k M20 ar First name : Name :

Name :

First name : k

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Examination duration: 1h50 – Write your answer to the questions on the examination paper.

Only the results are required.

Apart from this examination paper, no other document should be given back.

Answer the following questions on mechanics:

- On mechanics, when modelling a multi-technology system, what is the “ideal core”?

- On mechanics, which reasons justify to change the textual model into the vector model?

- What is the vector model of a kinematic body?

- What is the vector model of the helical joint with a reduced pitch λ and an axis ?

- Which unvarying vector elements allow removing the three degrees of freedom of rotation in a joint between two bodies Si and Sj?

- Which quantities allow introducing the time variable in the study of kinematics?

- Which conditions must be checked to ascertain a minimum path between bases?

- What is the meaning of the mass moment of inertia?

Signature

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Exercise:

Owing to a hydraulic power steering, a driver needn’t develop an important torque on the steering wheel in order to turn the wheels. This small force is always the same, no matter what the car’s speed is, may causes safety problems (high speed swerving).

To solve this problem, Citroen has introduced a return torque proportional to the speed of the car.

In order to take into account this speed, engineers decided to amplify this load, using a hydraulic distributor as a load intensifier associated with a centrifugal governor, which allows the return torque at the steering wheel varying in terms of the speed of the car.

The purpose of this centrifugal governor, driven by the drive hose from the output of the gearbox, is to provide a chamber in fluid under a controlled pressure as an increasing function of its spin, therefore the speed of the car.

The motion of the centrifugal governor shown below will be studied, after the body (S₂) initially in contact with the body (S1) has left the stop (translation on page 7/7).

1. Textual dynamic model and kinematic sketch 1.1. Geometry and distributed mass modelling The system identifies:

- Four rigid bodies: (S₀), (S₁), (S₂) and (S₃):

- The ground (S₀),

- The input shaft and its housing (S₁), a body of revolution,

- The centrifugal weight (S₂) (only one centrifugal weight is taken into account in the study) which plane (G₂,normal z: v₁₂₃

) is a plane of symmetry, - The plate (S₃), a body of revolution,

- A compressive spring (R), of negligible mass, located between the two bodies (S₁) and (S₃);

- Six rigid joints:

(S₀ - S1): revolute (S₁ - S2): revolute (S₂ - S3): point surface (S₁ - S3): prismatic (R – S1): sphere (R - S3): sphere

- A drive hose, located in parallel with the joint (S₀ - S₁), imparts the input shaft (position, velocity and acceleration are known functions of time).

B12

G2

A01

D3 C01

xr013

zr123

y13

r S1

S₃

S2

S₀

Drive hose

Distributor

y2

r

x₀₁₃ θ z0

z₁₂₃

y₀ y₁₃

y13

z123

α

x013

x2

y2 Points A, B, C, D and G2 are

located in the plane of the sketch.

Stop

(0; 1,3)

(1,3; 2)

R

1.2. Forces modelling

The spring is supposed to be a linear elastic spring of stiffness k and natural length l0.

The actions of the drive hose on the input shaft (S₁) are represented by a couple twistor of momentC_m xr₀₁₃

.

The actions of the distributor on the plate (S₃) are represented by a resultant twistor of axis of the wrench (D₃, xr₀₁₃

) and known resultantFxr₀₁₃

, defined in terms of the distributor characteristics.

The revolute, prismatic and sphere joints are supposed to be perfect joints unlike the point surface joint. For that joint, the sliding friction is taken into account through the Coulomb’s model defined in the plane (B₁₂,normal z: v₁₂₃

).

The system moves in the gravitational field which is defined by the upward vertical η

µ λ η

µ λ η

µ

λxr₀₁₃+ yr₀+ zr₀ ^{with 2}+ ²+ ² =^{1and , ,}

are supposed to remain constant during the study.

1.3. Galilean reference frame

During the study, the fixed body (S₀) frame is supposed to be a Galilean reference frame.

2. Construct the vector geometric model 2.1. Model the joints

- Draw the sketch of the joints

- Write the vector models of the joints

2.2. Model the rigid bodies

( )

[ ]

( )

[ ]

( )

[ ]

( )

[

]

3 3

123 2 2 2

2 2

123 13 013 1 1

1

0 0 013 0

0

, ,

; ,

, ,

; ,

, ,

; , , ,

, ,

; ,

z y x G

D R R

z y x G

B R R

z y x G C B A R R

z y x C

A R R

r r r

r r r

r r r

r r r

=

=

=

=

S2

S1

S0

S3

2.3. Define the parameters

- use a minimum path between the bases

- use a minimum path between the points

013 3 013

2 2 1 013

013

13 AC ax AG bx BG cy DG dx CD xx

y r

AB r r r r r r

=

−

=

=

−

=

−

=

=

2.4. Find the equations of constraint - take the joints not used yet into account

- take the specifications of some joints not used yet into account

- take the kinematical behaviour laws of the motors into account

2.5. Define the kinematically independent parameters

3. Express the laws of behaviour in terms of the geometric model 3.1. The compressive spring (R)

3.2. The twistor applied by the drive hose

{

}

3.3. The twistor applied by the distributor

{

}

B A

G2 G3

b0 b_{1, 3} b₂

θ α

G₁

C D

3.4. The point surface joint (S₂ - S3)

Let I be the contact point between (S₂) and (S₃) such as IB exr₂

=

- express the sliding velocity of (S2) with respect to (S3) at the contact point I between these two bodies:

( )

G_3,2 r

- Express the resultant of the twistor of the interaction forces (S₃) acting upon (S₂)

{

}

sr

- Condition of existence of the point surface joint:

- Coulomb’s law

3.5. The zero components of the interaction forces, according to the assumptions of perfect joints and the nonperfect joints with the rigid bodies.

3.6. The gravitational field

4. Gather the unknowns of the study

* Case of rolling, no sliding:

° Condition of existence

° Equation

* Case of rotation and sliding:

° Condition of existence

° Equation

5. Write the dynamic equations 5.1. Define the cut

5.2. Define the sketch of the characteristics

5.3. Write the scalar consequences of the dynamic equations

= with E =

= with E =

S2

S1

S0

S3 + Unknowns

S2

S1

S0

S₃

+ Unknowns

+ Unknowns

(In the following you keep the initial parameters) 5.4. Compute the components of the external forces

=

=

5.5. Compute the components of kinetics

=

=

Grâce à une direction assistée hydraulique, le conducteur d’un véhicule n’a pas besoin de produire un couple très important sur le volant pour orienter les roues. Ce faible effort, indépendant de la vitesse du véhicule, peut poser un problème de sécurité (coup de volant à grande vitesse).

Pour remédier à ce problème, Citroën a introduit un couple de rappel lié à la vitesse du véhicule.

Afin de prendre en compte cette vitesse, les concepteurs ont décidé d’amplifier l’effort résistant en utilisant un distributeur hydraulique multiplicateur d’effort associé à un régulateur centrifuge, qui permet de faire varier le couple de rappel du volant en fonction de la vitesse du véhicule.

Le rôle du "régulateur centrifuge", entraîné par flexible depuis la sortie de la boîte de vitesses, consiste à alimenter une chambre en fluide sous une pression régulée qui est une fonction croissante de sa vitesse de rotation, donc de la vitesse du véhicule.

On propose d'étudier le mouvement du régulateur centrifuge représenté ci-dessous, à partir du moment où le solide S2 en contact avec le solide S1 aura quitté la butée de repos.