Class LagrangianLinearTIDS¶

class LagrangianLinearTIDS : public LagrangianDS

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

The class LagrangianLinearTIDS allows to define and compute a generic ndof-dimensional Lagrangian Linear Time Invariant Dynamical System of the form :

$$M \ddot q + C \dot q + K q = F_{ext}(t,z) + p,$$

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.

The equation of motion is also shortly denoted as: $$M(q,z) \dot v = F(v, q, t, z) + p$$

where

• $$F(v, q, t, z) \in R^{ndof}$$ collects the total forces acting on the system, that is $$F(v, q, t, z) = F_{ext}(t, z) - Cv - Kq$$ This vector is stored in the SiconosVector forces()

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:

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]
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:

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

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

Attributes access

const SimpleMatrix getK() const

get a copy of the stiffness matrix

Return
SimpleMatrix

SP::SiconosMatrix K() const

Return
pointer on a SiconosMatrix

void setK(const SiconosMatrix &K)

set (copy) the value of the stiffness matrix

Parameters
• K: new stiffness matrix

void setKPtr(SP::SiconosMatrix newPtr)

Parameters
• newPtr: pointer to the new Stiffness matrix

const SimpleMatrix getC() const

get a copy of the damping matrix

Return
SimpleMatrix

SP::SiconosMatrix C() const

Return
pointer on a SiconosMatrix

void setC(const SiconosMatrix &C)

set (copy) the value of the damping matrix

Parameters
• C: new damping matrix

void setCPtr(SP::SiconosMatrix newPtr)

Parameters
• newPtr: pointer to the new damping matrix

SP::SiconosMatrix jacobianqForces() const

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

Return
pointer on a SiconosMatrix

SP::SiconosMatrix jacobianqDotForces() const

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

Return
pointer on a SiconosMatrix

Miscellaneous public methods

virtual bool isLinear()

Return
true if the Dynamical system is linear.

void display() const

print the data onto the screen

Public Functions

LagrangianLinearTIDS(SP::SiconosVector q0, SP::SiconosVector v0, SP::SiconosMatrix M, SP::SiconosMatrix K, SP::SiconosMatrix C)

constructor from initial state and all matrix operators.

Parameters
• q0: initial coordinates
• v0: initial velocity
• M: mass matrix
• K: stiffness matrix
• C: damping matrix

LagrangianLinearTIDS(SP::SiconosVector q0, SP::SiconosVector v0, SP::SiconosMatrix M)

constructor from initial state and mass matrix only.

Leads to $$M\dot v = F_{ext}(t,z) + p$$.

Parameters
• q0: initial coordinates
• v0: initial velocity
• M: mass matrix

~LagrangianLinearTIDS()

destructor