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Delay Differential Equations: Suitcase Model
The "Suitcase Model" describes correction in the side-to-side motion of a two-wheeled suitcase caused by a human delay in the response time.
The delay differential equation model (DDE) is as follows:
Where:
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Effective moment of inertia of suitcase rocking about either wheel
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Product of weight and the effective width of the suitcase between wheels
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Product of weight and height of suitcase
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Coefficient of the restoring moment
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Amplitude of excitation moment
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Frequency of excitation moment
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Phase of excitation moment
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In addition, when the angle passes through 0, there is a loss of energy when one of the wheels impacts the ground, and this is described by a decrease in the velocity based on a coefficient of restitution, , which we choose to have the value .
We choose the following parameter values and initial conditions:
where the delay has been left unspecified.
The energy loss of the wheel striking the ground is handled through the following event that states that when passes through 0, the velocity is reduced by :
Now consider the behavior of the system if there is no delay in the response time:
From this plot, it can be observed that the angle varies between approximately -0.92 and 1.16.
However, if a 0.1 sec. delay in introduced in the response time, the situation is quite different:
From this plot, it can be observed that with the presence of a delay, the system is visibly unstable.
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