First name: Name:

Name:

First name:

Examination duration: 2h – Write your answer to the questions on the examination paper.

All the results will be proved

No document allowed - SMARTPHONES, GSM and calculators prohibited First part:

The motion of the device shown below and the contact maintaining condition between the bodies (S₀) and (S₁) will be studied.

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

The system modelled as a planar motion, identifies:

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

- (S₀) the ground,

- (S_*) of negligible mass,

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

- (S₂) a punctual body of centre of inertia G₂ and mass m₂,

- a tensile-compressive spring (R01), of negligible mass, located between the bodies (S₀) at point O0 and (S₁) at point B1;

- a damper (Am01), of negligible mass, located between the bodies (S₀) at point O0

and (S₁) at point B1;

G1

A*1

0 12

*

0 z

z r

r =

y2

r

S_*

S₂

S₀

*

y0

r

G2

B1

S1

xr1

gr

O0

C12

Spring (R01) Damper (Am01)

Signature

2/10 M. Meyer, D. Sauhet

- six kinematic joints :

(S₀ – S*): prismatic (S_* – S1): revolute (S₀ – S1): point surface (S₁ – S2): revolute (R01 – S0) : sphere (R01 – S1) : sphere (Am₀₁ – S0): sphere (Am₀₁ – S1): sphere 1.2. Forces

The spring R₀₁ is supposed to be a linear elastic spring of stiffness k and natural length l₀.

The damper Am01 is supposed to be a viscous damper of viscous damping coefficient b.

All the joints are supposed to be perfect joints.

The system moves in the gravitational field which is defined by the place upward verticalyr_0*

.

1.3. Galilean reference frame

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

2. Construct the vector geometric model

Since the vector geometric model construction was the subject of the mid-term exam, the first steps results of the system model are given so as to principally devote oneself, in the impart time, to the study of dynamics

2.1. Model the joints

- draw the sketch of the joints

2.2. Model the rigid bodies

( )

[ ]

( )

[ ]

( )

[ ]

( )

[

]

2 2

12

* 0 1 1 1 1

1

12

* 0

* 0

* 0

*

*

12

* 0

* 0

* 0 0

0

, ,

; ,

, ,

; , , ,

, ,

;

, ,

;

z y x G

C R R

z y x G C B A R R

z y x A

R R

z y x O

R R

r r r

r r r

r r r

r r r

=

=

=

=

S₂ S1

S_*

S₀

Sphere

( )

Sphere

( )

Revolute

(

)

Point surface

(?)

Revolute

(

)

Prismatic

( )

0 , , ,

* 0

* 0

* 0

* 0

* 0

*

0 r r

r

r r

=

∧

∧O A y y

z y A O

2.3. Parameterize

- use a minimum path between the bases

- use a minimum path between the points

2 2

1 1 1

1 1

*

0 AB 2ax AC ax ay AG ax CG 2ay

y y

OA r r r r r r

−

=

= +

=

=

=

BEWARE YOUR WORK STARTS HERE

☺

2.4. Study the equations of constraint

- use the joints not taken yet into account through their vector model o Consequence of the point surface joint (S₀ − S₁)

- use the geometric condition of some joints not taken yet into account o Consequence of the prismatic joint (S₀ − S_*)

z0*12

α y_0*

y1

x0*

x1

(0*; 1)

z0*12

β y0*

y2

x0*

x₂

(0*; 2)

A G₁

O C

G2

B

(S1)

(S1)

(S2) (S1)

b1 b0* b2

α _β

4/10 M. Meyer, D. Sauhet

2.5. Define the kinematically independent parameters

o Number of kinematically independent parameters : Number = o Define the kinematically independent parameters :

In the following, as part of this final-exam and so as to write equations independent of the equations of constraint, one retains the initial parameters.

3. Express the laws of behaviour in terms of the geometric model 3.1. The spring (R₀₁): Express the resultant sr

{

}

{ }



= 







= 

→

_ 0 R _ 1

01 0

x S F

R r

r

*

In the following, one retains sr

{

}

=

→

3.2. The viscous damper (Am01): Express the resultant sr

{

}

{ }



= 







= 

→

_ 0 A _ 1

01 0

x S F

Am r

r

*

In the following, one retains sr

{

}

=

→ 3.3. The point surface joint(S₀ − S₁)

Condition of existence of this one-sided joint: 3.4. The perfect joints between the rigid bodies

Fill up the sketch with the characteristics of the perfect joints.

S2

S1

S*

S0

3.5. The gravitational field

4. Gather the unknowns of the study

- the kinematically independent parameters:

- the components of the interaction 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 new components of interaction forces not listed yet as the unknowns of the study in question 4.

5.2. Define the new unknowns of the study

5.3. Define the sketch of the characteristics

S2

S₁ S*

S₀

S₂ S1

S_*

+ +

+

S₀

6/10 M. Meyer, D. Sauhet

5.4. Write the scalar consequences of the dynamic equations

= with E =

= with E =

= with E =

In the following one retains the initial parameters and the spring (R₀₁) and the viscous damper (Am01) forces as:

{ }







= 

→ 0

x S F

R ^{R 0}

_ 1

01 r

r

* and

{ }







= 

→ 0

x S F

Am ^{A 0}

_ 1

01 r

r

*

5.5. Compute the components of the forces

=

=

=

5.6. Compute the components of kinetics

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

{

}

sr

8/10 M. Meyer, D. Sauhet

- Compute the dynamic moment about the point C of the body (S₂) in its motion with respect to (S₀)

( )

2 ,

δ^r0

- Write, without computing, the relations to compute the dynamic moment about the point A of the bodies (S₁) and (S₂) in their motion with respect to (S₀)

( )

U 2 1 ,

δ^r0

Second part:

Study, with SIMULINK, the motion of a viscously damped spring-mass system is subjected to the displacement d(t) of the support as shown in the figure below.

Build, page 10, a Simulink model that solves the following equation of motion of the system:

The variable x denotes the displacement of the mass from its equilibrium position.

Given informations:

- m = 20 kg: mass of the system, - k = 17,5 kN/m: spring stiffness,

- b = 240 N.s/m: viscous damping coefficient,

- d(t) = 50 sin (10t) mm: displacement of the support.

Initial conditions:

- x(t = 0) = x₀ = 0, - x’(t = 0) = x’0 = 0.

Display of results:

The displacement x will be displayed as:

- a curve x = x(t), - a matrix x = x(t).

Spring (k)

Mass (m)

Displacement d(t)

Displacement x

) ( )

(t bd t d

k kx x b x

m&&+ &+ = + &

Viscous damper (b)

10/10 M. Meyer, D. Sauhet

SIMULINK model: