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DOMV No 7 MATH MODELLING Lagrange Equations.pdf
1. Mathematical Modelling
Physical Principle
1) Newtonian Mechanics
(conservation of Momentum) in
direct form, or using 'Equilibrium'
Concepts based on d'Alembertâs
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'Alembertâs
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.
2. Mathematical Modelling
THE LAGRANGE EQUATIONS.
What are they?
We have seen that it is possible to use a
different coordinate system to describe the
motion of a beam e.g.:
⢠A SDOF Lumped-mass model of a beam
(then using Newtonian Mechanics)
⢠A SDOF Generalised Displacement model of
beam structure motion obtained via the
Principle of Virtual Work.
3. THE LAGRANGE EQUATIONS.
i) Lumped mass
model
Z1(t) is the
generalised
coordinate
ii) Generalised
displacement
model
Z(t) is the
generalised
coordinate
The displacement of the system is only defined at the lumped mass by Z1;
The displacement is defined at all values of x , but Z(t) is still only a single variable.
4. THE LAGRANGE EQUATIONS.
We might ask: is it possible to write down the equations
of motion in terms of âgeneralised coordinatesâ without
actually specifying what they are?
The answer is yes, i.e.: using Lagrange Equations.
The Lagrange Equations describe the dynamic behaviour
of systems with N-degrees-of-freedom in terms of energy
and (unspecified) âgeneralised coordinatesâ: q1,q1,âŠ,qn.
5. THE LAGRANGE EQUATIONS.
In a particular application, say in modelling a
structure (whether lumped or distributed mass),
the particular coordinate system being used is
specified.
Note: if we take an N degree-of-freedom system
and obtain the equations of motion using different
coordinates, then although the resulting equations
will look different, the predicted (physical) motion
of the system will be identical.
6. THE LAGRANGE EQUATIONS.
Generalised coordinates
The term generalised coordinates can refer to any set of (coordinates), including
commonly used coordinates, which serve to specify the configuration of a system.
There is a relationship between the Number of Degrees of freedom and the
number of generalised coordinates i.e.:
The number of degrees-of-freedom (DOF) = The number of independent
coordinates needed to configure a system without violating any constraints.
or
The number of DOF = The number of Cartesian coordinates need to specify
system â the number of Cartesian equations of constraint.
7. THE LAGRANGE EQUATIONS.
Example: The Simple Pendulum (1-DOF system)
Æ is a generalised coordinate, whereas the Cartesian position (x,y) is subject to the constraint:
ð¥ð¥2 + ðŠðŠ2 = ðð2
8. THE LAGRANGE EQUATIONS.
Lagrange Equations - detail
In defining the Lagrange equations an abstract set of independent generalised
coordinates is used: ðð1, ðð2, ðð3, ⊠, ðððð, where N is the number of degrees-of-
freedom. We can write down a set of N dynamical equations i.e. for each
coordinate ðððð in terms of the kinetic energy T and potential energy V:
ðð
ðððð
ðððð
ðð Ì
ðððð
â
ðððð
ðððððð
+
ðððð
ðððððð
= ðððð ðð = 1,2, ⊠, ðð
where the ðððð are known as the generalised forces representing non-
conservative or external forces (which cannot be represented in a particular
form i.e. cannot be derived from a potential function e.g. dissipative (damping
forces)). Absolute ðð and ðð, or changes in V and T (Îðð and Îðð) can be used.
If we have an N-degree-of-freedom system, we can write down N different
Lagrange equations, which apply to large (often nonlinear) displacements.
9. THE LAGRANGE EQUATIONS.
Special Cases
Conservative Systems:
This is the simplest case i.e.: when ðððð = 0 for all ðð.
Fixed Supports (Holonomic-scleronomic constraints):
For fixed constraints, the potential energy and kinetic
energy functions can usually be written: ðð =
ðð(ðð1, ðð2, ðð3, ⊠, ðððð) and ðð = ðð( Ì
ðð1, Ì
ðð2, Ì
ðð3, ⊠, Ì
ðððð) where
the generalised forces ðððð are defined such that a virtual
displacement of each generalised coordinate will produce
the correct total virtual work i.e. ð¿ð¿ð¿ð¿ = âðð=1
ðð
ððððð¿ð¿ðððð (but
here treated as positive!).
