# File kernel/src/modelingTools/LagrangianLinearDiagonalDS.hpp¶

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class LagrangianLinearDiagonalDS : public LagrangianDS
#include <LagrangianLinearDiagonalDS.hpp>

Lagrangian Linear Systems with time invariant and diagonal coefficients - $$M\dot v + Cv + Kq = F_{ext}(t,z) + p$$.

The class LagrangianLinearDiagonalDS allows to define and compute a generic ndof-dimensional Lagrangian Linear Time Invariant Dynamical System of the form : where $$M \ddot q + C \dot q + K q = F_{ext}(t,z) + p,$$ where

• $$q \in R^{ndof}$$ is the set of the generalized coordinates,
• $$\dot q \in R^{ndof}$$ the velocity, i. e. the time derivative of the generalized coordinates.
• $$\ddot q \in R^{ndof}$$ the acceleration, i. e. the second time derivative of the generalized coordinates.
• $$p \in R^{ndof}$$ the forces due to the Non Smooth Interaction. In particular case of Non Smooth evolution, the variable p contains the impulse and not the force.
• $$M \in R^{ndof \times ndof}$$ is Mass matrix stored in the SiconosMatrix mass().
• $$K \in R^{ndof \times ndof}$$ is the stiffness matrix stored in the SiconosMatrix K().
• $$C \in R^{ndof \times ndof}$$ is the viscosity matrix stored in the SiconosMatrix C().
• $$z \in R^{zSize}$$ is a vector of arbitrary algebraic variables, some sort of discret state.
• all matrix coefficients are diagonal

If required (e.g. for Event-Driven like simulation), reformulation as a first-order system (DynamicalSystem) is possible, with:

• $$n= 2 ndof$$
• $$x = \left[\begin{array}{c}q \\ \dot q\end{array}\right]$$
• rhs given by:

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

Its jacobian is:

$\begin{split}\nabla_{x}rhs(x,t) = \left[\begin{array}{cc} 0 & I \\ -M^{-1}K & -M^{-1}C \\ \end{array}\right]\end{split}$
The input due to the non smooth law is:

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

Attributes access

const SiconosVector get_stiffness() const

get a copy of the stiffness matrix (diagonal only)

Return
SiconosVector

SP::SiconosVector stiffness() const

get stiffness matrix (diagonal only, pointer link)

Return
pointer on a SiconosVector

const SiconosVector get_damping() const

get a copy of the damping matrix (diagonal only)

Return
SiconosVector

SP::SiconosVector damping() const

get damping matrix (diagonal only, pointer link)

Return
pointer on a SiconosVector

Right-hand side computation

void initRhs(double t)

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

Parameters
• t: time of initialization

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

Compute $$F(v,q,t,z)$$.

Parameters
• time: the current time
• q: SP::SiconosVector: pointers on q
• velocity: SP::SiconosVector: pointers on velocity

Miscellaneous public methods

bool isLinear()

Return
true if the Dynamical system is linear.

void display() const

print the data of the dynamical system on the standard output

Public Functions

LagrangianLinearDiagonalDS(SP::SiconosVector q0, SP::SiconosVector v0, SP::SiconosVector stiffness, SP::SiconosVector damping, SP::SiconosVector mass)

constructor from initial state and all operators.

Parameters
• q0: initial coordinates
• v0: initial velocity
• stiffness: diagonal of the stiffness matrix
• damping: diagonal of the damping matrix
• mass: diagonal of the mass matrix

LagrangianLinearDiagonalDS(SP::SiconosVector q0, SP::SiconosVector v0, SP::SiconosVector stiffness, SP::SiconosVector damping)

constructor for complete system with identity mass matrix

Parameters
• q0: initial coordinates
• v0: initial velocity
• stiffness: diagonal of the stiffness matrix
• damping: diagonal of the damping matrix

LagrangianLinearDiagonalDS(SP::SiconosVector q0, SP::SiconosVector v0, SP::SiconosVector stiffness)

constructor for undamped system and identity mass matrix

Parameters
• q0: initial coordinates
• v0: initial velocity
• stiffness: diagonal of the stiffness matrix

~LagrangianLinearDiagonalDS()

destructor

ACCEPT_STD_VISITORS()

Protected Functions

LagrangianLinearDiagonalDS()

default constructor

ACCEPT_SERIALIZATION(LagrangianLinearDiagonalDS)

Protected Attributes

SP::SiconosVector _damping

damping matrix

double _mu

mass density

SP::SiconosVector _stiffness

stiffness matrix