Equilibrium and stability condition
Equilibrium A “configuration” at which a mechanical system stays at rest as long as there is no external
disturbances (excitations).
Determination of equilibrium configurations
Example ml2θ&&+ mglsinθ = M (t)
= 0
=θ
θ&& &
EOM:
At equilibrium configuration θ =θ0 System stays at rest
m θ l
g M(t)
No external force and moment
M ( t ) = 0
) ( sin 0
2 mgl M t
ml θ&&+0 θ = 0
0
sinθ0 = θ0 = ±nπ n = 0,1,2K This pendulum has 2 equilibrium configs. at θ0 = 0 and π
Example: Equilibrium config.
m1 l1 θ1 g
θ2
m2 l2
Given EOM
0 )
sin (
cos
) cos (
sin sin
1 1
2 2
1 1
2 2
1 1
1 1
2 1 1
= +
− +
θ θ
θ θ
θ θ
l g
m
l g
m gl
m l
m &&
0 sin
)
( 1 1 2 2 2
2 2 2
2 2
2l θ + m l l θ + m gl θ =
m && &&
(assume small angle θ1 and θ2, and force in rod l2 ≅ m2g)
Determine equilibrium configuration.
Stability condition
A mechanical system disturbed from a “stable” equilibrium configuration oscillates about and/or returns to the same equilibrium configuration.
Equi.
Equi.
Stable at θ = 0 Unstable at θ = π
Determine stability condition (1)
Consider EOM
0 )
( )
( )
( t + c x t + kx t = x
m && &
The solution is, If m, c, k > 0
) sin(
)
(t = Ae−ζω ω t +φ
x nt d
) (
)
(t e nt a1e n 2 1t a2e n 2 1t x = −ζω −ω ζ − + +ω ζ −
nt
e t a a
t
x( ) = ( 1 + 2 ) −ω or
or
1
2
3
Determine stability condition (2)
0 )
( )
( )
( t + c x t + kx t = x
m && &
x(t) approaches zero as t becomes large If m, c, k > 0
Asymptotically stable If c, k < 0 but m > 0 Unstable, divergent
Flutter instability
Example: pendulum
m θ l
g
0
2θ + mgl sinθ =
ml &&
) cos(
) (
) sin(
)
sin(θ ≅ θ0 + θ −θ0 θ0 ) (
) 1 )(
( 0 )
sin(
EOM:
This pendulum has 2 equilibrium configs.
at θ0 = 0 and π
Consider at θ0 = π, linearize term sinθ θ
π π
θ
θ ≅ + − − = −
Sub. sinθ in EOM, yield
ml
2θ && − mgl θ = − mgl π
<0, unstable
Example: Inverted pendulum
Conclusion modeling
Assumptions
1. Neglect small effects
Reduces number and complexity of differential equations 2. Lumped characteristics
Leads to ordinary (rather than partial) differential equations 3. Linearity
Makes equations linear, allow superposition of solutions 4. Constant parameters
Leads to constant coefficients in differential equations 5. Neglecting uncertainly and noise
Avoids statistical treatment
Home work
Consider the disk of the figure connected to two springs.
Derive EOM for small angle θ (t). [Inman/1.82]
θ (t)
