1. Mathematical Modelling
Consider this system displaced by variable X from the
free spring length. Then consider a virtual displacement
𝛿𝑋 from variable X: 𝛿𝑋 is arbitrary but not zero.
Example: SDOF Linear Oscillator
k
x
f(t)
m
c
Application of the Principle of Virtual Work

2. REMINDER!
The Principle of Virtual Work 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
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.

3. Mathematical Modelling
Equilibrium Diagram: (Using d'Alembert's principle)
Virtual Work
kx
x
f(t)
Applying the sign convention, the (virtual) work done by each
force is as obtained as follows:
Source of Virtual Work Virtual Work
Spring: +𝒌𝑿𝜹𝒙
Damper: +𝒄 ሶ
𝑿𝜹𝒙
‘Inertial’ force: +𝒎 ሷ
𝑿𝜹𝒙
External force: −𝒇(𝒕)𝜹𝒙

4. Mathematical Modelling
෍ 𝛿𝑤𝑖 = 0
Now the Principle of Virtual Work state that:
Virtual Work
−𝑓 𝑡 𝛿𝑋 + 𝑚 ሷ
𝑋𝛿𝑋 + 𝑐 ሶ
𝑋𝛿𝑋 + 𝑘𝑋𝛿𝑋 = 0
−𝑓 𝑡 + 𝑚 ሷ
𝑋 + 𝑐 ሶ
𝑋 + 𝑘𝑋 𝛿𝑋 = 0
But since 𝛿𝑋 ≠ 0
−𝑓 𝑡 + 𝑚 ሷ
𝑋 + 𝑐 ሶ
𝑋 + 𝑘𝑋 = 0
Rearranging we get the familiar second order oscillator model :
𝑚 ሷ
𝑋 + 𝑐 ሶ
𝑋 + 𝑘𝑋 = 𝑓 𝑡
or:
Applying the Principle to the Virtual Work Terms obtained we get:

5. Mathematical Modelling
Application to a 2-DOF System
Virtual Work
k1 k
2
x1 x2
f1 f2
m
2
m
1
Now consider the work done in a virtual displacement of the two
variables 𝑋1 and 𝑋2 by independent and arbitrary amounts 𝛿𝑋1 and
𝛿𝑋2. First draw (two) 'Equilibrium' diagrams:

6. Mathematical Modelling
f1
X1
m1
f2
X2
m2
Virtual Work
'Equilibrium' diagrams:
So the total work done, (taking into account) whether +ve or -ve i.e.
those derived from external forces f1 and f2, the ‘inertia’ forces
(d’Alembert), damping, and elastic forces (which for each mass sum
to F1 and F2 respectively, the Principle of Virtual Work states that:
𝛿𝑤 = σ 𝐹1 𝛿𝑋1 + σ 𝐹2 𝛿𝑋2= 0

7. Mathematical Modelling
−𝑓1 + 𝑘1𝑋1 + 𝑚1
ሷ
𝑋1 − 𝑘2 𝑋2 − 𝑋1 𝛿𝑋1 + −𝑓2 + 𝑘2 𝑋2 − 𝑋1 + 𝑚2
ሷ
𝑋2 𝛿𝑋2=0
Virtual Work
but since 𝛿𝑋1 and 𝛿𝑋2 are arbitrary, independent, and non-zero,
this requires that both terms in the brackets must be zero i.e.:
൯
𝑚1
ሷ
𝑋1 + 𝑘1 + 𝑘2 𝑋1 − 𝑘2𝑋2 = 𝑓1(𝑡
And:
൯
𝑚2
ሷ
𝑋2 + 𝑘2𝑋2 − 𝑘2𝑋1 = 𝑓2(𝑡
These are exactly the same equations we obtained earlier using
Newtonian mechanics.

8. Mathematical Modelling
Virtual Work
Application of Virtual Work to beam structures.
The real power of Virtual Work becomes
apparent when trying to model distributed mass
structures like beams (not necessarily uniform).

9. Mathematical Modelling
𝜕𝑦

10. Mathematical Modelling
Application of Virtual Work to beam structures.

11. Mathematical Modelling
Application of Virtual Work to beam structures.

12. Mathematical Modelling
Application of Virtual Work to beam structures.
𝑑𝜃
R
d𝑠 = 𝑅𝑑𝜃

13. Mathematical Modelling
Application of Virtual Work to beam structures.
𝑥 𝑑𝑥
(′inertia′ force) 𝑚 ሷ
𝑦
Consider element of length dx, the local Virtual Displacement is:
𝛿𝑦(𝑥, 𝑡)
The virtual work done by the ‘inertia’ force
𝛿𝑦 = 𝜓(𝑥)𝛿𝑧
mass of
element
= 𝑚 𝑥 𝑑𝑥
𝑦

14. Mathematical Modelling
Application to Virtual Work beam structures.

15. Mathematical Modelling
Application of Virtual Work to beam structures.

16. Mathematical Modelling
Application of Virtual Work to beam structures.

17. Mathematical Modelling
Application of Virtual Work to beam structures.

18. Mathematical Modelling
Application of Virtual Work to beam structures.