"""Implementations of actuators for linked force and torque application.""" from abc import ABC, abstractmethod from sympy import S, sympify from sympy.physics.mechanics.joint import PinJoint from sympy.physics.mechanics.loads import Torque from sympy.physics.mechanics.pathway import PathwayBase from sympy.physics.mechanics.rigidbody import RigidBody from sympy.physics.vector import ReferenceFrame, Vector __all__ = [ 'ActuatorBase', 'ForceActuator', 'LinearDamper', 'LinearSpring', 'TorqueActuator', 'DuffingSpring' ] class ActuatorBase(ABC): """Abstract base class for all actuator classes to inherit from. Notes ===== Instances of this class cannot be directly instantiated by users. However, it can be used to created custom actuator types through subclassing. """ def __init__(self): """Initializer for ``ActuatorBase``.""" pass @abstractmethod def to_loads(self): """Loads required by the equations of motion method classes. Explanation =========== ``KanesMethod`` requires a list of ``Point``-``Vector`` tuples to be passed to the ``loads`` parameters of its ``kanes_equations`` method when constructing the equations of motion. This method acts as a utility to produce the correctly-structred pairs of points and vectors required so that these can be easily concatenated with other items in the list of loads and passed to ``KanesMethod.kanes_equations``. These loads are also in the correct form to also be passed to the other equations of motion method classes, e.g. ``LagrangesMethod``. """ pass def __repr__(self): """Default representation of an actuator.""" return f'{self.__class__.__name__}()' class ForceActuator(ActuatorBase): """Force-producing actuator. Explanation =========== A ``ForceActuator`` is an actuator that produces a (expansile) force along its length. A force actuator uses a pathway instance to determine the direction and number of forces that it applies to a system. Consider the simplest case where a ``LinearPathway`` instance is used. This pathway is made up of two points that can move relative to each other, and results in a pair of equal and opposite forces acting on the endpoints. If the positive time-varying Euclidean distance between the two points is defined, then the "extension velocity" is the time derivative of this distance. The extension velocity is positive when the two points are moving away from each other and negative when moving closer to each other. The direction for the force acting on either point is determined by constructing a unit vector directed from the other point to this point. This establishes a sign convention such that a positive force magnitude tends to push the points apart, this is the meaning of "expansile" in this context. The following diagram shows the positive force sense and the distance between the points:: P Q o<--- F --->o | | |<--l(t)--->| Examples ======== To construct an actuator, an expression (or symbol) must be supplied to represent the force it can produce, alongside a pathway specifying its line of action. Let's also create a global reference frame and spatially fix one of the points in it while setting the other to be positioned such that it can freely move in the frame's x direction specified by the coordinate ``q``. >>> from sympy import symbols >>> from sympy.physics.mechanics import (ForceActuator, LinearPathway, ... Point, ReferenceFrame) >>> from sympy.physics.vector import dynamicsymbols >>> N = ReferenceFrame('N') >>> q = dynamicsymbols('q') >>> force = symbols('F') >>> pA, pB = Point('pA'), Point('pB') >>> pA.set_vel(N, 0) >>> pB.set_pos(pA, q*N.x) >>> pB.pos_from(pA) q(t)*N.x >>> linear_pathway = LinearPathway(pA, pB) >>> actuator = ForceActuator(force, linear_pathway) >>> actuator ForceActuator(F, LinearPathway(pA, pB)) Parameters ========== force : Expr The scalar expression defining the (expansile) force that the actuator produces. pathway : PathwayBase The pathway that the actuator follows. This must be an instance of a concrete subclass of ``PathwayBase``, e.g. ``LinearPathway``. """ def __init__(self, force, pathway): """Initializer for ``ForceActuator``. Parameters ========== force : Expr The scalar expression defining the (expansile) force that the actuator produces. pathway : PathwayBase The pathway that the actuator follows. This must be an instance of a concrete subclass of ``PathwayBase``, e.g. ``LinearPathway``. """ self.force = force self.pathway = pathway @property def force(self): """The magnitude