- 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.