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Sect. 1.4: D’Alembert’s Principle & Lagrange’s Eqtns. Virtual ( infinitesimal ) displacement  Change in the system configuration as result of an arbitrary infinitesimal change of coordinates δ r i , consistent with the forces & constraints imposed on the system at a given time t.

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sect 1 4 d alembert s principle lagrange s eqtns
Sect. 1.4: D’Alembert’s Principle & Lagrange’s Eqtns
  • Virtual (infinitesimal)displacementChange in the system configuration as result of an arbitrary infinitesimal change of coordinates δri, consistent with the forces & constraints imposed on the system at a given time t.
  • “Virtual” distinguishes it from an actual displacement dri, occurring in small time interval dt(during which forces & constraints may change)
slide3
Consider the system at equilibrium:The total force on each particle is Fi = 0. Virtual work done by Fiin displacement δri:

δWi = Fiδri = 0. Sum over i:

 δW = ∑iFiδri = 0.

  • Decompose Fiinto applied forceFi(a) & constraint force fi: Fi = Fi(a) + fi

 δW= ∑i (Fi(a) + fi )δri δW(a)+ δW(c) = 0

  • Special case (often true, see text discussion): Systems for which the net virtual work due to constraint forces is zero: ∑ifiδri δW(c) = 0
principle of virtual work
Principle of Virtual Work

 Condition for system equilibrium:Virtual work

due to APPLIED forces vanishes:

δW(a)= ∑iFi(a)δri = 0 (1)

 Principle of Virtual Work

  • Note: In general coefficients of δri , Fi(a)  0 even though ∑iFi(a)δri = 0 because δriare not independent, but connected by constraints.
    • In order to have coefficients of δri = 0, must transform Principle of Virtual Work into a form involving virtual displacements of generalized coordinates q , which are independent. (1) is good since it does not involve constraint forces fi . But so far, only statics. Want to treat dynamics!
d alembert s principle
D’Alembert’s Principle
  • Dynamics: Start with Newton’s 2nd Law for particle i: Fi= (dpi/dt) Or: Fi- (dpi/dt) = 0

 Can view system particles as in “equilibrium” under a force

= actual force + “reversed effective force” = -(dp/dt)

  • Virtual work done is

δW = ∑i[Fi- (dpi/dt)]δri = 0

  • Again decompose Fi: Fi = Fi(a) + fi

 δW= ∑i[Fi(a)- (dpi/dt) + fi]δri = 0

  • Again restrict consideration to special case: Systems for which the net virtual work due to constraint forces is zero: ∑i fiδri δW(c) = 0
slide6
 δW = ∑i[Fi- (dpi/dt)]δri = 0 (2)

 D’Alembert’s Principle

    • Dropped the superscript (a)!
  • Transform (2) to an expression involving virtual displacements of q(which, for holonomic constraints, are indep of each other). Then, by linear independence, the coefficients of the δq= 0
slide7
δW = ∑i[Fi- (dpi/dt)]δri = 0(2)
  • Much manipulation follows! Only highlights here!
  • Transformation eqtns:

ri = ri(q1,q2,q3,.,t) (i = 1,2,3,…n)

  • Chain rule of differentiation (velocities):

vi (dri/dt) = ∑k(ri/qk)(dqk/dt) + (ri/t) (a)

  • Virtual displacements δriare connected to virtual

displacements δq: δri= ∑j (ri/qj)δqj(b)

generalized forces
Generalized Forces
  • 1st term of (2)(Combined with (b)):

∑i Fiδri = ∑i,j Fi(ri/qj)δqj ∑jQjδqj (c)

Define Generalized Force (corresponding to Generalized Coordinate qj): Qj ∑iFi(ri/qj)

    • Generalized Coordinates qj need not have units of length!

 Corresponding Generalized ForcesQj need not have units of force!

    • For example: If qj is an angle, corresponding Qj will be a torque!
slide9
2nd term of (2) (using (b) again):

∑i(dpi/dt)δri = ∑i[mi(d2ri/dt2)δri ] =

∑i,j[mi(d2ri/dt2)(ri/qj)δqj](d)

  • Manipulate with (d):∑i[mi(d2ri/dt2)(ri/qj)] =

∑i[d{mi(dri/dt)(ri/qj)}/dt] – ∑i[mi(dri/dt)d{(ri/qj)}/dt]

Also: d{(ri/qj)}/dt = {dri/dt}/qj (vi/qj)

Use (a):(vi/qj) = ∑k(2ri/qjqk)(dqk/dt) + (2ri/qjt)

From (a):(vi/qj) = (ri/qj)

So: ∑i[mi(d2ri/dt2)(ri/qj)]

= ∑i[d{mivi(vi/qj)}/dt] - ∑i[mivi(vi/qj)]

slide10
More manipulation  (2) is: ∑i[Fi-(dpi/dt)]δri = 0

∑j{d[(∑i (½)mi(vi)2)/qj]/dt - (∑i(½)mi(vi)2)/qj - Qj}δqj= 0

  • System kinetic energy is: T  (½)∑imi(vi)2

 D’Alembert’s Principle becomes

∑j{(d[T/qj]/dt) - (T/qj) - Qj}δqj= 0 (3)

    • Note: If qjare Cartesian coords, (T/qj) = 0

 In generalized coords, (T/qj) comes from the curvature of the qj. (Example: Polar coords, (T/θ) becomes the centripetal acceleration).

  • So far, no restriction on constraints except that they do no work under virtual displacement. qj are any set.

Special case:Holonomic Constraints It’s possible to find sets of qj for which each δqjis independent.

 Each term in (3) is separately 0!

slide11
Holonomic constraints D’Alembert’s Principle:

(d[T/qj]/dt) - (T/qj) = Qj (4)

(j = 1,2,3, … n)

  • Special case: A Potential ExistsFi= - iV
    • Needn’t be conservative! V could be a function of t!

Generalized forces have the form

Qj ∑i Fi(ri/qj) = - ∑i iV(ri/qj)  - (V/qj)

  • Put this in (4):(d[T/qj]/dt) - ([T-V]/qj) = 0
  • So far, V doesn’t depend on the velocities qj

 (d/dt)[(T-V)/qj] - (T-V)/qj = 0 (4´)

lagrange s equations
Lagrange’s Equations
  • Define:The LagrangianL of the system:

L  T - V

Can write D’Alembert’s Principle as:

(d/dt)[(L/qj)] - (L/qj) = 0 (5)(j = 1,2,3, … n)

(5)  Lagrange’s Equations

lagrange s eqtns
Lagrange’s Eqtns
  • Lagrangian:L  T - V
  • Lagrange’s Eqtns:

(d/dt)[(L/qj)] - (L/qj) = 0 (j = 1,2,3, … n)

  • Note:L is not unique, but is arbitrary to within the addition of a derivative (dF/dt). F = F(q,t) is any differentiable function of q’s & t.
  • That is, if we define a new Lagrangian L´

L´= L + (dF/dt)

It is easy to show that L´satisfies the same Lagrange’s Eqtns (above).