1

Proceedings of the ISRM 2009

1. International Symposium on Robotics and Mechatronics

September 21 - 23, 2009, Hanoi, Vietnam

Problem of Determining the Reaction Forces of Mechanical Constraints

Do Sanh1) , Dinh Van Phong1) , Phan Dang Phong 2)

Do Dang Khoa 3) , Nguyen Cao Thang 2)

1)Hanoi University of Technology

2)National Research Institute of Mechanical Engineering, Hanoi, Vietnam

3)University of Austin, Texas, USA

Abstract. In the paper it is presented a method of determining separately the

reaction forces of each constraint. For this purpose the Principle of Compatibility

and the method of the transformation of coordinates are applied together. In

accordance with this, each constraint is represented by a pseudo-coordinate,

which takes part in description of system motion. The zero value of this

coordinate is the condition of realization of the corresponding constraint. By

such a way, it is possible to define separately the reaction force of each

constraint.

Keywords: Principle of Compatibility, Pseudo-Coordinate, Motion Equation in

Matrix Form, Method of Transformation of Coordinate

1. Introduction

As known, the Euler- Newton Principle can be used

to find the reaction forces of constraints. For this aim

let exert the reaction forces of constraints to the

system under consideration. By such a way we are

engaged to the problem of investigating an

unconstrained system. However, this method is not

easy to determine the reaction force of each

constraint, which is of an interesting problem in

practical. This problem has been investigated in

many works, for example, in [2, 7]. However, the

proposed methods are not comfortable yet. For the

aim of determining the reaction force of constraint

let use the pseudo-coordinates to replace the

constraints of interest. These pseudo-coordinates are

in place of the dependent coordinates. The constraint

realized conditions are the magnitudes of these

coordinates equal to zero. In the other hand the latter

coordinates are just the constraints whose the

reaction forces are also the reaction forces of the

original constraint.

2. Background

Let us consider the mechanical system, the position

of which located by generalized coordinates

( 1, )i

q i m= . In addition the considered system

subjected the holonomic, stationary constraints the

kinetic energy of the system is of the form

, 1

1

2

m

ij i j

i j

T a q q=

= ∑ ɺ ɺ (1)

Where, ( , 1, )ij

a i j m= are the functions only of

generalized coordinates.

The expression of the kinetic energy can be written in

the matrix form

1

2T = T

q Aqɺ ɺ (2)

Where, qɺ is an (mx1) column matrix of generalized

velocities, but A is a matrix of (mxm) size. Let us

assume that A is a square, symmetric, nonsingular

matrix. The letter T located at right high corner

denotes the transpose of matrix. The generalized

forces corresponding to the generalized coordinate

( 1, )i

q i m= are denoted by ( 1, )i

Q i m= . The (mx1)

matrix of generalized forces are denoted by Q, that is

[ ]1 2. . .

T

mQ Q Q=Q .

From now on the matrices are written by the bold

letters, but the vector is treated by the column matrix.

Let the constraints imposing on the system under

consideration are of the form

2

1, 2( ,..., ) 0 ; ( 1, )

mf q q q rα α= = (3)

The system has then n=m-r degrees of freedom

By the Principle of Compatibility [13, 14, 15, 16] the

equations of the system can be written in the form

; ( 1, )i i

i i

d T TQ R i m

dt q q

∂ ∂− = + =

∂ ∂ɺ (4)

Where, ( 1, )i

R i m= are the generalized reaction

forces of the constraints Eq. (3) corresponding to the

coordinatei

q ,

1

Nk

i k

k i

rR R

q=

∂=

∂∑

��

(5)

Where, k

R�

is the reaction force of the constraints Eq.

(3) on the particle k

M having the position vector

denoted byk

r�

( 1, )k N= . Due to the constraints in Eq.

(3) to be ideal, the generalized reaction forces

( 1, )i

R i m= must satisfy the condition [14, 15, 16]:

1

0 ; ( 1, , )m

ki i

i

d R k n n m r=

= = = −∑ (6)

Where, ( 1, ; 1, )ki

d k n i m= = are the coefficients in the

expressions of the generalized accelerations

represented in the terms of pseudo-accelerations [5].

