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Example 2

Finding the governing equation for the example at hand using Polar coordinate, r, θ.

Kinetic energy of the system is

(10) KE = (1)/(2)m(r’(t)² + r(t)²θ’(t)²)

The potential energy of the system is due to gravity and spring. The potential energy due to the gravity field is:

(11) PE_m = mg(y₀ − r(t)cos(θ(t)))

Where y₀ is y coordinate at θ = 0.

The potential energy due to springs is given by:

(12) PE_s = (1)/(2)k(r(t) − y₀)²

Neglecting the boundary terms, the variational of kinetic energy is

(13) ^t₂⌠⌡_t₁(δKEdt) = δ^t₂⌠⌡_t₁(1)/(2)m(r’(t)² + r(t)²θ’(t)²)dt = ^t₂⌠⌡_t₁(mr’(t)δr’ + mr²(t)θ’δθ’ + mθ’²(t)r(t)δr)dt = ^t₂⌠⌡_t₁( − mr’’(t)δr − m(r²(t)θ’)’δθ + mθ’²(t)r(t)δr)dt = ^t₂⌠⌡_t₁( − mr’’(t)δr − mr²(t)θ’’δθ − 2mr’(t)r(t)θ’δθ + mθ’²(t)r(t)δr)dt

Similarly, the variational of the potential energy is

(14) δ^t₂⌠⌡_t₁( PE_s + PE_m)dt = ^t₂⌠⌡_t₁δ⎛⎝(1)/(2)k(r(t) − y₀)² + mg(y₀ − r(t)cos(θ(t)))⎞⎠dt = ^t₂⌠⌡_t₁( − mgcos(θ(t))δr + mgr(t)sin(θ(t))δθ + k(r(t) − y₀)δr)dt

Hamilton’s formulation is then applied:

There is no non-conservative force in this problem; therefore, δW_nc = 0. Finally

(15) ^t₂⌠⌡_t₁(δKE − δPE + δW_nc)dt = ^t₂⌠⌡_t₁( − mr’’(t)δr − mr(t)²θ’’δθ − 2mr’(t)r(t)θ’δθ + mθ’(t)²r(t)δr)dt + ^t₂⌠⌡_t₁(mgcos(θ(t))δr − mgr(t)sin(θ(t))δθ − k(r(t) − y₀)δr)dt

Since r(t) and θ(t) are independent, coefficients of δθ and δr should be zero independently. So the governing equations will be

(16) − mr’’(t) + mθ’(t)²r(t) + mgcos(θ(t)) − k(r(t) − y₀) = 0 − mr(t)²θ’’ − 2mr’(t)r(t)θ’ − mgr(t)sin(θ(t)) = 0

Which can be simplified to

(17) mr’’(t) − mθ’(t)²r(t) − mgcos(θ(t)) + k(r(t) − y₀) = 0 mr(t)θ’’ + 2mr’(t)θ’ + mgsin(θ(t)) = 0

Question

Compare the two sets of generalized coordinate that were used to derive the equations of motion for the example at hand? What is the advantage of each set?

Common Mistake

One common mistake is to assume r(t) is a constant. For example in eq. 13↑, using integration by parts to simplify δ∫^t₂_t₁(1)/(2)m(r(t)²θ’(t)²)dt, if r is assumed to be a constant, the derivation becomes

(18) ^t₂⌠⌡_t₁mr(t)²θ’δθ’dt = − ^t₂⌠⌡_t₁mr(t)²θ’’δθdt

This mistake leads to a missing 2mr’(t)θ’ term in the governing equation for θ (eq. 17↑).