Alazard, Daniel and Sanfedino, Francesco A short course on TITOP models for space system modelling. (2021) In: IFAC Workshop on Aerospace Control Education WACE 2021, 9

(1)

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible

Any correspondence concerning this service should be sent to the repository administrator: tech-oatao@listes- diff.inp-toulouse.fr

This is a publisher’s version published in: http://oatao.univ-toulouse.fr/28758

To cite this version:

Alazard, Daniel and Sanfedino, Francesco A short course on TITOP models for space system modelling. (2021) In: IFAC Workshop on Aerospace Control Education WACE 2021, 9

September 2021 - 10 September 2021 (Virtual event, Italy).

Official URL: https://doi.org/10.1016/j.ifacol.2021.11.002

Open Archive Toulouse Archive Ouverte

(2)

IFAC PapersOnLine 54-12 (2021) 7–13

ScienceDirect ScienceDirect

Peer review under responsibility of International Federation of Automatic Control.

10.1016/j.ifacol.2021.11.002

10.1016/j.ifacol.2021.11.002 2405-8963

A short course on TITOP models for space system modelling

D. Alazard^∗F. Sanfedino^∗∗

∗ISAE-SUPAERO, Toulouse, FRANCE (e-mail:

[email protected]).

∗∗ISAE-SUPAERO, Toulouse, FRANCE (e-mail:

[email protected])

Abstract: The Two-Input Two-Output Port model approach was proposed as a general tool to model flexible multi-body systems with varying parameters. The associated MAT- LAB/SIMULINK toolbox (SDTlib) now includes some tutorials for space system engineer training. This paper focuses on the well-known spring-mass systems to illustrate the main properties of this approach before to consider the 6 degrees of freedom case.

Keywords: Multi-body, Flexible structure, Linear Parameter Varying, Space engineering.

1. INTRODUCTION

The design and control of spacecraft require more and more accurate knowledge-based models, from the pre- liminary design phase till the V&V (Validation & Ver- ification) phase of the ACS (Attitude Control System).

Parametric sensitivity analyzes are the core of the various tasks to be performed during such a development: (i) mechanical/control co-design (Murali et al. (2015); Gonzalez et al. (2016b)), (ii) pointing error budget (Sanfedino et al.

(2017)-, (iii) robustness analyses, ... . Although non-linear model still required to very last step V&V process, LPV (Linear Parameter Varying) models are quite representa- tive for most of the analyses and design phases. Indeed the linearity (or small-motion) assumption is fully justified for space systems and is commonly adopted.

To fullfill the model needs, the SDT (Satellite Dynamics Toolbox, Alazard et al. (2008); Alazard and Cumer (2011)) was developed and recently enriched with a SIMULINK libray (SDTlib Alazard and Sanfedino (2020)). The objective is to build the 6 d.o.f. (degrees of reedom) LPV model of complex flexible multi-body systems fully parameterized according the system geometric configuration and the size- able mechanical and/or control parameters. The SDT is based on the TITOP (Two-Input Two-Output Port) model of a flexible body where the 2 ports are the 6 d.o.f wrench and acceleration vectors at the connection point of this body with the parent and child sub-structures (Chebbi et al. (2017); Gonzalez et al. (2016a)). The user-guide of the SDTlib (Alazard and Sanfedino (2021)) includes also several tutorials to cover most of the dynamic behaviours requiring parametric models: (i) flexible rotating solar arrays, (ii) flexible robotic arm, (iii) gyroscopic stiffness due to on-board angular momentum, mechanism with closed-kienematic chains,...). The tutorial proposed in this paper aims to better understand the basic principles of the proposed modelling approach.

This work was supported by ESA (European Space Agency).

Fig. 1. The spring-mass system.

In section 2, the very well known model of the spring- mass system is revisited under the TITOP form (see also:

Alazard et al. (2015)). The model channel inversion is then used to build model of any kinds of open or closed kinematic chains of elementary spring-mass sub-systems.

Section 3 propose a summary of the approach in the 6 dofs case.

2. ILLUSTRATION IN THE SINGLE-AXIS CASE 2.1 Elementatry TITOP model of the spring-mass system Let us consider the spring-mass systemAbetween points P and C depicted in Figure 1. Only motions along x- axis are considered.k,mP,mCare the stiffness, the mass attached to the pointP and the mass attached to the point Cof the system A, respectively.