of the force produced by the actuator.""" return self._force @force.setter def force(self, force): if hasattr(self, '_force'): msg = ( f'Can\'t set attribute `force` to {repr(force)} as it is ' f'immutable.' ) raise AttributeError(msg) self._force = sympify(force, strict=True) @property def pathway(self): """The ``Pathway`` defining the actuator's line of action.""" return self._pathway @pathway.setter def pathway(self, pathway): if hasattr(self, '_pathway'): msg = ( f'Can\'t set attribute `pathway` to {repr(pathway)} as it is ' f'immutable.' ) raise AttributeError(msg) if not isinstance(pathway, PathwayBase): msg = ( f'Value {repr(pathway)} passed to `pathway` was of type ' f'{type(pathway)}, must be {PathwayBase}.' ) raise TypeError(msg) self._pathway = pathway def to_loads(self): """Loads required by the equations of motion method classes. Explanation =========== ``KanesMethod`` requires a list of ``Point``-``Vector`` tuples to be passed to the ``loads`` parameters of its ``kanes_equations`` method when constructing the equations of motion. This method acts as a utility to produce the correctly-structred pairs of points and vectors required so that these can be easily concatenated with other items in the list of loads and passed to ``KanesMethod.kanes_equations``. These loads are also in the correct form to also be passed to the other equations of motion method classes, e.g. ``LagrangesMethod``. Examples ======== The below example shows how to generate the loads produced by a force actuator that follows a linear pathway. In this example we'll assume that the force actuator is being used to model a simple linear spring. First, create a linear pathway between two points separated by the coordinate ``q`` in the ``x`` direction of the global frame ``N``. >>> from sympy.physics.mechanics import (LinearPathway, Point, ... ReferenceFrame) >>> from sympy.physics.vector import dynamicsymbols >>> q = dynamicsymbols('q') >>> N = ReferenceFrame('N') >>> pA, pB = Point('pA'), Point('pB') >>> pB.set_pos(pA, q*N.x) >>> pathway = LinearPathway(pA, pB) Now create a symbol ``k`` to describe the spring's stiffness and instantiate a force actuator that produces a (contractile) force proportional to both the spring's stiffness and the pathway's length. Note that actuator classes use the sign convention that expansile forces are positive, so for a spring to produce a contractile force the spring force needs to be calculated as the negative for the stiffness multiplied by the length. >>> from sympy import symbols >>> from sympy.physics.mechanics import ForceActuator >>> stiffness = symbols('k') >>> spring_force = -stiffness*pathway.length >>> spring = ForceActuator(spring_force, pathway) The forces produced by the spring can be generated in the list of loads form that ``KanesMethod`` (and other equations of motion methods) requires by calling the ``to_loads`` method. >>> spring.to_loads() [(pA, k*q(t)*N.x), (pB, - k*q(t)*N.x)] A simple linear damper can be modeled in a similar way. Create another symbol ``c`` to describe the dampers damping coefficient. This time instantiate a force actuator that produces a force proportional to both the damper's damping coefficient and the pathway's extension velocity. Note that the damping force is negative as it acts in the opposite direction to which the damper is changing in length. >>> damping_coefficient = symbols('c') >>> damping_force = -damping_coefficient*pathway.extension_velocity >>> damper = ForceActuator(damping_force, pathway) Again, the forces produces by the damper can be generated by calling the ``to_loads`` method. >>> damper.to_loads() [(pA, c*Derivative(q(t), t)*N.x), (pB, - c*Derivative(q(t), t)*N.x)] """ return self.pathway.to_loads(self.force) def __repr__(self): """Representation of a ``ForceActuator``.""" return f'{self.__class__.