Class LagrangeRollerSim

Rollercoaster simulation that uses Lagrangian method of finding equations of motion. The shape of the roller coaster path is defined by HumpPath.

The Lagrangian method of finding the equations of motion is very different from the methods used in the other roller coaster simulations such as RollerSingleSim. For example, the NumericalPath is used here only for finding the initial conditions such as the path length position corresponding to the starting X value. Whereas in Roller1Sim the NumericalPath is used in the evaluate() method to find the rates of change.

Variables and Parameters

The variables stored in the VarsList are:

vars[0] = x position of the ball
vars[1] = v = dx/dt = x velocity of the ball
vars[2] = s = position measured along length of track
vars[3] = s' = ds/dt = velocity measured along length of track

The independent variables are the X position and X velocity. The position along the track s and velocity along the track s' are derived from the X position and X velocity of the ball.

Parameters are: text g = gravity m = mass Equation of Motion

The equation of the HumpPath is

y = 3 - (7/6) x^2 + (1/6) x^4

The equations of motion are derived from the HumpPath as shown in the file [RollerCoaster Lagrangian(../RollerCoaster_Lagrangian.pdf). They turn out to be:

x' = v

     -(x * (-7 + 2*x^2) * (3*g + (-7 + 6*x^2) * v^2))
v' = ------------------------------------------------
            (9 + 49*x^2 - 28*x^4 + 4*x^6)

Track Position and Velocity

The track position s and velocity s' = ds/dt are derived from the X position and X velocity of the ball.

s = position along length of track
ds/dt = (ds/dx) (dx/dt)
s = integral (ds/dt) dt
s = integral (ds/dx) (dx/dt) dt

A fundamental result of calculus relates the path length and slope of the path (this is only valid for curves where the slope is finite everywhere):

ds/dx = sqrt(1 + (dy/dx)^2)

From the definition of the HumpPath we can easily find the slope dy/dx as a function of x:

dy/dx = -(7/3) x + (2/3) x^3

Putting these together we get the track velocity as a function of x and v:

ds/dt = sqrt(1 + (-(7/3) x + (2/3) x^3)^2) v

The track position s is then found by integrating ds/dt over time.

Y Velocity

The vertical velocity of the ball, y' = dy/dt, can be found as a function of the independent variables x and v as follows:

dy/dt = (dy/dx) (dx/dt)
dy/dt = (-(7/3) x + (2/3) x^3) v

Note that this should agree with:

s' = ds/dt = (+/-)sqrt((dx/dt)^2 + (dy/dt)^2)

Hierarchy (view full)

Implements

Constructors

constructor

Properties

initialState_ lowestPoint_ potentialOffset_

Methods

addObserver addParameter broadcast broadcastParameter evaluate finishDrag getBroadcast getClassName getEnergyInfo getEnergyInfo_ getGravity getMass getName getObservers getPEOffset getParameter getParameterBoolean getParameterNumber getParameterString getParameters getPath getSimList getTime getVarsList handleKeyEvent modifyObjects mouseDrag moveObjects removeObserver removeParameter reset restoreState saveInitialState saveState setBroadcast setGravity setMass setPEOffset setTerminal setVarsList startDrag toString toStringShort

Constructors

constructor

Properties

Protected initialState_

initialState_: null | number[] = null

Initial values.

Private lowestPoint_

lowestPoint_: number

lowest possible y coordinate of path

Private potentialOffset_

potentialOffset_: number = 0

potential energy offset

Methods

addObserver

  • addObserver(observer): void

  • Adds the given Observer to this Subject's list of Observers, so that the Observer will be notified of changes in this Subject. An Observer may call Subject.addObserver during its observe method.

    Parameters

    Returns void

addParameter

broadcast

broadcastParameter

evaluate

  • evaluate(vars, change, _timeStep): null | object

  • Defines the differential equations of this ODESim; for an input set of variables, returns the current rate of change for each variable (the first derivative of each variable with respect to time).

    The timeStep is the time since the state variables were last fully calculated, which can be and often is zero. The current time can be regarded as getTime() + timeStep. The input variables correspond to the Simulation state at that time. Note that timeStep is different from the time step used to advance the Simulation (as in AdvanceStrategy.advance). The timeStep is typically used when finding collisions in CollisionSim.findCollisions.

