Generalized State Variables

MODELING MECHANICAL AND ELECTRICAL

2.5 VARIATIONAL PRINCIPLE OF SYSTEMS IN STATIC EQUILIBRIUM The definitions of variational motion and its relationship to work enables the devel-

2.5.1 Generalized State Variables

In our derivation of the variational principle, the variation has been applied to dis- placement of the mechanical system,u, and the charge associated with the electrical system. For systems with multiple electrical elements, this is written as the vectorq.

Thus, the variational principle has functional dependence on both the displacement

vector and the vector of charge coordinates. In many instances it will be useful or re- quired to rewrite the displacement vector in terms of a set ofgeneralized coordinates, ri, and to express the variational principle with respect to the generalized coordinates instead of with respect to the displacement vector. We define a set of generalized coor- dinates as the coordinates that locate a system with respect to a reference frame. Thus far in our discussion we have defined the displacement vector in a Cartesian frame of reference, but the generalized coordinates do not necessarily have to be expressed in one particular reference frame. They may have components in a Cartesian frame, a spherical frame of reference, or a mixture of multiple reference frames.

For the solution of problems in mechanics and in the mechanics of smart material systems, it is assumed that the variational principle is expressed as a set of generalized coordinates that are complete and independent. A set of coordinates is said to be completeif the coordinates chosen are sufficient to fix the locations of the parts of the system for an arbitrary configuration that is consistent with the geometric constraints.

A set of coordinates is said to beindependentif when all but any one of the coordinates is fixed, there remains a continuous range of values for the unfixed coordinate for all configurations, consistent with the geometric constraints.

Consider the case of specifying the generalized coordinates for a bar pivoting about a point. First let us assume that the bar is rigid and we define the displacement vector in terms of a Cartesian frame of reference that has the origin at the pivot point of the bar. Defineu₁ as displacement in the 1 direction andu₂ as displacement in the 2 direction. The question is whether this choice of coordinates is complete and independent. To check completeness, we see determineu₁andu₂ are sufficient to fix the displacement of the system for arbitrary configurations that are consistent with the geometric constraints. In this case the choice of coordinates is complete since specifyingu1 andu2 will fix the location of the bar and the mass. To check whether the choice of coordinates is independent, we fixu1and determine if there is a continuous range of values foru2that are consistent with the geometric constraints.

This check fails, because if we specify the locationu1, the location ofu2is also fixed since the bar is assumed to be rigid. Thus, this choice of generalized coordinates is complete but not independent.

Now let us consider the case in which the bar is assumed to be elastic. The choice of the coordinatesu1 andu2 is complete for the same rationale as for the case of a rigid bar. To check whether the coordinates are independent, fixu1 and determine if there is a continuous range of values foru₂. In this case,u₂is independent ofu₁ since the elasticity of the bar allows us to varyu₂ even for the case in whichu₁is fixed. Thus, for the case of an elastic bar, the generalized coordinatesu₁andu₂form a complete and independent set.

Two questions now arise. The first is how to choose generalized coordinates when the obvious choice does not form a complete and independent set. To answer this, let us return to the example of the rigid bar pivoting about a point. From the geometry of the problem we can write

u1=lsinψ

(2.174) u2= −lcosψ.

Since the length of the bar is fixed, we see that choosingψas the generalized coor- dinate forms a complete and independent set. Thus, for the case in which the bar is rigid, the generalized coordinateψcan be used to specify the work and energy terms in the variational principle.

The second question that arises is how to choose the generalized coordinates when more than one set of variables is complete and independent. Returning to the example of the elastic bar with the end mass, we see that the choice ofu₁andu₂is a complete and independent set; thus, these two coordinates could be used as the generalized coor- dinates of the problem. For the same reason, the choice of the coordinateslandθcould also form a complete and independent set. The decision to choose one set of general- ized coordinates over another is somewhat problem dependent and is often dictated by the geometry of the problem. Unfortunately, there are no specific rules that define which is a better choice, but some of these issues are illustrated in upcoming examples, and experience in using the variational principle often helps in the decision process.

The definition of generalized coordinates allows us to rewrite the variational prin- ciple in terms of these variables. The displacementuis assumed to be a function of the Nr generalized coordinates,u=u(r1,r2, . . . ,rN). The variation of the displacement is then written as

δu= ∂u

∂r1

δr1+ ∂u

∂r2

δr2+ · · · + ∂u

∂rN

δrN. (2.175)

Substituting this result into equation (2.172) yields

N_r

i=1

fM· ∂u

∂ri δri+

N_q

j=1

vjδqj =δVE+δVM. (2.176)

Let us denote the term in parentheses as thegeneralized mechanical force,Fi, Fi =fM· ∂u

∂ri

. (2.177)

The term on the right-hand side is written as a variation of the total potential energy, δVT. After expressing displacement in terms of the generalized coordinates, the total potential energy is written as the function

δVM+δVE =δVT(r₁, . . . ,rNr,q₁, . . . ,qNq). (2.178) The variation of the total potential energy is written as

δVT = ∂VT

∂r1

δr1+ · · · +∂VT

∂rNr

δrN_r+∂VT

∂q1

δq1+ · · · + ∂VT

∂qNq

δqN_q

=

N_r

i=1

∂VT

∂ri

δri+

Nq

j=1

∂VT

∂qj

δqj. (2.179)

Combining equations (2.177) and (2.179) with equation (2.176) produces an expres- sion for the work and energy balance as a function of the variations in the generalized state variablesri andqj:

Nr

i=1

Fiδri+

N_q

j=1

vjδqj =

Nr

i=1

∂VT

∂ri

δri+

N_q

j=1

∂VT

∂qj

δqj. (2.180)

This expression can be rewritten as

N_r

i=1

Fi−∂VT

∂ri δri+

N_q

j=1

vj−∂VT

∂qj δqj=0. (2.181) Recall that the generalized coordinates are assumed to be independent of one another.