The objective is to compute the dynamic modelZ(s) ofA between 2 inputs (in blue):

• the force f./A,C applied by the external world onA at the pointC,

• the inertial acceleration ¨xP prescribed at the pointP, and 2 outputs (in red):

• the inertial acceleration ¨xC at the pointC,

• the force f_A/.,P applied byA on the external world at the pointP.

Only variations around the equilibrium conditions are considered.δxis the variation on the length of the vector

A short course on TITOP models for space system modelling

1. INTRODUCTION

A short course on TITOP models for space system modelling

1. INTRODUCTION

A short course on TITOP models for space system modelling

1. INTRODUCTION

A short course on TITOP models for space system modelling

1. INTRODUCTION

A short course on TITOP models for space system modelling

1. INTRODUCTION

• the force f_./A,C applied by the external world onA at the pointC,

(3)

8 D. Alazard et al. / IFAC PapersOnLine 54-12 (2021) 7–13

Fig. 2. Block-diagram model of the spring-mass system TITOP model.

Fig. 3. Spring-mass model dialog box.

−−→P C w.r.t its lengthl₀ at rest. The equilibrium conditions are:xP =const,xC =xp+l0, ˙xp= ˙xC= 0.

The Newton principle applied on each of the 2 masses leads to:

• mPx¨p=−f_A/.,P+kδx,

• mCx¨C =f_./A,C−kδx.

Considering thatδ¨x= ¨xC−x¨P, a state space representation ofZ(s), associated with the state vectorx= [δx δx]˙ ^T, reads:

δx˙ δx¨

= 0 1

−m^kC 0 δx δx˙

+ 0 0

1

mC −1 f./A,C

¨ xP

(1) x¨C

f_A/.,P

=

−m^kC 0 k 0 δx

δx˙

+ ₁

mC 0 0 −mP

f./A,C

¨ xP

(2), and the 2×2 transfer matrixZ(s) reads:

X¨C(s) F_A/.,P(s)

=

s² k

k −mPmC(s²+ω²_f) mC(s²+ω²_c,P)

_Z(s)

F./A,C(s) X¨P(s)

,

with: (3)

• ωf =_k(m

P+mC)

mPmC is the free frequency ofA,

• ωc,P =

k

mC is the cantilever frequency of Awhen it is cantilevered at pointP.

The TITOP model Z(s) can also be represented by a SIMULINK block diagram model, depicted in the Fig- ure 2 and embedded in a masked sub-system (Figure 3):

Note that the interest of the block-diagram representation,

w.r.t. to the state space or transfer matrix representation is that the number of occurences of the 3 mechanical parameters ofA(k,mP,mC) is minimized: each parameters is associated with a single gain. Thus the uncertain system which can be derived from thisSIMULINKmodel considering uncertain parameters fork,mP,mCis guaranteed to be minimal in terms of order and parametric occurences.

Numerical application: k = 1 (N m), mP = 2 (Kg), mC= 3 (Kg).

MK_elem

[a,b,c,d]=linmod(’MK_elem’);

Z=ss(a,b,c,d);

2.2 Channel (port) inversion

The open-loop dynamics of the model Z(s) , i.e. when the 2 inputs (f_./A,C and ¨xP) are null, corresponds to the clamped atP / free at Cboundary conditions, i.e. the roots of s²+ω_c,P² . SinceZ(s) is a square ( 2×2) system with an invertible direct-freedthrough, it is possible to inverse 1 or 2 input/output channel(s) :

• the channel # 1 is the channel fromf./A,C to ¨xC,

• the channel # 2 is the channel from ¨xP tof_A/.,P. That can be done thanks to the function invio of the SDTLIB.See also Chebbi et al. (2017)

Let us note Z⁻¹^I(s) the model Z(s) where the channels numbered in the vector of indexesI are inverted, then:

• the open-loop dynamics of Z⁻¹²(s) is the free at P / free at C dynamics: s²+ω_f²,

• the open-loop dynamics ofZ⁻¹^[1,2](s) =Z⁻¹(s) is the free atP / clamped atCdynamics: s²+ω_c,C² where ωc,C =

k

mP is the cantilever frequency of A when it is cantilevered at pointC,

• the open-loop dynamics of Z⁻¹¹(s) is theclamped at P / clamped at Cdynamics: s².

