S S S z = xx r rr

Name : First name :

Examination duration: 2h

This examination paper must be returned with your copy sheet All the results will be proved

No document allowed - SMARTPHONES, GSM and calculators prohibited

The motion of the experimental device shown below will be studied.

1. Textual dynamic model and kinematic sketch 1.1. Geometry and mass

The system, modelled as a planar motion, identifies:

- an actuator (Ac01) which applies to the body (S₁) a known explicit function of time horizontal force F(t),

- a tensile - compressive spring (R₁₂), of negligeable mass, located between the bodies (S₁) at the attachment point B₁ and (S₂) at the attachment point D₂,

- a damper (Am₁₂), of negligeable mass, located between the bodies (S₁) at the attachment point B1 and (S₂) at the attachment point D2,

Signature

Actuator (AC₀₁)

0

01

x

x r r

23

=

y r

S

z

r

S

S

Damper (Am₁₂)

S

c a b

d

B1

D2

G1 C13

G3

g

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

- (S₀) the ground,

- (S₁) of centre of inertia G1 and m1, - (S₂) of negligeable mass,

- (S₃) of centre of inertia G₃ and m₃, - six kinematic joints :

(S₀ – S1): prismatic (S₁ – S2): prismatic

(S₁ – S3): revolute (S₂ – S3): two sided point surface (R₁₂Am₁₂ – S1): sphere (R₁₂Am₁₂ – S2): sphere

1.2. Forces

All the joints are supposed to be perfect joints.

The spring (R12)is supposed to be a linear elastic spring, of stiffness k and natural length l0.

The damper (Am₁₂) is supposed to be a viscous damper of viscous damping coefficient ν.

The system moves in the gravitational field which is defined by the vertical upward direction zr₀

.

1.3. Galilean reference frame

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

Reminder : _dt^{d tan}

( )

[

( )

]

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

- draw the sketch of the joints

S0

S2

[ ]

0 y B A y

x y B A

Prismatic

01 1 0 01

01 01 1

0 r r

r

r r

=

×

×

[ ]

0 y D B y

x y D B

Prismatic

12 2 1 12

12 12 2

1 r r

r

r r

=

∧

∧ S1

S3

[

]

Revolute r

[ ]

Point

[ ]

Sphere

[ ]

Sphere

2.2. Model the rigid bodies

( )

[ ]

( )

[ ]

( )

[ ]

( )

[

]

3 3

012 012 0 2

2

012 012 0 1 1

1

012 012 0 0

0

z , y , x

; G C R R

z , y , x

; D R R

z , y , x

; G C B R R

z , y , x

; A R R

r r r

r r r

r r r

r r r

=

=

=

=

2.3. Parameterize

- use a minimum path between the bases

- use a minimum path between the points

3 3

1

z d CG z

b y a BC

z b y c BG y

y AD y

y AB

012 012

012 012

012 2 012

1

r r r

r r

r r

−

=

−

=

−

=

=

=

ATTENTION: YOUR WORK STARTS HERE

2.4. Study the equations of constraint

- use the joints not taken yet into account through their vector model Consequence of the point surface joint (S2 − S3)

- check the geometric condition of the joints not taken yet into account Consequence of the prismatic joint (S0 – S1)

Consequence of the prismatic joint (S1 – S2)

- use the kinematic laws of behaviour introduced by the actuators xr0

zr012

z3

r

y012

r y3

r

(012 - 3)

α

b₀₁₂ α b₃

xr0

(S₁)

D A B G1 C G3

(S1)

(S3)

2.5. Define the kinematically independent parameters - Number of kinematically independent parameters: Number = - Define the kinematically independent parameters:

In the following, as part of this final exam, one must retain the parameters y₁ and α. Resolve y₂ andy&₂ in terms of ^y₁^{, y&}₁^,α^etα^& from the equation of constraint below:

0 b α

a y

y₂ − ₁ − + tan =

3. Express the laws of behaviour in terms of the geometric model - the actuator (Ac₀₁)

{ }



= 

→

B

01 S

Ac ₁

- the spring (R12)

{ }



= 

→

D 2

12 S

R

In the following, one must retain: sr

{

}

=

→ - the damper (Am₁₂)

{ }



= 

→

D

S2

Am₁₂

In the following, one must retain: sr

{

}

=

→

- the zero components of the interaction forces in the perfect joints

0 0

0 0

0

=

=

=

=

=

- the gravitational field

{ } { }



= 

 →



=

→

G3

3 G

1 g S

S g

1

4. Gather the unknowns of the study

- the kinematically independent parameters:

- the components of the intercation forces introduced from the laws of behaviour:

- the components of the interaction forces unknowns of the study :

5. Write the dynamic equations 5.1. Define the cut

Locate the cut and define the number of new components of interaction forces not listed yet as the unknowns of the study in section 4.

The cut is located at the joint level:

Unknowns of the study:

5.2. Define the sketch of the characteristics

5.3. Write the scalar consequences of the dynamic equations S₀

S2

S₁

S3

+ S0

S2

S1

S₃ +

+

=

=

=

=

=

=

E with

E with

E with

5.4. Compute the components of forces

5.5. Compute the components of kinetics

One recalls that the dynamic twistor of a massless body is the zero twistor.

- Compute the dynamic resultant of the bodies (S₁), (S₂) and (S₃) in their motion with respect to (S₀)

- Compute the dynamic moment about the point C f the body (S₃) in its motion with respect to (S₀)

5.6. Write the dynamic equations

=

=

=

=

=

=