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by: Lyda Ryan I

DynamicEngineeringSystems ENGR232

Marketplace > Drexel University > Engineering and Tech > ENGR232 > DynamicEngineeringSystems
Lyda Ryan I
GPA 3.98


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Class Notes
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This 10 page Class Notes was uploaded by Lyda Ryan I on Wednesday September 23, 2015. The Class Notes belongs to ENGR232 at Drexel University taught by ThomasChmielewski in Fall. Since its upload, it has received 118 views. For similar materials see /class/212340/engr232-drexel-university in Engineering and Tech at Drexel University.

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Date Created: 09/23/15
ENGR 232 Dynamic Engineering Systems Week 1 Lecture 1 Introduction to Problem Solving with Differential Equations Dr Thomas Chmielewski June 222u1u ENGR 232 so moan Week 1 7 Lecture 1 Staff o Lecturer Dr Thomas Chmielewski o Recitation instructors Dr B uce Char Dr Vasileios Nassis Dr Baris Taskin 0 Teaching assistants Mr Donald Bucci Mr Ryan Measel Mr Feiyu Xiong 0 See web site for syllabus contact information and hours June 22 2mm ENGR 232 511 moan Week 1 7 Lecture 1 Textbook o TextJames R Brannan and William E Boyce Differential Equations An Introduction to Modern Methods amp Applications JohnWiley amp Sons Inc 2007 ISBN10 0471 9 Hill lll Tl l l ll 39ll June 222u1u ENGR 232 Su noon Week 1 7 Lecture 1 Course Formalities Course materials will be on the learningdrexeledu web site Two inclass midterms Week 4 and 7 Two Matlab exams Week 5 and 8 Nine homework assignments posted on Tuesday due next Tuesday by 4 PM at the ECE Lab window on the second floor of Bossone No HW will be collected at lecture Eight prelabs and labs second half of recitation You may not change labrecitation sections N mework labs or exams except with valid prior notification or with documented medical emergenc Reference Syllabus for grading policy and details for lab procedures June 22 2mm ENGR 232 Su noon Week 1 7 Lecture 1 How To Succeed Read the syllabus and follow instructions 0 Download and print lecture notes prior to class follow and annotate during lecture No annotated notes will be posted 0 Read and study the text do the homework o Read labs prior to recitationlab session 0 Complete the prelabs and bring hardcopy stapled to verification sheet to lab sessions 0 If you have any questions see me or any of the instructors or TA39s We are available by email always and by appointment See syllabus for TA office hours and appointment days Help is available from the Drexel Learning Center httptutorsdlcdrexeledu June 22 2mm ENGR 232 So mean Week 1 7 Lecture 1 5 Introduction 0 Differential equations are equations containing 39 es derivatlv Some examples of physical phenomena involving rates fchange Motion of mechanical systems Population dynamics Electrical circuits 0 A differential equation that describes a physical process is often called a mathematical model 0 This lecture focuses on Section 11 and 12 of the text June 22 2mm ENGR 232 Ed norm Week 1 7 Lecture 1 5 Problem 0 We drop a stone from the top of the CN Tower in Toronto 553m How long before it hits the ground Time impact How fast will it go Terminal speed 0 General plot of what will happen FE v 539 1 velnctv position I l time time June 222u1u ENGR 232 So neon Week 1 7 Lecture 1 Example 1 Free Fall Section 11 Formulate a differential equation describing motion of an object falling near sea level neglect the force of the air Variables time t velocity v position x dXdt a dvdt d2ltdt2 Law F ma mdvdt net force Force of gravity F mg downward force At t 0X 0 v 0 initial condition 0000 2lt gli n O x we quot2 Plan Find equation for vt Find equation forxt Solve for t when xt 0 Compute vt June 22 2mm ENGR 232 511 mean Week 1 7 Lecture 1 Example 1 Free Fall Model June 222u1u ENGR 232 so mean Week 1 7 Lecture 1 9 Example 2 Increased Model Complexity A Falling Hailst one 1 of4 o A hailstone has mass m0025 kg and drag coefficient kgs1 0 Taking g 98 msecz the differential equation for the falling hailstone is dv m mgi 7v dt 9870281 1r June 22 2mm ENGR 232 511 mean Week 1 7 Lecture 1 1t Example 2 Sketching Direction Field 2 of 4 0 Using differential equation and table plot slope estimates on axes below The resulting graph is called a direction field values of v do not depend on t v39 987 028v June 222u1u ENGR 232 ad Dean Week 1 7 Lecture 1 Example 2 Computer Plot of Direction 3 of 4 httpmathriceedudfielddfpphtml June 22 2mm ENGR 232 Ed neon Week 1 7 Lecture 1 Example 2 Direction Field amp Equilibrium Solution 4 of 4 o Arrows give tangent lines to solution curves and indicate where soln is increasing amp decreasing and by how much Horizontal solution curves are called equilibrium solutions Use the graph below to solve for equilibrium solution and then determine analytically by setting 11 0 Setv390 vl 98702812 June 22 2mm ENGR 232 Su ugrm Week 1 7 Lecture 1 13 Mice and Owls A Model 0 Consider a mouse population that reproduces at a rate proportional to the current population assuming no owls Let 1 represent time pt represent the mouse population and r represent the growth rate micetime Then 0 When owls are present they eat the mice If the predation rate is a constant k micetime then June 22 2mm ENGR 232 Su ugrm Week 1 7 Lecture 1 14 Example 3 Mice and Owls 1 of 2 2 dt rpik 0 Consider a mouse population pt is the numberof mice at time t that reproduces at a rate prOportional to the current population with a rate constant equal to 05 micemonth assuming no owls present 39039 When owls are present they eat the mice SuppOse that39the owls eat 15 per day average Write a differential equation describing mouse population in the presence of owls Assume that there are 30 daysin a month 0 dp 05 450 cit p 39 June 22 2010 39 ENGR 232 30 0910 Week 1 7 Lecture 1 39 15 Example 3 Direction Field 2 of2 Discuss solution curve behavior and nd equilibrium soln p3905p 450 mm m 39 June 22 2010 39 ENGR 232 30 0910 Week 1 7 Lecture 1 39 16 Solutions of Some Differential Equations Section 12 0 Recall the examples from the last section 0 Free fall In ngiw dt 0 Owls and mice P e k dt VP 0 These equations have the general form y39 ay b 0 We can use methods of calculus to solve differential equations of this form June 222u1u ENGR 232 Su ugrm Week 1 7 Lecture 1 17 Example 1 Mice and Owls 1 of 3 For the differential equation p 05127 450 the solution is p 900 Ice where kis a consmnt Verify solution is correct June 22 2mm ENGR 232 Su ugrm Week 1 7 Lecture 1 1E Example 1 Integral Curves 2 of 3 Thus we haveinfinitely many solutions tooour equation k t 5 05197450 p900 9 since k is an arbitrary constant Graphs of solutions integral curves for several values of k and direction field for differential equation are given below 0 Choosing k 0 We obtain the equilibrium solution while for k 7 0 the solutions diverge from equilibrium solution lznn Mun mun 12m MUD June 22 2010 ENGR 232 Sn 0910 Week 1 Lecture 1 Example 1 InitialConditions 3 of 3 o A differential equation usually has in nitely many solutions If a point on the solution curve is known such as an initial condition then this determines a unique solution In the miceowl differential equation suppose we know that the mice population starts out at 850 Then p0 850 and pt 900 keo 5 June 22 2010 ENGR 232 Sn 0910 Week 1 Lecture 1


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