Indeed:

damp(Z)

Pole Damping Frequency Time Constant

(rad/seconds) (seconds) 0.00e+00 + 5.77e-01i 0.00e+00 5.77e-01 Inf 0.00e+00 - 5.77e-01i 0.00e+00 5.77e-01 Inf

damp(invio(Z,2))

damp(inv(Z))

damp(invio(Z,1))

(rad/seconds) (seconds)

0.00e+00 -1.00e+00 0.00e+00 Inf

It is surprising and interesting to note that the dynamics of Z⁻¹¹(s) corresponds to a double integrator s². Indeed, the two inputs of Z⁻¹¹(s) are ¨xC and ¨xP. Obviously, the steady state response ofZ⁻¹¹(s) to 2 independent steps on these 2 inputs creates infinite outputs (the forces f./A,C

andf_A/.,P). Thus the DCgain ofZ⁻¹¹(s)is:

Z⁻¹¹(0) =

−∞+∞

and the state equation for the model Z⁻¹¹(s) reads (ob- sviously):

δx˙ δ¨x

=0 1 0 0 δx

δx˙

+0 0 1 −1 x¨C

¨ xP

(4) f./A,C

f_A/.,P

= k 0 k 0 δx

δx˙ +

mC 0 0 −mP

¨ xC

¨ xP

. (5) If the 2 inputs are imposed to be equal: ¨xP = ¨xC = ¨x_A, then the 2 integrators must be reduced such that the transfer from ¨x_A tof./A,C andf_A/.,P is:

f_./A,C f_A/.,P

= mC

−mP

x¨_A. (6) Indeed, one can recover this resuts using the function minreal (minimal realization):

minreal(invio(Z,1)*[1;1])

2 states removed. D = u1

y1 3 y2 -2 Static gain.

One can also consider the "dual" property which consists to consider a single outputf./A=f./A,C−f_A/.,P(i.e. the total force applied toAby the external world) in order to recover the quite obvious static model:

f_./A= [mC mP]x¨C

¨ xP

: (7)

minreal([1 -1]*invio(Z,1))

2 states removed. D = u1 u2

y1 3 2

Static gain.

Conclusion: the main interest of the TITOP modelZ(s) is that it can be used to model the system A for any arbitrary boundary conditions at its tipsP andC, thanks to simple channel operations.

2.3 Assembling open-kinematic chains of spring-mass systems

The previous conclusion works not only for clamped or free boundary conditions but also for any dynamical systems connected toAat the point P or at the pointC. Indeed, let us consider the systemsSdepicted in the Figure 4 and

Fig. 4. An open kinematic chain of spring-mass systems.

Fig. 5. The block diagram sketch of the system described in Figure 4.

composed of 4 spring-mass sub-systems Aⁱ, i = 1,· · ·4. The objective is to find the model between the inputu(N), a force applied to the point P1, and the variations of positions of the pointsP₁(δxP1) andC₄(δxC4). For each sub-systemAⁱ, let us denote:

• Pi,Ci: the 2 points at the tips of Aⁱ,

• ki, mPi, mCi: its 3 mechanical parameters,

• Zi(s): its TITOP model associated with state vector xi= [δxi δx˙i]^T,

whose numerical values are provided in Table 1 Table 1. Numerical application

i ki(N/m) mPi(Kg) mCi(Kg)

1 1 2 3

2 4 5 6

3 7 8 9

4 10 11 12

The model of the whole systemS can then be defined by the interconnection of the 4 blocksZ₁⁻¹²(s), Z2(s),Z3(s) andZ₄(s) according to theSIMULINKmodel depicted in the Figure 5 and considering the action/re-action relations: u=−f_A₁/.,P₁; f./A1,C₁ =f_A₂/.,P₂ +f_A₃/.,P₃; f./A2,C₂ = f_A₄/.,P4; f_./A₃,C3 = f_./A₄,C4 = 0, and the accelaration constraints: ¨xP2 = ¨xP3= ¨x_C1, ¨xP4 = ¨xC2.

