Vibrations of Structures

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1 Vibrations of Structures Modue I: Vibrations of Strings and Bars Lesson : The Initia Vaue Probem Contents:. Introduction. Moda Expansion Theorem 3. Initia Vaue Probem: Exampes 4. Lapace Transform Method Keywords: Initia vaue probem, Expansion theorem, Coapse of bar, Striking of string, Lapace transform

2 The Initia Vaue Probem Introduction The initia vaue probem concerns the determination of the evoution of a system given some initia dispacement and veocity conditions. For sefadjoint systems, this probem may be convenienty approached using the moda expansion theorem. Another approach is using the Lapace transformation. Moda Expansion Theorem Consider the free axia vibration probem of a bar with varying crosssection given by µ(x)u,tt + K[u] = () where µ(x) = ρa(x) and K = [EA(x)( ),x ],x. Assume a soution as an expansion in terms of the eigenfunctions in the form u(x, t) = k= k= p k (t)u k (x), () where p k (t) is the unknown moda coordinate, and U k (x) are mutuay orthogona eigenfunctions satisfying K[U k (x)] = ω kµ(x)u k (x), k =,,...,. (3)

3 Substituting the expansion () in () yieds [ k= k= k= p k (t)u k (x) + K k= p k (t)u k (x) ] =. (4) Using the inearity property of the operator K[ ] and (3), one can rewrite (4) as k= k= [ pk (t) + ω kp k (t) ] µ(x)u k (x) =. (5) Taking the inner product on both sides with U j, j =,,...,, and using the orthogonaity condition, we have p j (t) + ω j p j(t) =, j =,,...,. (6) The genera soution of the j th moda coordinate is, therefore, obtained as p j (t) = C j cos ω j t + S j sin ω j t, and, the genera soution of the free vibration probem can be written in the form u(x, t) = (C k cos ω k t + S k sin ω k t)u k (x). k= Thus, we have reconstructed back the soution of the free vibration probem using the eigenfunction expansion (). The fundamenta requirement for the expansion method to work is that any physicay possibe shape of the system, say a bar, shoud be expandabe as a inear combination of the eigenfunctions U k (x) in the form (). In other words, U k (x) shoud form a basis of the space of a physicay possibe shapes 3

4 u(x, t) ρ, A, E T x Figure : An axiay stretched bar of the string. The set of a eigenfunctions of the string is indeed a basis of the function space under consideration, and this foows from the sef-adjointness of the differentia operator K[ ]. This statement is referred to as the expansion theorem. 3 Initia Vaue Probem: Exampes Coapse of stretched bar: Consider a uniform bar stretched by a string under a tension T, as shown in Fig.. We wi study the axia vibrations of the bar when the string suddeny snaps. The initia conditions are given by u(x, ) = T x/ea and u,t (x, ) =. The fina soution is given by the expansion u(x, t) = k (C k cos ω k t + S k sin ω k t) sin(k )πx/, where ω k = (k )πc/. Using the initia conditions, we have C k sin k (k )πx = T x EA, S k ω k sin k (k )πx =. (7) The veocity condition impies S k = for a k. The coefficients C K can be determined by taking inner-product with sin(k )πx/ on both sides of 4

5 the dispacement condtion. This gives C k = Thus, the compete soution is given by 8T (k )π sin (k ) π EA u(x, t) = k= 8T (k )πx (k ) π EA ( )(k ) cos ω k t sin The successive configurations of the bar at certain time instants are shown in Fig.. A struck string: Consider a string struck in the midde which gives it an initia veocity profie in its initia equiibrium position (i.e., w(x, ) = ) given by w,t (x, ) = v [ ( x + cos π )], 5 x 3 5 as shown in Fig. 3. Using these initia conditions, the constants of integration are obtained as C k = for a k, and S k = v sin kπx [ ( x + cos π ω k )] = v πkω k dx ( ) cos kπ, k =,,...,. k It may be noted that S k = for a even vaues of k. The shapes of the string at certain seected time points are shown in Fig. 4. 5

6 EAu/T EAu/T t =. t =.5/c EAu/T t = /c EAu/T t =./c EAu/T EAu/T t =./c t =.5/c Figure : Axia vibrations of a bar reeased from rest from an initiay tensioned state at seected times 6

7 w,t (x, )/v Figure 3: Initia veocity profie of a struck string 4 Lapace Transform Method The Lapace transform method is one of the standard methods of soving initia vaue probems. Consider the wave equation w,tt c w,xx =, (8) with homogeneous boundary conditions w(, t) and w(, t), and initia conditions w(x, ) = w (x), and w,t (x, ) = v (x). Taking the Lapace transform of both sides of (8) and the boundary conditions yieds w s c w = c [sw (x) + v (x)], (9) w(, s), and w(, s), () where w(x, s) represents the Lapace transform of w(x, t), and is defined as w(x, s) = w(x, t)e st dt. () 7

8 πcw/v πcw/v. t =. t =.75/c.. πcw/v πcw/v. t =.5/c. t =.95/c.. πcw/v πcw/v. t = /c. t =.5/c.. Figure 4: Transverse vibrations of a struck string at seected times 8

9 The homogeneous soution of (9) is obtained as w(x, s) = ae sx/c + be sx/c. () Using the boundary conditions () yieds [ ] { } a e s/c e s/c =. b For non-trivia soutions of (a, b), we must have e s/c = s = inπc, n =,,...,. For these vaues of s, one can easiy obtain (a, b) = (, ), and therefore, the genera soution of (9)-() can be written using () as w = n= A n (s) sin nπx, (3) where A n (s) are arbitrary constants. Using this soution expansion in (9), and taking inner product with sin mπ yieds on simpification A m (s) = s s + α m w (x) sin mπx dx + s + α m v (x) sin mπx dx, where α m = mπc /. Substituting this expression in (3) and taking the inverse Lapace transform yieds where w(x, t) = n= (C n cos α n t + S n sin α n t) sin nπx, (4) C n = w (x) sin nπcx dx, and S n = α n v (x) sin nπcx dx. 9

MA 201: Partial Differential Equations Lecture - 10

MA 201: Partial Differential Equations Lecture - 10 MA 201: Partia Differentia Equations Lecture - 10 Separation of Variabes, One dimensiona Wave Equation Initia Boundary Vaue Probem (IBVP) Reca: A physica probem governed by a PDE may contain both boundary