Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Some basic concept Vibration of Structures are Harmonic Waves, Influence of Axial Force, Initial Value Problem, Mathematical Modeling, Modal Analysis, Motion of Material Points, Orthogonality Relations, Projection Methods.Main points of this lecture are: Initial Value Problem, Modal Expansion Theorem, Laplace Transform Method, Collapse of Bar, Striking of String, Free Axial Vibration Problem, Modal Coordinate, Linearity Property of Operator, Eigenfunction Expansion
Typology: Study notes
1 / 9
This page cannot be seen from the preview
Don't miss anything!
Contents:
Keywords: Initial value problem, Expansion theorem, Collapse of bar, Strik- ing of string, Laplace transform
The initial value problem concerns the determination of the evolution of a system given some initial displacement and velocity conditions. For self- adjoint systems, this problem may be conveniently approached using the modal expansion theorem. Another approach is using the Laplace transfor- mation.
Consider the free axial vibration problem of a bar with varying cross- section given by μ(x)u,tt + K[u] = 0 (1)
where μ(x) = ρA(x) and K = −[EA(x)(·),x],x. Assume a solution as an expansion in terms of the eigenfunctions in the form
u(x, t) =
k∑=∞ k=
pk(t)Uk(x), (2)
where pk(t) is the unknown modal coordinate, and Uk(x) are mutually orthog- onal eigenfunctions satisfying
K[Uk(x)] = ω k^2 μ(x)Uk(x), k = 1, 2 ,... , ∞. (3)
2
l
u(x, t)
x
ρ, A, E T
Figure 1: An axially stretched bar
of the string. The set of all eigenfunctions of the string is indeed a basis of the function space under consideration, and this follows from the self-adjointness of the differential operator K[·]. This statement is referred to as the expan- sion theorem.
Collapse of stretched bar: Consider a uniform bar stretched by a string under a tension T , as shown in Fig. 1. We will study the axial vibrations of the bar when the string suddenly snaps. The initial conditions are given by u(x, 0) = T x/EA and u,t(x, 0) =
Ck sin (2k^ − 2 l1) πx= (^) EAT x, ∑ k
Skωk sin (2k^ − 2 l1) πx= 0. (7)
The velocity condition implies Sk = 0 for all k. The coefficients CK can be determined by taking inner-product with sin(2k − 1)πx/ 2 l on both sides of 4
the displacement condtion. This gives
Ck = (^) (2k −^8 1)T l (^2) π (^2) EA sin (2k^ − 2 1)π
Thus, the complete solution is given by
u(x, t) =
k=
8 T l (2k − 1)^2 π^2 EA(−1)
(k−1) (^) cos ωkt sin (2k^ −^ 1)πx 2 l
The successive configurations of the bar at certain time instants are shown in Fig. 2.
A struck string: Consider a string struck in the middle which gives it an initial velocity profile in its initial equilibrium position (i.e., w(x, 0) = 0) given by
w,t(x, 0) = v 20
1 + cos 10π
(x l −^
, 25 ≤ x l ≤ (^35)
as shown in Fig. 3. Using these initial conditions, the constants of integration are obtained as Ck = 0 for all k, and
Sk = (^) lωv^0 k
∫ (^) l 0 sin^
kπx l
1 + cos 10π
(x l −^
dx
= (^100) πkωvk^0
(cos kπ − 1 k^2 − 100
, k = 1, 2 ,... , ∞.
It may be noted that Sk = 0 for all even values of k. The shapes of the string at certain selected time points are shown in Fig. 4.
5
1
1
w,t(x, 0)/v 0
x/l
Figure 3: Initial velocity profile of a struck string
The Laplace transform method is one of the standard methods of solving initial value problems. Consider the wave equation
w,tt − c^2 w,xx = 0, (8)
with homogeneous boundary conditions w(0, t) ≡ 0 and w(l, t) ≡ 0, and initial conditions w(x, 0) = w 0 (x), and w,t(x, 0) = v 0 (x). Taking the Laplace transform of both sides of (8) and the boundary conditions yields
w˜′′^ − s 2 c^2 w˜^ =^ −^
c^2 [sw^0 (x) +^ v^0 (x)]^ ,^ (9)
w˜(0, s) ≡ 0 , and w˜(l, s) ≡ 0 , (10)
where ˜w(x, s) represents the Laplace transform of w(x, t), and is defined as
w˜(x, s) =
0 w(x, t)e
−st (^) dt. (11) 7
1
1
1
1
1
1
t = 0. 05
t = 0. 25 l/c
t = 0. 5 l/c
t = 0. 75 l/c
t = 0. 95 l/c
t = 1. 05 l/c
πcw/v 0 l
πcw/v 0 l
πcw/v 0 l
πcw/v 0 l
πcw/v 0 l
πcw/v 0 l
x/l
x/l
x/l
x/l
x/l
x/l
Figure 4: Transverse vibrations of a struck string at selected times
8
