2.1.4.1 External Forces and Reaction Forces
External forces are forces (torques) acting upon the components of a system. In this form they constitute the cause of motion in accordance with the Newton’s second law. Reaction forces of constraints (Fig. 2.4) form the internal forces act- ing along the applied constrains and operate so that the system preserves the state which results from the imposed constraints. Hence, reaction forces of constraints do not constitute the cause of the motion but result in the preservation of the sys- tem in conformity with the constraints. In ideal circumstances the forces of constraint reactions do not exert any work associated with the motion of a system, which is applied in d’Alembert principle discussed later.
Fig. 2.4 Equilibrium between reaction forces of constraints R1,R2 resulting in constant dis- tance l between balls in motion
2.1.4.2 Virtual Displacements
The introduction of the notion of virtual displacements, i.e. ones that are compati- ble with constraints is indispensable in analytical dynamics due to their role in elimination of constraint reaction forces occurring in constrain based systems [12,13,16]. The vector of virtual displacements is denoted analogically to the con- struction of the position vector (2.2)
) , , , ,
( r1 r2 r3 rN
r δ δ δ δ
δ = … (2.20) where:
) , , ( ) , ,
( i i i 3i 2 3i1 3i
i δx δy δz δξ δξ δξ
δr = = − −
The vector of virtual displacements is constructed by the increments of variables which fulfill the following conditions:
1° - possess infinitesimal value 2° - are compatible with constraints
3° - their displacements occur within fixed a moment of time
These conditions also mean that virtual displacements are also referred to as in- finitesimal displacements, i.e. small testing displacements which occur consis- tently with applied constraints without accounting for their duration. As a result, it is possible to compare work exerted by a system for various vectors of virtual dis- placements. Virtual displacements do not necessarily have to overlap with sections of actual paths of motion but need to be consistent with potential paths from the kinematics perspective. From the statement of consistency between virtual dis- placements and constraints the following relation can be established:
0 ) ( )
(r+ r − f r =
f δ
which upon resolving into Taylor series relative to δr and omission of higher powers (δr)2, (δr)3,… leads to the statement of the relation between virtual dis- placements for a given j-th equation of constraints
0
1
∂ =
∑∂
= i N
i i
fj
r δr or 3 0
1
∂ =
∑∂
=
k N
k k
fj
ξ δξ (2.21) The relation (2.21) also means that any equation for holonomic constraints, fj, j=1,…,h enables one to find an expression for a particular virtual displacement by use of the remaining ones
) 1 (
, 1
1 , 1 1 , 2
2 , 1 1 , ,
n n j k
k j k k j j
j k j
k a a a a a
a δξ δξ δξ δξ δξ
δξ =− + +…+ − − + + + +…+ (2.22)
where:
k j k j
a f ξ
∂
= ∂
,
2.1.4.3 Perfect Constraints
It is only possible to define perfect constraints in a system in which friction forces are either missing or in the case where the inherent friction forces can be consid- ered as external forces. After this prerequisite is fulfilled, it is possible to define perfect constraints. Such constraints satisfy the condition that total work exerted on the virtual displacements is equal to zero:
∑=
=
N i
i i 1
0 r
Rδ (2.23) An example of perfect constraints include a rigid connection of material points which is not subjected to tension or bending. Historically, the concept of perfect constraints originates from d’Alembert principle and forms a postulate confirmed by numerous examples.
2.1.4.4 d’Alembert Principle
It constitutes the first analytical statement of the motion of a system in which par- ticles are constrained. In order to eliminate forces of constraint reactions the prin- ciple applies the notion of perfect constraints. For a material point (particle) with mass m the equation of motion directly results from Newton’s second law of motion:
F r= m
For a system with N material points limited by constraints, the above equation can be restated for every material point to account for the resulting force of constraint reactions R beside the external force F
N i
miri =Fi+Ri =1… (2.24)
The unknown constraint reaction forces do not yield it possible to directly apply equations (2.24). After summation of the equations it is possible to eliminate con- straint reaction forces on the basis of the notion of the perfect constraint (2.23)
0 ) (
1
=
−
∑ −
=
i i i i i N i
mr F R δr
which gives: ( ) 0
1
=
∑ −
=
i i i i N i
mr F δr (2.25) Virtual displacements are not separate entities but are related to one another by equations resulting from the constraints. Hence, d’Alembert principle is expressed by the system of equations:
⎪⎪
⎩
⎪⎪
⎨
⎧
=
∂ =
∂
=
−
∑
∑
=
=
h f j
m
i N
i i
j
i i i i N i
1… 0
0 ) (
1 1
r r
r F r
δ δ
(2.26)
This is a set of differential - algebraic equations on the basis of which it is possible to obtain equations of motion e.g. using Lagrange indefinite multiplier method.
