ESE 318-02, Fall 2014Lecture 1, Aug. 26 2014The Laplace Transform and Its InverseDefinition of the Laplace transform ""0pairtransformLaplacesFtfsFdttfetfstLProvided the integral converges (stays finite for proper s). We are only interested in t > 0. Notation: lower case for the time function; corresponding upper case for transform, which is a function of s. For the integral to converge, we usually have to constrain s such that s > k, where k is particular to the f(t).In Chapter 4 you will be dealing only with functions for which the Laplace integral converges (for proper s). You will not be asked to prove whether that is the case. But for your information, sufficient conditions are that f be piecewise continuous and of “exponential order”, meaning TtallforMetfct, where M, c and T are some finite constants.The Laplace transform of the constant function, f(t) = 1. (Also applies to the unit step function which starts at t = 0, U(t), to be covered later.) 0110110limlim10000sssifsssesesedtedttfetftortfsttsttstststLUWe understand the evaluation bounds as limits, but we will not always show that extra step. In the above case, to get the limit as t goes to infinity to converge, we needed s > 0 (or more accurately, the real part of s must be > 0, since s can be complex in general). You should be able to describe such conditions when doing similar problems.ESE 318-02, Fall 2014The Laplace transform of the function, f(t) = t. We’ll use integration by parts, and recall the evaluation of the integral from the above case of f(t) = 1. 011111''0100limlim2210000000sstsabovecasefromssRulesHopitalLsifdtesdtsesetsetdtsesetdttevduuvudvdtedvdtdusevtudttetstststtsttststststststLLLThe Laplace transform of a function multiplied by eat. asFtfdttfedtetfetfeasstasstatatLL00That is, we find the Laplace transform of f(t), and simply replace s by s-a.The Laplace transform of eat. asseeassassatat1111 LLLWe could have also evaluated it directly, as in Example 3 of Section 4.1 of Zill. But you will see that we often tend to find Laplace transforms by using properties rather than evaluating the integral in the definition.ESE 318-02, Fall 2014Linearity. sbGsaFtgbtfadttgebdttfeadttbgtafetbgtafstststLLL000The Laplace transforms of sine and cosine. We will make use of the Euler formula for sine and cosine. 2222222222sincossincos11sincossincoskskktkssktkskikssktiktksiksiksiksiksiksektiktEulerktikteiktiktLLLLLLThe Laplace transforms of hyperbolic sine and cosine. 22212222212121212121coshcosh11sinhsinhksskteektforSimilarlykskkskskskskseekteektktktktktktktLLLLThe Laplace transforms of t. Students will do problem 41(b) of Zill 4.1, which finds the transform for t, which can be used to find transforms for powers of t. (See Appendix II of Zill for more on the gamma function. You’ll see that it is a generalization of a factorial.)ESE 318-02, Fall 2014The Laplace transform tables. So far we have found a few Laplace transforms and properties, summarized below. A more complete table is in the inside back cover of Zill and in Appendix III of Zill. sbGsaFtbgtafasFtfekssktkskktasestsntstsatatnnLL2222112cossin11!111Class participation exercise. What are the indicated Laplace transforms? (Pretend we donot have access to the complete Table, only the abbreviated table above.) ?cos?sin ktekteatatLLRecall from above: 2222sincoskskktkssktasFtfeat LLLApply that property to those Laplace transforms: 222222222222222coscos2sinsinkaassaskasaskssktktekaasskkaskkskktkteassassatassassatLLLLThese are actually common cases and are indeed solutions for many physical phenomena involving oscillation/vibration. For example, a mass on a spring, with damping/fraction (or a real spring with friction).ESE 318-02, Fall 2014Inverse Laplace transforms. We called F(s) the Laplace transform of f(t). We therefore call f(t) the inverse transform of F(s). Thus the left sides of the above table are the inverseLaplace transforms of the corresponding right sides. Examples. .cos111122112111etcktksseastsstfsFatLLLLL(There is actually an integral formula for the inverse Laplace transform. It is shown on page 729 of Zill. It is actually a contour integral that requires the use of complex variables. The good news: we do not use that formula in this course. Essentially, we find inverse transforms by assuming we can break things down into a number of known functions with known transforms and using “the tables”.)Finding Inverse Laplace transforms in simple cases. If we need to find an inverse Laplace transform, and the formula is already in “The Table” (Appendix III), then we are done. In some cases, simple arithmetic can get you there, as in the following examples. You need to know what’s in The Table to do this.ttssssstss2sin32cos24226424627sin7177717122222 Here is another example of a fraction that splits up fairly easily.22532333232212251215252ttsssssssssssPartial Fraction Expansion (PFE). Unfortunately, it’s not always easy to find inverse Laplace transforms. You must learn to do PFE by
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