Group global stiffness

X.H.2 Group global stiffness

We have seen in section X.H.1 that the parts are assumed infinitely stiff compared to the conenctions. The only source of flexibility in the assembly is the flexibility of connections. Here, to fix the ideas, we consider that part “A” is fixed, and that only part “B” moves slightly. This means that only the motions of part “B” must be considered in the development of expressions leading to the estimation of group global stiffnesses.

As part “B” is rigid, its motion can be characterized by a vectorial translation $T$ and a vectorial rotation $Ω$ of components $T_{j}$ and $Ω_{j}$ respectively. These components are expressed in bolt group center of gravity Cartesian coordinate system defined in section X.H.2. The motion of “B” side of connection $i$ is then

Δ x_{j}^{i} = T_{j} + ϵ_{j k l} Ω_{j} x_{l}^{i},

in which we use the Bose-Einstein convention for the notation of components, except that no distinction is done between the covariant and contravariant types of components as coordinate system is a Cartesian one. Previous expression can also simply be written:

{Δ x}^{i} = T + Ω \times x^{i} .

(X.H.5)

This equation shows that the translational elongation of each connection depends on its location wrt bolt group center of gravity. On the other hand, all the rotational deformations are exactly equal to part “B” global rotation:

{Δ ω}^{i} = Ω .

(X.H.6)

To these connection deformations correspond connection forces and moments:

f^{i} = k_{T}^{i} {Δ x}^{i} = k_{T}^{i} (T + Ω \times x^{i}) .

(X.H.7)

m^{i} = k_{R}^{i} {Δ ω}^{i} = k_{R}^{i} Ω .

(X.H.8)

With these expressions for connection deformations, bolt group total strain energy can be developed as follows:

\begin{align} U & = \frac{1}{2} \sum_{i = 1}^{N} {Δ x}^{i} \cdot f^{i} + \frac{1}{2} \sum_{i = 1}^{N} {Δ ω}^{i} \cdot m^{i}, \\ = \frac{1}{2} \sum_{i = 1}^{N} k_{T}^{i} {({Δ x}^{i})}^{2} + \frac{1}{2} \sum_{i = 1}^{N} k_{R}^{i} {({Δ ω}^{i})}^{2}, \\ = \frac{1}{2} \sum_{i = 1}^{N} k_{T}^{i} [T + Ω \times x^{i}) \cdot (T + Ω \times x^{i}] + \frac{1}{2} \sum_{i = 1}^{N} k_{R}^{i} Ω^{2}, \\ = \frac{1}{2} \sum_{i = 1}^{N} k_{T}^{i} [T \cdot T + 2 T \cdot (Ω \times x^{i}) + (Ω \times x^{i}) \cdot (Ω \times x^{i})] + \frac{1}{2} \sum_{i = 1}^{N} k_{R}^{i} Ω^{2}, \\ = \frac{1}{2} \sum_{i = 1}^{N} k_{T}^{i} T \cdot T + \sum_{i = 1}^{N} k_{T}^{i} (T \times Ω) \cdot x^{i} + \frac{1}{2} \sum_{i = 1}^{N} k_{T}^{i} (Ω \times x^{i}) \cdot (Ω \times x^{i}) + \frac{1}{2} \sum_{i = 1}^{N} k_{R}^{i} Ω^{2}, \\ = \frac{1}{2} \sum_{i = 1}^{N} k_{T}^{i} T \cdot T + (T \times Ω) \cdot \sum_{i = 1}^{N} k_{T}^{i} x^{i} + \frac{1}{2} \sum_{i = 1}^{N} k_{T}^{i} (Ω \times x^{i}) \cdot (Ω \times x^{i}) + \frac{1}{2} \sum_{i = 1}^{N} k_{R}^{i} Ω^{2}, \\ = \frac{1}{2} \sum_{i = 1}^{N} k_{T}^{i} T \cdot T + \frac{1}{2} \sum_{i = 1}^{N} k_{T}^{i} (Ω \times x^{i}) \cdot (Ω \times x^{i}) + \frac{1}{2} \sum_{i = 1}^{N} k_{R}^{i} Ω^{2} . & (X.H.9) \end{align}

In last expression, one term has been set to zero because $\sum_{i = 1}^{N} k_{T}^{i} x^{i} = 0$ (origin of coordinate system located at bolt group center of gravity). Finally, one develops this expression following similar development for the definition of interia tensor in chapter 32 of [eEL94]:

\begin{align} U & = \frac{1}{2} \sum_{i = 1}^{N} k_{T}^{i} T \cdot T + \frac{1}{2} \sum_{i = 1}^{N} k_{T}^{i} (Ω \times x^{i}) \cdot (Ω \times x^{i}) + \frac{1}{2} \sum_{i = 1}^{N} k_{R}^{i} Ω^{2}, \\ = \frac{1}{2} \sum_{i = 1}^{N} k_{T}^{i} T \cdot T + \frac{1}{2} \sum_{i = 1}^{N} k_{T}^{i} [Ω^{2} (x^{i} \cdot x^{i}) - {(Ω x^{i})}^{2}] + \frac{1}{2} \sum_{i = 1}^{N} k_{R}^{i} Ω^{2}, \\ = \frac{1}{2} (\sum_{i = 1}^{N} k_{T}^{i}) T^{2} + \frac{1}{2} Ω \cdot {\sum_{i = 1}^{N} k_{T}^{i} [(x^{i} \cdot x^{i}) δ - x^{i} x^{i}]} \cdot Ω + \frac{1}{2} (\sum_{i = 1}^{N} k_{R}^{i}) Ω \cdot δ \cdot Ω . & (X.H.10) \end{align}

Last expression allows us to introduce one scalar and one tensorial quantities characterizing bolt group total stiffness:

k_{T}^{B G} = \sum_{i = 1}^{N} k_{T}^{i},

(X.H.11)

J^{B G} = \sum_{i = 1}^{N} k_{T}^{i} [(x^{i} \cdot x^{i}) δ - x^{i} x^{i}] + (\sum_{i = 1}^{N} k_{R}^{i}) δ .

(X.H.12)

The scalar quantity corresponds to a interface global translational stiffness. The tensorial quantity is the interface total rotational stiffness. Using these expressions, the bolt group strain energy is given by

U = \frac{1}{2} k_{T}^{B G} T \cdot T + \frac{1}{2} Ω \cdot J^{B G} \cdot Ω .

The strain energy is also given by:

U = \frac{1}{2} F \cdot T + \frac{1}{2} M \cdot Ω .

This allows to write an expression that relates the interface global force and moment to interface deformation:

(\begin{matrix} F \\ M \end{matrix}) = (\begin{matrix} k_{T}^{B G} δ & 0 \\ 0 & J^{B G} \end{matrix}) (\begin{matrix} T \\ Ω \end{matrix}),

that can be inverted as follows:

(\begin{matrix} T \\ Ω \end{matrix}) = (\begin{matrix} \frac{1}{k_{T}^{B G}} δ & 0 \\ 0 & \frac{1}{J^{B G}} \end{matrix}) (\begin{matrix} F \\ M \end{matrix}) .

(X.H.13)

We have assumed here that $J^{B G}$ can be inverted.

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