APPLIED MECHANICS

Lecture 02

Slovak University of TechnologyFaculty of Material Science and Technology in Trnava

ENERGY METHODS OF APPLIED MECHANICS

Energy Methods An alternative way to determine the equation of motion

and an alternative way to calculate the natural frequency of a system

Useful if the forces or torques acting on the object or mechanical part are difficult to determine

ENERGY METHODS OF APPLIED MECHANICS

Quantities used in these methods are scalars - scalar dynamics

Method provides a very powerful tool for two main reason: It considerably simplifies the analytical formulation of the motion

equations for a complex mechanical system It gives rise to approximate numerical methods for the solution for

both discrete and continuous systems in the most natural manner

ENERGY METHODS OF APPLIED MECHANICS Potential energy - the potential energy of mechanical systems Ep is

often stored in “springs” - (remember that for a spring F = kx0)

Kinetic energy - the kinetic energy of mechanical systems Ek is due to the motion of the “mass” in the system

20

00 2

100

kxkxdxFdxExx

p

202

1xmEk

Conservation of mechanical energy - for a simply, conservative (i.e. no damper), mass spring system the energy must be conserved.

max,max,0)(. pkpkpk EEEEdt

dconstEE

Principle of virtual work

Dynamic equilibrium of the particle (d’Alembert)

Let us consider that the particle follows during the time interval [t1, t2] a motion trajectory distinct from the real one ui. This allows us to define the virtual displacement of the particle the relationship

,,,, 3210 iXum iii

*iu

where ui represents the displacement of the particle, Xi are forces.

iii uuu *

021 )()( tutu ii

Principle of virtual work

The virtual work principle for the system of particles

Multiplying of equation of motion by associated virtual displacement and sum over the components

,)( 03

1

i

iiii uXum

„The virtual work produced by the effective forces acting on the particle during a virtual displacement ui is equal to zero“.

,)( 01

3

1

N

k iikikikk uXum

„If the virtual work equation is satisfied for any virtual displacement compatible with the kinematical constraints, the system is in dynamic

equilibrium“.

Principle of virtual work

The kinematical constraints the state of the system would be completely defined by the 3N

displacement components uik, the particles are submitted to kinematic constraints which

restrain their motion, they define dependency relationship between particles, they represent the instantaneous configuration, starting from the reference configuration xik, instantaneous configuration determined by

ik(t) = xik + uik(t) The kinematical constraints are divided on:

holonomic constraints - defined by f(ik(t)) = 0 Non-holonomic constraints

Principle of virtual work

The kinematical constraints are divided on: holonomic constraints - defined by

f(ik, t) = 0

scleronomic - constraints not explicitly dependent on time rheonomic – constraints explicitly dependent on time

non-holonomic constraints - defined by

0 ),,( tf ikik

These equations are generally not integrable.

Principle of virtual work

),,...,(),( tqqUtxu nikik 1

Generalized coordinates and displacements If s holonomic constraints exist between the 3N displacements

of the system, the number of DOF is then reduced to 3N - s. It is then necessary to define n = 3N - s generalized coordinates, noted in terms of which the displacements of the system of particles are expressed in the form

The virtual displacement compatible with the holonomic constraints may be expressed in the form

n

ss

s

ikik q

q

Uu

1

Principle of virtual work

Virtual work equation becomes

01 1

3

11 1

3

1

s

n

ss

N

k s

ik

iikks

n

s

N

k i s

ikikikk qQ

q

Uumq

q

UXum )(

where

is the generalized force

N

k i s

ikiks q

UXQ

1

3

1

Hamilton´s Principle Hamilton´s principle - time integrated form of the virtual work

principle obtained by transforming the expression

02

1 1

3

1

t

t

N

k iikikikk dtuXum )(

Applied forces Xik can be derived from the potential energy - virtual work is expressed in the form

p

n

sss

n

ss

N

k iikik EqQquX

11 1

3

1

The generalized forces are derived from the potential energy

s

ps q

EQ

Hamilton´s Principle

The term associated with inertia forces

2ikikk

ikikkikikkikikkikikkuum

uumuumuumuumdt

d )(

The kinetic energy

N

k iikikkk uumE

1

3

12

1

02

1

2

11

3

1

t

tpk

t

t

N

k iikikk dtEEuum )(

Then, time integrated form of the virtual work principle

Hamilton´s Principle In terms of generalized coordinates is expressed

021 )()( tqtq ss

n

ss

s

ikikik q

q

U

t

Uu

1

),,( tqqEE kk ),( tqEE pp

“Hamilton´s principle:

2

1

t

tpk dtEE )(The real trajectory of the system is such as the integral

remains stationary with respect to any compatible virtual displacement arbitrary between both instants t1, t2 but vanishing at the ends of the interval

02

1

2

1

t

t

t

tpk LdtdtEE )(

where L – is a kinetic potential or Lagrangian

Lagrange´s Equations of 2nd Order Using expression

n

ss

s

ks

s

kk q

q

Eq

q

EE

1

in equation for Hamilton´s principle

02

1 1

t

t

n

ss

s

kss

s

k dtqq

EqQ

q

E

The second term can be integrate by parts

2

1

2

1

2

1

t

ts

s

kt

ts

s

kt

ts

s

k dtqq

E

dt

dq

q

Edtq

q

E

Lagrange´s Equations of 2nd Order

Taking into account the boundary conditions the following is equivalent to Hamilton´s principle

2

1 1

t

ts

n

ss

s

k

s

k dtqQq

E

q

E

dt

d

The variation qs is arbitrary on the whole interval and the equations of motion result in the form obtained by Lagrange

nsQq

E

q

E

dt

ds

s

k

s

k ,...,,, 21

Lagrange´s Equations of 2nd Order

Classification of generalized forces Internal forces

Linking force - connection between two particles

Elastic force - elastic body - body for which any produced work is stored in a recoverable form - giving rise to variation of internal energy

03

1211

3

12211

iiii

iiiii uuXuXuXA )()(

n

sss

i

N

kik

ik

pp qQu

u

EE

1

3

1 1int,

with the generalized forces of elastic origin

s

ps q

EQ

int,

Lagrange´s Equations of 2nd Order

Dissipative force - remains parallel and in opposite direction to the velocity vector and is a functions of its modulus. Dissipative force acting of a mass particle k is expressed by

k

ikkkikik v

vvfCX )(

where Cik is a constant fk(vk) is the function expressing velocity dependence, vk is the absolute velocity of particle k

3

1

2

iikkk uv || v

The dissipative force

N

k s

kkkks q

vvfCQ

1 )(

The dissipative function

N

k

v

kkkdis

k

dvvfCE1 0

)(

Lagrange´s Equations of 2nd Order

External conservative force - conservative - their virtual work remains zero during a cycle

0 ss qQA

generalized force is expressed s

extp

q

EQ

,

External non-conservative force

generalized force is expressed

3

1 1i

N

k s

ikiks q

uXQ

General form of Lagrange equation of 2nd order of non-conservative systems with rheonomic constraints

.,...,,),( nstQq

E

q

E

q

E

q

E

dt

ds

s

dis

s

p

s

k

s

k 21