# DOMV No 6 MATH MODELLING Virtual Work.pdf

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### DOMV No 6 MATH MODELLING Virtual Work.pdf

• 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).
• 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.
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