__name__}({self.force}, {self.pathway})' class LinearSpring(ForceActuator): """A spring with its spring force as a linear function of its length. Explanation =========== Note that the "linear" in the name ``LinearSpring`` refers to the fact that the spring force is a linear function of the springs length. I.e. for a linear spring with stiffness ``k``, distance between its ends of ``x``, and an equilibrium length of ``0``, the spring force will be ``-k*x``, which is a linear function in ``x``. To create a spring that follows a linear, or straight, pathway between its two ends, a ``LinearPathway`` instance needs to be passed to the ``pathway`` parameter. A ``LinearSpring`` is a subclass of ``ForceActuator`` and so follows the same sign conventions for length, extension velocity, and the direction of the forces it applies to its points of attachment on bodies. The sign convention for the direction of forces is such that, for the case where a linear spring is instantiated with a ``LinearPathway`` instance as its pathway, they act to push the two ends of the spring away from one another. Because springs produces a contractile force and acts to pull the two ends together towards the equilibrium length when stretched, the scalar portion of the forces on the endpoint are negative in order to flip the sign of the forces on the endpoints when converted into vector quantities. The following diagram shows the positive force sense and the distance between the points:: P Q o<--- F --->o | | |<--l(t)--->| Examples ======== To construct a linear spring, an expression (or symbol) must be supplied to represent the stiffness (spring constant) of the spring, alongside a pathway specifying its line of action. Let's also create a global reference frame and spatially fix one of the points in it while setting the other to be positioned such that it can freely move in the frame's x direction specified by the coordinate ``q``. >>> from sympy import symbols >>> from sympy.physics.mechanics import (LinearPathway, LinearSpring, ... Point, ReferenceFrame) >>> from sympy.physics.vector import dynamicsymbols >>> N = ReferenceFrame('N') >>> q = dynamicsymbols('q') >>> stiffness = symbols('k') >>> pA, pB = Point('pA'), Point('pB') >>> pA.set_vel(N, 0) >>> pB.set_pos(pA, q*N.x) >>> pB.pos_from(pA) q(t)*N.x >>> linear_pathway = LinearPathway(pA, pB) >>> spring = LinearSpring(stiffness, linear_pathway) >>> spring LinearSpring(k, LinearPathway(pA, pB)) This spring will produce a force that is proportional to both its stiffness and the pathway's length. Note that this force is negative as SymPy's sign convention for actuators is that negative forces are contractile. >>> spring.force -k*sqrt(q(t)**2) To create a linear spring with a non-zero equilibrium length, an expression (or symbol) can be passed to the ``equilibrium_length`` parameter on construction on a ``LinearSpring`` instance. Let's create a symbol ``l`` to denote a non-zero equilibrium length and create another linear spring. >>> l = symbols('l') >>> spring = LinearSpring(stiffness, linear_pathway, equilibrium_length=l) >>> spring LinearSpring(k, LinearPathway(pA, pB), equilibrium_length=l) The spring force of this new spring is again proportional to both its stiffness and the pathway's length. However, the spring will not produce any force when ``q(t)`` equals ``l``. Note that the force will become expansile when ``q(t)`` is less than ``l``, as expected. >>> spring.force -k*(-l + sqrt(q(t)**2)) Parameters ========== stiffness : Expr The spring constant. pathway : PathwayBase The pathway that the actuator follows. This must be an instance of a concrete subclass of ``PathwayBase``, e.g. ``LinearPathway``. equilibrium_length : Expr, optional The length at which the spring is in equilibrium, i.e. it produces no force. The default value is 0, i.e. the spring force is a linear function of the pathway's length with no constant offset. See Also ======== ForceActuator: force-producing actuator (superclass of ``LinearSpring``). LinearPathway: straight-line pathway between a pair of points. """ def __init__(self, stiffness, pathway, equilibrium_length=S.Zero): """Initializer for ``LinearSpring``. Parameters ========== stiffness : Expr The spring constant. pathway : PathwayBase The pathway that the actuator follows. This must be an instance of a concrete subclass of ``PathwayBase``, e.g. ``LinearPathway``. equilibrium_length : Expr, optional The length at which the spring is in equilibrium, i.e. it produces no force. The default value is 0, i.e. the spring force is a linear function of the pathway's length with no constant offset. """ self.stiffness = stiffness self.pathway = pathway self.equilibrium_length = equilibrium_length @property def force(self): """The spring force produced by the linear spring.""" return -self.stiffness*(self.pathway.length - self.equilibrium_length) @force.setter def force(self, force): raise AttributeError('Can\'t set computed attribute `force`.') @property def stiffness(self): """The spring constant for the linear spring.""" return self._stiffness @stiffness.setter def stiffness(self, stiffness): if hasattr(self, '_stiffness'): msg = ( f'Can\'t set attribute `stiffness` to {repr(stiffness)} as it ' f'is immutable.' ) raise AttributeError(msg) self._stiffness = sympify(stiffness, strict=True) @property def equilibrium_length(self): """The length of the spring at which it produces no force.""" return self._equilibrium_length @equilibrium_length.setter def equilibrium_length(self, equilibrium_length): if hasattr(self, '_equilibrium_length'): msg = ( f'Can\'t set attribute `equilibrium_length` to ' f'{repr(equilibrium_length)} as it is immutable.' ) raise AttributeError(msg) self._equilibrium_length = sympify(equilibrium_length, strict=True) def __repr__(self): """Representation of a ``LinearSpring``.""" string = f'{self.__class__.__name__}({self.stiffness}, {self.pathway}' if self.equilibrium_length == S.Zero: string += ')' else: string += f', equilibrium_length={self.equilibrium_length})' return string class LinearDamper(ForceActuator): """A damper whose force is a linear function of its extension velocity. Explanation =========== Note that the "linear" in the name ``LinearDamper`` refers to the fact that the damping force is a linear function of the damper's rate of change in its length. I.e. for a linear damper with damping ``c`` and extension velocity ``v``, the damping force will be ``-c*v``, which is a linear function in ``v``. To create a damper that follows a linear, or straight, pathway between its two ends, a ``LinearPathway`` instance needs to be passed to the ``pathway`` parameter. A ``LinearDamper`` is a subclass of ``ForceActuator`` and so follows the same sign conventions for length, extension velocity, and the direction of the forces it applies to its points of attachment on bodies. The sign convention for the direction of forces is such that, for the case where a linear damper is instantiated with a ``LinearPathway`` instance as its pathway, they act to push the two ends of the damper away from one another. Because dampers produce a force that opposes the direction of change in length, when extension velocity is positive the scalar portions of the forces applied at the two endpoints are negative in order to flip the sign of the forces on the endpoints wen converted into vector quantities. When extension velocity is negative (i.e. when the damper is shortening), the scalar portions of the fofces applied are also negative so that the signs cancel producing forces on the endpoints that are in the same direction as the positive sign convention for the forces at the endpoints of the pathway (i.e. they act to push the endpoints away from one another). The following diagram shows the positive force sense and the distance between the points:: P Q o<--- F --->o | | |<--l(t)--->| Examples ======== To construct a linear damper, an expression (or symbol) must be supplied to represent the damping coefficient of the damper (we'll use the symbol ``c``), alongside a pathway specifying its line of action. Let's also create a global reference frame and spatially fix one of the points in it while setting the other to be positioned such that it can freely move in the frame's x direction specified by the coordinate ``q``. The velocity that the two points move away from one another can be specified by the coordinate ``u`` where ``u`` is the first time derivative of ``q`` (i.e., ``u = Derivative(q(t), t)``). >>> from sympy import symbols >>> from sympy.physics.mechanics import (LinearDamper, LinearPathway, ... Point, ReferenceFrame) >>> from sympy.physics.vector import dynamicsymbols >>> N = ReferenceFrame('N') >>> q = dynamicsymbols('q') >>> damping = symbols('c') >>> pA, pB = Point('pA'), Point('pB') >>> pA.set_vel(N, 0) >>> pB.set_pos(pA, q*N.x) >>> pB.pos_from(pA) q(t)*N.x >>> pB.vel(N) Derivative(q(t), t)*N.x >>> linear_pathway = LinearPathway(pA, pB) >>> damper = LinearDamper(damping, linear_pathway) >>> damper LinearDamper(c, LinearPathway(pA, pB)) This damper will produce a force that is proportional to both its damping coefficient and the pathway's extension length. Note that this force is negative as SymPy's sign convention for actuators is that negative forces are contractile and the damping force of the damper will oppose the direction of length change. >>> damper.force -c*sqrt(q(t)**2)*Derivative(q(t), t)/q(t) Parameters ========== damping : Expr The damping constant. pathway : PathwayBase The pathway that the actuator