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Slovak University of Technology
Faculty of Material Science and Technology in Trnava
APPLIED MECHANICS
Lecture 02
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)
x0
x0
0
0
E p   Fdx   kxdx 

1 2
kx0
2
Kinetic energy - the kinetic energy of mechanical systems Ek is due to
the motion of the “mass” in the system
1 2
Ek  mx0
2

Conservation of mechanical energy - for a simply,
conservative (i.e. no damper), mass spring system the
energy must be conserved.
Ek  E p  const . 
d
( Ek  E p )  0  Ek ,max  E p,max
dt
Principle of virtual work

Dynamic equilibrium of the particle (d’Alembert)
mi ui  X i  0, i  1,2,3,
where ui represents the displacement of the particle, Xi are forces.

Let us consider that the particle follows during the time
interval [t1, t2] a motion trajectory u i* distinct from the real
one ui. This allows us to define the virtual displacement
of the particle the relationship
ui  ui*  ui
ui (t1 )  ui (t 2 )  0
Principle of virtual work

Multiplying of equation of motion by associated virtual
displacement and sum over the components
3
 (mi ui  X i )ui  0,
i 1
„The virtual work produced by the effective forces acting on the particle
during a virtual displacement ui is equal to zero“.

The virtual work principle for the system of particles
N
3
 (mk uik  X ik )uik
 0,
k 1 i 1
„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
xik(t) = xik + uik(t)
The kinematical constraints are divided on:
 holonomic constraints - defined by f(xik(t)) = 0
 Non-holonomic constraints
Principle of virtual work

The kinematical constraints are divided on:

holonomic constraints - defined by
f(xik, t) = 0



scleronomic - constraints not explicitly dependent on time
rheonomic – constraints explicitly dependent on time
non-holonomic constraints - defined by
f (x ik , x ik , t )  0
These equations are generally not integrable.
Principle of virtual work

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
uik ( x, t )  U ik (q1 ,..., qn , t )

The virtual displacement compatible with the holonomic constraints
may be expressed in the form
n
U ik
q s
s 1 q s
uik  
Principle of virtual work

Virtual work equation becomes
N 3
U ik


  (mk uik  X ik ) q
s
s 1  k 1 i 1
n
n N 3


U ik


 Qs q s  0
q s    mk u ik
q s


s 1 
k 1 i 1
where
N 3
U ik
Qs   X ik
q s
k 1 i 1
is the generalized force
Hamilton´s Principle

Hamilton´s principle - time integrated form of the virtual work
principle obtained by transforming the expression
t2
N 3



  (mk uik  X ik )uik dt  0

t1 k 1 i 1

Applied forces Xik can be derived from the potential energy - virtual
work is expressed in the form
N 3
   X ik uik
s 1  k 1 i 1
n

n

q s   Qs q s  E p

s 1

The generalized forces are derived from the potential energy
Qs  
E p
q s
Hamilton´s Principle

The term associated with inertia forces
d
 mk uik uik 
(mk uik uik )  mk uik uik  mk uik uik  mk uik uik  

dt
2



The kinetic energy
1 N 3
Ek   mk uik uik
2 k 1 i 1

Then, time integrated form of the virtual work principle
t
t2
 N 3
2
  mk uik uik     ( Ek  E p )dt  0
 k 1 i 1
 t
t1
1
Hamilton´s Principle

In terms of generalized coordinates is expressed
uik
n
U ik
U ik


q s
t
s 1 q s
Ek  Ek (q, q, t )

q s (t1 )  q s (t 2 )  0
E p  E p ( q, t )
“Hamilton´s principle:
t2
The real trajectory of the system is such as the integral
 ( Ek  E p )dt
t1
remains stationary with respect to any compatible virtual displacement
arbitrary between both instants t1, t2 but vanishing at the ends of the interval
t2
t2
t1
t1
  ( Ek  E p )dt    Ldt  0
where L – is a kinetic potential or Lagrangian
Lagrange´s Equations of 2nd Order

Using expression
n
 E

E
Ek    k q s  k q s 
q s

s 1  q s
in equation for Hamilton´s principle
t2 n
 Ek


Ek




Q

q


q
   qs s  s q s s dt  0

t1 s 1 
The second term can be integrate by parts
t2
t
t
 E k
 2 2 d  E k
E k
 q s q s dt   q s q s    dt  q s
t1
t1
t1

q s dt

Lagrange´s Equations of 2nd Order
Taking into account the boundary conditions the following is equivalent
to Hamilton´s principle
t2 n
 d  Ek
   dt  q s
t1 s 1 

 Ek
 
 Qs q s dt

 q s
The variation qs is arbitrary on the whole interval and the equations of
motion result in the form obtained by Lagrange
d  Ek

dt  q s
 Ek
 
 Qs , s  1,2,..., n
 q s
Lagrange´s Equations of 2nd Order

Classification of generalized forces

Internal forces

Linking force - connection between two particles
3
3
i 1
i 1
A   ( X i1ui1  X i 2 ui 2 )   X i1 (ui1  ui 2 )  0

Elastic force - elastic body - body for which any produced work is
stored in a recoverable form - giving rise to variation of internal energy
3 N
E p,int  
E p
i 1 k 1 uik
n
uik   Qs q s
s 1
with the generalized forces of elastic origin
Qs  
E p ,int
q s
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
X ik
vik
 Cik f k (vk )
vk
where Cik is a constant fk(vk) is the function expressing velocity
dependence, vk is the absolute velocity of particle k
vk  | v k |
3
2

u
 ik
i 1
N
The dissipative force
vk
Qs   Ck f k (vk )
q s
k 1
N vk
The dissipative function
E dis    C k f k (v k )dv
k 1 0
Lagrange´s Equations of 2nd Order

External conservative force - conservative - their virtual work remains
zero during a cycle
A   Qs q s  0
generalized force is expressed

Q
E p ,ext
q s
External non-conservative force
3 N
generalized force is expressed
Qs   X ik
i 1 k 1
uik
q s
General form of Lagrange equation of 2nd order of non-conservative
systems with rheonomic constraints
d  Ek

dt  q s
 Ek E p Edis
 


 Qs (t ),

q s
 q s q s
s  1,2,..., n.