Delay DE: Suitcase Model - Maple Help

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:

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Where:

 $M$ Effective moment of inertia of suitcase rocking about either wheel $\mathrm{M__b}$ Product of weight and the effective width of the suitcase between wheels $\mathrm{M__h}$ Product of weight and height of suitcase $\mathrm{k__0}$ Coefficient of the restoring moment $A$ Amplitude of excitation moment $\mathrm{\omega }$ Frequency of excitation moment $\mathrm{\eta }$ Phase of excitation moment

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, $e$, which we choose to have the value $0.913$.

We choose the following parameter values and initial conditions:

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 ${\mathrm{ddesys}}{≔}\left\{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}{}{\mathrm{θ}}{}\left({t}\right){+}{0.2400000000}{}{\mathrm{signum}}{}\left({\mathrm{θ}}{}\left({t}\right)\right){}{\mathrm{cos}}{}\left({\mathrm{θ}}{}\left({t}\right)\right){-}\frac{{1}}{{2}}{}{\mathrm{sin}}{}\left({\mathrm{θ}}{}\left({t}\right)\right){+}{\mathrm{θ}}{}\left({t}{-}{\mathrm{τ}}\right){=}{0.75}{}{\mathrm{sin}}{}\left({1.37}{}{t}{+}{0.6944982656}\right){,}{\mathrm{θ}}{}\left({0}\right){=}{0}{,}{\mathrm{D}}{}\left({\mathrm{θ}}\right){}\left({0}\right){=}{0}\right\}$ (1)

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 $\mathrm{θ}\left(t\right)$ passes through 0, the velocity is reduced by $0.913$:

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Now consider the behavior of the system if there is no delay in the response time:

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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:

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From this plot, it can be observed that with the presence of a delay, the system is visibly unstable.

 References The model described above is from the paper: S. Suherman, R.H. Plaut, L.T. Watson, S. Thompson, "Human delayed response time in correcting the side-to-side motion of a two wheeled suitcase." J. Sound Vibration 207 (1997). Link: http://www.researchgate.net/publication/243364611_EFFECT_OF_HUMAN_RESPONSE_TIME_ON_ROCKING_INSTABILITY_OF_A_TWO-WHEELED_SUITCASE