Determination of Kelvin-Voigt Damping Coefficients

Chapter VIII: Slew Maneuver Dynamics

8.3 Determination of Kelvin-Voigt Damping Coefficients

is due to either the available torque or momentum, not the structure, regardless of the requirement on the residual angular velocity. Similarly, with the larger reaction wheel and the coarser pointing requirement, Fig. 8.5b shows that the available torque again drives the minimum slew time. The only exception is the case with the larger reaction wheels and the finer pointing requirement; see Fig. 8.5d. In this case, the structure-based performance limit constrains the minimum slew time for spacecraft at length scales below approximately 40 m. Above 40 m, the slew times are again torque-constrained. Even so, below the crossover point, the structure-based performance limit results in minimum slew times on the same order of magnitude as those from the torque limit.

Decreasing either the fraction of the momentum and torque available for slews or the maximum momentum and torque shifts the corresponding curves in Fig. 8.5 up. Similarly, decreasing the requirement on the amplitude of the residual angular velocity also shifts the corresponding curves in Fig. 8.5 up. Increasing the maximum momentum and torque is likely to require replacing reaction wheels with control moment gyroscopes.

Based on Fig. 8.5, the capabilities of each spacecraft’s attitude control system are often significantly more limiting than the dynamics of the structure. When this is the case, the results suggest that a lighter-weight, less-stiff, and lower-cost structure can be used to shift the structure-based performance limit closer to that of the attitude control system, at least as far as slewing is concerned. The figure likewise emphasizes that SSPP-like flexible spacecraft can likely achieve slew times on the order of 10 min or less for 90 deg, single-axis maneuvers at length scales as large as 50 m. If this is indeed the case, these slew times are realistically an order of magnitude or more faster than the existing state-of-practice. To gain confidence in these results, Sec. 8.4 compares the predictions from the reduced-order model for the 24 m×24 m spacecraft with those from geometrically nonlinear simulations of the corresponding full finite element model. In the interim, Sec. 8.3 develops the optimization approach for determining the viscoelastic damping coefficients that are inputs to the full finite element model.

simulations in Sec. 8.4.

Unlike linear finite element formulations, where it is straightforward to determine a global damping matrix, geometrically exact finite element formulations require modifications at the element level to guarantee that the damping formulation is in- variant to superposed rigid body motions. Invariant damping formulations only di- rectly dissipate energy associated with the elastic motion. A recent numerical study demonstrates that these formulations are important for correctly modeling damping effects in large-deformation simulations [259]. However, these damping models are often troublesome due to the difficulties associated with determining appropriate damping coefficients [260]. To that end, this section proposes an optimization-based approach for determining these coefficients. Even though this section focuses on geometrically exact beam finite elements, the approach readily generalizes to other types of geometrically exact finite elements (e.g., plates or shells) by appropriately modifying the tangent damping matrix.

The simplest damping model for geometrically exact finite elements is referred to as Kelvin-Voigt damping. For geometrically exact beams, Kelvin-Voigt damping augments the constitutive relation [Eq. (4.42)] for the force resultantNand moment resultantM with terms proportional to the material strain rate 𝚪¤ and the material curvature rateK¤ [148]. In other words,

S=CE+DE¤ (8.1)

whereS^𝑇 = N^𝑇,M^𝑇

, E^𝑇 = 𝚪^𝑇,K^𝑇

, C∈ R^6×6is the sectional stiffness matrix, and D ∈ R^6×6 is the matrix of to-be-determined sectional damping coefficients.

To the author’s knowledge, there are two systematic approaches in the literature for determiningD, although most studies instead tend to use “reasonable guesses”

[213] or sensitivity studies. The first approach [261] derives closed-form expres- sions for the damping coefficients for geometrically exact beams with homogeneous, isotropic material properties. However, these expressions assume the availability of viscoelastic material properties, specifically the viscoelastic bulk and shear vis- cosities. These material properties are unavailable for the equivalent beam models of the strips, and hence, this approach is not applicable here. The second approach [213] applies modal analysis to the linearized partial differential equations govern- ing the dynamics of geometrically exact beams with simple boundary conditions to derive expressions for the unknown damping coefficients. The optimization- based approach developed here generalizes this approach to finite element models of arbitrary complexity.

The formulation of the optimization problem starts from the dynamic equilibrium equations for a nonlinear finite element model:

Finer(g,g¤,g¥) +Fint(g,g¤) =Fext (8.2) whereg∈R^𝑛is the vector of generalized coordinates,F_iner(g,g¤,g) ∈¥ R^𝑛is the vector of generalized inertia forces,F_int(g,g) ∈¤ R^𝑛is the vector of generalized viscoelastic forces, and F_ext ∈ R^𝑛 is the vector of generalized external forces. The tangent damping matrixC𝑇 ∈R^𝑛×𝑛then follows as

C𝑇(g) = 𝜕F_int(g,g¤)

𝜕g¤ . (8.3)

For simplicity (and without any loss of generality), these developments only consider generalized coordinates in a vector space and neglect external constraints, e.g., due to the joints in flexible multibody systems. The treatment of generalized coordinates in a Lie group entails straightforward modifications to Eq. (8.3) and what follows.

The formulation is independent of any external constraints on the system.

