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 θt−τ = A sinω t+η:
Effective moment of inertia of suitcase rocking about either wheel
Product of weight and the effective width of the suitcase between wheels
Product of weight and height of suitcase
Coefficient of the restoring moment
Amplitude of excitation moment
Frequency of excitation moment
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
ddesys ≔ ⅆ2ⅆt2⁢θ⁡t+0.2400000000⁢signum⁡θ⁡t⁢cos⁡θ⁡t−12⁢sin⁡θ⁡t+θ⁡t−τ=0.75⁢sin⁡1.37⁢t+0.6944982656,θ⁡0=0,D⁡θ⁡0=0
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.
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).
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