D Alembert s principle of virtual work

Transcription

1 PH101 Lecture 9 Review of Lagrange s equations from D Alembert s Principle, Examples of Generalized Forces a way to deal with friction, and other non-conservative forces

2 D Alembert s principle of virtual work If virtual work done by the constraint forces is ( =) (from eq.-1), = D Alembert s principle of Virtual work Now, for a general system of particles having virtual displacements,,,..,, ( ) = Applied force on particle Does not necessarily means that individual terms of the summation are zero as are not independent, they are connected by constrain relation

3 Lagrange s equation from D Alembert s principle D Alembert s principle, ( ) = Constraint forces are out of the game! Now, no need of additional subscript, we shall simply write instead of But How to express this relation so that individual terms in the summation are zero? Switch to generalized coordinate system as they are independent! Let s take the 1 st term = = = Generalized force = Dimensions of is not always of force! Dimensions of is always of work!

4 Lagrange s equation from D Alembert s principle 2 nd Term: = =, Bit of rearrangement in derivatives = Time and coordinate derivative can be interchanged! = = dot cancellation! = =

5 Lagrange s equation from D Alembert s principle Thus 2 nd term becomes =, = = The 1 st term =

6 Lagrange s equation from D Alembert s principle D Alembert s principle in generalized coordinates becomes = =0 Well, we are very close to Lagrange s equation! Since generalized coordinates are all independent each term in the summation is zero = = + + If all the forces are conservative, then = = = = = Total potential =

7 Lagrange s equation from D Alembert s principle Hence, = = Assume that does not depend on, then =0 = = Where, (,,)=(,,) (,) We have reached to Lagrange s equation from D Alembert s principle.

8 Review of the steps we followed Started from Newton s law = + Taken dot product with virtual displacement to kick out constrain force from the game by using =0; Arrive at D Alembert s principle = Extended D Alembert s principle for a system of particles; ( ) = Converted this expression in generalized coordinate system that every term of this summation is zero to get = This is a more general expression! Now, with the assumptions: i) Forces are conservative, =, hence = and ii) potential does not depend on, then =0 We get back our Lagrange s eqn., =

9 Discussion on generalized force A system may experience both conservative, non-conservative forces i,e. = + Hence generalized force for the system = = + = = + + = = Generalized force corresponding to conservative part Generalized force corresponding to non-conservative part

10 Lagrange s equation with both conservative and nonconservative force If system may experience both conservative, non-conservative forces = + Generalized force corresponding to conservative force can be derived from potential = = + = = Assume that does not depend on, then = =

11 More on Lagrange s equations

12 Example-5 Example 5: A mass slides down a frictionless plane inclined at angle. A pendulum, with length, and mass, is attached to. Find the equations of motion. For small oscillation

13 Example-5 X (, ) (, ) Four constrains equations =0; =0 = tan ( ) +( ) = Step-1: Find the degrees of freedom and choose suitable generalized coordinates Y Two particles =2,. =4 =32 4=2 Hence number of generalized coordinates must be two. and can serve as generalized coordinates (they are independent nature)

14 Example-5 continued. Step-2: Find out transformation relations = cos; = sin = cos+sin ; = sin+cos Step-3: Write in Cartesian All the constrains relations have been included in the problem through these relationship = V= Step-4:Convert From transformation equation = 1 2 [ + +2cos(+)] V= (sin+cos) sin = cos; =sin = cos+cos; =sin sin

15 Example-5 continued. Step-5: Write down Lagrangian = = 1 2 [ + +2cos(+)] (sin+cos)+ sin Step-5: Write down Lagrange s equation for each generalized coordinates =0 =0 From 1 st eqn [+ cos + +] gsin sin=0 + + cos + + sin(+) + sin=0 From 2 nd eqn [ + cos(+)]+sin + +sin=0 +cos + +sin=0

16 Problems with generalized force

17 Example-6 Y X

18 Example-7; Ring & mass on horizontal plane? Y = C R X

19 Example-8; Wedge & Block under friction, f Y The Table is friction less! X Generalized coordinate (,)

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