" 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)
• Consider the system at equilibrium: The total
force on each particle is Fi = 0. Virtual work done by Fi in displacement δri:
δWi = Fi•δri = 0. Sum over i: ⇒ δW = ∑iFi•δri = 0. • Decompose Fi into applied force Fi(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
⇒ 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 δri are 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
• 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
⇒ δW = ∑i[Fi - (dpi/dt)]•δri = 0 ≡ D’Alembert’s Principle
(2)
– 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
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) displacements δq : δri = ∑j (∂ri/∂qj)δqj (a) (b)
• Virtual displacements δri are connected to virtual
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 Forces Qj need not have units of force! – For example: If qj is an angle, corresponding Qj will be a torque!
• 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)]
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
– Note: If qj are Cartesian coords, (∂T/∂qj) = 0
(3)
⇒ 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 δqj is independent.
• Holonomic constraints ⇒ D’Alembert’s Principle:
(d[∂T/∂qj]/dt) - (∂T/∂qj) = Qj
• Special case: A Potential Exists ⇒ Fi = - ∇ iV
(4)
(j = 1,2,3, … n)
– 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
• Define: The Lagrangian L of the system:
L≡T-V
⇒ Can write D’Alembert’s Principle as:
(d/dt)[(∂L/∂qj)] - (∂L/∂qj) = 0 (5)
1,2,3, … n)
(j =
(5) ≡ Lagrange’s Equations
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).
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