follows. This must be an instance of a concrete subclass of ``PathwayBase``, e.g. ``LinearPathway``. See Also ======== ForceActuator: force-producing actuator (superclass of ``LinearDamper``). LinearPathway: straight-line pathway between a pair of points. """ def __init__(self, damping, pathway): """Initializer for ``LinearDamper``. Parameters ========== damping : Expr The damping constant. pathway : PathwayBase The pathway that the actuator follows. This must be an instance of a concrete subclass of ``PathwayBase``, e.g. ``LinearPathway``. """ self.damping = damping self.pathway = pathway @property def force(self): """The damping force produced by the linear damper.""" return -self.damping*self.pathway.extension_velocity @force.setter def force(self, force): raise AttributeError('Can\'t set computed attribute `force`.') @property def damping(self): """The damping constant for the linear damper.""" return self._damping @damping.setter def damping(self, damping): if hasattr(self, '_damping'): msg = ( f'Can\'t set attribute `damping` to {repr(damping)} as it is ' f'immutable.' ) raise AttributeError(msg) self._damping = sympify(damping, strict=True) def __repr__(self): """Representation of a ``LinearDamper``.""" return f'{self.__class__.__name__}({self.damping}, {self.pathway})' class TorqueActuator(ActuatorBase): """Torque-producing actuator. Explanation =========== A ``TorqueActuator`` is an actuator that produces a pair of equal and opposite torques on a pair of bodies. Examples ======== To construct a torque actuator, an expression (or symbol) must be supplied to represent the torque it can produce, alongside a vector specifying the axis about which the torque will act, and a pair of frames on which the torque will act. >>> from sympy import symbols >>> from sympy.physics.mechanics import (ReferenceFrame, RigidBody, ... TorqueActuator) >>> N = ReferenceFrame('N') >>> A = ReferenceFrame('A') >>> torque = symbols('T') >>> axis = N.z >>> parent = RigidBody('parent', frame=N) >>> child = RigidBody('child', frame=A) >>> bodies = (child, parent) >>> actuator = TorqueActuator(torque, axis, *bodies) >>> actuator TorqueActuator(T, axis=N.z, target_frame=A, reaction_frame=N) Note that because torques actually act on frames, not bodies, ``TorqueActuator`` will extract the frame associated with a ``RigidBody`` when one is passed instead of a ``ReferenceFrame``. Parameters ========== torque : Expr The scalar expression defining the torque that the actuator produces. axis : Vector The axis about which the actuator applies torques. target_frame : ReferenceFrame | RigidBody The primary frame on which the actuator will apply the torque. reaction_frame : ReferenceFrame | RigidBody | None The secondary frame on which the actuator will apply the torque. Note that the (equal and opposite) reaction torque is applied to this frame. """ def __init__(self, torque, axis, target_frame, reaction_frame=None): """Initializer for ``TorqueActuator``. Parameters ========== torque : Expr The scalar expression defining the torque that the actuator produces. axis : Vector The axis about which the actuator applies torques. target_frame : ReferenceFrame | RigidBody The primary frame on which the actuator will apply the torque. reaction_frame : ReferenceFrame | RigidBody | None The secondary frame on which the actuator will apply the torque. Note that the (equal and opposite) reaction torque is applied to this frame. """ self.torque = torque self.axis = axis self.target_frame = target_frame self.reaction_frame = reaction_frame @classmethod def at_pin_joint(cls, torque, pin_joint): """Alternate construtor to instantiate from a ``PinJoint`` instance. Examples ======== To create a pin joint the ``PinJoint`` class requires a name, parent body, and child body to be passed to its constructor. It is also possible to control the joint axis using the ``joint_axis`` keyword argument. In this example let's use the parent body's reference frame's z-axis as the joint axis. >>> from sympy.physics.mechanics import (PinJoint, ReferenceFrame, ... RigidBody, TorqueActuator) >>> N = ReferenceFrame('N') >>> A = ReferenceFrame('A') >>> parent = RigidBody('parent', frame=N) >>> child = RigidBody('child', frame=A) >>> pin_joint = PinJoint( ... 