DOMV No 5 MATH MODELLING Newtonian d'Alembert Virtual Work (1).pdf
1. Mathematical Modelling
We have now seen the process of creating a physical
discretisation of a real structure or machine, with
distributed mass (and therefore, a potentially large
number of degrees of freedom (i.e. using the
Lumped-Mass Procedure, the Generalised
Displacement model, and the Finite-Element
Concept).
The next step is to construct equations of motion,
which involves application of physical laws.
2. Mathematical Modelling
To construct the equations of motion, three
different (but very commonly used) approaches
are described , based on different physical
principles.
The choice of approach depends on how easy it is
to use, but the resulting mathematical model
should be similar regardless of which physical
principle is used. These approaches do NOT solve
the equations โ that involves analysis โ later!
3. Mathematical Modelling
Physical Principle
1) Newtonian Mechanics
(conservation of Momentum) in
direct form, or using 'Equilibrium'
Concepts based on d'Alembertโs
principle.
Comments
Involves application of Newton's
second law, therefore requires vector
operations (mainly useful for lumped
mass models).
2) The Principle of Virtual Work using
virtual displacements (an energy
principle using d'Alembertโs
principle).
Work terms are obtained through
vector dot products but they may be
added algebraically.
3) Lagrange Equations (an energy-
based Variational method - a
corollary of Hamilton's Principle).
This approach is developed entirely
using energy (i.e. scalar quantities)
which can therefore be added
algebraically.
4. Mathematical Modelling
Newtonian Methods
Application of Newtonโs 2nd law to a discrete mass m, which has an
applied force f(t), gives rise to the statement:
๐ ๐ก =
๐
๐๐ก
๐ แถ
๐ฅ = rate of change of momentum
where แถ
๐ฅ is the absolute velocity of the mass (i.e. vector differential
of position). If the mass is constant, i.e. แถ
๐=0, then:
๐(๐ก) = ๐ แท
๐ฅ
This equation states that the total external force ๐ ๐ก is equal to
the mass times the acceleration. This can be used to directly
construct the equations of motion for a discrete dynamic system.
5. Mathematical Modelling
Example: Consider a 2DOF system (undamped Lumped Mass model):
k1
k2
x1 x2
F2(t)
F1(t)
X1 and X2 are displacements from the equilibrium position.
First, assume X2 > X1 (in general, assume XN > XN-1 > ... > X1).
Then draw free body diagrams for each mass, and apply
Newtonโs 2nd law to each mass.
7. Mathematical Modelling
Application of Newtonโs 2nd Law to the two masses:
ฯ ๐ = ๐ แท
๐ฅ: ๐1 ๐ก + ๐2 ๐2 โ ๐1 โ ๐พ1๐1 = ๐1
แท
๐1
ฯ ๐ = ๐ แท
๐ฅ: ๐2 ๐ก โ ๐2 ๐2 โ ๐1 = ๐2
แท
๐2
and
๐1
แท
๐1 + (๐1 + ๐2)๐1 โ ๐พ2๐2 = ๐1 ๐ก
๐2
แท
๐2 + ๐2๐2 โ ๐พ2๐1 = ๐2 ๐ก
A coupled
system of linear
differential
equations
and
8. Mathematical Modelling
The coupled system model can
be put into matrix form i.e.:
เตฏ
๐ แท
๐ + ๐พ ๐ = ๐(๐ก
where the mass matrix is:
๐ =
๐1 0
0 ๐2
and the stiffness mass matrix is:
๐พ =
๐พ1 + ๐พ2 โ๐พ2
โ๐พ2 ๐พ2
9. Mathematical Modelling
d'Alembert's Principle
Note that Newton's 2nd law is written:
๐ ๐ก = ๐ แท
๐ฅ
but can be rearranged in the form:
๐ ๐ก โ ๐ แท
๐ฅ = 0
So the term ๐ แท
๐ฅ can be thought of as an 'inertia' force
which, when included on an 'equilibrium diagramโ
(rather than a free-body diagram), reduces the problem
to one of 'equilibrium'.
10. Mathematical Modelling
The concept of introducing an inertia force on a mass
which is proportional to its acceleration, and which
opposes the motion, is called d'Alembert's Principle,
and can be very useful in modelling continuous
systems. The inertial force is of course fictitious (it
doesn't really exist) but it is helpful (for modelling
purposes) to think of the system as being in
โequilibriumโ where the 'inertia force' is included.
d'Alembert's Principle
11. Mathematical Modelling
An example: a SDOF problem.
k1
x1
f1
k1x1
f1
๐น๐ผ = ๐ แท
๐ฅ1
Equilibrium diagram using
d'Alembert's principle:
d'Alembert's principle: ๐1โ๐1๐ฅ1 โ ๐ แท
๐ฅ1 = 0
๐ แท
๐ฅ1 + ๐1๐ฅ1 = ๐1(t)
And therefore:
No advantage of using d'Alembert's principle, on Lumped-Mass
systems since Newton's 2nd law can be applied directly. The real
advantage is derived when we use Virtual Work principles.
12. Mathematical Modelling
The Principle of Virtual Work
Again, the focus is on constructing a discrete model of the form:
๐ แท
๐ + ๐ แถ
๐ + ๐ ๐ = ๐(๐ก).
The Principle states that when a system is in โequilibriumโ (in
the sense of d'Alembert) under the action of external forces,
and is forced to move through a virtual displacement,
without violating the system constraints, and without the
passage of time, at the same time as adhering to a sign
convention, then the total virtual work done is zero i.e.:
เท ๐ฟ๐ค๐ = 0
13. Mathematical Modelling
The Sign Convention:
๏ท Forces acting in the direction of a Virtual
Displacement do โve (negative) Virtual Work.
๏ท Strain energy put into a system is always
deemed to be positive.
14. Mathematical Modelling
Virtual Displacements and Virtual Work
Consider a system with N
degrees-of-freedom, with
corresponding
coordinates (X1, X2, ..., XN)
used to specify the
position. Assume forces
F1, F2, ...,FN are applied at
each coordinate in the
(+ve) direction of each
coordinate.
15. Mathematical Modelling
Now if we imagine the system is given an arbitrary set of
small displacements ๐ฟ๐1, ๐ฟ๐2, โฆ , ๐ฟ๐๐ then the magnitude
of the work done by these applied forces will be:
๐ฟ๐ค = โ เท
๐
๐
๐น๐๐ฟ๐๐
The small displacements are imaginary and are therefore
virtual because they occur without the passage of time, and
are different from small changes dx which occur in time dt
(i.e. real ones). The virtual displacements conform to the
kinematic constraints which apply.
Virtual Displacements and Virtual Work
16. Mathematical Modelling
In general, forces will occur in arbitrary directions and thus
the work done is expressed as a dot or vector product i.e.
๐ฟ๐ค = โ ฯ๐
๐
๐น๐ โ ๐ฟ๐๐
where ๐ฟ๐๐ are the small changes in position vectors.
Virtual Work
Sign Convention and itโs impact on the sign of the Virtual Work
Adherence to the Sign convention will always produce the
correct sign for all the Virtual terms. Some text books also
define Virtual Work as being Internal (suffix I) or external (suffix
E). The Principle of Virtual Work can then be stated as:
๐จ(๐ฟ๐ค๐ธ + ๐ฟ๐ค๐ผ) = 0
17. Example: SDOF Linear Oscillator
k
x
f(t)
m
c
Next Lecture!
A simple example of applying the Principle of Virtual Work.