The motion equations of the system Eq. (4) can be

written in the matrix form

d T T

d

∂ ∂− =

∂ ∂Q + R

q qɺ (7)

Where, the matrix of the reaction forces R being the

(mx1) matrix, in accordance with Eq. (6), is written

as follows

D R = 0 (8)

The (n x m) matrix is of the form [14, 15, 16]

[ ]1, , 1,ki k n i m

d= =

=D (9)

The numerical method of determining the matrix D

is constructed by the software [3, 4]. The motion

equations of the system under consideration in

accordance with Eq. (2) can be written in the matrix

form [14, 15, 16]

o *Aq = Q + Q + Q + Rɺɺ (10)

Where, Q is the matrix of the acting forces, *

o

Q , Q are the (mx1) matrices, which are determined

only by means of the matrix of inertia A, but R-the

matrix of reaction forces, qɺɺ - the (mx1) matrix of

generalized accelerations, that is

[ ]1 2...

T

mq q q=qɺɺ ɺɺ ɺɺ ɺɺ

To find the reaction forces of Eq. (3), let us introduce

the pseudo-coordinates ( 1, ) :u mα α =

1 2( , ,..., ) ; ( 1. )

mu f q q q rα α α≡ = (11)

In order to define the position of the system under

consideration instead of the coordinates

{ }1 2, ,...

mq q q we use the set of coordinates

{ } { }1 2 1 2, ,.... , , ,..., , ; ( 1, ; 1, )

n r kq q q u u u q u k n rα α≡ = =

(12)

Where, the coordinates { }kq are the independent

ones chosen among the coordinates of the system

1 2( , ,..., ).

mq q q In other words, the coordinates

1 2( , ,..., )

mq q q are the coordinates , ( 1,

kq u k nα = ;

1, )rα = . By such a way the motion of the system is

described in the new coordinates

{ }, ; ( 1, ; 1, )k

q u k n rα α= = with the constraints of the

form [12]

0 ; ( 1, )u rα α= = (13)

It is noticed that the coordinates { }( 1, )k

q k n= are

the independent ones among the

coordinates { } ( 1, )i

q i m= . For the purpose to

separate the independent coordinates let us introduce

the (nx1) matrix q(k)

[ ]1 2( ) . . .

T

nk q q q=q (14)

In addition we apply the notation ( )αu to be a (rx1)

matrix, that is

[ ]1 2( ) . . .

T

ru u uα =u (15)

Let assume that by mean of the transformation from

the set of coordinates { }1 2, ,...,

mq q q to the new

coordinates { }1 2 1 2, ,..., , , ,...,

n rq q q u u u it may express

the generalized velocities in the old variables through

the ones of new variables. In other worlds we assume

that

1

; ( 1, )m

i ij j

j

q h i mσ=

= =∑ɺ ɺ (16)

Where, , ; ( 1, ; 1, )k k

q u k n rα ασ σ α≡ ≡ = =

The relations Eq. (16) may be written as follows

q = Hσɺ ɺ (17)

3

Where, H is a (mxm) matrix, its elements are the

coefficients in right hands of Eq. (16). The kinetic

energy of the system in the new coordinates will take

the form

1

2

TT = σ Aσɺ ɺ (18)

Where, the matrix A is of the following form

T=A H AH (19)

The matrix A in the form Eq. (19) can be rewritten

in the form

1 3

3 2

T

=

A AA

A A (20)

Where, the matrices 1 2

A , A are the (n x n) and (r x r)

matrices, respectively, but the (n x r) matrix3

A . It is

noticed that n is number of independent coordinates,

but r is corresponding to the dependent ones. In the

new coordinates the motion equations of the system

under consideration take the following form

0 *σ = +A Q Q + Q + Rɺɺ (21)

Where, by means of the transformation matrix H we

calculate

;T T=Q H Q R = H R (22)

For calculating the quantities *

0

Q , Q let us compute

the (mxm) matricesi

∂ A , which are of the form [14,

15, 16]

, 1,

rs

i

i r s m

a

σ=

∂∂ =

∂ A (23)

By means Eq. (23) we get

0 0 0 0

1 2 . . . mQ Q Q = Q (24)

Where,

0 1; ( 1, )

2

T

i iQ i mσ σ= ∂ =Aɺ ɺ (25)

