Endogenous Force Analysis: A Justification for the Method

The Principle of Minimum Potential Energy says that any system in stable equilibrium is at a local minimum in its potential energy. Theodore Tauchert³ gives the following formal statement of this principle:

Of all displacement fields which satisfy the prescribed constraint conditions, the correct state is that which makes the total potential energy of the structure a minimum.

3Tauchert74, p. 74.

7.2. ENDOGENOUS MEMBER FORCES 157 In a tensegrity system, the potential energy is the energy bound up in the tendons and struts. When a member changes length, its potential energy changes according to how much work is done on it:⁴

de_im = f_imdl_im

where deim is the change in potential energy of the imth member, fim is the signed magnitude of the force on the member and dl_im is the change in length of the member. The usual convention that f_im is negative when the force is compressive and positive when the force is tensile applies here. If the system is in equilibrium, a small feasible⁵ change in the lengths of all the members should result in a zero change in the aggregate potential energy of the system since that potential energy must be at a minimum.⁶ The condition for zero aggregate energy change can be summarized as:

0 = de₁+ de₂+ · · · + de_nm where, as in Chapter 3, n_m is the number of members.

Using the other formula, this can be rewritten as:

0 = f₁dl₁+ f₂dl₂+ · · · + f_nmdl_nm

How does this relate to the mathematical programming problem of Chapter 3? Since members 1 through n_o appear in the objective function and members n_o+ 1 through n_m appear as constraints, and using ^∂o

∂(l²_i˜

o) to denote the amount the objective function changes in response to a change in the second power of the length of the i_o˜th constrained

member, it must be that the response of the objective function to an arbitrary change in the lengths of the constrained members is:

do = ∂o

∂(l²_no+1)d(l²_no+1) + · · · + ∂o

∂(l²_nm)d(l²_nm)

4The members are assumed to be linearly elastic.

5Feasible here means that all constraint equations continue to be satisfied. In contrast to the situation in Chapter 3 however, all member lengths may change. This means ln_o+1, . . . , ln_m may change. In addition the constraints are met with equality

6A negative change directly violates the assumption that the original configuration is a minimum. A positive change indirectly violates the assumption since a point displacement which results in the change can be negated resulting in a negative change from the original configuration.

= 2 ∂o

∂(l²_no+1)l_no+1dl_no+1+ · · · + 2 ∂o

∂(l²_nm)l_nmdl_nm

The formula for o says, for the objective members, it is also true that:

do = w1d(l^∗₁²) + w2d(l^∗₂²) + · · · + wnod(l^∗_no²) which reduces to:

do = 2w₁l^∗₁dl^∗₁+ 2w₂l^∗₂dl^∗₂+ · · · + 2w_nol_n^∗_odl_n^∗_o where l_i^∗_o is the minimizing length of the i_oth unconstrained member.

If all the constraints are changed by an arbitrary amount, then it must be true that:

2w₁l^∗₁dl₁^∗+ · · · + 2w_nol^∗_nodl^∗_no = 2 ∂o

∂(l²_no+1)l_no+1dl_no+1+ · · · + 2 ∂o

∂(l²_nm)l_nmdl_nm or (using the fact that the constraints are met with equality, canceling the common factor of two and collecting terms):

0 = w₁l₁^∗dl₁^∗+ · · · + w_nol^∗_nodl^∗_no + − ∂o

∂(l²_no+1)l_no+1dl_no+1+ · · · + − ∂o

∂(l²_nm)l_nmdl_nm

The similarity of this formula to the formula for potential energy minimization indicates a conclusion is almost at hand. The only complication is that in this latter formula,

although the changes in the lengths of the constrained members may be considered

arbitrary, the changes in the lengths of members included in the objective function must be regarded as changes in the minimizing tendon lengths and are not arbitrary feasible

changes. This complication can be disposed of by noticing that it is assumed feasible displacements from a minimizing solution are being examined. Since the objective function is at a minimum, any feasible displacement of the objective members’ lengths away from their minimizing values will have no effect on the objective function value.

Thus, a feasible displacement of the member lengths is broken into two parts. First, the lengths of the constrained members are displaced. That displacement will result in a

corresponding minimizing displacement of the unconstrained member lengths such that the

7.2. ENDOGENOUS MEMBER FORCES 159 equation just set forth is satisfied. Then an additional displacement is added to the lengths of the unconstrained members so that the total displacement is equal to the initial

arbitrary feasible displacement. The additional effect of this displacement on the objective function value must be zero since it is a feasible displacement from a minimum with no change in the constraints. Therefore, the change in the objective function resulting from the arbitrary displacement is the same as the result obtained when the unconstrained members change in a minimizing manner.

So it is verified that for an arbitrary feasible deviation from a minimizing solution:

w₁l₁dl₁+ · · · + w_nol_nodl_no = w₁l^∗₁dl^∗₁+ · · · + w_nol_n^∗_odl_n^∗_o Thus:

0 = w₁l₁dl₁+ · · · + w_nol_nodl_no + − ∂o

∂(l²_n

o+1)l_no+1dl_no+1+ · · · + − ∂o

∂(l²_n

m)l_nmdl_nm So, if

f₁ = λw₁l₁

· · ·

f_no = λw_nol_no f_no+1 = −λ ∂o

∂(l²_no+1)l_no+1

· · ·

f_nm = −λ ∂o

∂(l²_nm)l_nm

where λ is some positive constant, the system will be in stable equilibrium. These are precisely the formulas described in Section 7.2.1. Notice that since for a strut ^∂o

∂(l²_i˜

o) is positive, f_i˜o will be negative, a compressive force. And since ^∂o

∂(l²_i˜

o) is negative for a tendon, fio˜ is positive, a tensile force.

This manner of computing the member forces is very convenient since it derives from the method for computing member lengths. These force computations can be used to check proposed solutions of the mathematical programming problem which characterizes a given

tensegrity. If tendons are not in tension, or struts are not in compression, the solution is not valid. (Perhaps some constraints which have been assumed to hold with equality are actually not effective.) In more complex structures, such a check is almost obligatory since some adjustments may need to be made for a valid solution to be attained. Thus, the processes of length computation and endogenous force computation are highly

interdependent.