Lecture #6 Lagrange's method (examples)
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ME 563 Mechanical Vibrations Lecture #6 Lagrange’s Method for Deriving Equations of Motion (examples)
Transcript of Lecture #6 Lagrange's method (examples)
ME 563 Mechanical Vibrations
Lecture #6 Lagrange’s Method for
Deriving Equations of Motion (examples)
Example Rolling Disc on a Moving Cart
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K1
f(t)
No slip 3D/4
D
K2
M1,Icm
M2
g
x2
θ1
A
B
2
Eliminate constraints (keep x2 and θ1)
Example Rolling Disc on a Moving Cart
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€
Qθ1
* = Fi,nc( )i=1
N
∑ ⋅∂ri∂θ1
= f (t)ii=1
N
∑ ⋅∂ x2 +
D2θ1
i
∂θ1
= f (t)i ⋅ D2i
=D2f (t)
€
Qx2* = Fi,nc( )
i=1
N
∑ ⋅∂ri∂x2
= f (t)ii=1
N
∑ ⋅∂ x2 +
D2θ1
i
∂x2= f (t)i ⋅1i= f (t)
Example Rolling Disc on a Moving Cart
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Example Rolling Disc on a Moving Cart
Torque Example Rolling Disc on Incline
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Torque, τ(t), acting about point C
τ(t)
In the last slide, we considered forces but what about a torque applied at the center of mass of the rolling disc? How do we compute the generalized “force.”
We need to think in terms of torque acting about a point
Explanation Rolling Disc on Incline
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There are several things to keep in mind with a torque:
- A torque is not a force; therefore, the generalized coordinate we need to use is φ not x. -Our intuition about torques and where they are applied can let us down; when we apply a torque, we say it is applied about a point in the plane, not at a point x. -To understand the solution, we can think in terms of the work done by the generalized “forces”; the work done by these forces must be equal to the actual work done by the forces and torques applied to the body.
Option #1 Rolling Disc on Incline
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So the work done by a force p(t) applied at c is:
And the work done by a torque τ(t) applied about c is:
Then we can find the generalized “force” by equating the last two expressions in the these two equations.
Option #2 Rolling Disc on Incline
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Or, we can solve for the generalized “force” by applying the equation we used in all the other examples we have done:
Force p(t) at c, Torque τ(t) about c,
Example Two Degree of Freedom System
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Example Two Degree of Freedom System
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Raleigh’s dissipation function
Example Two Degree of Freedom System
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Check for accuracy through symmetry and positive definiteness of element matrices.
Example Two Degree of Freedom System
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Use of relative coordinates will distort symmetry.
Sprung Pendulum 13
r
θ
g
Sprung Pendulum 14
Rate of change of unit vector
Sprung Pendulum 15
Sprung Pendulum 16
r
θ
g
What if r is fixed? What if θ is fixed?