The tangent damping matrix C𝑇 for a finite element model with 𝑛_𝑒 elements is defined by a standard finite assembly step. Thus,

C𝑇(g) =

𝑛_𝑒

Õ

𝑒=1

L^𝑇𝑒C_𝑇^𝑒(L𝑒g)L𝑒 (8.4) where𝑒∈ {1, 2, . . . , 𝑛_𝑒};L𝑒∈R^𝑛

𝑒 𝑐×𝑛

is the Boolean matrix that indexes the element nodal coordinatesg𝑒 ∈R^𝑛

𝑒

𝑐 from the generalized coordinatesg, i.e., g𝑒 = L𝑒g; and C_𝑇^𝑒 ∈ R^𝑛

𝑒

𝑐×𝑛^𝑒𝑐 is the elemental tangent damping matrix. For a geometrically exact beam element of lengthℓ_𝑒,C_𝑇^𝑒 is given by

C^𝑒𝑇(g𝑒) =

∫ ℓ𝑒

0

B^𝑇(g𝑒, 𝑠)D𝑒B(g𝑒, 𝑠)d𝑠 (8.5) whereB∈R^6×12is the discrete strain gradient matrix [Eq. (5.48)],D𝑒 ∈R^6×6is the unknown matrix of viscoelastic damping coefficients for element𝑒, and𝑠is the arc length coordinate along the element’s reference axis. The integral in Eq. (8.5) is normally approximated by a quadrature rule with𝑛_𝑞 weights𝑤_𝑖 at arc lengths𝑠_𝑖. It follows that

C𝑇(g) =

𝑛_𝑒

Õ

𝑒=1

L^𝑇𝑒

ℓ_𝑒 2

𝑛𝑞

Õ

𝑖=1

𝑤_𝑖B^𝑇(L𝑒g, 𝑠_𝑖)D𝑒B(L𝑒g, 𝑠_𝑖)

!

L𝑒 (8.6)

which is a linear function ofD𝑒.

From here, an optimization problem determines the 𝑛_𝑒 unknown matrices of ele- mental damping coefficientsD𝑒 that minimize the error between the global tangent damping matrix C𝑇 and a reference damping matrix Cref. In doing so, the finite element model inherits the damping characteristics ofCref in the specified config- uration (here, the initial configuration). A particularly simple choice for C_ref is stiffness-proportional damping; i.e.,

C_ref= 𝛽K𝑇 (8.7)

where𝛽=2𝜁₁/𝜔₁,𝜔₁is the first-mode natural frequency,𝜁₁is the desired fraction of critical damping in the first mode, andK𝑇 =𝜕F_int/𝜕gis the tangent stiffness matrix.

With stiffness-proportional damping, successive modes are more heavily damped.

Specifically, the fraction of critical damping in the𝑖th mode is𝜁_𝑖 =𝜁₁(𝜔_𝑖/𝜔₁)where 𝜔_𝑖 is the𝑖th mode’s natural frequency;𝑖 =1, . . . , 𝑛; and𝑛is the number of modes.

In the undeformed configuration, only the material tangent stiffness [Eq. (5.55)]

contributes to K𝑇, from which it follows that C𝑇 and K𝑇 share the same matrix structure (sparsity pattern). This implies that there is an optimal set of elemental damping coefficients that results in negligible errors betweenC𝑇 andCref. For this reason, stiffness-proportional damping is used to determine the elemental damping coefficients for the dynamic simulations in Sec. 8.4.

Following Sec. 7.3.1 and [243], the optimization problem enforces constraints on the structure of D𝑒. In general, D𝑒 is a symmetric positive definite matrix, i.e., D𝑒 > 0 [148, 261, 262]. For simplicity, however, D𝑒 is taken to be diagonal, i.e., D𝑒 =diag{d𝑒}, where the coefficients ofd𝑒are strictly positive. With diagonalD𝑒, the optimization problem takes the form

minimize

d𝑒∀𝑒∈ {1,2, ..., 𝑛_𝑒}

kC^𝑇(g0,d^𝑒) −Crefk_𝐹 (8.8) subject tod𝑒 > 06×1. Here,

C𝑇(g₀,d𝑒) =

𝑛_𝑒

Õ

𝑒=1

L^𝑇𝑒

ℓ_𝑒 2

𝑛_𝑞

Õ

𝑖=1

𝑤_𝑖B^𝑇(L𝑒g₀, 𝑠_𝑖)diag{d𝑒}B(L𝑒g₀, 𝑠_𝑖)

!

L𝑒, (8.9) the subscript 0 denotes the initial (undeformed) configuration, and 𝐹 denotes the Frobenius norm.

To solve Eq. (8.8), the constraintd𝑒 > 0_6×1is relaxed from strictly positive to simply nonnegative, i.e., d𝑒− 𝜖1_6×1 > 0_6×1 where1_6×1 is a vector of ones and 𝜖 1 is a small positive scalar. This results in a convex program with a unique, globally

optimal solution. SinceC𝑇(g₀,d𝑒)is linear in d𝑒, the convex program can then be manipulated into the more standard form of a constrained least squares problem, i.e., an optimization problem of the form

minimize

d

kAd−bk_𝐹 (8.10)

subject tod−𝜖1₆𝑛𝑒×1 > 0₆𝑛𝑒×1 whered^𝑇 = d^𝑇₁, . . . , d^𝑇𝑒

∈R^6𝑛𝑒. Equation (8.10) is solved numerically using convex programming, in this case implemented using CVX [244, 245] with the SDPT3 solver [246, 247].