MULTI-DIMENSIONAL SYSTEM: Ship Stability

Size: px

Start display at page:

Download "MULTI-DIMENSIONAL SYSTEM: Ship Stability"

Raymond Black
5 years ago
Views:

1 MULTI-DIMENSIONAL SYSTEM: I this computer simulatio we will explore a oliear multi-dimesioal system. As before these systems are govered by equatios of the form = f ( x, x,..., x ) = f ( x, x,..., x where is the dimesio of the system. A model for ship stability M = f ( x, x,..., x ) ) I this lab we'll be usig a simplified model of a ship i rollig seas i order to study the ship s stability ad fid out how likely it is that the ship may capsize. This system is a example of a forced oliear oscillator. We ll use Matlab to iteractively ivestigate this model ad explore the differet types of dyamics it ca exhibit. You should thik of the equatio for the model as the experimetal system ad Matlab as the apparatus you are usig to measure its properties. This lab aims to emphasize that, firstly, although a ship i the sea is a very complicated system, may of its features ca be captured usig a simple (ofte empirical) model. Secodly, eve though the model is simple, the behavior is ot. I fact, you will fid that, due to the oliearities, the ship may capsize eve though the sea coditios are far from where oe would expect capsize to occur. The goverig equatio I this study we'll cosider the simple case of a ship broadside to the waves so that the wave actio causes the ship to roll from side to side. Figure shows a ship sittig i still water. There are two forces actig o the ship that try to make it "sit up straight". These are the weight ad the buoyacy which act together to reduce the agle x. If the ship is displaced to oe side it tries to restore itself ad oscillates util the dampig of the water brigs it to rest. x buoyacy still water level weight Figure : Schematic of ship i still water showig the actio of buoyacy ad weight to right the ship. Rev: 3 April 00

2 Noliear Dyamical Systems Lab Page of 6 While experimets have bee used to obtai equatios of motio empirically for some hull types (see the refereces give below) a more geeral approach is to cosider the simplest possible equatio that gives qualitatively sigificat results. We choose a model that represets a ship that is biased to capsize to oe side, i this case the right (+ x) side. This bias could be due to wid, ice loadig or movemet of the cargo to oe side. The simplest equatio we ca have i this case is the damped, oliear, forced oscillator, with & x = β ( x x ) + F si( ωt) () where x is the agle of icliatio of the ship, β is the dampig costat, F is the amplitude of the forcig (the size of the waves) ad ω is the frequecy with which the waves arrive. I the absece of the quadratic term this equatio is that of a forced damped harmoic oscillator. It is the quadratic term that makes this a oliear equatio ad allows the possibility of capsize ad other iterestig dyamics.. Overdamped ad oliear with o forcig I order to see how the oliear term ca lead to capsize we'll first set the forcig to zero ad let β be very large. You ca thik of this as a model for a ship i a perfectly calm sea of treacle syrup. For large β the system is overdamped ad we ca igore the & term. Strippig the model dow i this way allows us to isolate the effect of the oliear term. The equatio the becomes β = ( x x ) f ( x) () (a) Sketch the curve f(x), locate the fixed poits, classify their stability ad show the vector field. Sketch the correspodig effective potetial. (Remember f (x) = dv (x)/dx ). The overdamped equatio () tells us that the system will seek out a local miimum of the effective potetial. Lookig at the effective potetial we ca begi to uderstad the capsizig. For sufficietly small x the ship fids a miimum at x = 0. For sufficietly large x the ship "goes over the edge" of the potetial ad, because it moves i the directio of decreasig potetial, it does ot recover.. Uderdamped ad oliear with o forcig Now that we have a sese of the effective potetial we will keep the iertial term & so that we ca iclude overshoot ad oscillatory effects. We'll cotiue to igore the forcig term, so that ow our model correspods to a ship i a perfectly calm sea of water. Our secod-order equatio ow reads & x + β + x x = 0 As we did for the damped harmoic oscillator, we ow trasform this ito two first-order equatios by defiig a ew variable x & = v so that the secod-order equatio becomes

