Items tagged with dynamics


In this file you will be able to observe and analyze how the exercises and problems of Kinematics and Dynamics are solved using the commands and operators through a very well-structured syntax; Allowing me to save time and use it in interpretation. I hope you can share and spread to break the traditional and unnecessary myths. Only for Engineering and Science. Share if you like.

In Spanish.

Lenin Araujo Castillo

Ambassador of Maple

Here, I have a 3D map T=(T1,T2,T3) with




How can I use  IterativeMaps:-Attractor obtain the attractor for T.

How do we write code for optimal problem using Pontryagin's maximum principle for simulation.

I'd like to pay attention to an article J, B. van den Berg and J.-P. Lessard, Notices of the AMS, October 2015, p. 1057-1063.  We know numerous  applications of CASes to algebra. The authors present such  applications to dynamics. It would be interesting and useful to obtain  opinions of Maple experts on this topic.

Here is its introduction:

"Nonlinear dynamics shape the world around us, from the harmonious movements of celestial bod-
ies,  via  the  swirling  motions  in  fluid  flows,  to the  complicated  biochemistry  in  the  living  cell.
Mathematically  these  beautiful  phenomena  are modeled by nonlinear dynamical systems, mainly
in  the  form  of  ordinary  differential  equations (ODEs), partial differential equations (PDEs) and
delay differential equations (DDEs). The presence of nonlinearities severely complicates the mathe-
matical analysis of these dynamical systems, and the difficulties are even greater for PDEs and DDEs,
which are naturally defined on infinite-dimensional function spaces. With the availability of powerful
computers and sophisticated software, numerical simulations have quickly become the primary tool
to study the models. However, while the pace of progress increases, one may ask: just how reliable
are our computations? Even for finite-dimensional ODEs, this question naturally arises if the system
under  study  is  chaotic,  as  small  differences  in initial conditions (such as those due to rounding
errors  in  numerical  computations)  yield  wildly diverging outcomes. These issues have motivated
the development of the field of rigorous numerics in dynamics"

ABSTRACT. In this paper we demonstrate how the simulation of dynamic systems engineering has been implemented with graphics software algorithms using maple and MapleSim. Today, many of our researchers the computational modeling performed by inserting a piece of code from static work; with these packages we have implemented through the automation components of kinematics and dynamics of solids simple to complex.

It is very important to note that once developed equations study; recently we can move to the simulation; to thereby start the physical construction of the system. We will use mathematical and computational methods using the embedded buttons which lie in the dynamics leaves and viewing platform cloud of Maplesoft and power MapleNet for online evaluation of specialists in the area. Finally they will see some work done; which integrate various mechanical and computational concepts implemented for companies in real time and pattern of credibility.



(in spanish)


Lenin Araujo Castillo



Derive the orbit of the Moon around the Earth by doing a Verlet algorith of Molecular Dynamics simulation. Use one hour for your step τ. Place the stationary Earth at the origin of the Cartesian system. For initial conditions, use the position and the speed of the Moon when it is at its apogee (furthest from Earth). Plot the orbit.

Hi! I'm having trouble with the MapleSim "Equation Extraction" template. I have an extremely simple dynamics scenario with RigidBody at a particular position in a world; when the simulation starts, the body should simply start fallin with a constant acceleration, and I'm looking to extract kinematic equations for this situation.

However, the problem I have with the extracted equations is that they contain some rotational dependent terms that I wish to avoid - a lot of sines and cosines with arguments that I actually cannot match up to any variables in the model, or when I enable quaternion representation, terms like this:

What I don't understand is what exactly these rotation terms represent. The object should be falling straight downwards, and gravitation should be applied straight downwards, and when I start the simulation it does run through, also indicating that there should be no unbound variables. I'm not sure exactly what parameters in the Modelica model do they correspond to? I did not find any such angles in the object settings. And can I ask MapleSim to generate equations without these unbound rotation variables?

The equations of motion for a rigid body can be obtained from the principles governing the motion of a particle system. Now we will solve with Maple.

(in spanish)


Lenin Araujo Castillo

Corrección ejercico 4


4.- Cada una de las barras mostradas tiene una longitud de 1 m y una masa de 2 kg. Ambas giran en el plano horizontal. La barra AB gira con una velocidad angular constante de 4 rad/s en sentido contrario al de las manecillas del reloj. En el instante mostrado, la barra BC gira a 6 rad/s en sentido contrario al de las manecillas del reloj. ¿Cuál es la aceleración angular de la barra BC?


restart; with(VectorCalculus)



m := 2

L := 1

theta := (1/4)*Pi

a[G] = x*alpha[BC]*r[G/B]-omega[BC]^2*r[G/B]+a[B]NULL


a[B] = x*alpha[AB]*r[B/A]-omega[AB]^2*r[B/A]+a[A]


aA := `<,>`(0, 0, 0)

`&alpha;AB` := `<,>`(0, 0, 0)

rBrA := `<,>`(1, 0, 0)

`&omega;AB` := `<,>`(0, 0, 4)

aB := aA+`&x`(`&alpha;AB`, rBrA)-4^2*rBrA

Vector[column](%id = 4411990810)


`&alpha;BC` := `<,>`(0, 0, `&alpha;bc`)

rGrB := `<,>`(.5*cos((1/4)*Pi), -.5*sin((1/4)*Pi), 0)

aG := evalf(aB+`&x`(`&alpha;BC`, rGrB)-6^2*rGrB, 5)