This can be performed as follows: each of h algebraic equations in (2.26) is multi- plied by indefinite factor λj and summed up:
∑∑= =
∂ =
∂
h j
i N
i i
j j
f
1 1
0 rδr
λ (2.27) The expression in (2.27) is subsequently subtracted from the equation of motion, thus obtaining:
0 ) (
1 1
∂ =
− ∂
− ∑
∑= =
i h
j i
j j i i i N i
m f r
F r
r λ δ (2.28) For the resulting sum of N parenthetical expressions multiplied by subsequent vir- tual displacements δri, the following procedure is followed: for the first h expres- sions in parentheses i=1,…,h the selection of multipliers λj should be such that the value of the expression in parenthesis is equal to zero. In consequence, for the remaining parenthetical expressions the virtual displacements δri, i=h+1,…,N are already independent, hence, the parenthetical expressions must be equal to zero.
The final equations of motion take the form:
N h f i
m
h
j i
j j i i
i 0 1…
1
+
=
∂ = + ∂
= ∑
= r
F
r λ (2.29)
The term thereof: ∑
= ∂
∂
h
j i
j j
f
1λ r constitutes the reaction force of nonholonomic constraints for equations (2.24). In a similar manner it is possible to extend d’Alembert principle to cover systems limited by nonholonomic constraints (2.11-2.12). As a result the following expression is obtained:
N h f i
m
nh
l i
l l h
j i
j j i i
i 0 1…
1 1
+
=
∂ = + ∂
∂ + ∂
= ∑ ∑
=
= r r
F
r λ μ ϕ (2.30)
where: ϕl - nonholonomic functions of constrain type (2.12) μl - indefinite multipliers for nonholonomic constraints
d’Alembert principle leads to the statement of a system of equations with constraints; however, this procedure is time-consuming and quite burdensome since the obtained forms of equations are extensive and complex due to the selec- tion of coordinates of motion that is far from optimum. This can be demonstrated in a simple presentation.
Example 2.3. In a planar system presented in Fig. 2.5 a set of two balls of mass m1
and m2 are connected by a stiff rod. They are put in motion under the effect of ex- ternal forces F1 and F2, in which gravity pull and friction force are already ac- counted for. The equation of motion are subsequently stated in accordance with d’Alembert principle.
Solution: the single equation of constraints stated in accordance with (2.13) takes the form:
0 )
( ) (
: 1 2 2 1 2 2 2
1 x −x + y −y −l =
f
The vector of Cartesian coordinates for this system is as follows:
) , , , ( ) ,
(1 2 =Ξξ1ξ2 ξ3 ξ4
= r r r
Fig. 2.5 Set of two balls connected by a stiff rod
The system of differential - algebraic equations written in accordance with (2.29) takes the form:
⎪⎪
⎪
⎭
⎪⎪
⎪
⎬
⎫
=
−
− +
−
+
− +
=
+
− +
=
− +
=
− +
=
0 )
( ) (
) (
2
) (
2
) ( 2
) ( 2
2 2 2 1 2 2 1
2 1 1 2 2 2
2 1 1 2 2 2
2 1 1 1 1 1
2 1 1 1 1 1
l y y x x
y y F
y m
x x F
x m
y y F
y m
x x F
x m
y x y x
λ λ λ λ
(2.31)
The equation of holonomic constraints eliminates one of the Cartesian coordinates since it is an dependent variable in the description of dynamics. At this points, let us assume that it is y2, hence:
2 2 1 2 1 2 2
1 2 2 2
1 ; ( )
) (
2 y y l x x
y y
y m F y
−
−
− =
= − ∓
λ
By introducing these variables into (2.31) we obtain
⎪⎪
⎪
⎭
⎪⎪
⎪
⎬
⎫
−
−
=
−
=
−
+
= +
+
= +
2 2 1 2 1 2
2 2 2 2 2 2
2 1 2 2 1 1
2 1 2 2 1 1
) (x x l
y y
F F y m x m
F F y m y m
F F y m x m
y x
y y
y x
∓
α α
α α
(2.32)
where:
2 1
2
) 1
( y y
x x
−
= −
=α r α
The resulting system of equations of motion still requires the elimination of y2,y , 2 which is only an algebraic problem. This set of equations is very complex in its analytical notation despite the fact that it presents a very simple mechanical sys- tem. This is associated with the necessity of application of Cartesian coordinates, which is not the most adequate choice for the case of equations containing constraints, in particular from the point of view of simple notation of dynamic equations. A favorable option in this respect is offered by the introduction of gen- eralized coordinates and expression of the equations of motion in the form of Lagrange’s equations.