3 Noliear Dyamical Systems Lab Page 3 of 6 = v v& = βv x + x (3) With the -dimesioal overdamped system described by Eq. () we were able to ispect the effective potetial to see what values of x will lead to capsize. This is ot so easy ow that we have a -dimesioal system. It is o loger eough to just kow the iitial value x 0, we also eed to kow how fast ad i what directio the ship is iitially rockig (v 0 ). Our tactic is ow to umerically itegrate the above set of equatios for ad v& see which iitial coditios (x 0 ad v 0 ) lead to capsize. The umerical itegratio is doe by extedig the Ruge-Kutta method to -dimesios (see Strogatz page 33 for -D ad page 47 for -D). (a) Before you do ay computig you should first sketch what the phase portrait looks like usig the guidelies provided i class ad also show i Strogatz Sectios 6.3 ad 6.4 i.e., idetify ay fixed poits, classify their stability to plot the phase portrait i their viciity. The use commo sese to fill i the rest. Show your calculatios. We will ow use the Matlab program pplae7.m to get the phase portrait ad the use it to ivestigate the behavior of the system from a umber of differet iitial coditios. The Matlab program pplae7.m is ru from withi Matlab. Istructios for usig pplae7.m are attached to this write-up. (b) Use pplae7.m with a value of β=0. to plot ad prit out the phase portrait. Use a rage of [-,] for x ad [-,] for v. Does the phase portrait agree qualitatively with the oe you sketched above? O your phase portrait try to locate the bouds of stability, that is, idetify the iitial coditios that lead to capsize ad those that do ot. The set of iitial coditios that do ot lead to capsize is ofte referred to as the basi of attractio. You have bee supplied with a Matlab program (stable.m) that you ca use to map out the basi of attractio. This program calls a adaptive fourth/fifth order Ruge-Kutta algorithm to perform the umerical itegratio of Eqs. (3). It sets the iitial values of v ad x ad the keeps track of which iitial coditios lead to capsize ad which result i the ship settlig dow. It the plots the basi of attractio. Quit pplae7.m. Ru the program stable.m. A value of β=0. has bee set for dampig. (c) Obtai ad prit out the plot showig the basi of attractio. (d) Look at the program ad idetify the criteria that is used to assess whether a startig poit is outside the basi of attractio. Use pplae7.m to assess ad explai whether it is sesible. (e) The program has bee preset with a fial time, t f, over which the Ruge-Kutta itegratio will take place. Is this value a good choice? What are the drawbacks of decreasig or icreasig this time? (f) Without ruig the program agai, ca you predict what the effect of chagig β will be? Now check the predictio by tryig β= 0.0 ad β=0.5. You ca chage the value of β by editig

4 Noliear Dyamical Systems Lab Page 4 of 6 the program stable.m ad chagig the parameter beta ear the top. Prit out each basi of attractio for these other values of β ad explai qualitatively what happes as β is icreased. 3. Damped ad forced without the oliear term Let's ow go back to our forced equatio ad remove the oliearity. This will allow us to gauge the effect of the forcig term. We ow have a equatio for a damped, drive ad liear oscillator: & x = β x + F si( ωt) or as a coupled system, = v v& = β v x + F si( ωt) We'll ow examie how x ad v evolve i time for differet forcig frequecies ω. This equatio is very well kow i physics ad exhibits a pheomeo kow as resoace. That is, at a specific forcig frequecy (the atural frequecy of the system) the respose of the system is a maximum. Now use the program forcig.m to explore this behavior. First set the oliear term to zero by settig the parameter c = 0. You ca also modify the forcig amplitude, F, ad the frequecy, ω (omega) ear the top of the program. Ru the umerical itegratio for may forcig periods so that the system has settled dow after the iitial trasiet. You may wat to start with about 0 periods. You ca set t fial to a suitable value by trial ad error, but it may be quicker to figure out how log 0 periods is i terms of ω ad the put that i for t fial i the program. We will quatify the system respose as the amplitude of the ship s roll (x) at the ed of the forcig. (a) Set F = 0.05, β=0., with the iitial coditios x = v = 0. Use the graph of x vs. t to obtai the respose for each frequecy (for ω = 0.4 to. i steps of 0.). Plot this respose as a fuctio of frequecy (you ca do this by had, or with Excel, or by modifyig the forcig.m program appropriately, or ). Record the frequecy at which the respose of the system is a maximum. We call this the resoat frequecy. 4. Damped, forced ad oliear Now we will put back i the oliearity by settig c = i forcig.m. (a) What do you thik its effect will be o the resoat frequecy? Obtai the ew resoat frequecy with F = 0.05, β=0., by repeatig the steps above, but with the oliear term icluded. What is the ew resoat frequecy? Now you see that the oliearity has the effect of "softeig" the resoace (that is, makig the resoace occur at a lower frequecy). Note also that there is ow a sharp discotiuity. The ship's amplitude jumps suddely. I this particular case the jump is ot fatal, but it shows that i the presece of oliearity the behavior of the system ca chage dramatically with a small chage i the iputs.

