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.

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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 [email protected] 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.

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