This document discusses different approaches for constructing mathematical models from physical systems: 1) Newtonian mechanics uses Newton's second law to directly obtain equations of motion for lumped mass systems. 2) D'Alembert's principle allows inertia forces to be included in equilibrium diagrams, making it useful for continuous systems. 3) The principle of virtual work equates the total virtual work done by internal and external forces during virtual displacements to zero, providing another approach for developing equations of motion. Examples are provided to illustrate Newtonian mechanics and the principle of virtual work.

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 – that involves analysis – later!

3. Mathematical Modelling
Physical Principle
1) Newtonian Mechanics
(conservation of Momentum) in
direct form, or using 'Equilibrium'
Concepts based on d'Alembert’s
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'Alembert’s
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 Newton’s 2nd law to a discrete mass m, which has an
applied force f(t), gives rise to the statement:
𝑓 𝑡 =
𝑑
𝑑𝑡
𝑚 ሶ
𝑥 = rate of change of momentum
where ሶ
𝑥 is the absolute velocity of the mass (i.e. vector differential
of position). If the mass is constant, i.e. ሶ
𝑚=0, then:
𝑓(𝑡) = 𝑚 ሷ
𝑥
This equation states that the total external force 𝑓 𝑡 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
Newton’s 2nd law to each mass.

6. Mathematical Modelling
F1(t)
k2(x2 – x1)
k1 x1
Free body diagrams
FBD for Mass 1:
FBD for Mass 2:

7. Mathematical Modelling
Application of Newton’s 2nd Law to the two masses:
σ 𝑓 = 𝑚 ሷ
𝑥: 𝑓1 𝑡 + 𝑘2 𝑋2 − 𝑋1 − 𝐾1𝑋1 = 𝑚1
ሷ
𝑋1
σ 𝑓 = 𝑚 ሷ
𝑥: 𝑓2 𝑡 − 𝑘2 𝑋2 − 𝑋1 = 𝑚2
ሷ
𝑋2
and
𝑚1
ሷ
𝑋1 + (𝑘1 + 𝑘2)𝑋1 − 𝐾2𝑋2 = 𝑓1 𝑡
𝑚2
ሷ
𝑋2 + 𝑘2𝑋2 − 𝐾2𝑋1 = 𝑓2 𝑡
A coupled
system of linear
differential
equations
and

8. Mathematical Modelling
The coupled system model can
be put into matrix form i.e.:
൯
𝑀 ሷ
𝑋 + 𝐾 𝑋 = 𝑓(𝑡
where the mass matrix is:
𝑀 =
𝑚1 0
0 𝑚2
and the stiffness mass matrix is:
𝐾 =
𝐾1 + 𝐾2 −𝐾2
−𝐾2 𝐾2

9. Mathematical Modelling
d'Alembert's Principle
Note that Newton's 2nd law is written:
𝑓 𝑡 = 𝑚 ሷ
𝑥
but can be rearranged in the form:
𝑓 𝑡 − 𝑚 ሷ
𝑥 = 0
So the term 𝑚 ሷ
𝑥 can be thought of as an 'inertia' force
which, when included on an 'equilibrium diagram’
(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
‘equilibrium’ where the 'inertia force' is included.
d'Alembert's Principle

11. Mathematical Modelling
An example: a SDOF problem.
k1
x1
f1
k1x1
f1
𝐹𝐼 = 𝑚 ሷ
𝑥1
Equilibrium diagram using
d'Alembert's principle:
d'Alembert's principle: 𝑓1−𝑘1𝑥1 − 𝑚 ሷ
𝑥1 = 0
𝑚 ሷ
𝑥1 + 𝑘1𝑥1 = 𝑓1(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:
𝑚 ሷ
𝑍 + 𝑐 ሶ
𝑍 + 𝑘 𝑍 = 𝑝(𝑡).
The Principle states that when a system is in ‘equilibrium’ (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.:
෍ 𝛿𝑤𝑖 = 0

13. Mathematical Modelling
The Sign Convention:
 Forces acting in the direction of a Virtual
Displacement do –ve (negative) Virtual Work.
 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 𝛿𝑋1, 𝛿𝑋2, … , 𝛿𝑋𝑁 then the magnitude
of the work done by these applied forces will be:
𝛿𝑤 = − ෍
𝑗
𝑁
𝐹𝑗𝛿𝑋𝑗
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.
𝛿𝑤 = − σ𝑗
𝑁
𝐹𝑗 ∙ 𝛿𝜏𝑗
where 𝛿𝜏𝑗 are the small changes in position vectors.
Virtual Work
Sign Convention and it’s 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:
𝝨(𝛿𝑤𝐸 + 𝛿𝑤𝐼) = 0

17. Example: SDOF Linear Oscillator
k
x
f(t)
m
c
Next Lecture!
A simple example of applying the Principle of Virtual Work.