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# TheoryofElasticityII MEM661

Drexel

GPA 3.65

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This 4 page Class Notes was uploaded by Jada Daniel on Wednesday September 23, 2015. The Class Notes belongs to MEM661 at Drexel University taught by Tein-MinTan in Fall. Since its upload, it has received 22 views. For similar materials see /class/212382/mem661-drexel-university in Mechanical Engineering at Drexel University.

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Date Created: 09/23/15

THEORY OF ELAS TI CI TY Review of Fourier Series Review ofFourier series Fig 1 A periodic function Piecerwise Continuous Functions A function is said to be pieceewise continuous in an interval if 1 the interval can be divided into a finite number of subintervals in each of whidi is continuous and 2 the limits of as x approadies the endpoints of eadi subinterval are finite Periodic Functions lf fxP fx P gt 0 then the smallestP is said to be the period of De nition ofFourier series If a function fx defined in the interval cc 2L is piecewise continuous then fx2L x and can be expressed in the following Fourier series quotx 041261 cos l71 sin a where the Fourier coefficients 110 11L and l71 are given by 1 5 2L LLferx w 1 cZL n x u x cos dx c LL A L m 1 cZL n x l7 x sin dx d LL R L ltgt It is noted that 202 is the average value of fx and the series converges to the average value at every discontinuity By locating the origin at the midpoint of a period then the function is defined in the interval iL L and the coefficients of Fourier series can be expressed in the following form wm a an JLLfxcosdx f l71 foxsindx g HalfRange Fourier Sine or Cosine Series lf fx defined in the interval 7L L is an even function ie if ex in the interval then we can express the function in the following halfrange Fourier cosine series x 110 nILx 11 cos h 2 n L TM Tan Drexel University 1 November 30 2008 THEORY OF ELAS TI CI TY Review of Fourier Series where 7 2 L 7 2 L n x 110 ifquot fxdx an 7 Iquot fxcosde 1 Similarly if the function is an odd function ie if ex 7fx in the interval iLL then we can express the function in the followinghalfrange Fourier sine series x Eb sin 139 11 L where l71 fxsindx k Example 1 Expand the following function as shown in Fig 2 in the interval 75 5 into a Fourier series 0 7 5 lt x lt 0 wila Oltxlt5 Since the function is neither even nor odd we must use the expressions given in b c and d to obtain the Fourier coefficients 1 5 1 5 110 Elis x x E10 3dx 3 5 an if fxcos dx EKism j 0 5 5 5 5 mt 5 0 5 b lJs fltxgtsjn nlzxdx3 ficosn x 317cosn7r 5 5 5 5 117 5 0 and the Fourier series representation of the function is given by 3 3licosn7r rim 3 6 71x 1 371x 1 57m fx l f i i i j 2 W1 117 5 2 IIK u l u l The Fourier series representations of the function in the interval 75 5 using various numbers 717 5 3 555 of terms are shown in Fig 3 TM Tan Drexel University 2 November 30 2008 THEORY OF ELAS TI CI TY Review of Fourier Series Fig 3 Example 2 Onedimensional Heat Conduction Problems The governing equation for onedimensional heat transfer problems is given by 014xt K 62u3t 1 at 6x where uxt is the temperature distribution along a onedirnensional bar at time t and K is called dijj iAsivity Using the method of separation of variables ie by assuming that uxt UxTt m we can rewrite l in the following form wltxgt 1m um K Tt 0 Since the lefthand side of n is a function of x while the righthand side is a function of t both sides therefore must be equal to a constant say 712 Consequently n becomes U AZU 0 and TKAZT 0 The solutions of the above two equations are given by U AlcosAxBlsinAx and T Cle39mzt 0 Substituting 0 in m yields uxt Clequot 12A1 cositx B1 sin Ax e39mzt A cos Ax Bsin Ax p Constants A and B may be determined by using the boundary and initial conditions Consider for instance the following boundary condition u0tuLt0 q where L is the length of the bar Substituting p in q yields A 0 and Be W sin AL 0 r Since B 0 is trivial we have sinAL 0 TM Tan Drexel University 3 November 30 2008 THEORY OF ELASTI CI TY and the solution p becomes Review of Fourier Series m0r1r2 s L t mlzx xt ZBme L sin L t m1 in which constants 3 may be determined by using the initial condition eg ux0 By expanding into a Fourier sine series as x 2F sin quotMquot u m1 L then we have by comparing eadi term in t and Bm Fm For instance let ux0 25 then we have 25 2F sin m where coefficients Fm can be obtained by using d ie Fm EILZS Sin m x dx 5017 cosmlr Bm L 0 L m7 and the final solution is given by xt 2 5017 cosmlr ETI sin mlzx W1 m7 L 100 jth x 1 37m 1 Z 2t 571x E m l L m l m l 7r L 3 L 5 L TM Tan Drexel University November 30 2008

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