Finally: 2 double integators was added on ¨xP₁ and ¨xC₄

to obtain the position outputsδxP1 andδxC4 . Note that

(4)

0.00e+00 -1.00e+00 0.00e+00 Inf

It is surprising and interesting to note that the dynamics of Z⁻¹¹(s) corresponds to a double integrator s². Indeed, the two inputs of Z⁻¹¹(s) are ¨xC and ¨xP. Obviously, the steady state response ofZ⁻¹¹(s) to 2 independent steps on these 2 inputs creates infinite outputs (the forces f./A,C

andf_A/.,P). Thus the DCgain ofZ⁻¹¹(s)is:

Z⁻¹¹(0) =

−∞+∞

and the state equation for the model Z⁻¹¹(s) reads (ob- sviously):

δx˙ δ¨x

=0 1 0 0 δx

δx˙

+0 0 1 −1 x¨C

¨ xP

(4) f./A,C

f_A/.,P

= k 0 k 0 δx

δx˙ +

mC 0 0 −mP

¨ xC

¨ xP

. (5) If the 2 inputs are imposed to be equal: ¨xP = ¨xC = ¨x_A, then the 2 integrators must be reduced such that the transfer from ¨x_A tof./A,C andf_A/.,P is:

f_./A,C f_A/.,P

= mC

−mP

x¨_A. (6) Indeed, one can recover this resuts using the function minreal (minimal realization):

minreal(invio(Z,1)*[1;1])

2 states removed.

D = u1 y1 3 y2 -2 Static gain.

One can also consider the "dual" property which consists to consider a single outputf./A=f./A,C−f_A/.,P(i.e. the total force applied toAby the external world) in order to recover the quite obvious static model:

f_./A= [mC mP]x¨C

¨ xP

: (7)

minreal([1 -1]*invio(Z,1))

2 states removed.

D = u1 u2

y1 3 2

Static gain.

Conclusion: the main interest of the TITOP modelZ(s) is that it can be used to model the system A for any arbitrary boundary conditions at its tipsP andC, thanks to simple channel operations.

2.3 Assembling open-kinematic chains of spring-mass systems

The previous conclusion works not only for clamped or free boundary conditions but also for any dynamical systems connected toAat the point P or at the pointC. Indeed, let us consider the systemsSdepicted in the Figure 4 and

Fig. 4. An open kinematic chain of spring-mass systems.

composed of 4 spring-mass sub-systems Aⁱ, i = 1,· · ·4.

The objective is to find the model between the inputu(N), a force applied to the point P1, and the variations of positions of the pointsP₁(δxP1) andC₄(δxC4).

For each sub-systemAⁱ, let us denote:

• Pi,Ci: the 2 points at the tips ofAⁱ,

• ki, mPi, mCi: its 3 mechanical parameters,

• Zi(s): its TITOP model associated with state vector xi= [δxi δx˙i]^T,

whose numerical values are provided in Table 1 Table 1. Numerical application

i ki(N/m) mPi(Kg) mCi(Kg)

1 1 2 3

2 4 5 6

3 7 8 9

4 10 11 12

The model of the whole systemS can then be defined by the interconnection of the 4 blocksZ₁⁻¹²(s),Z2(s), Z3(s) andZ₄(s) according to theSIMULINKmodel depicted in the Figure 5 and considering the action/re-action relations:

u=−f_A₁/.,P₁; f./A1,C₁ =f_A₂/.,P₂ +f_A₃/.,P₃; f./A2,C₂ = f_A₄/.,P4; f_./A₃,C3 = f_./A₄,C4 = 0, and the accelaration constraints: ¨xP2 = ¨xP3 = ¨x_C1, ¨xP4 = ¨xC2.

Finally: 2 double integators was added on ¨xP₁ and ¨xC₄

to obtain the position outputsδxP1 andδxC4 . Note that

(5)

10 D. Alazard et al. / IFAC PapersOnLine 54-12 (2021) 7–13

Fig. 6. A closed-kinematic chain of spring-mass systems.

one of the 2 double integrators is non-miminal and can be removed considering the geometric constraints between the outputs and the internal state variables:

• δxC₄ =δxP₁+δx1+δx2+δx4,

• δx˙C4 =δx˙P1+δx˙₁+δx˙₂+δx˙₄.

