Hi Everyone

Just to put you in the context: during an internal turning operation, the overhang (ratio Length/Diameter of the tool [L/D]) is really important to guarantee the stability of the process (minimal vibration as possible). Having said that, it is desirable to increases the overhang to do deep holes, because of this the ratio L/D varies depending on the necessity and consequently the natural frequency of the tool will change.

As you can see in the attached Picture_A and B, I am trying to find the Eigenvalues when the overhang (ratio L/D) changes. Is it possible like in Figure 2 in the attached paper (link below)?

https://www.mdpi.com/2076-3417/9/15/2996/pdf

Consider:

L1 = ratio L/D (changeable);
L2 = Fixation of the tool (content)
L3 = the remaining part of the tool out of the fixation (changeable)

Tool length is content = L1 + L2 + L3

13_07_2020_Eigenvalues_3_span_beam.mw

How do I write this in logical notation in Maple? Is it even possible?

Show logical notation that expresses the following statement: If one dice shows an even number of spots and the second dice shows an odd number of spots, then the total for the pair is less than or equal to 9.

I've somehow understood that sequences are extremly powerful so far. However I have to admit that they are quite complicated to read and understand, and I still am a bit confused how to work with them properly.

The attached example shows a very simple sequence.

My question is - how can I use its results further on?

What I for example would like to have, is a plot, where the first results in the array are the x- values, and the second are used as the y-values. Before that I need to extract both of them into vectors.

Could this be done in one operation, or can I somehow iterate throught the members of the returned values?

exprseq.mw

Hello to all, does anyone create an example for me that how to calculate and plot streamline of fluid flow.

Hello

I solved these equations numerically but I need to solve it by Runge-Kutta fourth order Method. Kindly help me in the coding of the same.

restart;
N1 :=1:N2 :=1: N3 :=0.1 :R := -1:
EQ:={(1+N1)*diff(f(x),x$4)-N1*diff(g(x),x$2)-R*(-diff(f(x),x)*diff(f(x),x$2)+f(x)*diff(f(x),x$3))=0, N2*diff(g(x),x$2)+N1*(diff(f(x),x$2)-2*g(x))-N3*R*(f(x)*diff(g(x),x)-diff(f(x),x)*g(x))=0}:

IC:={D(D(f))(0)=0, D(f)(1)=0,f(0)=0,f(1)=1,g(0)=0,  g(1)=0}:

sol:= dsolve(EQ union IC, numeric,output=Array([0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1])):
 

Some rewriting for a general equation , how to do ?

Download vraag1-herleidingconics2.mw

I need to check if expression is "special" kind of polynomial. Where powers are allowed to be non-integers and can be fractions. This is not polynomial in the mathematical sense ofcourse so can not use type(expr,polynom) on it, and did not see a way to tell type(expr,polynom) to accept non-integer exponents.

For an example, given p(x):=x^2+x^(1/3)+3+sqrt(x)+x^Pi+1/x*sin(x): it will return false, due to sin(x) term there. Without this term, it will return true, since all other terms have the form x^anything.

Currently, this is what I do

expr:=x^2+x^(1/3)+3+sqrt(x)+x^Pi+1/x*sin(x):

if type(expr,polynom(anything,x)) then
   print("Yes, normal polynomial");
else
   what_is_left:=remove(Z->type(Z,{`^`('identical'(x),algebraic),'identical'(x)}),expr);
   if has(what_is_left,x) then
      print("Not special polynomial");
   else
      print("Yes, special polynomial");
   fi;
fi;

While with

expr:=x^2+x^(1/3)+3+sqrt(x)+x^Pi+1/x:

It will print "Yes, special polynomial"

Is the above a good way to do this, or do you suggest a better way? It seems to work on the few tests  I did on it so far.

It is always "polynomial" in one symbol, such as x. So if it contains any function of x, other than  x^exponent, where exponent can only be numeric, or other symbol, it will fail the test. So this below will pass the a above test as well

expr:=x^2+x^(1/3)+3+sqrt(x)+x^a+1/x:

I was just praising Maple for not rewriting/simplifying  expressions automatically without the explicit user asking for it, when I found the following strange result

expr:=arccos(a-p) does not cause any change in the input. Good.

But when I change the letter ordering to expr:=arccos(p-a) now Maple changed it to Pi - arccos(a - p)

I have no idea why. Is there an option to tell Maple not to do that, even if it is mathematically correct? 

restart;
expr:=arccos(a-p);

restart;
expr:=arccos(p-a);

It seems to use lexicographical ordering to re-write things. Is it possible to turn that off?  Notice that I did not ask for simplification or anything. 

