FOUNDATION of MECHANICS 1FOUNDATION of MECHANICS 1diem1.ing.unibo.it/personale/troncossi/FOM1/... · 2018-04-05 · ELEMENTS OF DYNAMICS Energy equation (I)– Special formulation

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Presentation04: Elements of Statics and Dynamics

Outline

• Elements of Statics: Cardinal Equations; analytical and graphical methods for static analysis of mechanisms; Principle of Virtual Work.

• Elements of Dynamics: definitions; D’Alembert principle (Cardinal Equations for dynamics); Principle of Virtual Work; Energy equations

ELEMENTS OF STATICS

Moment of a force F with Cardinal Equations for Statics

Fn

P O M F

respect to a given pole O A rigid body is in static equilibrium conditions if the following eqns. hold:

1j

j

R F 0

Fn

O P O M F

1

F

O j jj

P O

M F 0Moment of a force F with respect to a given pole O’

O

0Fn

F 0Fn

P O F

' '

'O P O

P O O O

M F

F F

1

0

0

F

jxj

n

F

F

1

0

0F

j j xj

n

P O

OP

F

F

P O O O F F1

0

0Fn

jyj

F

F

1

0

0F

j j yj

n

P

O

O

P

F

F1

0jzj

F

1

0j j zj

P O

F

ELEMENTS OF STATICS: Example of analytical methodExample of analytical method

Known: a, b, c, dB, dD N

Q3, Q1, Q2, Wa, WB, WD

Unknowns:

1 21

0 0x x

N

jxj

N

F Q QA C

Unknowns:P, Cx, Cz, Ax, Ay, Az

10 0

0 0

yjy a b dj

N

F W W W

F Q

A

A C P

31

3

0 0

0 ( ) ( )(2 ) 0

z zjzj

N

j

F Q

M a b Q a b

A C P

C P

31

2 1 3

0 ( ) ( )(2 ) 0

0 ( ) ( ) 02 2

jxj

z

NB D

jy

M a b Q a b

d dM Q Q

C P

P Q

1

1 21

2 2

0 ( ) ( ) 0X

jyj

N

jzj

CM a b Q Q a

1j

ELEMENTS OF STATICS

VFree body diagram

M

VV

MM

V

+M +Linear systems Principle of Effect Superposition

ELEMENTS OF STATICS: GRAPHICAL METHOD

Case 1 Two forcesCase 1

C = | F1 | a

lC = | F1 | a

Pure Moment(Couple)

Case 2 F1 a (Torque)

l1

F2

P

Couple of Forces

l2

F2

1

Fn

jj

R F 0

Fn

Couple of Forces

1

F

O Ojj

M M 0

ELEMENTS OF STATICS

Cardinal Equations for Statics

Fn

1

jj

R F 0

CF nn

1 1

CF

O j j jj jP O

M F C 0 O

To highlight the torques

ELEMENTS OF STATICS: GRAPHICAL METHOD

Three forcesCase 3Case 3

l1l3

l2

F1

P

l1l A

1

Fn

jj

R F 0

Fn l2 Al31

F

O Ojj

M M 0

ELEMENTS OF STATICS: GRAPHICAL METHOD

Case 3′ Three forces

F1A

F2 F3

a b

l1 l2 l31

Fn

jj

R F 0

Fn

1

F

O Ojj

M M 0

ELEMENTS OF STATICS: GRAPHICAL METHOD

Case 4 Four forces

l1

l2 ll42 l3laux

1

Fn

jj

R F 0

Fn

1

F

O Ojj

M M 0

ELEMENTS OF STATICS: GRAPHICAL METHOD

Case 4′ Four forces

l2

l1 l4l

Impossible problem

l3

1

Fn

jj

R F 0

Fn

1

F

O Ojj

M M 0

ELEMENTS OF STATICS: GRAPHICAL METHOD

Case 4′′ Four forces

l1 l2

l3

l4

Indefinite problem1

Fn

jj

R F 0

Fn

1

F

O Ojj

M M 0

ELEMENTS OF STATICS: ENERGETIC METHODS

Principle of Virtual Works

An ideal mechanical system is in equilibrium if and only if the net virtual work of all the active actions vanishes

for every set of reversible virtual displacements.

Ideal mechanical system = system where constraints do no workActive actions= all forces and torques which do non-zero virtual workVirtual displacements = a set of ideal infinitesimal displacements which are

i t t ith th t i tconsistent with the constraintsVirtual work = work done by specified actions on virtual displacements

NN

P.V.W.1 1

0CF NN

j j j jj j

F r C

ELEMENTS OF STATICS: ENERGETIC METHODS

Principle of Virtual Works

j j j jd dt d dt r v ω

The statement of the principle remains valid if the terms virtual work and virtual displacements are replaced by the terms virtual power and virtual velocity respectively.

