# Class SecondOrderDS¶

class SecondOrderDS : public DynamicalSystem

Second Order non linear dynamical systems - $$M(q,z) \dot v = F(v, q, t, z) + p$$.

This class defines and computes a generic ndof-dimensional second order Non Linear Dynamical System of the form :

$\begin{split}M(q,z) \dot v = F(v, q, t, z) + p \\ \dot q = G(q,v)\end{split}$

where

• $$q \in R^{ndof}$$ is the set of the coordinates,

• $$\dot q =v \in R^{ndof}$$ the velocity,

• $$\ddot q =\dot v \in R^{ndof}$$ the acceleration, i. e. the second time derivative of the generalized coordinates.

• $$p \in R^{ndof}$$ the reaction forces due to the Non Smooth Interaction.

• $$M(q) \in R^{ndof \times ndof}$$ is the inertia term (access : mass() method).

• $$F(\dot q , q , t) \in R^{ndof}$$ are the forces (access forces() method).

• $$z \in R^{zSize}$$ is a vector of arbitrary algebraic variables, some sort of discrete state.

q[i] is the derivative number i of q. Thus: q[0]= $$q$$, global coordinates, q[1]= $$\dot q$$, velocity, q[2]= $$\ddot q$$, acceleration.

The following operators (and their jacobians) can be plugged, in the usual way (see User Guide, ‘User-defined plugins’)

• $$M(q)$$ (computeMass())

• $$F(v , q , t, z)$$ (computeF()) 3 If required (e.g. for Event-Driven like simulation), formulation as a first-order system is also available, and writes:

• $$n= 2 ndof$$

• $$x = \left[\begin{array}{c}q \\ \dot q\end{array}\right]$$

• rhs given by:

$\begin{split}\dot x = \left[\begin{array}{c} \dot q\\ \ddot q = M^{-1}(q)\left[F(v, q , t, z) + p \right]\\ \end{array}\right]\end{split}$

• jacobian of the rhs, with respect to x

$\begin{split}\nabla_{x}rhs(x,t) = \left[\begin{array}{cc} 0 & I \\ \nabla_{q}(M^{-1}(q)F(v, q , t, z)) & \nabla_{\dot q}(M^{-1}(q)F(v, q , t, z)) \\ \end{array}\right]\end{split}$

with the input due to the non smooth law:

$\begin{split}\left[\begin{array}{c} 0 \\ p \end{array}\right]\end{split}$

In that case, use the following methods:

Subclassed by LagrangianDS

Right-hand side computation

virtual void resetToInitialState() = 0

reset the state to the initial state

virtual void initRhs(double time) = 0

allocate (if needed) and compute rhs and its jacobian.

Parameters
• time: of initialization

virtual void initializeNonSmoothInput(unsigned int level) = 0

set nonsmooth input to zero

Parameters
• level: input-level to be initialized.

virtual void computeRhs(double time) = 0

update right-hand side for the current state

Parameters
• time: of interest

virtual void computeJacobianRhsx(double time) = 0

update $$\nabla_x rhs$$ for the current state

Parameters
• time: of interest

virtual void resetAllNonSmoothParts() = 0

reset non-smooth part of the rhs (i.e.

p), for all ‘levels’

virtual void resetNonSmoothPart(unsigned int level) = 0

set nonsmooth part of the rhs (i.e.

p) to zero for a given level

Parameters
• level:

virtual void setRhs(const SiconosVector &newValue)

set the value of the right-hand side, $$\dot x$$

Parameters

virtual void setRhsPtr(SP::SiconosVector newPtr)

set right-hand side, $$\dot x$$ (pointer link)

Parameters
• newPtr: SP::SiconosVector

virtual void computeForces(double time, SP::SiconosVector q, SP::SiconosVector velocity) = 0

function to compute $$F(v,q,t,z)$$ for the current state

Parameters
• time: the current timeCompute $$F(v,q,t,z)$$

• time: the current time

• q: SP::SiconosVector: pointers on q

• velocity: SP::SiconosVector: pointers on velocity

virtual void computeJacobianqForces(double time) = 0

Compute $$\nabla_qF(v,q,t,z)$$ for current $$q,v$$ Default function to compute forces.

Parameters
• time: the current time

virtual void computeJacobianvForces(double time) = 0

Compute $$\nabla_{\dot q}F(v,q,t,z)$$ for current $$q,v$$.

