DIFFERENTIAL EQUATIONS OF MOTION OF A CONSTRAINED PARTICLE
Kalju Kenk
kkenk@staff.ttu.ee
Department of Mechatronics
Tallinn University of Technology
Introduction
To solve the problems of a constrained particle dynamics the mixed systems of
differential and algebraic equations (differential-algebraic equations – DAE) have been
usually used. But many mathematical difficulties arise by direct solving of DAEs. If the
index of DAE is greater than one, the index reduction operations and the changes of the
coordinates are recommended [1 ]. Often the singularities arise.
In this paper the differential equations of motion in Cartesian coordinates are derived to
solve the problems of dynamics of a particle having the holonomic and ideal constraints.
Any singularity does not arise. To become the solutions the Matlab ODE solvers can be
direct used. The motion of the particles on the frictionless cylindrical surface and on the
frictionless parabolic curve are examined for example.
Differential equations of motion
Let the constraints of the particle are holonomic and workless.
It is wellknown, that the dynamics of this particle can be examined by use of the
equations
(1)
mx1  F1a  F1c
mx2  F2a  F2c
(2)
mx3  F3a  F3c
 1 x1 , x 2 , x 3 , t   0
 2 x1 , x 2 , x 3 , t   0
(3)
(4)
(5)
where
m - mass of the particle
xi - Cartesian coordinates i  1,2,3 in an inertial frame
xi - second time derivate of xi
Fia - component on the i - axis of the resultant of the active forces
Fic - component on the i - axis of the resultant of the constraint forces
   0 - scalar constraints conditions   1,2
The equations from (1) to (5) constitute a DAE.
The condition    0 represent a surface and consider the workless of the constraints
1
 
(6)
xi
where  - respective scalar Lagrange factor.
The equations from (1) to (3) can yet be rewritten in form
 
(7)
mxi  Fia  
xi
and the repetition of the index  means the summation from 1 to 2.
The equations (7) consist the unknown factors 1 and 2 . Does not exist any
differential equation to calculate ones. Thus is suitable to eliminate them.
The second time derivates of the equations from (4) to (5) give respectively
 1
 2 1
 2 1
 2 1
xi 
x i x j  2
x i 
0
(8)
x i
x i x j
x i t
t 2
Fci  
 2
 2 2
 2 2
 2 2
xi 
x i x j  2
x i 
0
x i
x i x j
x i t
t 2
where the repetition of the indices i and j means the summation from 1 to 3.
From equations (7)
Fia
  
xi 

m
m x i
The substitution of (10) into (8) and (9) gives
1 1 1 2 1  2

 f1
m xi xi
m xi xi
1 1  2 2  2  2

 f2
m xi xi
m xi xi
where
 Fia 1  21
 21
 21 




f1  

x x 2
xi  2
 m xi xi x j i j
xi t
t 

 F a  2  2  2
 2 2
 2  2 
f 2   i

x i x j  2
x i 
 m x i
x i x j
x i t
t 2 

From (11) and (12)
1 D1

m D
 2 D2

m
D
where
 2  2
1  2
D1  f 1
 f2
xi xi
xi xi
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
2
D2  f 2
1 1
1  2
 f1
xi xi
xi xi
(18)
2
 1  1  2  2   1  2 

(19)
D

x i x i x j x j  x i x i 
Thus the differential equations of motion (10) can be rewritten in form
F a D  
xi  i  
(20)
m
D x i
If the DOF of the particle is equal to one, i.e. both  1 and  2 do not be identical to
zero, only one of equations (20) is independent. Regardless of this, these equations can be
numerically solved by use for example the ODE solvers of Matlab.
If the DOF of the particle is equal to two, i.e. for example  2 is identical to zero, two of
equations (20) are independent. In this case the equations (20) can be written as
Fa
f1
 1
xi  i 
(21)
 1  1 x i
m
x j x j
and again direct solved by ODE solvers.
Example 1.
The particle of mass m is constrained to move on the smooth cylindrical surface given
by equation
1  x12  x 22  r 2  0
where r is the radius of the surface and the active force is the gravitation given by
equations
F1  mg, F2  0, F3  0
The initial values are given as
x1  0, x 2  r, x 3  0
x1  0, x 2  0, x 3  v 0
Solution
It is easy to become
 1  2  1  2  1


 0,
t
x i t
t 2
 1
 2 x1 ,
x1
 1
 2 x2 ,
x 2
 1
0
x 3


 F a  1
 2 1
 2 1
 2  1 
f 1   i

x i x j  2
x i 
 2 gx1  x12  x 22
2 
 m x i

x

x

x

t
t 
i
j
i

 1  1
 4 x12  x 22  4r 2
x l x l
and the differential equations of the motion (21) in form



3
x1  g 

gx12  x1 x12  x 22

r2

gx1 x 2  x 2 x12  x 22
x2  
r

2
x3  0
In following the new coordinates     1,2,3,...,6
1  x1 ,  2  x 2 ,  3  x 3 ,  4  x1 ,  5  x 2 ,  6  x 3
and the correspond system of first order equations
1   4
  
2
5
3   6
4  g 
5  

g12  1  42   52
r
2

g1 2   2  42   52


r2
6  0
are used.
To numerical solve of these equations with initial values at t  0 and date values
10  0,  20  r,  30  0,  40  0,  50  0,  60  v0
g  9.81 ms-2, r  2 m, v0  1 ms-1
the Matlab solver ODE45 was used.
The results are presented on the following figure 1.
Figure 1
4
Example 2.
The particle of mass m is constrained to move along the smooth parabolic curve given
by equations
 1  2 x 2  x12  0
 2  x3  0
and the active force is the gravitation given by equations
F1  mg, F2  0, F3  0
The initial values are given as
x1  0, x 2  0, x 3  0
x1  0, x 2  0, x 3  0
Solution
 1  2  1  2  1


 0,
t
x i t
t 2
 1
 1
 1
 2 x1 ,
 2,
0
x1
x 2
x 3
 2
 0,
x1

 2
 0,
x 2

 2
1
x 3


D1  2 gx1  x12 ,
D2  0,
D  4 1  x12
Consider above results the equations (20) can be written in form
gx12  x1 x12
x1  g 
1  x12

x2 

gx1  x12 
1  x12
x3  0
It is obviously that x 3 is identical to zero.
In the following the new coordinates     1,2,3,4
1  x1 ,  2  x 2 ,  3  x1 ,  4  x 2
and the correspond system of first order equations
1   3
 2   4

g12  112 
 3  g 
1  12

g1  12 
 4 
1  12
with initial values at t  0
5
10   20   30   40  0
and the Matlab solver ODE45 are used.
The results are presented on the following figure 2
.
Figure 2
Conclusions
The differential equations of motion of constrained particle are presented in Carthesian
coordinates. Any change of the coordinates and usually recommended [2] determine of
dependent and independent variables do not be necessary. The using of the proposed
method to solve the problems of dynamics of body and body systems is a matter of
following .papers.
References
1. K.E. Brenan,, S.L. Campbell and L.R. Petzold. Numerical Solution of Initial-Value
Problems in Differential-Algebraic Equations.SIAM, Philadelphia,1996
2. E. J. Haug. Intermediate Dynamics. Prentice Hall, New Jersey, 1992
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