'pin', ... parent, ... child, ... joint_axis=N.z, ... ) Let's also create a symbol ``T`` that will represent the torque applied by the torque actuator. >>> from sympy import symbols >>> torque = symbols('T') To create the torque actuator from the ``torque`` and ``pin_joint`` variables previously instantiated, these can be passed to the alternate constructor class method ``at_pin_joint`` of the ``TorqueActuator`` class. It should be noted that a positive torque will cause a positive displacement of the joint coordinate or that the torque is applied on the child body with a reaction torque on the parent. >>> actuator = TorqueActuator.at_pin_joint(torque, pin_joint) >>> actuator TorqueActuator(T, axis=N.z, target_frame=A, reaction_frame=N) Parameters ========== torque : Expr The scalar expression defining the torque that the actuator produces. pin_joint : PinJoint The pin joint, and by association the parent and child bodies, on which the torque actuator will act. The pair of bodies acted upon by the torque actuator are the parent and child bodies of the pin joint, with the child acting as the reaction body. The pin joint's axis is used as the axis about which the torque actuator will apply its torque. """ if not isinstance(pin_joint, PinJoint): msg = ( f'Value {repr(pin_joint)} passed to `pin_joint` was of type ' f'{type(pin_joint)}, must be {PinJoint}.' ) raise TypeError(msg) return cls( torque, pin_joint.joint_axis, pin_joint.child_interframe, pin_joint.parent_interframe, ) @property def torque(self): """The magnitude of the torque produced by the actuator.""" return self._torque @torque.setter def torque(self, torque): if hasattr(self, '_torque'): msg = ( f'Can\'t set attribute `torque` to {repr(torque)} as it is ' f'immutable.' ) raise AttributeError(msg) self._torque = sympify(torque, strict=True) @property def axis(self): """The axis about which the torque acts.""" return self._axis @axis.setter def axis(self, axis): if hasattr(self, '_axis'): msg = ( f'Can\'t set attribute `axis` to {repr(axis)} as it is ' f'immutable.' ) raise AttributeError(msg) if not isinstance(axis, Vector): msg = ( f'Value {repr(axis)} passed to `axis` was of type ' f'{type(axis)}, must be {Vector}.' ) raise TypeError(msg) self._axis = axis @property def target_frame(self): """The primary reference frames on which the torque will act.""" return self._target_frame @target_frame.setter def target_frame(self, target_frame): if hasattr(self, '_target_frame'): msg = ( f'Can\'t set attribute `target_frame` to {repr(target_frame)} ' f'as it is immutable.' ) raise AttributeError(msg) if isinstance(target_frame, RigidBody): target_frame = target_frame.frame elif not isinstance(target_frame, ReferenceFrame): msg = ( f'Value {repr(target_frame)} passed to `target_frame` was of ' f'type {type(target_frame)}, must be {ReferenceFrame}.' ) raise TypeError(msg) self._target_frame = target_frame @property def reaction_frame(self): """The primary reference frames on which the torque will act.""" return self._reaction_frame @reaction_frame.setter def reaction_frame(self, reaction_frame): if hasattr(self, '_reaction_frame'): msg = ( f'Can\'t set attribute `reaction_frame` to ' f'{repr(reaction_frame)} as it is immutable.' ) raise AttributeError(msg) if isinstance(reaction_frame, RigidBody): reaction_frame = reaction_frame.frame elif ( not isinstance(reaction_frame, ReferenceFrame) and reaction_frame is not None ): msg = ( f'Value {repr(reaction_frame)} passed to `reaction_frame` was ' f'of type {type(reaction_frame)}, must be {ReferenceFrame}.' ) raise TypeError(msg) self._reaction_frame = reaction_frame def to_loads(self): """Loads required by the equations of motion method classes. Explanation =========== ``KanesMethod`` requires a list of ``Point``-``Vector`` tuples to be passed to the ``loads`` parameters of its ``kanes_equations`` method when constructing the equations of motion. This method acts as a utility to produce the correctly-structred pairs of points and vectors required so that these can be easily concatenated with other items in the list of loads and passed to ``KanesMethod.kanes_equations``. These loads are also in the correct form to also be passed to the other equations of motion method classes, e.g. ``LagrangesMethod``. Examples ======== The below example shows how to generate the loads produced by a torque actuator that acts on a pair of bodies attached by a pin joint. >>> from sympy import symbols >>> from sympy.physics.mechanics import (PinJoint, ReferenceFrame, ... RigidBody, TorqueActuator) >>> torque = symbols('T') >>> N = ReferenceFrame('N') >>> A = ReferenceFrame('A') >>> parent = RigidBody('parent', frame=N) >>> child = RigidBody('child', frame=A) >>> pin_joint = PinJoint( ... 