Under this condition, the only way for equation (2.181) to holdfor arbitrary variations in the generlized state variables is for the following two sets of equations to be satisfied:

Fi−∂VT

∂ri

=0 i=1, . . . ,Nr

(2.182) vj−∂VT

∂qj

=0 j =1, . . . ,Nq.

Rewriting the expressions, we have Fi = ∂VT

∂ri

i =1, . . . ,Nr

(2.183) vj = ∂VT

∂qj

j =1, . . . ,Nq.

The final result of this analysis is a set of Nr+Nq governing equations in terms of the generalized state variables. These governing equations must be satisfied for equilibrium to be satisfied, which is equivalent to saying that satisfying the set of equations in equation (2.183) is identical to satisfying the energy balance expressed in the variational principle.

Example 2.8 Derive the equilibrium expressions for the system consisting of three springs as shown in Figure 2.18. Assume that the nodes are massless and that the coordinates are the displacement of the two nodes. The stiffness of the left and right springs iskand the stiffness of the middle spring isαk, whereαis a positive constant.

Solution The displacements of the system are defined asu1 andu2. Writing the potential energy of the system requires that we add the potential energies for each of

u₂ u₁

f₁ f₂

k αk k

Figure 2.18 Three-spring system for static analysis.

the three springs:

VT = 1

2ku²₁+1

2αk(u2−u₁)²+1 2ku²₂. The variation of the mechanical work is defined as

δWM = f₁δu₁+ f₂δu₂.

Taking the variation of the potential energy function and combining it with the vari- ation of the mechanical work yields

δVT +δWM =ku₁δu1+αk(u2−u₁)δ(u2−u₁)+ku₂δu2+ f₁δu1+ f₂δu2. Grouping the terms according to their variational displacement yields

[f₁−ku₁+αk(u2−u₁)]δu1+[f₂−ku₂+αk(u1−u₂)]δu2=0.

Since the variational displacements are independent, the terms in brackets must each be equal to zero for the system to be in equilibrium for arbitrary choices of the variational displacements. This produces two equations for static equilibrium:

f1−ku1+αk(u2−u1)=0 f2−ku2+αk(u1−u2)=0.

These equations can be rewritten in matrix form as f1

f2

=k

1+α −α

−α 1+α u1

u2

.

Notice that theαterm produces the off-diagonal coefficients in the matrix. Physically, this represents that fact that the middle springcouplesthe motion of the two nodes. If the middle spring was not there,α=0 and the two nodes would move independently.

Increasing the stiffness of the middle spring compared to the other two springs (α1) makes the system move as a rigid body.

f k_l

x y

l₁ l₂

f k_l

k_t k_t

ψ

Figure 2.19 Mechanical lever with linear and torsional stiffness.

Example 2.9 A mechanical lever with linear and torsional stiffness is shown in Figure 2.19. The force is applied at the right end of the bar, which is at a distance l₂ from the pivot. A torsionsal spring with stiffness k_tis located at the pivot and a linear spring of stiffness k_lis located at the left end at a distance l₁ from the pivot. Derive the equlibrium equations for small anglesψ.

Solution The first step in applying the variational principle is to choose a set of independent coordinates. In this example we see that it is natural to choose three coordinates to represent the motion of the system: the motion at the left end of the bar, x, the motion at the right end of the bar,y, and the rotational angleψ. Examining the geometry we see that there are kinematic constraints among these three coordinates,

x= −l₁sinψ y=l₂sinψ x= −l1

l2

y.

Now we can write the potential energy terms as a function of our choice of coordinates, VT = 1

2klx²+1 2ktψ²,

where the first term is the potential energy due to the linear spring and the second term is the potential energy due to the rotational spring. The variation of the external work performed by the force is

W_M= f δy

To continue with the analysis, we need to choose a single coordinate to represent all of the work and energy terms. We have the freedom to choose the coordinate;

therefore, it is best to choose the coordinate that will simplify the following analysis.

Due to the fact that we have to take derivatives with respect to the coordinates, let us choose to represent the work and energy terms as a function of the rotational angle ψ. Substituting the expressions forxandyas a function of the rotational angle yields the potential energy term

VT = 1

2kl(−l1sinψ)²+1 2ktψ².

Applying the variation to the energy and work terms produces δVT =kll₁²sinψcosψδψ+ktψδψ δWM = f l2cosψδψ.

Combining these two terms according to the variational principle, equation (2.173), yields

fl₂ cosψ−kll²₁sinψcosψ−ktψ

δψ=0.

For this expression to be valid, the term in parentheses must be equal to zero for arbitrary variational displacements. This leads to the equilibrium expression

kll₁²sinψcosψ+ktψ=fl₂ cosψ.

For small angles, sinψ≈ψand cosψ≈1, leading to the expressions kll²₁+kt

ψ=fl₂.

This equation represents the equilibrium expression for small angles. Note that the stiffness of the system is a combination of the stiffness due to the torsional spring and the stiffness due to the liner spring. The stiffness due to the linear spring is modified by the square of the distance due to the lever.