Numerical Methods. ME Mechanical Lab I. Mechanical Engineering ME Lab I
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1 5 9 Mechancal Engneerng -.30 ME Lab I ME.30 Mechancal Lab I Numercal Methods Volt Sne Seres SIN(X) Normalzed Squared Functon *PI Dr. Peter Avtable Unverst of Massachusetts Lowell Numercal Methods Coprght 00
2 Mechancal Engneerng -.30 ME Lab I Some bref notes on numercal methods are ncluded n ths secton Root Mean Square (RMS) Dfferentaton of a Sgnal Integraton of a Sgnal Least Squares Ft of Data Dr. Peter Avtable Unverst of Massachusetts Lowell Numercal Methods Coprght 00
3 Mechancal Engneerng -.30 ME Lab I Root Mean Square (RMS) of a Sne Wave RMS T T 0 (t) dt T Perod Peak to Peak Peak Dr. Peter Avtable Unverst of Massachusetts Lowell Numercal Methods Coprght 00
4 Mechancal Engneerng -.30 ME Lab I A sne wave can be wrtten as a contnuous functon. Ths sne wave can be wrtten n dscrete form for small delta ncrements To evaluate the RMS, the sne wave needs to be frst squared and then multpled b each of the delta ncrements. These values are summed, dvded b the total tme and then a square root taken. Volt Sne Seres.5 Normalzed Squared Functon SIN(X) *PI Dr. Peter Avtable Unverst of Massachusetts Lowell Numercal Methods Coprght 00
5 Mechancal Engneerng -.30 ME Lab I In Ecel, the data values are created and used to form a sne. These values are squared and dvded b the spacng (ncrement). A summaton of the values are dvded b the entre tme length. The square root of ths elds the RMS value of the sgnal 3 Dr. Peter Avtable Unverst of Massachusetts Lowell Numercal Methods Coprght 00
6 Mechancal Engneerng -.30 ME Lab I Numercal Dfferentaton needs to be performed n man cases SLOPE d d + + There are man dfferent methods avalable for numercal processng Dr. Peter Avtable Unverst of Massachusetts Lowell Numercal Methods Coprght 00
7 Mechancal Engneerng -.30 ME Lab I st Forward Dfferentaton SLOPE d d st Backward Dfferentaton SLOPE d d st Central Dfference SLOPE d d + + Dr. Peter Avtable Unverst of Massachusetts Lowell Numercal Methods Coprght 00
8 Mechancal Engneerng -.30 ME Lab I nd Central Dfference Dfferentaton Equal Spacng d d + + Intal condtons need to be specfed for the start of the numercal process. Ths ma have an effect on the accurac of the results obtaned. Dr. Peter Avtable Unverst of Massachusetts Lowell Numercal Methods Coprght 00
9 Mechancal Engneerng -.30 ME Lab I Numercal Integraton - Rectangular Rule I I + + ( ) The smaller the ncrement, the closer the result approaches the actual theoretcal value Dr. Peter Avtable Unverst of Massachusetts Lowell Numercal Methods Coprght 00
10 Mechancal Engneerng -.30 ME Lab I Numercal Integraton - Trapezodal Rule I + I ( ) The trapezodal approach s more accurate than the rectangular approach and s the preferred method Dr. Peter Avtable Unverst of Massachusetts Lowell Numercal Methods Coprght 00
11 Mechancal Engneerng -.30 ME Lab I Least Squares Ft of Data - Regresson Analss Y Man tmes t s necessar to ft the best lne to a set of collected data. Ths s tpcall performed usng a least squares error mnmzaton of ths data to appromate the parameters that best descrbe the lne. A straght lne s shown net - ths can be etended to an order polnomal. X Dr. Peter Avtable Unverst of Massachusetts Lowell Numercal Methods Coprght 00
12 Mechancal Engneerng -.30 ME Lab I Least Squares Ft - Straght Lne --- a+b An error term s generated that descrbes the data n relaton to the lne descrbng the data. ( a b) e + The sum of the errors wll, n the lmt, approach zero. Therefore, the sum of the square of the error s tpcall used z e + e + e3 +L z n [ ( a + b) ] Dr. Peter Avtable Unverst of Massachusetts Lowell Numercal Methods Coprght 00
13 Mechancal Engneerng -.30 ME Lab I Least Squares Ft - Straght Lne --- a+b To mnmze the error, take the dervatve of z WRT a and b n z [ ( a + b) ] 0 a n z [ ( a + b) ] 0 b Ths can be recast as a a n + bn n n n + b n Dr. Peter Avtable Unverst of Massachusetts Lowell Numercal Methods Coprght 00
14 Mechancal Engneerng -.30 ME Lab I Least Squares Ft - Straght Lne --- a+b The Sum of the Squares Error (SSE) s an ndcator of the goodness of the ft. Smaller SSE ndcates a better ft SSE n [ f ( )] Another ndcator s the R-Squared Value. Values approachng.0 ndcate a good ft r SST n SSE SST [ ] Dr. Peter Avtable Unverst of Massachusetts Lowell Numercal Methods Coprght 00
15 Mechancal Engneerng -.30 ME Lab I Least Squares Ft - Straght Lne usng Alternate Coordnates The data ma not alwas be best descrbed b a lnear ft n recatngular coordnates. A hgher order model ma be needed. However, man tmes a change n the coordnate sstem ma result n a form that s best descrbed b a straght lne. Equaton Tpe Equaton Coordnate Sstem Eponental b ae Log Y vs X Logarthmc a ln + b Y vs Log X Power b a Log Y vs Log X Dr. Peter Avtable Unverst of Massachusetts Lowell Numercal Methods Coprght 00
16 Mechancal Engneerng -.30 ME Lab I Least Squares Ft - Matr Formulaton The same least squares mnmzaton problem can be formulated wth a matr approach usng MATLAB. The basc equatons can be cast as 3 a a a M 3 + b + b + b to fnd the coeffcents a and b a b {} [{} {} ] [ coef ] a b a b [ T ] [ ] T coef coef [ coef ] {} Dr. Peter Avtable Unverst of Massachusetts Lowell Numercal Methods Coprght 00
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