Maple 2020.1, Physics 724

I was checking my solution against Maple. Maple gives solution with integral and RootOf. I am not able to simplify it to better compare.

restart;
ode:=y(x)=ln(cos(diff(y(x),x)))+diff(y(x),x)*tan(diff(y(x),x));
sol:=dsolve(ode);

It is the second solution above I want to simplify/evaluate. So I tried

restart;
ode:=y(x)=ln(cos(diff(y(x),x)))+diff(y(x),x)*tan(diff(y(x),x));
sol:=dsolve(ode);
sol:=[sol][2];
DEtools:-remove_RootOf(sol)

But this did nothing. Then I tried adding useint in the dsolve command

restart;
ode:=y(x)=ln(cos(diff(y(x),x)))+diff(y(x),x)*tan(diff(y(x),x));
sol:=dsolve(ode,useint);

But this also kept the integral there. Then I tried eval

restart;
ode:=y(x)=ln(cos(diff(y(x),x)))+diff(y(x),x)*tan(diff(y(x),x));
sol:=[dsolve(ode,'explicit')];
eval(sol[2])

The solution I obtained is 

y(x) = -ln(_C1^2 - 2*_C1*x + x^2 + 1)/2 + ((-2*x + 2*_C1)*arctan(-x + _C1))/2

Which does verify to zero OK using odetest.

Any suggestions/tricks to use to get an more explicit solution from Maple I am overlooking?

Maple 2020.1 , Physics 724

I can post/attach my full solution if needed. 

Hello,

I have a particular problem that I haven't managed to find the solution to by googling.

9/10 times that I open Maple 2020 I can't seem to open any of my saved documents. it opens the start tab, and I'm still able to open a new document or worksheet, but then when I try to change to text instead of math, it won't let me. I can type, but I can't press backspace or enter.

I have tried restarting and shutting down my computer, but every time I have to just force quit Maple because it won't respond half of the time and it won't quit normally.

I'm running the latest version of Maple and using macOS Catalina version 10.15.5.

How to plot the following functions?

plot(4*sqrt(L/g)*EllipticF(Pi/2, sin(theta0/2)), theta0 = 1 .. 20)

Hi. I have a relatively complicated inequality with three real variables. I was wondering how I can plot it three-dimensionally. The inequality is

where \(a\) and \(m\) and non-negative, while \(alpha\) could have any value. I am using Maple 2017.

 Suppose that A is an m×n real matrix. The function TA:ℝn→ℝm   defined by

TA(x)=Ax,  for all 𝕩∈ℝn

     is a linear transformation.

let A=<< 87, -66, -90, 12, 48>|<-36, -40, -82, -54, 15>|<99, 79, 76, -31, 27>|<-69, 15, -10, 45, -9>>

and x= <9,-4,2,-17>.

Then TA(x)=      

   Alternatively, you can copy your answer from your Maple worksheet and paste it to the answer box.

(b) Suppose now that the linear map T:ℝ2→ℝ3  is defined by, for all 𝕩=(x1,x2)∈ℝ2

 T(x)=  <<x1+6x2>|<−2x2>|<−4x1+2x2>>

     Enter the matrix M , in Maple syntax, in the box below such that T(x)=Mx  for all x∈ℝ2,  

  M=       

Can someone pls help me with this question. TIA.

I have no problems assigning simple plots to variables and then displaying all in a single plot using the display command.

However, I am now trying to do the same using implicitplot without success.

This is my code;

restart;
f := 3*(x^2 + y^2)^2 = 100*x*y;

with(plots, implicitplot);
p1 := implicitplot(f, x = -4 .. 4, y = -4 .. 4, rangeasview = true):

p2 := pointplot([[1, Pi/2], [-1, -Pi/2]], color = red):

display({p1,p2})

What I get is a small implicit plot and the code for the pointplot back at me.

Is there any way to make this work?

Thank you for your help.

Jose

I solved this ODE and got a solution and wanted to compre it with Maple. This is initial value first oder ODE. So it should have no constants in it. But Maple's solution contains something I never seen before _B1~

I wonder what it means? And odetest did not verify Maple's solution. 

restart;
ode:=diff(y(x),x)-y(x)/x+csc(y(x)/x)=0;
sol:=dsolve([ode,y(1)=0]);
simplify(odetest(sol,ode));

odetest does not gives zero.

This is my solution

mysol:=y(x)=x*arccos(ln(x)+1);
odetest(mysol,ode)

   0

Any idea what _B1~ means? The ~ looks like it is an assumed variable? may be leaked from inside Maple.

Maple 2020.1 on windows 10