The matrix*

Q is defined by the formula

* *

1

m

i

i

σ=

= ∂∑Q A ɺ (26)

Where,

[ ]*

1 2. . .

i i i m iσ σ σ σ σ σ σ=ɺ ɺ ɺ ɺ ɺ ɺ ɺ (27)

Let us write these matrices in the form

= (k) (α) ; = (k) (α)

= (k) (α) ; = (k) (α)

TT

T T

o o o

* * *

Q Q Q Q Q Q

Q Q Q R R R

(28)

The matrices with the bracket including k and α

have n columns and r rows respectively. In the new

coordinates, the equations of constraints are of the

form Eq. (3). Therefore, the matrix D in Eq. (8) will

take the form

[ ]( ) ( )k α=D D D (29)

Where D(k) is the identity matrix of the dimension (n

x n), but D( )α is the null matrix of the dimension (n

xr). By taking account to the construction of the

matrix D and the condition of ideality of the

constraints we get

( ) 0k ≡R (30)

Therefore,

0 ; ( 1, )k

R k n≡ = (31)

It is necessary to notice that the motion equations of

the system in new coordinates are of the form Eq.

(21). In addition the matrices

( )α α α= ≡u u( ) = u( ) 0ɺ ɺɺ

The motion equations Eq. (10) are being divided in

two groups:

-First group

k k k k ko *

1A ( )q( ) = Q( ) + Q ( ) + Q ( )ɺɺ (32)

-Second group

k α α α αo *

3A q( ) = Q( ) + Q ( ) + Q ( ) + R( )ɺɺ

ɺ

(33)

The first group does not contain the reaction forces of

constraints. In addition this is a system of closed

differential equation, which allows finding the

motion of the system under consideration by the

defined initial conditions. In the other words, we get

( ), ( ), ( ) ; ( 1, )k k k

q t q t q t k n=ɺ (34)

Together with the equations of constraints in Eq. (3)

we calculate

4

( ) ; ( 1, )q t rα α = (35)

By such a way the motion of the system with the

constraints in Eq. (3) is defined. The reaction forces

of constraints now are found by means of the second

group

*

3( ) ( ) ( )okα α α αR( ) = A q( ) - Q - Q - Qɺɺ (36)

Where,

( ), ( ), ( ) ; ( 1, )k k k

q t q t q t k n=ɺɺ ɺ

are calculated by means Eq. (32)

3. Application

For the aim of illustrating let us consider the

following example.

Example. Zhukovsky’s problem [2]. A beam AB of

the length 2L and mass M (P=Mg) slides with its

ends A and B on a perfectly smooth vertical wall and

a perfectly smooth horizontal floor (Fig.1). An

animal of mass m (p=mg) runs along the beam, its

motion is described by the law s = s (t). Find the

reaction force at the beam acting on the wall as well

the reaction force at the beam acting on the ground.

Let us consider the system consisting the beam and

the animal. Let choose the coordinate to be x, y,θ

and s, where, x, y are the coordinates of the mass

centre of the beam, θ is the angle between the beam

and the vertical. The kinetic energy of the system is

of the form

2 2 2 2 21 1 1( ) ( )

2 2 2

2 ( ) cos

2 sin 2 ( ) 2 cos )

T M x y J m x y

m L s x

m xs m L s y m y

θ

θ θ

θ θ θ θ

= + + + +

− −

+ − − −

ɺɺ ɺ ɺ ɺ

ɺɺ

ɺ ɺɺɺ ɺ ɺ

The matrix of inertia of the system then has the form

[ ]

2

1

2

where,

0

0 ( )

sin ( ) cos

cos ( )sin

sin cos

( ) cos ( )sin

( ) 0

0 ( )

m

J m L s

m m L s

m m L s

m m

m L s m L s

m M

m M

θ θ

θ θ

θ θ

θ θ

=

+ − = − − − − −

− − − − − = +

+

1 2A A , A

A

A

(36)

The generalized forces corresponding to generalized

co-ordinates will be

[ ]cos (2 )sin 0To mg mg L s Mgθ θ= − −Q (37)

The system under consideration is of two degrees of

freedom. Two constraint equations are of the form

1 2sin 0 ; cos 0f x L f y Lθ θ≡ − = ≡ − = (38)