5 Noliear Dyamical Systems Lab Page 5 of 6 At this stage of our ivestigatio we could either elect to ivestigate stability as a fuctio of iitial coditios or as a fuctio of forcig frequecy ad amplitude. For ow we'll attempt to sketch the regimes of capsize ad stability as a fuctio of forcig amplitude (F) ad forcig frequecy (ω), for fixed iitial coditios v =0 ad x= 0. (c) Use the program forcig.m for values of forcig frequecy from ω = 0.3 to.5 (i steps of 0.) to fid the value of forcig amplitude where the system becomes ustable, i.e. where x diverges. Note that evidece of the divergece of x may be a sigularity i your plot of x vs. t. For each frequecy obtai the critical forcig amplitude ad make a graph separatig the regimes of stability ad istability. The graph should look roughly like figure. forcig amplitude ievitable capsize regio of stability forcig frequecy Figure : Schematic of regios of stability ad istability. You ca ow also start to see other strage behavior i this system. For example, as you sca i amplitude you'll see regios where the respose of the system is o loger at the same period as the forcig. I fact, the period is sometimes twice that of the forcig, a pheomeo called period doublig. You ca fid such a place at ω = 0.3 ad F =0.4. (d) Show that the period is i fact doubled. 5. What's happeig i the phase space? You may have bee woderig what is happeig i phase space. The phase space is ow threedimesioal ad is spaed by (x,v,t). The reaso we eed t is because it appears explicitly o the right had side of our set of differetial equatios i the forcig term. The trajectories look like the wires of a complicated cable ad wrap aroud each other. If we plot the phase space (x,v), the plot looks tagled. This makes visualizatio quite difficult. We ca gai some sese of what is happeig by lookig at 3-dimesioal trajectories. Have a look at the behavior i phase space by usig forcigphase.m. You should use β=0. ad c=. This will geerate phase plots of v vs. x ad v vs. x vs. t, usig our differetial equatios with forcig. You'll see the trajectories o loger have that "well groomed" look to them that Strogatz describes (page 49). (a) Geerate ad prit the phase plots whe you are (i) far from resoace, (ii) ear to resoace ad (iii) whe the system is period doubled.

6 Noliear Dyamical Systems Lab Page 6 of 6 Report I the lab report you should complete all the tasks highlighted by i the above text. Make sure that you correctly label each oe (e.g..(b) ) ad order your prit outs/sketches accordigly. (b) Other ship stability studies have used a potetial of the form V = x 4 x 4 + αx where α is a costat. Sketch this potetial ad the associated restorig force, give by f (x) = dv (x) / dx, for several values of α i the rage 0 < α <. Idetify ay fixed poit(s). POSSIBLE PROJECT: Here are refereces for further readig. Let your lab/lecture istructor kow if you would like to cosider extedig this aalysis as part of your project. Refereces Solima, M. S. ad J. M. T. Thompso. "Trasiet ad steady state aalysis of capsize pheomea." Applied Ocea Research, 3(), 8-9, (99) Bishop, S. R. ad M. S. Solima, "The predictio of ship capsize: ot all fractals are eviromet friedly." Applicatios of Fractals ad Chaos. Crilly, Earshaw ad Joes (Eds). Spriger-Verlag 993

Figure 1: Schematic of ship in still water showing the action of bouyancy and weight to right the ship.

Figure 1: Schematic of ship in still water showing the action of bouyancy and weight to right the ship. MULTI-DIMENSIONAL SYSTEM: In this computer simulation we will explore a nonlinear multi-dimensional system. As before these systems are governed by equations of the form x 1 = f 1 x 2 = f 2.. x n = f n