16 Jun Vibration Analysis of Structures Assignment 4 (5 points)
TMME40 Vibration Analysis of Structures
Assignment 4 (5 points)
Jonas St̊alhand
Division of Solid Mechanics, Linköping University
581 83 Linköping, Sweden
December, 2016
A bar with variable cross section (A and 2A) is axially loaded by the time dependent force F = F (t), see Fig. 1. The bar has the homogeneous elastic modulus E and density ρ. It is discretised into five elements, each of length L. The axial displacement inside an element is approximated by u(x, t) = N1(x)u1(t) +N2(x)u2(t), where the shape functions are given by
N1(x) = L− x L
, N1(x) = x
L ,
and 0 ≤ x ≤ L.
Figure 1: The axially loaded bar with variable cross section.
(a) Show that the element stiffness and element mass matrix are, respectively,
[k] = EA′
L
[ 1 −1
−1 1
] and [m] =
ρA′L
6
[ 2 1 1 2
] , (1)
where A′ is the cross section area of the element.
(b) Use [k] and [m] to assemble the global stiffness matrix [K] and the lumped global mass matrix [M ].
(c) Let the displacement of the nodes be given by the vector [u] = (u1, u2, u3, u4) T.
1
The undamped vibration is then governed by the differential equation
Mü + Ku = F , (2)
where F is the vector of external forces. For the problem to be solvable, we also need initial conditions which are assumed to be
u(0) = 0 and u̇(0) = 0. (3)
Derive the time-discrete version of Eq. (2) by replacing the time derivatives of u for their central difference approximations. Assume a constant time step ∆t such that un = u(n∆t) = u(t).
(d) The central difference approximation requires the displacement vector u−1 = u(−∆t). If the bar is initially at rest, this value can be approximated by
u−1 = (∆t)2
2 M−1F 0, (4)
where F 0 = F (0). Use a Taylor expansion of u about t = 0 to show Eq. (4).
Write a short report where you present the solution to the questions above. Make sure to write your name and personal identity number (or LiU-id) at the top of each page. Send your report as a PDF via email to jonas.stalhand@liu.se no later than January 8, 2017. Hand-written reports are accepted if they are neatly written and the PDF is created using a proper scanner.
2
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