Xp=
X1 X2 X3T
(3.4) A material point in the beam neutral fibre is described by setting Yp=
0 0 0T . Accordingly, its reference position, displaced position and displacement vector become Xk =
X1 0 0T
, xk and uk, respectively. The tangent vector to the neutral fibre is given by
t= dxk
dX1 (3.5)
This vector, also of unit magnitude due to assumption (iii), does not coincide with vector e1owing to the shear deformation of Timoshenko beam theory.
In order to specify the position of point p in the inertial reference frame, {OI, XIYIZI}, defined as rp, the position of the body reference frame relative to the inertial reference frame, rn, must be known. These two vectors are considered here with components along the inertial reference frame. Finally, the basic kinematic equation for a material point p of the flexible link is
rp= rn+ Rnxp
= rn+ Rn
xk+ RekYp
= rn+ Rn
Xk+ uk+ RekYp
(3.6)
where because of deformation alone one has
xp= Xk+ uk+ RekYp (3.7)
In this equation, the only terms that are owing to the deformation of the link are the displacement vector of the material points of the neutral axis, uk, and the rotation matrix of a cross section Rek. These are deduced in the next section.
3.3 The strain–displacement relations
The strain measures that are most commonly applied to the treatment of flexible beams are the Green–Lagrange strain measures (Sharf, 1996) and the displacement gradient measure of deformation (Géradin and Cordona, 2001). The former provides a general means of measuring the deformation inside a three-dimensional continuum whereas the latter is a simplification specific to beam kinematics. Both of these are invariant under rigid-body motion, therefore, they can be calculated from vectors expressed in
the body reference frame or in the inertial reference frame. Here, the body reference frame is adopted. Considering Figure 3.1 they are respectively given as
G
The Green–Lagrange strain tensor, G, is calculated by considering the change during deformation of the squared length of an infinitesimal fibre and expressing the dif-ference in the body redif-ference frame (the redif-ference un-deformed configuration). On the other hand, the displacement gradient strain vector, D, is obtained by calculating the position gradient with respect to the parameter describing the neutral axis, X1, in the cross-section reference frame, and subtracting from it the position gradient in the reference un-deformed configuration.
G is a 3× 3 symmetric matrix, and it expresses the strain state of a small material volume. D is a 3×1 vector, and it expresses the strain state at a specific cross section.
Defining the components of the Green–Lagrange strain tensor as
G= and the remaining strain components are reduced to three, namelyε11,ε12= ε21and ε13 = ε31 since the tensor must be symmetric. The Green–Lagrange strains at a beam cross section may now be obtained by multiplying the strain tensor by the unit vector normal to the cross section expressed in the same reference frame as the tensor.
Multiplying the shear components by 2 to give engineering strains, the result is Gb=
ε11 2ε12 2ε13
T
(3.11) The component form of the displacement gradient strain vector, on the other hand, is defined as
D=
d1 d2 d3
T
(3.12) At this point, assuming small displacement vectors in equations (3.8) and (3.9) in order to lead to small strain components may be premature and lead to incorrect models as mentioned in Section 3.2. Therefore, it is best to rewrite these strain vectors in terms of the variables that express deformation in the basic kinematic equation (3.7), namely, the displacement vector of the material points on the neutral axis, uk(or their displaced positions xk), and the rotation matrix of a cross section Rek. Substituting for xpfrom equation (3.7) into equation (3.9) the displacement gradient strains become
54 Flexible robot manipulators
where, equation (3.5) has been used. The vectork =
1 2 3
T
represents the strains of neutral axis of the beam.1contemplates two phenomena, the exten-sional strain of the neutral axis owing to a longitudinal force, and the longitudinal strain induced by transverse shear (Simo and Vu-Quoc, 1986, 1987);
1= eT1t− 1 (3.14)
The former has been neglected by settingt = 1 in equation (3.5) owing to assump-tion (iii), and the latter will be neglected in Secassump-tion 3.3.1 on account of assumpassump-tion (vi).2and3are the transverse Timoshenko shears,
i= eTit, i= 2, 3 (3.15)
and will also be considered according to assumption (vi).
The skew symmetric matrix Kkrepresents the curvature of the beam cross sections from which the curvature vector with components along the cross-section reference frame may be extracted, Kk= vect(Kk) =
K1is the torsional strain of the neutral axis and K2and K3are the bending strains, all expressed in the cross-section reference frame. Similar to the shear strains, these bending strains will also be considered according to assumption (vi). Note that the beam curvature is not given by this vector owing to shear deformation.
The above quantities provide an alternative representation of the strain vector that is more adequate for expansion than the representation in equation (3.9). Assumptions at the local deformation level are more realistic than at the displacement level. The latter depend on the length of the beam whereas the former do not.
Similar to the displacement gradient strains, it is shown in Géradin and Cardona (2001) that the Green–Lagrange strains are given by
Gb
It is verified that the Green–Lagrange strains yield the same shear strains but different longitudinal strains than the displacement gradient. Expanding the second term in equation (3.17) yields longitudinal strain dependence on quadratic bending curvature terms Kk, quadratic shear termskand coupling shear and bending terms. All these
terms are neglected on the basis of assumption (vi). Therefore, the two strain measures are approximate, and are explicitly written as
Gb
3.3.1 Parameterisation of the rotation matrix
Equation (3.13) implies that for infinitesimal strain measures one has infinitesimal shear strains, k, and infinitesimal bending strains, Kk. However, equation (3.16) implies that infinitesimal bending strains along the beam, Kk, do not necessarily yield infinitesimal cross-section rotations. Similar to Bremer and Pfeiffer (1992) and Schwertassek and Wallrapp (1999), expanding Rekand Kkin Taylor series in equation (3.16), and retaining only the first-order term of the bending strains results
dRek1 term of the bending strains. The rotation matrix can thus be calculated up to order m through integration of the same-order terms in equation (3.19):
Rek = I + Rek1+ Rek2+ · · · + Rekm (3.20)
3.3.2 Parameterisation of the neutral axis tangent vector
On introducing the shear anglesϕ12 andϕ13, as defined by the classical theory of elasticity (Novozhilov, 1961; Schwertassek and Wallrapp, 1999), equation (3.15) becomes
56 Flexible robot manipulators
where assumption (iii),t = 1, has been considered. Furthermore, vector t may be written with components along the cross-section reference frame as
kt= (
1− sin2(ϕ12) − sin2(ϕ13) e1+ sin (ϕ12) e2+ sin (ϕ13) e3 (3.24) Considering now assumption (vi), yields
i ≈ ϕ1i, i= 2, 3 (3.25)
and
1≈ 0 (3.26)
and finally
kt= e1+ ϕ12e2+ ϕ13e3 (3.27)
The resulting vector of infinitesimal shear strains is finally
k ≈ k1 =
0 ϕ12 ϕ13
T
(3.28) The strain state of the beam is described by the variables K1, K2, K3,ϕ12andϕ13. With the Rayleigh–Ritz approach to spatially discretise the system equations these are the variables for which trial functions have to be chosen that obey the imposed boundary conditions at the ends of the beam.
3.3.3 Displacement of the neutral axis
To complete the kinematic description of equation (3.7), the displacement vector of the points on the neutral axis must be specified. To this end, note that
dxk
uk can now be calculated through integration resulting
uk =
Even though the bending strains are infinitesimal, equation (3.19), and the shear strains are infinitesimal, equation (3.28), the strain–displacement relations, equation (3.9), may be non-linear if the rotation matrix Rekis expanded up to non-linear terms, equation (3.21), in the kinematics equation (3.7).