Dynamics of Machines - Part II - IFS.pdf

1.5 Calculating Stiffness Matrices – Beam Theory
Two Different Lengths for Applying Forces – To facilitate the understanding of steps
which will be presented, one can introduce the follow nomenclature (see figure 3(b)):
• L ∗ = L1 or L ∗ = L2 – length where the force F is applied.
• x = L1 or x = L2 – length where the displacement is measured.
Taking into account two different points for applying the forces and measuring the displacements
of the beam, one works with the following set of equations
x [0, L ∗ ]
and
x [L ∗ , L]
⎧
⎪⎨
⎪⎩
⎧
⎪⎨
⎪⎩
y(x) = − F
E I
dy(x)
dx
= − F
E I
x3 6 − L∗ x2
2
x2 2 − L∗
x
y(x) = y(L ∗ ) + dy(x)
dx
dy(x)
dx
= dy(x)
dx
x=L ∗
x=L∗ · (x − L∗ )
which are responsible for describing the deflection of the beam, considering the loading on
different coordinates.
Let us introduce an example of a system with two points of force application. Assuming in case
(I) the force F is applied to the first coordinate L ∗ = L1. One can measure and/or calculate
the beam deflection at the coordinates x = L1 and x = L2 through equations (10) and (11):
y1 = y(L1) = F · L3 1
3 · EI
and
y2 = y(L2) = F
6EI · (2L3 1 + 3L 2 1(L2 − L1)) (13)
Assuming in case (II) that the force F is applied to the second coordinate L ∗ = L2, one can
measure and/or calculate the follow beam displacements at the coordinates x = L1 and x = L2
through the equations (10) and (11):
y1 = y(L1) = F
6EI · (2L3 1 + 3L 2 1(L2 − L1)) (14)
and
y2 = y(L2) = F · L3 2
3 · EI
7
(10)
(11)
(12)
(15)