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Re: Animation = Translation + Vibration, But How?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg95192] Re: Animation = Translation + Vibration, But How?
  • From: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>
  • Date: Fri, 9 Jan 2009 06:24:39 -0500 (EST)

Hi, Gidil,
there was a nice joke (though I would not take a risk of repeating it here) 
ending up with a dialog: "But why?" "But how?". Did you mean it?  :-) 

To be serious, why not to take a well-known solution of a cantilever and animate it? It will then at least 
behave as it should.

See for instance, Landau, L. D. & Lifshitz, E. M. Theory of Elasticity (Pergamon Press, Oxford, 1986) 
the chapter on bending of rods. Take the one with one end clumped, another loaded by a point force. You will 
easily find it in problems to the chapter on small bending. 

Just to give an example:

(* Begin of the example 1 *)
(* This is the solution I mean for a cantilever, length 1, clumped at its left end and *)
(* loaded by a force 0.1*Sin[t] at its another end *)

z[x_, t_] := x^2*(3 - x)*0.1*Sin[t];

(* This shows vibration of its free end. Play here with the option Thickness[n] *)
(* to display the rod as a rod, rather than a line *)

Animate[Plot[z[x, t], {x, 0, 1}, PlotStyle -> Thickness[0.02], 
  PlotRange -> {-1, 1}, Frame -> False, Ticks -> None, 
  Axes -> None], {t, 0, 10 Pi, 0.1}, Paneled -> False]

(* End of the example 1 *)


Try this. If you need to show a more complex motion you may again play with equation of motion of its left end
for which you can give any function.  
For example, try this:

(* Begin of example 2 *)

(* this is the law of motion of the left end *)
x0[t_] := Cos[t/5];

(* This we already had in the previous example, but now the left end moves *)

z[x_, t_] := (x - x0[t])^2*(3 - (x - x0[t]))*0.1*Sin[t];

Animate[Plot[z[x, t], {x, x0[t], 1 + x0[t]}, 
  PlotStyle -> Thickness[0.02], PlotRange -> {{-1, 2}, {-1, 1}}, 
  Frame -> False, Ticks -> None, Axes -> None], {t, 0, 10 Pi, 0.1}, 
 Paneled -> False]

(* End of Example 2 *)

You may want to attach any object to the oscillating cantilever end. Try this:

(* Begin of the example 3  *)

x0[t_] := Cos[t/5];
z[x_, t_] := (x - x0[t])^2*(3 - (x - x0[t]))*0.1*Sin[t];
Animate[Show[{Plot[z[x, t], {x, x0[t], 1 + x0[t]}, 
    PlotStyle -> Thickness[0.02], PlotRange -> {{-1, 2.3}, {-1, 1}}, 
    Frame -> False, Ticks -> None, Axes -> None], 
   Graphics[Disk[{x0[t] + 1, z[x0[t] + 1, t]}, 0.1]]}], {t, 0, 10 Pi, 
  0.1}, Paneled -> False]

(* End of the example 3 *)


.... and so on. 
 

Finally, if you need to to have a self-standing movie, have a look into the today-posted discussion 
Re: [mg95123]  Manipulate, Export, .avi, forward run without the slider in the...

Have fun  :-) , Alexei 




GidiL wrote:

> Dear All!

> 

> I created a cantilever in Mathematica (nothing fancy, a Graphics 3D

> object created with Polygon).

> The only thing that I want now is to simulate its movement. I thought

> it would be easy, but it's proving to be diabolically difficult.

> Boundary conditions: the cantilever should be fixed in one end, and

> allowed to oscillate in the other (the oscillations are predetermined

> by some simple trigonometric function).

> This system should be allowed to translate in space (a moving beam, so

> to speak).

> So it should be allowed to move in the X-Y plane and oscillate along

> the Z- axis.

> 

> Moving it in the X-Y plane is accomplished with the Translate

> function. But how can I make it oscillate in a specific manner? How

> can I combine in one animation both movements?

> 

> Any help would be greatly apprerciated,

> 

> Gideon

-- 
Alexei Boulbitch, Dr., Habil.
Senior Scientist

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