Thus it is recommended to use minreal on the model derived from the SIMULINK model:

[a,b,c,d]=linmod(’MK_elem_x4’);

G=ss(a,b,c,d);

G=minreal(G);

2 states removed.

damp(G)

(rad/seconds) (seconds) -5.67e-18 + 6.54e-09i 8.67e-10 6.54e-09 1.76e+17 -5.67e-18 - 6.54e-09i 8.67e-10 6.54e-09 1.76e+17 -3.82e-16 + 4.90e-01i 7.79e-16 4.90e-01 2.62e+15 -3.82e-16 - 4.90e-01i 7.79e-16 4.90e-01 2.62e+15 2.50e-16 + 7.27e-01i -3.44e-16 7.27e-01 -4.00e+15 2.50e-16 - 7.27e-01i -3.44e-16 7.27e-01 -4.00e+15 2.78e-16 + 1.15e+00i -2.42e-16 1.15e+00 -3.60e+15 2.78e-16 - 1.15e+00i -2.42e-16 1.15e+00 -3.60e+15 -1.67e-16 + 1.27e+00i 1.32e-16 1.27e+00 6.00e+15 -1.67e-16 - 1.27e+00i 1.32e-16 1.27e+00 6.00e+15

Conclusion: One of the main interest in the proposed TITOP modeling approach is that the model of a flexible multi-body system is highly structured and based on the interconnection of the TITOP models of the various flexible bodies. Each TITOP model depends only on its own mechanical parameters. Thus this approach is suitable to derive minimal LFT (Linear Fractionar Transformation) model when parametric variations are taken into account.

2.4 Assembling closed-kinematic chains of spring-mass systems

Let us consider the systemS depicted in the Figure 6 and composed of 3 spring-mass sub-systems Aⁱ, i = 1,· · ·3.

Inside the subsytem A¹, a controlled force u(N) can be applied between the 2 tips masses. The objective is to find the model between the input u(N) and the variations of acceleration ¨xP1 of the pointP₁.

Numerical application: ki = 1 (N m), mPi = 2 (Kg), mCi= 3 (Kg), ∀i= 1,2,3. ThusZi(s) =Z(s).

Such a system is a closed kinematic chain mechanism since:

xP1 =xP3 and xC2 =xC3 at any time. The model of the

whole systemScan then be defined by the interconnection of the 3 blocks Z₁⁻¹²(s), Z₂(s), and Z₃⁻¹¹(s) according to the SIMULINK model depicted in the Figure 7 and considering the action/re-action relations:

f_./A₁_,P₁ =−u+f_A₃_/.,P3;f_./A₁_,C₁ =f_A₂_/.,P₂+u;

f./A2,C2 = −f./A3,C3 and the acceleration constraints:

¨

xP1 = ¨xP3, ¨xC2 = ¨xC3, and ¨xP2 = ¨xC1. MK_elem_cl

[a,b,c,d]=linmod(’MK_elem_cl’);

G=ss(a,b,c,d);

damp(G)

(rad/seconds) (seconds) 5.77e-17 + 6.17e-09i -9.35e-09 6.17e-09 -1.73e+16 5.77e-17 - 6.17e-09i -9.35e-09 6.17e-09 -1.73e+16 -4.90e-18 + 7.38e-01i 6.64e-18 7.38e-01 2.04e+17 -4.90e-18 - 7.38e-01i 6.64e-18 7.38e-01 2.04e+17 6.26e-18 + 8.30e-01i -7.54e-18 8.30e-01 -1.60e+17 6.26e-18 - 8.30e-01i -7.54e-18 8.30e-01 -1.60e+17

The dyamics is characterized by 2 flexible modes (at 0.738 (rad/s) and 0.83(rad/s)) and 2 eigenvalues around 0. These 2 eigenvalues corresponds to the geometric constraints between the internal state variables to maintain closed the kinematic chain for any value of the deforma- tion:

• δx₁+δx₂−δx₃= 0,

• δx˙1+δx˙2−δx˙3= 0.

They can be reduced using a minimal realization of the transfer usingminreal:

G=minreal(G);

2 states removed.

damp(G)

(rad/seconds) (seconds) 1.11e-16 + 7.38e-01i -1.51e-16 7.38e-01 -9.01e+15 1.11e-16 - 7.38e-01i -1.51e-16 7.38e-01 -9.01e+15

0.00e+00 + 8.30e-01i 0.00e+00 8.30e-01 Inf

0.00e+00 - 8.30e-01i 0.00e+00 8.30e-01 Inf

More complex example of closed-loop mechanism involving 2D plates and several closed kinematic chains or the 4 bar mechanism are proposed in the tutorial 9 A solar

Fig. 8. The flexible substructureA.

array with 3 pannels and the tutorial 11 the 4 bar mechanism of Alazard and Sanfedino (2021).