10. THE LAGRANGE EQUATIONS.
Special Cases
If, in fact, we have force vectors ð¹ð¹ðð defined in terms of
Cartesian Coordinates, which are related to the
generalised Coordinates as follows:
ð¥ð¥ðð = ðððð(ðð1, ðð2, ⊠, ðððð) ðð = 1,2, ⊠, 3ðð
then it can be shown that:
ðððð = ï¿œ
ðð=1
3ðð
ð¹ð¹ðð
ððð¥ð¥ðð
ðððððð
(ðð = 1,2, ⊠, ðð) (3ð·ð·)
where ð¹ð¹ðð are the components of the force vectors
described in Cartesian Coordinates.
11. THE LAGRANGE EQUATIONS.
Example: Conservative 2DOF System
Here N= 2, the Generalised Coordinates are ðð1 â¡ ð¥ð¥1 , ðð2 â¡ ð¥ð¥2.
Kinetic energy ðð =
1
2
ðð1 Ì
ð¥ð¥1
2
+
1
2
ðð2 Ì
ð¥ð¥2
2
and
Potential energy ðð =
1
2
ðð1ð¥ð¥1
2
+
1
2
ðð(ð¥ð¥2 â ð¥ð¥1)2
There are no external forces or dissipative forces ⎠ðððð = 0
13. THE LAGRANGE EQUATIONS.
Giving the usual equations of motion:
ðð1 Ì
ð¥ð¥1 + ðð1ð¥ð¥1 â ðð2 ð¥ð¥2 â ð¥ð¥1 = 0
and
ðð2 Ì
ð¥ð¥2 + ðð2 ð¥ð¥2 â ð¥ð¥1 = 0
14. THE LAGRANGE EQUATIONS.
EXTERNAL FORCES
How are the generalised forces ðððð established? If, in the
previous example, external forces ðð1(ð¡ð¡) and ðð2(ð¡ð¡) had been
applied to masses ðð1 and ðð2 then a virtual displacement ð¿ð¿ð¿ð¿
would be made up of ð¿ð¿ð¥ð¥1 and ð¿ð¿ð¥ð¥2 so the total Virtual Work
done would be:
ð¿ð¿ð¿ð¿ = ðð1 ð¡ð¡ ð¿ð¿ð¥ð¥1 + ðð2(ð¡ð¡)ð¿ð¿ð¥ð¥2
since ð¥ð¥1 = ðð1 ð¥ð¥2 = ðð2, this gives:
ð¿ð¿ð¿ð¿ = ï¿œ
ðð
ðð
ððððð¿ð¿ðððð = ðð1ð¿ð¿ðð1 + ðð2ð¿ð¿ðð2
We can identify the Generalised Forces ðððð (by association)
as ðð1= ðð1(ð¡ð¡) and ðð2 = ðð2(ð¡ð¡) appearing on the rhs of the
equations of motion.
15. THE LAGRANGE EQUATIONS.
Dissipative Forces (Damping)
These are non-conservative forces which can actually be absorbed
into the generalised forces ðððð. It is possible to write down part of
the generalised forces in terms of a so called dissipation function:
ð·ð· = ð·ð· Ì
ðð1, Ì
ðð2, ⊠, Ì
ðððð
which contributes to the generalised force a term:
ðððð
ðð Ì
ðððð
Modified
Lagrange equations can then be written:
ðð
ðððð
ðððð
ðð Ì
ðððð
â
ðððð
ðððððð
+
ðððð
ðððððð
+
ðððð
ðð Ì
ðððð
= ðððð
Examples include the Rayleigh Dissipation function
ð·ð· =
1
2
ðð1 Ì
ðð1
2
+ ðð2 Ì
ðð2
2
+. .
which can be used to model linear viscous dampers (creating a
diagonal damping matrix).