Vector[column](%id = 4412052178)


usando "(&sum;)M[G]=r[BC] x F[xy]"

rBC := `<,>`(.5*cos((1/4)*Pi), -.5*sin((1/4)*Pi), 0)

Fxy := `<,>`(Fx, -Fy, 0)


`&x`(rBC, Fxy) = (1/12*2)*1^2*`&alpha;bc`

(.2500000000*sqrt(2)*(-.70710*`&alpha;bc`-25.456)+(.2500000000*(57.456-.70710*`&alpha;bc`))*sqrt(2))*e[z] = (1/6)*`&alpha;bc`



"(&sum;)Fx:-Fx=m*ax"           y             "(&sum;)Fy:Fy=m*ay"

ax := -28.728+.35355*`&alpha;bc`



ay := .35355*`&alpha;bc`+12.728



Fx := -2*ax



Fy := 2*ay



`&x`(rBC, Fxy) = (1/12*2)*1^2*`&alpha;bc`

(.2500000000*sqrt(2)*(-.70710*`&alpha;bc`-25.456)+(.2500000000*(57.456-.70710*`&alpha;bc`))*sqrt(2))*e[z] = (1/6)*`&alpha;bc`


.2500000000*sqrt(2)*(-.70710*`&alpha;bc`-25.456)+(.2500000000*(57.456-.70710*`&alpha;bc`))*sqrt(2) = (1/6)*`&alpha;bc`

.2500000000*2^(1/2)*(-.70710*`&alpha;bc`-25.456)+(14.36400000-.1767750000*`&alpha;bc`)*2^(1/2) = (1/6)*`&alpha;bc`



[[`&alpha;bc` = 16.97068481]]





I have been having problems with using the BodePlot function with units:


R1 := 18.2*10^3*Unit('Omega');

R2 := 10^3*Unit('Omega');

C1 := 470*10^(-12)*Unit('F');

C2 := 4.7*10^(-9)*Unit('F');

# wo is in hertz

wo := 1/sqrt(R1*R2*C1*C2);

# Q is unitless

Q := wo*R1*R2*C2/(R1+R2)



sys := TransferFunction(wo^2/(s^2+wo*s/Q+wo^2));


This is the error message I got:

Error, (in Units:-Standard:-+) the units `1` and `Hz` have incompatible dimensions


I think the problem is that the BodePlot function doesn't expect 'wo' to have units.  

So I tried to work around the issue by using the loglogplot but it doesn't seem to like 

complex function even when I used abs to find the magnitude (with or without units).


 Any workaround is appreciated.

Is it possible to solve piecewise differential equations directly instead of separating the pieces and solving them separately.

like for example if i have a two dimensional function f(t,x) whose dynamics is as follows:

dynamics:= piecewise((t,x) in D1, pde1, pde2); where D1 is some region in (t,x)-plane

now is it possible to solve this system with one pde call numerically?

pde(dynamics, boundary conditions, numeric); doesnot work

I am trying to solve a set of equations for a Fluid dynamics problem and I cannot get a result...Any ideas why?

rho := 1.184;
nu := 1.562*10^(-5);
ID := .15;
L := 24.5;
Kl := 12.69;
Ho := 50.52;
a := 2.1*10^(-5);
E := 0.1e-2; alpha := 1.05;

sys := {Re = ID*V/nu, hl = (f*L/ID+Kl)*V^2/(2*9.81), Vflow = (1/4)*Pi*ID^2*V, Hrequired = alpha*V^2/(2*9.81)+hl, Hrequired = -a*Vflow^2+Ho, 1/sqrt(f) = -1.8*log[10](6.9/Re+(E/(3.7))^1.11)};

solve(sys*{Re, V, f, hl, Vflow, Hrequired});
Error, (in unknown) invalid input: Utilities:-SetEquations expects its 2nd argument, equations, to be of type set({boolean, algebraic, relation}), but received {{Re = (9603072983/1000000)*V, hl = (5096839959/100000000000)*((1633333333/10000000)*f+1269/100)*V^2, Vflow = (9/1600)*Pi*V, Hrequired = (5351681957/100000000000)*V^2+hl, Hrequired = -(21/1000000)*Vflow^2+1263/25, 1/f^(1/2) = -(9/5)*ln((69/10)/Re+27367561/250000000000)/ln(10)}}




I am trying to solve rigid nody dynamics on Maplesim!! Trying to simulate Gyroscopic Effect.. I want to plot Angular Momentum of that rigid body!!


How do I do this??




 Sorry for my english, I'm french! :)

I try to make a modelisation of pedestrian dynamics.



There's someone, (a dot in Maple) at some randomn place in a square.


He want to go to another place.

So, there's 3 dot in Maple:

- S: The start fixed

- X: Mr X who is traveling from S to E

- E: The end fixed

so, without any rule, X is drawing a line from S to E.

I have uploaded to the Maplesoft Application Center a worksheet exploring the orbital dynamics of the recently discovered Kepler 16 system, where a planet orbits a double star. 

Your comments and suggestions will be appreciated.

And so with this provocative title, "pushing dsolve to its limits" I want to share some difficulties I've been having in doing just that. I'm looking at a dynamic system of 3 ODEs. The system has a continuum of stationary points along a line. For each point on the line, there exist a stable (center) manifold, also a line, such that the point may be approached from both directions. However, simulating the converging trajectory has proven difficult.

I have simulated as...

1 2 Page 1 of 2