3. SIX DOFS CASE

The previous TITOP modelling approach can be directly extented to the 6 d.o.fs case while taking into account:

• the dynamics coupling between translations and ro- tations due to the geometry (lever-arm effect),

• the various frames in which are described the various bodiesAⁱ ,

• the 6 d.o.f.s behavior of the flexible modes of each bodieAⁱ. The dynamic model of this flexible behavior can be:

· (i) computed analytically for very simple shaped body: uniform beam, uniform rectangular plate,

· (ii) provided by finite element method softwares.

A NASTRAN/SIMULINK interface is available in theSDTlibfor this purpose.

3.1 Six d.o.fs TITOP model of a flexible body

Let us consider a flexible substructureA(link) connected a the parent substructureP at the pointPand to the child substructureC at pointC as depicted in Figure 8. R^a is the body frame attached to A. The TITOP (Two-Input Two-Output Port) model [M^A_P,C]_Ra(s) is a linear dynamic model between 12 inputs:

• the 6 components inR^a of the wrench [W_C/A,C]_Ra = F_C/A

T_C/A,C

Ra

applied by the child substructureC to Aat point C,

• the 6 components in R^a of the acceleration twist [¨xP]Ra = aP

˙ ωP

Ra

(time-derivative of the twist) of the pointP,

and 12 outputs:

• the 6 components in R^a of the acceleration twist [¨xC]Ra =aC

˙ ωC

Ra

of pointC,

• the 6 components in F_A/P R^aof the wrench [W_A/P,P]_Ra = T_A/P,P

Ra

applied by A to the parent substruc- tureP at pointP,

Fig. 9. Block diagram of the 12×12 TITOP model. and can be represented by the block-diagram depicted in Figure 9. The way to obtain the TITOP modelM_P,C^A (s) in the general case is detailed in Alazard et al. (2015) and is based on the modal analysis of the body A with the clamped at P - free at C boundary conditions. Indeed, the Lagrange’s formulation of dynamics leads to the generalized 2-nd order differential equation (all the terms are projected in the frameR^a, omitted for brevity):

D^AP L^TP

LP 1n

¨ xP

¨ η

+0 0

0diag(2ξjωj)

˙ η

+ 0 0

0diag(ω²_j) η

=

−I6 τ^T_CP

0 Φ^T_C W_A/P,P

W_C/A,C

(8)

where:

• nis the number of flexible modes of bodyAcharacter- ized by the modal coordinates vectorη, the frequen- ciesωj and the damping ratioξj, withj= 1,· · ·, n,

• 1n is the identity matrix of sizen,

• LP is then×6 modal participation factor matrix of the body at pointP,

• ΦC is the 6×nprojection matrix of then clamped- free modal shapes on the 6 d.o.fs at pointC,

• τCP is the "rigid” kinematic model between point C and P: τCP =

13 (^∗−−→CP) 03×3 13

where (^∗−−→CP) is the skew-symmetric matrix associated with the vector fromC toP,

• D^A_P is the 6×6 rigid mass model of the body at point P.D^A_P =τ^T_AP

m^A13 03×3

03×3 [I^A_A]

τAP whereA,m^A and I^A_A are the centre of mass, the mass and the inertia tensor atA, respectively.

The acceleration twist ¨xC at pointC reads:

¨

xC = [τCP ΦC]¨xP

¨ η

. (9)

These equations can be expressed with a block-diagram representation and thus a SIMULINK model as it is proposed in theSDTlib. All the required data (n, ωj, ξj, LP, ΦC, −−→CP, −→AP, m^A, I^A_A) can be extracted from the NASTRANfile of the bodyA(files***.bdfand ***.f06). Finally, the TITOP model of a flexible body can be easily extended to NINOP (N-Input, N-Output Ports) model, considering several child points C₁, C₂, ...( Sanfedino et al. (2018); Preda et al. (2020); Sanfedino et al. (2021); Murdoch et al. (2018)) .

For a uniform beam and a uniform rectangular plate, all the required data are derived analytically from the geometry and material characteristics in the blocks TITOP flexible beam (Chebbi et al. (2017)) and NINOP flexible plate. Rigid bodies can also be handled under this formalism thanks to the block Multi-port rigid body.