    Parameters

    • vars: number[]

      the current array of state variables (input), corresponding to the state at getTime() + timeStep

    • change: number[]

      array of change rates for each variable (output), all values are zero on entry.

    • _timeStep: number

      the current time step (might be zero)

    Returns null | object

    null if the evaluation succeeds, otherwise an object relating to the error that occurred. The change array contains the output results.

finishDrag

  • finishDrag(_simObject, _location, _offset): void

  • Called at the end of a mouse drag operation, performs whatever action is appropriate. Only called if startDrag returned true.

    Parameters

    • _simObject: null | SimObject

      the SimObject being dragged, or null if no SimObject was found

    • _location: Vector

      the location of the mouse in simulation coordinates of the SimView where simObject was found, or in the focus view if simObject is null.

    • _offset: Vector

      distance from the initial object position to the mouse location at start of drag.

    Returns void

Protected getBroadcast

getClassName

getEnergyInfo

Private getEnergyInfo_

getGravity

getMass

getName

getObservers

getPEOffset

getParameter

getParameterBoolean

getParameterNumber

getParameterString

getParameters

getPath

getSimList

getTime

getVarsList

handleKeyEvent

  • handleKeyEvent(_evt, _pressed, _modifiers): void

  • Called when a key is pressed or released, performs whatever action is appropriate for that event.

    Parameters

    • _evt: KeyboardEvent

      the KeyboardEvent that happened

    • _pressed: boolean

      true means this is a key-down event; false means a key-up event

    • _modifiers: ModifierKeys

      the modifier keys down during event

    Returns void

modifyObjects

mouseDrag

  • mouseDrag(simObject, location, offset): void

  • Called at each movement during a mouse drag, performs whatever action is appropriate. Only called if startDrag returned true. The SimObject being moved is passed in, along with the current mouse position, in simulation coordinates, and an offset calculated at the start of the drag.

    Setting the SimObject position to (x - offsetX, y - offsetY) will move the SimObject smoothly along with the mouse movement.

    Parameters

    • simObject: null | SimObject

      the SimObject being dragged, or null if no SimObject was found

    • location: Vector

      the location of the mouse in simulation coordinates of the SimView where simObject was found, or in the focus view if simObject is null.

    • offset: Vector

      distance from the initial object position (from DisplayObject.getPosition) to the mouse location at start of drag.

    Returns void

Private moveObjects

removeObserver

removeParameter

reset

restoreState

saveInitialState

saveState

Protected setBroadcast

  • setBroadcast(value): boolean

  • Sets whether this Subject will broadcast events, typically used to temporarily disable broadcasting. Intended to be used in situations where a subclass overrides a method that broadcasts an event. This allows the subclass to prevent the superclass broadcasting that event, so that the subclass can broadcast the event when the method is completed.

    Parameters

    • value: boolean

      whether this Subject should broadcast events

    Returns boolean

    the previous value

setGravity

setMass

setPEOffset

setTerminal

Protected setVarsList

startDrag

  • startDrag(simObject, _location, _offset, _dragBody, _modifiers): boolean

  • Called at the start of a mouse drag. The nearest dragable SimObject is passed in, along with mouse position in simulation coordinates. If no dragable SimObject was found, null is passed for the first argument. If the EventHandler does not recognize the SimObject then it should return false.

    Parameters

    • simObject: null | SimObject

      the SimObject that is nearest to the mouse drag coordinates, or null if no SimObject was found

    • _location: Vector

      the location of the mouse in simulation coordinates of the SimView where simObject was found, or in the focus view if simObject is null.

    • _offset: Vector

      distance from the initial object position (from DisplayObject.getPosition) to the mouse location at start of drag

    • _dragBody: null | Vector

      location of 'drag point' on the SimObject in body coordinates of the SimObject; this is where for example a spring will be attached on the SimObject when dragging; or null when no SimObject was found

    • _modifiers: ModifierKeys

      the modifier keys down during event

    Returns boolean

    true if the EventHandler will handle dragging the SimObject

toString

toStringShort

  • toStringShort(): string

  • Returns a minimal string representation of this object, usually giving just identity information like the class name and name of the object.

    For an object whose main purpose is to represent another Printable object, it is recommended to include the result of calling toStringShort on that other object. For example, calling toStringShort() on a DisplayShape might return something like this:

    DisplayShape{polygon:Polygon{'chain3'}}

    Returns string

    a minimal string representation of this object.

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