Parameters
• time: the current time

Attributes access

unsigned int dimension() const

return the number of degrees of freedom of the system

Return

an unsigned int.

virtual SP::SiconosVector q() const = 0

generalized coordinates of the system (vector of size dimension())

Return

pointer on a SiconosVector

virtual void setQ(const SiconosVector &newValue) = 0

set value of generalized coordinates vector (copy)

Parameters
• newValue:

virtual void setQPtr(SP::SiconosVector newPtr) = 0

set value of generalized coordinates vector (pointer link)

Parameters
• newPtr:

SP::SiconosVector q0() const

Return

pointer on a SiconosVector

virtual void setQ0(const SiconosVector &newValue) = 0

set initial state (copy)

Parameters
• newValue:

virtual void setQ0Ptr(SP::SiconosVector newPtr) = 0

Parameters
• newPtr:

virtual SP::SiconosVector velocity() const = 0

Return

pointer on a SiconosVector

virtual void setVelocity(const SiconosVector &newValue) = 0

set velocity vector (copy)

Parameters
• newValue:

virtual void setVelocityPtr(SP::SiconosVector newPtr) = 0

Parameters
• newPtr:

virtual SP::SiconosVector velocity0() const = 0

get initial velocity (pointer)

Return

pointer on a SiconosVector

virtual void setVelocity0(const SiconosVector &newValue) = 0

set initial velocity (copy)

Parameters
• newValue:

virtual void setVelocity0Ptr(SP::SiconosVector newPtr) = 0

Parameters
• newPtr:

virtual SP::SiconosVector acceleration() const = 0

Return

pointer on a SiconosVector

virtual SP::SiconosVector forces() const = 0

get $$F(v,q,t,z)$$ (pointer link)

Return

pointer on a SiconosVector

virtual SP::SiconosMatrix jacobianqForces() const = 0

get $$\nabla_qF(v,q,t,z)$$ (pointer link)

Return

pointer on a SiconosMatrix

virtual SP::SiconosMatrix jacobianvForces() const = 0

get $$\nabla_{\dot q}F(v,q,t,z)$$ (pointer link)

Return

pointer on a SiconosMatrix

Memory vectors management

virtual const SiconosMemory &qMemory() = 0

get all the values of the state vector q stored in memory.

note: not const due to SchatzmanPaoliOSI::initializeWorkVectorsForDS

Return

a memory

virtual const SiconosMemory &velocityMemory() = 0

get all the values of the state vector velocity stored in memory.

note: not const due to SchatzmanPaoliOSI::initializeWorkVectorsForDS

Return

a memory

virtual const SiconosMemory &forcesMemory() = 0

get forces in memory buff

Return

pointer on a SiconosMemory

virtual void initMemory(unsigned int size) = 0

initialize the SiconosMemory objects with a positive size.

Parameters
• size: the size of the SiconosMemory. must be >= 0

virtual void swapInMemory() = 0

push the current values of x, q and r in the stored previous values xMemory, qMemory, rMemory,

Public Functions

virtual ~SecondOrderDS()

destructor

SP::BoundaryCondition boundaryConditions()

get Boundary Conditions

Return

SP::BoundaryCondition pointer on a BoundaryConditions

virtual void computeMass() = 0

default function to compute the mass

virtual void computeMass(SP::SiconosVector position) = 0

function to compute the mass

Parameters
• position: value used to evaluate the mass matrix

virtual void display(bool brief = true) const = 0

print the data of the dynamical system on the standard output

virtual void init_forces() = 0

Allocate memory for forces and its jacobian.

virtual void init_inverse_mass() = 0

Allocate memory for the lu factorization of the mass of the system.

Useful for some integrators with system inversion involving the mass

SP::SimpleMatrix inverseMass() const

get (pointer) inverse or LU-factorization of the mass, used for LU-forward-backward computation

Return

pointer SP::SimpleMatrix

SP::SiconosMatrix mass() const

Return

SP::SiconosMatrix

SP::SiconosVector p(unsigned int level = 2) const

get p

Return

pointer on a SiconosVector

Parameters
• level: unsigned int, required level for p, default = 2

SP::SiconosVector reactionToBoundaryConditions()

get Reaction to Boundary Conditions

Return

pointer on a BoundaryConditions

virtual void setBoundaryConditions(SP::BoundaryCondition newbd)

set Boundary Conditions

Parameters
• newbd: BoundaryConditions

void setMassPtr(SP::SimpleMatrix newPtr)

set mass to pointer newPtr

Parameters
• newPtr: a plugged matrix SP

void setReactionToBoundaryConditions(SP::SiconosVector newrbd)

set Reaction to Boundary Conditions

Parameters
• newrbd: BoundaryConditions pointer

virtual void update_inverse_mass() = 0

Update the content of the lu factorization of the mass of the system, if required.

virtual void updatePlugins(double time) = 0

default function to update the plugins functions using a new time:

Parameters
• time: the current time