1. PHYSICAL MODEL TYPES
(FOR DYNAMIC ANALYSIS)
Mathematical modelling of structures and
machines focuses on constructing practically
useful models for real structures, which in
general, have distributed mass.
Two stages: physical discretisation, then application
of physical laws to construct equations of motion

2. What are we trying to do?
The ultimate aim is to create a discrete dynamic model of a
real structure in the form:
𝑚 ሷ
𝑍 + 𝑐 ሶ
𝑍 + 𝑘 𝑍 = 𝑝(𝑡)
i.e. a (linear) system of differential equations. But in this set
of slides, no attempt will be made to construct the equations
of motion (that will happen in the next presentation).

3. Methods of Discretisation
Consider a real structure, for example, a uniform beam:
ν(x,t) (the displacement at
any point along the beam)
x
P(t)
The lateral dynamics (i.e. vertical motion) of a beam could be
analysed via the beam equation (a 4th-order Partial Differential
equation (PDE)). It has an infinite number of degrees-of-freedom.
This procedure is often not practical even for a single beam. It is very
impractical for treatment of multiple beams, connected to other
building blocks (plates and shells) in 2D and 3D (i.e. real structures).

4. Three Physical Model Types
Three (physical) procedures are available to directly
create a discrete model of a distributed-mass system
(which can then be used to apply laws of physics).
• The Lumped-Mass Procedure.
• The Generalised Displacement model.
• The Finite-Element Concept.

5. The Lumped-Mass Procedure
This is by far the simplest approximate model of a
distributed-mass dynamic system. It assumes that the mass
of a structure is concentrated at a finite number of discrete
points. At these discrete points, displacements are defined
as: Z1, Z2, Z3, …, ZN. Displacements at any other points, are
not defined.
When the number of discrete points in a Lumped-Mass
model of a uniform beam for example, is allowed to become
infinite, in the limit, application of Newton’s laws (plus Euler
Bernoulli Beam theory to obtain the correct stiffness values)
would result in the Beam Equation (a 4th-order PDE).

6. The Lumped-Mass Procedure
p(t)
m1 m2 m3
Z1
Z2
Z3
displacements:
Z1, Z2, Z3
Example: a uniform beam. Assume the mass of the beam is
concentrated at 3 discrete points only: e.g at Z1, Z2, and Z3 :
Modelling is much simplified because all forces are concentrated at
mass points. To model the system, only necessary to choose these
masses correctly, and to define displacements and accelerations at
discrete locations. Stiffness properties might be based on static analysis.
The number of displacement components needed to represent dynamic
effects is usually referred to as the number of degrees-of-freedom.

7. The Generalised Displacement
Model
(The aim is still to create a model of the
form: 𝑚 ሷ
𝑍 + 𝑐 ሶ
𝑍 + 𝑘 𝑍 = 𝑝(𝑡))
Lumped-mass models can be accurate discretisations for
structures with concentrated mass, i.e. where mass
behaves as rigid.
Where mass is distributed (i.e. not concentrated), then an
alternative approach to limiting the number of degrees-of-
freedom may be preferable. This is achieved with a
Generalised Displacement Model.

8. The Generalised Displacement Model
The Generalised Displacement model is based on the
assumption that the deflected shape of an (entire)
structure can be expressed as the sum of specified
displacement patterns, chosen to be compatible with
the prescribed Boundary Conditions, and which
maintain continuity of internal displacements, such
that displacements are given by:
𝜈 𝑥, 𝑡 = σ 𝑍𝑛 𝑡 𝜓𝑛(𝑥)
where shape functions 𝜓𝑛(𝑥) describe an assumed
displacement pattern.

9. The Generalised Displacement Model
For example, by just using one term it might be assumed (a guess)
that a beam, fixed at both ends, might vibrate in a sine wave shape
i.e. by choosing.
𝜓1 𝑥 = 𝑠𝑖𝑛
𝜇𝑥
𝐿
Just using one term, the displacement along the beam is given by:
𝜈 𝑥, 𝑡 ≃ 𝑍1 𝑡 𝜓1(𝑥)
and we can refer to 𝑍1 𝑡 as a single generalised coordinate i.e.
assuming we are able to describe the entire beam motion to a good
approximation with just one coordinate.

10. The Generalised Displacement Model
The true vibration shape in the previous 1-DOF example
might be rather different, in which case a better
approximation might be obtained via some other 𝜓1 𝑥 .
When more than one term is used, the variables 𝑍1 ,
𝑍2, … , 𝑍𝑛(𝑡) are then examples of generalised coordinates.
For example, approximating the beam displacement with 2
coordinates 𝑍1 and 𝑍2 i.e. 2-DOF, would take the form:
)
𝜈 𝑥, 𝑡 = 𝑍1 𝑡 𝜓1 𝑥 + 𝑍2 𝑡 𝜓2(𝑥
So this approximate description has 2 degrees-of-freedom.

11. The Generalised Displacement Model
But so far, we have not said anything about how the
coordinates 𝑍1 𝑡 , 𝑍2 𝑡 , … , 𝑍N 𝑡 are related to the
excitation, mass, stiffness, and external excitation. This
relationship will emerge when appropriate physical laws
are applied.
)
𝜈 𝑥, 𝑡 = 𝑍1 𝑡 𝜓1 𝑥 + 𝑍2 𝑡 𝜓2(𝑥 + … + 𝑍N 𝑡 𝜓N 𝑥
Approximating the beam displacement with N coordinates,
is achieved with an assumption of the form:

12. The Finite-Element Concept
Another way of expressing the displacement in terms of
discrete coordinates, involves combining certain features
of both the Lumped-Mass and Generalised Displacement
procedures. But again, the aim is still to create a model
of the form: 𝑚 ሷ
𝑍 + 𝑐 ሶ
𝑍 + 𝑘 𝑍 = 𝑝(𝑡).
The basic approach is to divide the structure into an
appropriate number of elements. The points where
elements are interconnected are referred to as nodal
points or nodes. The displacements of these nodal points
then become the generalised coordinates of the structure.

13. Finite-Element Concept
Nodes 1, 2, 3 and 4
3 elements.
One dimensional elements
1
ν(x,t)
2 3 4
Again, consider the single beam as an example:
The deflection shape of the complete structure can now
be expressed in terms of these generalised coordinates
assuming an appropriate set of displacement functions.

14. Finite-Element Concept
The beam response at any point can be written in the form:
i.e. a generalised displacement model, where 𝑍𝑛 𝑡 are the
generalised displacements. The functions 𝑁𝑛(𝑥) are used as
interpolation functions to describe the displacement within
each element (subject to continuity and geometric modal
displacement conditions). For a uniform beam these
interpolation functions can be derived exactly (i.e. cubic
Hermite polynomials).
𝜈 𝑥, 𝑡 = ෍
1
𝑁
)
𝑍𝑛(𝑡)𝑁𝑛(𝑥