ME 012 Engineering Dynamics

Transcription

1 ME 012 Engineering Dynamics Lecture 7 Absolute Dependent Motion Analysis of Two Particles (Chapter 12, Section 10) Tuesday, Feb. 05, 2013

2 Today s Objectives: Understand translating frames of reference. Use translating frames of reference to analyze relative motion. In-Class Activities: Applications Relative Position, Velocity and Acceleration Vector & Graphical Methods Examples 2

3 APPLICATIONS When you try to hit a moving object, the position, velocity, and acceleration of the object must be known. Here, the boy running on the ground at a constant speed starts at some distance d when thegirlinthewindowthrowstheballtohim. Howfastshouldtheballbethrown? When fighter jets take off or land on an aircraft carrier, the velocity of the carrier becomes an issue. The aircraft carrier travels at a forward velocity of 50 km/hr and plane A takes off at a horizontal air speed of200km/hr(measuredbysomeoneonthewater), How do we find the velocity of the plane relative to the carrier? HowwouldyoufindthesamethingforairplaneB? Howdoesthewindimpactthissortofsituation? 3

4 RELATIVE POSITION The absolute position of two particles and withrespecttothefixed,, referenceframeare givenby and : / = = i+ j+ k = i+ j+ k The position of relative to (i.e. position of particle B w.r.t. x -y -z reference frame) is then represented by: / =( )i+( )j+( )k The position of relative to is then represented by: / = /= / =( )i+( )j+( )k 4

5 RELATIVE POSITION Therefore, if: =10i+3j 9k =2i 6j+4k Then: / =(2 10)i+( 6 3)j+(4 ( 9))k / = 8i 9j+15k OR: / =8i+9j 15k 5

6 RELATIVE VELOCITY The relative motion equations of velocity and acceleration are applied in the same manner EXCEPT that the origin of the fixed axis does not have to be specified Recall our notations for and which are absolute velocities =! i+! i+! i=" i+"j+"k =(# $ ) i+(# % ) j+(# & ) k AND =! i+! i+! i=" i+" j+" k =(# $ ) i+(# % ) j+(# & ) k 6

7 The relative motion equations of velocity and acceleration are applied in the same manner EXCEPT that the origin of the fixed axis does not have to be specified RELATIVE VELOCITY The time derivative of the relative position equation is taken to determine the relative velocities: The velocity of relative to is then: / = / =[ # $ # $ ]i+[ # % # % ]j +[(# & ) (# & ) ]k With the velocity of relative to represented by: / = /= / = # $ # $ i+[ # % # % ]j +[(# & ) (# & ) ]k 7

8 RELATIVE ACCELERATION The relative motion equations of velocity and acceleration are applied in the same manner EXCEPT that the origin of the fixed axis does not have to be specified The time derivative of the relative velocity equation yields a similar acceleration relationship between the absolute and relative accelerations of particles and. The relative acceleration of with respect to is: ) / =) ) ) / =[ * $ * $ ]i+[ * % * % ]j +[(* & ) (* & ) ]k The relative acceleration of with respect to is: ) / = ) /=) ) ) / = * $ * $ i+[ * % * % ]j +[(* & ) (* & ) ]k 8

9 SOLVING PROBLEMS When applying relative position it is helpful to specify a translating axes, x -y -z. For the case illustrated here, the origin of x -y -z is placed onknownpositionofparticleaasweareinterestedinthe motionofbrelativea. The resulting relative velocity (v A/B ) and acceleration (a A/B ) equations are applied in a similar manner except that in this case the origin ofthe fixedx-y-z axis does not need to be specified. / = / = ) / =) ) 9

10 SOLVING PROBLEMS From this point, two approaches may be taken: (1)Thevelocityvectors,forexample,in = + / could be written as Cartesian vectors as was illustrated. The resulting scalar equations solved for up to two unknowns. (2) Alternatively, vector problems can be solved graphically by use of trigonometry. This approach usually makes use of thelawofsinesorthelawofcosines(seenextslide). Though both may be used in your assignments, the preferred approach is (1) as Cartesian vector forms can be coded and programmed with relative ease. 10

11 LAW OF SINES AND COSINES Since vector addition or subtraction forms a triangle, sineandcosinelawscanbeappliedtosolveforrelative or absolute velocities and accelerations. As review, their formulations are provided below: a C b Law of Sines: * sin = - sin =. sin/ B c A Law of Cosines: * 0 = cos - 0 =* *.cos. 0 =* *-cos/ 11

12 EXAMPLE 1 y Determine the relative velocity of particle B with respect to particle A using both vector and graphical analysis. B v B =100 km/h A 30 v A =60 km/h x 12

13 EXAMPLE 1: Solution 13

14 EXAMPLE 2 Two boats leave the shore at the same time and travel in directions and speeds as follows: # =20 ft/s,5 6 =30 deg # =15 ft/s,5 0 =45 deg Determine the speed of boat A with respect to boat B. How long after leaving the shore will the boatsbeatadistance800ftapart? 14

15 EXAMPLE 2: Solution 15

16 EXAMPLE 3 At the instant shown, the car at A is traveling at 10 m/s around the curve while increasing its speed at 5 m/s 2. The car at B is traveling at 18.8 m/s along thestraightawayandincreasingitsspeedat2m/s 2. Determine the relative velocity and relative acceleration of A with respect to B at this instant assuming: =100 mand5 =45. 16

17 EXAMPLE 3: Solution 17

18 EXAMPLE 4 The airplane has a speed relative to the wind of# =100 mi/hr. The speed of the wind relative to the ground is #? =10 mi/hr at an angle =20 deg. Determine the angle 5 at which the plane must be directed in order to travel in the direction of the runway. Also, what is its speed relative to the runway? 18

19 EXAMPLE 4: Solution 19