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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:

dde  M ⅆ2ⅆ t2 θt+signumθt M__b cosθt2  M__h sinθt2+k__0 θtτ = A sinω t+η:

Where:

M

Effective moment of inertia of suitcase rocking about either wheel

M__b

Product of weight and the effective width of the suitcase between wheels

M__h

Product of weight and height of suitcase

k__0

Coefficient of the restoring moment

A

Amplitude of excitation moment

ω

Frequency of excitation moment

η

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:

vals  M=1, M__b=0.48, M__h=1, k__0=1, A=0.75, ω=1.37, η=arcsinM__b/A: ics  θ0=0,Dθ0=0:

ddesys  evalevaldde,vals,vals,ics

ddesysⅆ2ⅆt2θt+0.2400000000signumθtcosθt12sinθt+θtτ=0.75sin1.37t+0.6944982656,θ0=0,Dθ0=0

(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 θt passes through 0, the velocity is reduced by 0.913:

evts  θt=0, ⅆⅆ t θt = 0.913 ⅆⅆ t θt:

 

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

dsn  dsolveevalddesys, τ=0, numeric, events = evts, maxfun=0: plotsodeplotdsn,0..400,size=800,golden

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:

dsn  dsolveevalddesys, τ=0.1, numeric, events = evts, maxfun=0: plotsodeplotdsn,0..400,size=800,golden

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

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