| Topic: |
Science > Physics |
| User: |
"GRad" |
| Date: |
12 Nov 2004 01:11:15 PM |
| Object: |
Lyapunov stability |
good afternoon
i wish to ask
1) what could be an apropriate candidate for a Lyapunov function, V(x), so
that one can determine the stability of a simple mass (m) -spring
(k) -damper (c)system
the model equation is: mx(ddot)+cx(dot)+kx = 0
V(x) must be positive definite & d/dt{V(x)} must be negative definite
2) Also how does one prove the above system stable using LaSalle's invariant
principle?
many thanks for any insight
d
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| User: "Dave Langers" |
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| Title: Re: Lyapunov stability |
14 Nov 2004 06:26:29 AM |
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1) what could be an apropriate candidate for a Lyapunov function, V(x), so
that one can determine the stability of a simple mass (m) -spring
(k) -damper (c)system
the model equation is: mx(ddot)+cx(dot)+kx = 0
V(x) must be positive definite & d/dt{V(x)} must be negative definite
Must V(x) be a function of x only? I don't think that can be done in
general (if you require d/dt{V(x)} to be negative). However, if x(dot)
can be used as a variable as well, it seems feasible. The system moves
in a spiral pattern in state space (in a coordiante system with axes x
and x(dot) I mean), so choose the equi-V contours to be ellipses.
I would suggest
V(x,x(dot)) = A * x^2 + B * x(dot)^2
with A and B yet unknown.
Now
d/dt{V(x)} = 2*A*x*x(dot) + 2*B*x(dot)*x(ddot)
= 2*x(dot) * (A*x + B*x(ddot))
If you take A = k and B = m, then this equals
d/dt{V(x)} = 2*x(dot) * -c*x(dot) = -2c*x(dot)^2 < 0
if c > 0, with the obvious exception of x(dot) = x = 0.
2) Also how does one prove the above system stable using LaSalle's invariant
principle?
I am not familiar with LaSalle's principle...
--
M.vr.gr.
Dave
("d-dot-langers-at-wxs-dot-nl")
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| User: "Arnold Neumaier" |
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| Title: Re: Lyapunov stability |
14 Nov 2004 07:09:02 AM |
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GRad wrote:
i wish to ask
1) what could be an apropriate candidate for a Lyapunov function, V(x), so
that one can determine the stability of a simple mass (m) -spring
(k) -damper (c)system
the model equation is: mx(ddot)+cx(dot)+kx = 0
V(x) must be positive definite & d/dt{V(x)} must be negative definite
2) Also how does one prove the above system stable using LaSalle's invariant
principle?
s.p.r is not the right forum for getting your exercises done.
Have you ever seen an athlete win a medal who had others do the
physical exercises for him? Science is not really different!
You can find background for the above in most introductions to
theoretical mechanics. Read it from several different perspectives
(i.e., authors) if you can't get the insight from the first one.
Arnold Neumaier
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| User: "Robert Israel" |
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| Title: Re: Lyapunov stability |
14 Nov 2004 06:27:45 AM |
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In article <cn0a3g$eue$1@nunki.unm.edu>, GRad <dchristo@unm.edu> wrote:
1) what could be an apropriate candidate for a Lyapunov function, V(x), so
that one can determine the stability of a simple mass (m) -spring
(k) -damper (c)system
the model equation is: mx(ddot)+cx(dot)+kx = 0
V(x) must be positive definite & d/dt{V(x)} must be negative definite
It's not a function of x, it's a function of x and x(dot).
Hint: since you're cross-posting to physics groups, think about energy.
Robert Israel
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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