# Class MoveableDoublePendulumSim

Simulation of a double pendulum hanging from a moveable anchor point.

The anchor point or 'cart' position can be given any program of motion, it is not affected by the pendulum movement at all. So you could regard the cart as a having infinite mass in comparison to the pendulums. Or that some outside entity is applying whatever forces are needed on the cart to keep it to the fixed program of motion.

The cart is both dragable by the mouse and/or can have a periodic up/down motion. With the periodic motion, this becomes a demonstration of an 'inverted pendulum': if the periodic motion is rapid enough, the pendulum position pointing straight up becomes stable.

There is a parallel but independent simulation for the movement of the cart. The cart is regarded as a point mass that is dragable by a spring force controlled by the user's mouse. Optionally, the periodic force moves the cart up and down.

Note that when changing the anchor amplitude or frequency, that we set the anchor vertical velocity such that the anchor stays centered at its current position. Otherwise, the anchor tends to move rapidly out of view.

Derivation of equations of motion is shown at https://www.myphysicslab.com/Moveable-pendulum.html.

## Variables Array

The variables are stored in the VarsList as follows

``````vars[0] = theta_1   angle of rod 1
vars[1] = omega_1 = theta_1'  angular velocity of rod 1
vars[2] = theta_2   angle of rod 2
vars[3] = omega_2 = theta_2'  angular velocity of rod 2
vars[4] = t  time
vars[5] = x_0  anchor X position
vars[6] = vx_0 = x_0'  anchor X velocity
vars[7] = y_0  anchor Y position
vars[8] = vy_0 = y_0'  anchor Y velocity
``````

## To Do

Make a Parameter for whether to limit angles to +/- Pi. For some graphs you want this, but for other graphs you don't want it.

The energy values are not correct. When the anchor is moving then energy is being added to the pendulum. The potential energy should change from moving up and down in gravitational field. The kinetic energy should include the motion added by the anchor.

TO DO add ParameterBoolean specifying whether to limit angles to +/-Pi.

## Properties

amplitude_: number = 200

amplitude of driving force on anchor to make periodic up/down motion

anchorDamping_: number = 0.8

damping applied to anchor

damping_: number = 0.5

damping of pendulum

frequency_: number = 30

frequency of driving force on anchor to make periodic up/down motion

initialState_: null | number[] = null

Initial values.

length1_: number = 1

length of pendulum rod 1

length2_: number = 1

length of pendulum rod 2

mouse_: PointMass = ...

Follows the mouse position while applying spring force to anchor

pendulumDragging_: boolean = false

true when dragging pendulum bob

potentialOffset_: number = 0

potential energy offset

running_: boolean = false

Whether the simulation is running; determines whether mouse dragging of anchor results in applying spring force or just moving the anchor directly.

springDragging_: boolean = false

true when applying spring force to anchor by mouse drag

springStiffness_: number = 3

stiffness of spring made for dragging anchor

## Methods

• 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.

#### Returns void

• Adds the Parameter to the list of this Subject's available Parameters.

#### Throws

if a Parameter with the same name already exists.

• Notifies all Observers that the Parameter with the given name has changed by calling observe on each Observer.

#### Parameters

• ##### name: string

the language-independent or English name of the Parameter that has changed

#### Throws

if there is no Parameter with the given name

• 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.

• 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

• Return amplitude of driving force on anchor to make periodic up/down motion

#### Returns number

amplitude of driving force on anchor

• Return frequency of driving force on anchor to make periodic up/down motion

#### Returns number

frequency of driving force on anchor

• Returns the Parameter with the given name.

#### Parameters

• ##### name: string

the language-independent or English name of the Parameter

#### Returns Parameter

the Parameter with the given name

#### Throws

if there is no Parameter with the given name

• Returns the ParameterBoolean with the given name.

#### Parameters

• ##### name: string

the language-independent or English name of the ParameterBoolean

#### Returns ParameterBoolean

the ParameterBoolean with the given name

#### Throws

if there is no ParameterBoolean with the given name

• Returns the ParameterNumber with the given name.

#### Parameters

• ##### name: string

the language-independent or English name of the ParameterNumber

#### Returns ParameterNumber

the ParameterNumber with the given name

#### Throws

if there is no ParameterNumber with the given name

• Returns the ParameterString with the given name.

#### Parameters

• ##### name: string

the language-independent or English name of the ParameterString

#### Returns ParameterString

the ParameterString with the given name

#### Throws

if there is no ParameterString with the given name

• Returns the current Simulation time.

#### Returns number

the current Simulation time.

#### Throws

if there is no time variable for the simulation

• 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

• 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

• Removes the Observer from this Subject's list of Observers. An Observer may call `removeObserver` during its `observe` method.

#### Parameters

• ##### observer: Observer

the Observer to detach from list of Observers

#### Returns void

• Removes the Parameter from the list of this Subject's available Parameters.

#### Parameters

• ##### parameter: Parameter

the Parameter to remove

#### Returns void

• Set anchor damping factor

#### Parameters

• ##### value: number

anchor damping factor

#### Returns void

• Calculates anchor Y velocity, so that the anchor stays visible, as though in a 'steady state'. Otherwise the anchor tends to quickly wander off screen.

Derivation:

``````y'' = a sin(frequency t)
y' = integral(y'' dt) = -(a/frequency) cos(frequency t) + C
y = integral(y' dt) = -(a/frequency^2) sin(frequency t) + C t + C_2
``````

To avoid the anchor wandering need `C = 0` therefore

``````at time t = 0, this gives y' = -(a/frequency)
``````

#### Returns void

• 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

• Set amplitude of of driving force on anchor to make periodic up/down motion

#### Parameters

• ##### value: number

amplitude of driving force on anchor

#### Returns void

• Set frequency of driving force on anchor to make periodic up/down motion

#### Parameters

• ##### value: number

driving force on anchor

#### Returns void

• Set length of pendulum rod 1

#### Parameters

• ##### value: number

length of pendulum rod 1

#### Returns void

• Set length of pendulum rod 2

#### Parameters

• ##### value: number

length of pendulum rod 2

#### Returns void

• Set mass of pendulum bob 1

#### Parameters

• ##### value: number

mass of pendulum bob 1

#### Returns void

• Informs the simulation of whether the clock is running, which determines whether mouse dragging of anchor results in applying spring force or just moving the anchor directly.

#### Returns void

• Sets spring stiffness for dragging the anchor mass

#### Parameters

• ##### value: number

spring stiffness for dragging the anchor mass

#### Returns void

• Sets the Terminal object that this simulation can print data into.

#### Parameters

• ##### terminal: null | Terminal

the Terminal object that this simulation can print data into.

#### Returns void

• Sets the VarsList for this simulation.

#### Parameters

• ##### varsList: VarsList

the VarsList to use in this simulation

#### Returns void

• 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

• 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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