For practical application in static analysis of mechanisms

CF NN

For practical application in static analysis of mechanisms, it is used as “Principle of Virtual Powers”

1 10

CF

j j j jj j

F v C ω

ELEMENTS OF STATICS: ENERGETIC METHODS

Principle of Virtual WorksProblem: Given the force Q acting on the slider, compute the crank torque M

0 0N

j j F r P δB M δφ

P v0 M P v 0 M PN

B k F v

10 0j j

j

F r P δB M δφ

11 1

0 M P v 0 M Pj j B Vj

k

F v

sin(2 )v (sin( ) )r A

B1

2M=?

1

1 2 2

( )v (sin( ) )2 1 sin ( )

B r

O

B3 P

vB

4

ELEMENTS OF DYNAMICS

Center of Mass (C.o.M.) Center of Mass Velocity XYdm

m

PO'

GO'P

m

dmV

V

Center of Mass (C.o.M.) Center of Mass Velocity

G

OY Y’P

dm

mGO G m

V

(Linear) Momentum Angular Momentum

O’Z X’

Z’

Gm

P mdm VVQ ( )O Pm

P O dm K V

in in P Gdd dm mdt

QF F a a

Resultant of the inertia forces

m m dt

( ) ( )P O d P O dm M F aResultant moment of the inertia forces with respect to a pole O

, ( ) ( )in O in Pm m

P O d P O dm M F a

ELEMENTS OF DYNAMICS

( ) ( )P O d P O d M F( )P O dK V , ( ) ( )in O in Pm m

P O d P O dm M F a( )O Pm

P O dm K V

d

K ( ) ( )OP P P P O P iO

m m m m

d P O dm P O dm dm dmdt

K V a V V V V M

, ( )O O Oin O O P O P O G

m m

d d ddm dm mdt dt dt

d

K K KM V V V V V V

KOO

ddt

K V Q = 0 if VO = 0 or O≡G or VO // VG

0J J J K J

;

M J J

0; ; 0

0

x x xy xz z y

y O xy y yz z x

z xz yz z y x

J J JJ J JJ J J

J O OK J

, ;in O O O M J J

ELEMENTS OF DYNAMICS

JInertia Tensor computed in a certain system of reference with origin O

OJInertia Tensor computed in a certain system of reference with origin O

ELEMENTS OF DYNAMICS

D’ALEMBERT principle (Cardinal Equations for Dynamics)

A rigid body is in dynamic equilibrium conditions if the following equations hold:

1

Fn

inj

F F 0

, ,1 1

CF nn

O in O j j j in in Gj jP O G O

M M F C F C 0OO

Principle of Virtual Works (for Dynamics)

1NN

1

, ,1 1 1

( ) 0CF

k

NN m

j j j j k k in G k kj j k

m

F r C a G C

ELEMENTS OF DYNAMICS

Energy equation (I)

– Special formulation of the P.V.W. when we consider real infinitesimal displacements instead of virtual displacements:

dd

r r

1

,1 1 1

( ) 0CF NN m

j j j j k k in k kj j k

m

F r C a G C

– if the system internal energy does not change and the mechanical system exchanges energy with the external environment only in the form of mechanical energy:form of mechanical energy:

where0 idLdW

• dW = infinitesimal work done by the external, active, non-conservative actions (forces and torques);

• dLi = infinitesimal work done by the inertia actions• dLi = infinitesimal work done by the inertia actions

ELEMENTS OF DYNAMICS

Energy equation (II)

21 ( ( ))T O P O d

1 if is fixed2 OT O J

Kinetic energy:

2( ( ))2 m

T O P O dm 221 1 if G

2 2G GT m O v J

2 21 12 2G Gz zT mv J Motion in the plane xy: ωx = ωy = 02 2

212 Gz zT J Body rotating around G (z axis)2

2 21 ( | | )2 Gz zT J m OG Body rotating around O G (z axis)2

ELEMENTS OF DYNAMICS

Energy equation (II)

The different contributions associated with driving and resistantexternal actions (dL and dL respectively) can be made explicit in dW

( )indW dL dT

external actions (dLm and dLr respectively) can be made explicit in dW.

Moreover the non ideal nature of the constraints (presence of friction)m rdW dL dL

Moreover, the non-ideal nature of the constraints (presence of friction) could be taken into account by introducing the energy loss due to the passive work (dLp) done by the internal reactions.

m r p indL dL dL dL dT (where the left-side terms must beconsidered as positive, e.g. dLr = |dLr |)

dLm := motor (or driving) workm ( g) |dLr |:= useful (resistant) work |dLp |:= passive (or lost or dissipated) work

m r p inL L L L T (Equation of Works)

ELEMENTS OF DYNAMICS

Energy equation (III)

d T T V

LAGRANGE equation

; 1,...,kk k

k

d T T V Q k ldt q qq

: DOFs of the mechanisml

: O s o e ec s: generalized coordinate

: generalized forcek

lq

Q

: generalized force

: kinetic energy of the mechanismt ti l f th h i

kQ

TV

: potential energy of the mechanismV