For this purpose of determining the reaction forces

between the beam and the wall as well the beam and

the ground let us introduce the new coordinate u, v ,s

and θ , where, u is the abscissa of the end A but v is

the ordinate of the end B, i.e.

sin ; cosu x L v y Lθ θ= − = −

The coordinates s and θ take the old meaning that

motion of the system restricted by the constraint

of form now

0 ; 0u v= = (39)

The matrix of the transformation from the old

coordinates to the new the ones

1 0 0 0

0 1 0 0

0 cos 1 0

0 sin 0 1

L

L

θ

θ

=

−

H (40)

The matrix of inertia in the new coordinates is of the

form

B x

p

P

A

y

θ

Fig.1. Zhukovsky’s problem

5

2 2

0

sin 2

sin 2 4 ( )sin

sin ( ) cos

cos ( 2 )sin

sin cos

( ) cos ( 2 )sin

( ) 0

0 ( )

m mL

mL J ms mL L s

m ML ms

m ML mL ms

m m

ML ms ML mL ms

M m

M m

θ

θ θ

θ θ

θ θ

θ θ

θ θ

=

+ + − = +

− − + −

−

+ − + − = +

+

T

1 2

1

2

A = H AH A , A

A

A

(41)

2

0 0 0 0

0 2 4 sin cos sin

0 cos 0 0

0 sin 0 0

s

ms mL m m

m

m

θ θ θ

θ

θ

− ∂ =

A

(42)

[ ]

0 2 cos2

2 cos2 4 ( )sin2

cos ( )sin

sin ( 2 )cos

cos sin

( )sin ( 2 )cos

0 0

0 0

mL

mL mL L s

m ML ms

m ML mL ms

m m

ML ms ML mL ms

θ

θ

θ θ

θ θ

θ θ

θ θ

θ θ

∂ =

− = − +

− + −

− + − + − =

A M,N

M

N

0

0

u

v

∂ =

∂ =

A

A

(43)

Let calculate the quantities , , o *

Q Q Q in the new

coordinates with taking into account of Eq. (39) that

are

1 0 0 0 cos

0 1 cos sin sin

0 0 1 0 0

0 0 0 1

mg

L L mgs

Mg

θ

θ θ θ

− =

−

TQ = H Q

(44)

cos

( ) sin;

0

mg

ms ML g

Mg

θ

θ

− + =

−

T

o o o o o

s u vQ Q Q Qθ = Q

1

2

Ts

o sQ s u v s u vθ θ = ∂ = Aɺ ɺɺ ɺ ɺ ɺ ɺ ɺ

2( 2 sin )m s L θ θ− ɺ

2

1

2

= 2 cos 2 2 ( )sin 2

ToQ s u v s u v

m s mL L s

θ θθ θ

θθ θθ

= ∂

+ −

ɺ ɺɺ ɺ ɺ ɺ ɺ ɺ

ɺ ɺɺ

A

10

2

To

u uQ s u v s u vθ θ = ∂ = Aɺ ɺɺ ɺ ɺ ɺ ɺ ɺ

1

02

To

v vQ s u v s u vθ θ = ∂ = Aɺ ɺɺ ɺ ɺ ɺ ɺ ɺ

2 2

2

( sin )

2 cos 2 ( )sin 2

0

0

o

m L s

mL s mL L s

θ θ

θ θ θθ

−

+ − =

Q

ɺ

ɺ ɺɺ

(45)

* * * *

s u v

s s u u u v

θ

θ θ= ∂ + ∂ + ∂ + ∂ɺ ɺ ɺ ɺ

* * * * *Q = Q + Q + Q + Q

Aq Aq Aq Aq, (46)

Where,

* 2 * 2

* 2 * 2

;

;

s

u v

s s us vs s u v

su u u vu sv v uv v

θθ θ θ θ θ

θ θ

= =

= =

ɺ ɺ ɺ ɺ ɺɺ ɺɺ ɺ ɺɺ ɺɺ ɺ ɺ ɺ

ɺ ɺɺ ɺɺ ɺ ɺ ɺ ɺ ɺ ɺɺ ɺ ɺ ɺ ɺ

q q

q q

By following (41), (45) and (46), we calculate

2

2

*

2

2

2 cos 2

4 ( )sin 2 2 cos 2

cos ( )sin

( 2 )cos sin

ms

mL L s mL s

m s ML ms

ML mL ms m s

θθ

θθ θθ

θ θ θθ

θθ θ θ

− + = − + − + − +

Q

ɺ

ɺ ɺɺ

ɺ ɺɺ

ɺ ɺɺ

(47)

Due to the constraints to be of the form Eq. (39), the

condition of ideality will be of the form Eq. (8),

where, the matrix D is written as follows

1 0 0 0

0 1 0 0

=

D (48)

By the condition of ideality of the constraints

restricted to the system

DR = 0 (49)

We get

0 ; 0s

R Rθ= = (50)

Motion equations of the system under consideration

by Eq. (28) now are written as follows

6

2 2 *

sin 2

sin 2 4 ( )sin

sin ( )cos 0

cos ( 2 )sin

sin cos

( ) cos ( 2 )sin

0 0

0 ( )

m mL s

mL mL L s ms J

m ML ms

m ML mL ms v

m m s

ML ms ML Lm ms

M m

M m v

θ

θ θ θ

θ θ

θ θ

θ θ

θ θ θ

− + + +

− − + −

−

+ − + − + +

+

ɺɺ

ɺɺ

ɺɺ

ɺɺ

ɺɺ

ɺɺ

=

2 2

2

2

2

0cos ( 2 sin ) 2 cos 2

02 ( )sin 2 ( ) sin

2 cos ( )sin

( 2 )cos sin

u

v

mg m s L mL

mL L s ms ML g

Rm s ML ms

RML mL ms m s

θ θ θθ

θθ θ

θ θ θθ

θθ θθ

− − −

− − + − + + − + +

− + − −

ɺ

ɺ

ɺ ɺɺ

ɺ ɺ ɺ

(51)

The reaction force by the vertical wall acting on the

beam AB, i. e. the reaction force at the end A is

calculated by the expression

2

sin ( )cos

2 cos ( )sin

uR m s ML ms

m s ML ms

θ θθ

θ θ θθ

= + +

+ − +

ɺɺɺɺ

ɺ ɺɺ

(52)

But the reaction force from the ground acting to the

beam at the end B will be

2

cos ( 2 )sin

sin ( 2 )cos

vR m s ML mL ms

m s ML mL ms

θ θθ

θθ θθ

= − − + −

+ + + −

ɺɺɺɺ

ɺ ɺɺ

(53)

This result is coincided with the one in [2]. By such a

way the reaction forcesu

R , v

R depend on the motion

of the system, which can be defined by the system of

following equations

2

2 * 2

sin 2 cos ;

sin 2 (4 ( )sin )

2 sin ( ) cos

( ) sin ;

ms mL mg ms

mL s mL L s J ms

ms s m sv ML ms v

ms ML gL

θθ θ θ

θ θ θ

θ θ θ θ

θ

+ = +

+ − + +

= + + +

− +

ɺɺ ɺɺɺ

ɺɺɺɺ

ɺ ɺɺ ɺɺ ɺ

We get a close system of two differential equations

including unknowns , s θ . Solving these equations

together with the defined initial conditions it is

possible to calculate

( ), ( )s s t tθ θ= =

By substituting the obtained quantities into the

expressions of the reaction forces Eq. (52) and Eq.

(53) we get

( ) ; ( )u u v v

R R t R R t= =

It is more interesting to find the condition for the

beam falling from the vertical wall. This condition

will be realized, if the magnitude of reaction

forceu

R equals zero. By this condition it is easy to

find the instant of time when the beam falls from the

vertical wall. The above mentioned method allows

determining easily the reaction forces between the

animal and the beam.

4. Conclusion

The presented method allows calculating easily the

reaction forces by means of a simple itinerary.

Especially, the introduced method is based on the

matrix calculations, which is easy to apply the

special software such as Maple, Mathcad, and

Matlab. The proposed method may be applied for the

problems of calculating the reaction forces in the

joints in mechanisms [1, 6, 7, 18], in industrial

robotics or in the systems with the unilateral

constraints [11].

Acknowledgement

The publication is completed thank to the financial

support from the National Basics Research Program

in Natural Sciences.

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