'pin', ... parent, ... child, ... joint_axis=N.z, ... ) >>> actuator = TorqueActuator.at_pin_joint(torque, pin_joint) The forces produces by the damper can be generated by calling the ``to_loads`` method. >>> actuator.to_loads() [(A, T*N.z), (N, - T*N.z)] Alternatively, if a torque actuator is created without a reaction frame then the loads returned by the ``to_loads`` method will contain just the single load acting on the target frame. >>> actuator = TorqueActuator(torque, N.z, N) >>> actuator.to_loads() [(N, T*N.z)] """ loads = [ Torque(self.target_frame, self.torque*self.axis), ] if self.reaction_frame is not None: loads.append(Torque(self.reaction_frame, -self.torque*self.axis)) return loads def __repr__(self): """Representation of a ``TorqueActuator``.""" string = ( f'{self.__class__.__name__}({self.torque}, axis={self.axis}, ' f'target_frame={self.target_frame}' ) if self.reaction_frame is not None: string += f', reaction_frame={self.reaction_frame})' else: string += ')' return string class DuffingSpring(ForceActuator): """A nonlinear spring based on the Duffing equation. Explanation =========== Here, ``DuffingSpring`` represents the force exerted by a nonlinear spring based on the Duffing equation: F = -beta*x-alpha*x**3, where x is the displacement from the equilibrium position, beta is the linear spring constant, and alpha is the coefficient for the nonlinear cubic term. Parameters ========== linear_stiffness : Expr The linear stiffness coefficient (beta). nonlinear_stiffness : Expr The nonlinear stiffness coefficient (alpha). pathway : PathwayBase The pathway that the actuator follows. equilibrium_length : Expr, optional The length at which the spring is in equilibrium (x). """ def __init__(self, linear_stiffness, nonlinear_stiffness, pathway, equilibrium_length=S.Zero): self.linear_stiffness = sympify(linear_stiffness, strict=True) self.nonlinear_stiffness = sympify(nonlinear_stiffness, strict=True) self.equilibrium_length = sympify(equilibrium_length, strict=True) if not isinstance(pathway, PathwayBase): raise TypeError("pathway must be an instance of PathwayBase.") self._pathway = pathway @property def linear_stiffness(self): return self._linear_stiffness @linear_stiffness.setter def linear_stiffness(self, linear_stiffness): if hasattr(self, '_linear_stiffness'): msg = ( f'Can\'t set attribute `linear_stiffness` to ' f'{repr(linear_stiffness)} as it is immutable.' ) raise AttributeError(msg) self._linear_stiffness = sympify(linear_stiffness, strict=True) @property def nonlinear_stiffness(self): return self._nonlinear_stiffness @nonlinear_stiffness.setter def nonlinear_stiffness(self, nonlinear_stiffness): if hasattr(self, '_nonlinear_stiffness'): msg = ( f'Can\'t set attribute `nonlinear_stiffness` to ' f'{repr(nonlinear_stiffness)} as it is immutable.' ) raise AttributeError(msg) self._nonlinear_stiffness = sympify(nonlinear_stiffness, strict=True) @property def pathway(self): return self._pathway @pathway.setter def pathway(self, pathway): if hasattr(self, '_pathway'): msg = ( f'Can\'t set attribute `pathway` to {repr(pathway)} as it is ' f'immutable.' ) raise AttributeError(msg) if not isinstance(pathway, PathwayBase): msg = ( f'Value {repr(pathway)} passed to `pathway` was of type ' f'{type(pathway)}, must be {PathwayBase}.' ) raise TypeError(msg) self._pathway = pathway @property def equilibrium_length(self): return self._equilibrium_length @equilibrium_length.setter def equilibrium_length(self, equilibrium_length): if hasattr(self, '_equilibrium_length'): msg = ( f'Can\'t set attribute `equilibrium_length` to ' f'{repr(equilibrium_length)} as it is immutable.' ) raise AttributeError(msg) self._equilibrium_length = sympify(equilibrium_length, strict=True) @property def force(self): """The force produced by the Duffing spring.""" displacement = self.pathway.length - self.equilibrium_length return -self.linear_stiffness * displacement - self.nonlinear_stiffness * displacement**3 @force.setter def force(self, force): if hasattr(self, '_force'): msg = ( f'Can\'t set attribute `force` to {repr(force)} as it is ' f'immutable.' ) raise AttributeError(msg) self._force = sympify(force, strict=True) def __repr__(self): return (f"{self.__class__.__name__}(" f"{self.linear_stiffness}, {self.nonlinear_stiffness}, {self.pathway}, " f"equilibrium_length={self.equilibrium_length})")