, I thought the best idea would be a simple “turning” test. Each trolley would be pushed with the same split in pushing force between the left and right handle frames.
As the castor models don’t feature a vertical model to support the trolley model under gravity, the trolleyFrame(s) are only allowed to move in a planar fashion.
With the British specification trolley on the right, and the American one on the left, the results are quite clear to see. Her Majesty’s shopping trolley is clearly more nimble, able to spin easily on its axis, compared to the American one.
Written by: Theodor Ensbury – Project Engineer
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