16. THE LAGRANGE EQUATIONS.
Equations of Motion of a MDOF Linear Dynamic System
(undamped)
When considering a linear MDOF dynamic model for a
structure in terms of displacements, a linear system can be
shown to belong to a particular form when certain terms
are considered for the kinetic energy ðð and potential
energy ðð. These forms can be established using either the
Lagrange equations or Virtual Work principles.
17. THE LAGRANGE EQUATIONS.
For small displacements (i.e. small amplitude oscillations) the
kinetic and potential energies can be expressed as so called
âquadratic formsâ (i.e. for small movements about equilibrium
points, these terms follow from Maxwellâs Reciprocal Theorem
see Newland p314). The kinetic and potential energies are
written:
ðð =
1
2
âðð=1
ðð âðð=1
ðð
ðððððð Ì
ðððð Ì
ðððð =
1
2
Ì
ðððð
ðð Ì
ðð
And
ðð =
1
2
âðð=1
ðð âðð=1
ðð
ð ð ðððððððððððð =
1
2
ðððð[ð ð ]ðð
where [ðð] is the mass matrix, and [ð ð ] is the stiffness matrix.
18. THE LAGRANGE EQUATIONS.
Now the usual Lagrange equations can be written in vector form
i.e.:
ðð
ðððð
ðð
ðð Ì
ðð
ðð â
ðð
ðððð
ðð +
ðð
ðððð
ðð = ðð
where
ðð
ðððð
=
ðð
ðððð1
ðð
ðððð2
ðð
ðððð3
âŠ
ðð
and
ðð
ðð Ì
ðð
=
ðð
ðð Ì
ðð1
ðð
ðð Ì
ðð2
ðð
ðð Ì
ðð3
âŠ
ðð
are âoperatorsâ, and ðð is a vector of generalised forces.
19. THE LAGRANGE EQUATIONS.
And since, as we have seen, the energy terms for a general linear
system in equilibrium, written in vector form, are:
ᅵ
ðð =
1
2
Ì
ðððð ðð Ì
ðð
ðð =
1
2
ðððð[ð ð ]ðð
It can be shown that
ðððð
ðð Ì
ðððð
= [ðð] Ì
ðð and therefore
ðð
ðððð
ðð Ì
ðð = [ðð] Ì
ðð and
that:
ðððð
ðððððð
= [ð ð ]ðð. And since for a system with fixed constraints
ðððð
ðððð
= 0, then on substitution into the Lagrange Equations we get
the usual form of matrix differential equation:
ðð Ì
ðð + ðð ðð = ðžðž
where all the non-conservative forces are absorbed into the
generalised force vector ðð including all damping forces.
20. THE LAGRANGE EQUATIONS.
If damping forces can be modelled as linearly-dependent on
velocity e.g. using a Rayleigh dissipation function then a general
linear model, via Lagrange equations, takes the form:
ðŽðŽ Ì
ðð + ðªðª Ì
ðð + ð²ð² ðð = ðžðž(ðð)
where ðð now only includes external force components, which
are obtained individually by considering the work done ð¿ð¿ðððð by
(non-conservative) forces in going through some arbitrary Virtual
Displacements of each of the coordinates ð¿ð¿ðð1, ð¿ð¿ðð2, ⊠, ð¿ð¿ðððð.
21. Properties of the damping matrix [C]
Properties of the damping matrix [C]
Constructing the damping matrix of a complete structure is one of
the most difficult parts of the modelling process (even when
damping is linear).
However in choosing a (viscous) damping model which depends
linearly on the velocity, then the form of the chosen (Rayleigh)
model implies that the damping matrix is symmetric i.e. ðððððð = ðððððð or
[ð¶ð¶]ðð= [ð¶ð¶].
But in practice, for real structures, damping matrices are rarely
obtained other than by some sort of experimental calibration
process which is often based on specific assumptions.
22. A very important assumption which allows the equations of
motion to be solved using so called Normal Coordinates, is to
assume damping is proportional to either the stiffness matrix
[K] or the mass matrix [M], or in general, both i.e. assume:
ð¶ð¶ = â ðð + ðœðœ[ðŸðŸ]
where â and ðœðœ are constant (scalars). This type of damping
model is known as âproportional dampingâ. We will return to
the use of this important damping model soon.
Proportional Damping