Relative motion. measurements by taking account of the motion of the. To apply Newton's laws we need measurements made

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Jacob Crawford
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1 Relative motion To apply Newton's laws we need measuements made fom a 'fixed,' inetial efeence fame (unacceleated, non-otating) In many applications, measuements ae made moe simply fom moving efeence fames We then need a way to ecove the inetial measuements by taking account of the motion of the measuement fame In othe situations, we may wish to ecove motions elative to a moving fame fom measuements of inetial motion L4:1

Fo example The flutte motion elative to axes fixed in the body of the aicaft is elatively simple: flapping 'up' and 'down' We seek to measue it by placing an acceleomete at the wing tip with its

2 Fo example The flutte motion elative to axes fixed in the body of the aicaft is elatively simple: flapping 'up' and 'down' We seek to measue it by placing an acceleomete at the wing tip with its sensitive axis aligned with the -axis of the aicaft Howeve, the acceleomete measues the -component of the inetial acceleation plus the acceleation due to gavity y Late, we will wok out how to ecove the simple flutte motion fom the acceleomete eadings G x L4:

3 Thee kinds of obseves Inetial obseve unacceleated non-otating O 1 Tanslating obseve acceleated but non-otating Rotating obseve acceleated and otating O A V L4:3 See Dynamics (vol 1) by R L Halfman Addison-Wesley (196) O 3 A V ω

4 Notation Tanslating obseve O O L4:4 O 1 R O O A O = R PO P = O 3 PO 3 Inetial obseve V O R O3 We will geneally use: Uppe case lettes (R, V, A) fo inetial measuements Lowe-case lettes (, v, a) fo noninetial measuements O 3 Position of P elative to O 3 A O3 V O3 Rotating obseve ω

5 K Motion elative to tanslating obseve Inetial position of paticle P O 1 R O J O X O Y O R Z O x y I k j i P R = RO + i.e., position of P elative to O 1 = position of P elative to O plus position of O elative to O 1 dr d R O d Inetial velocity of paticle P V dt dt dt = = + whee d/dt denotes time ate of change seen by inetial obseve Obseve may choose to descibe using Catesian coodinate system with unit vectos i j k, which he/she sees as constants: = xi+ yj+ k Velocity seen by is v = = x i+ y j + k t whee / t denotes time ate of change seen by moving obseve L4:5

6 K We have: Now, because i j k ae non-otating, also sees these unit vectos as constant; i.e., d O 1 R O J O X O Y O R Z O x y I k V d R O d = +, dt dt j i P = xi+ yj+ k Hence, inetial velocity of P V = VO + v, v = = x i+ y j + k t = xi+ yj+ k = = v dt t i.e., absolute velocity of P = absolute velocity of O plus velocity of P elative to tanslating obseve Inetial acceleation of P d dv dv A = = AO dt + dt v v = = xi + yj + k = a dt t A = AO + a L4:6

7 O 1 Tanslating efeence fames Conside 3 paticles A, B, C in motion Suppose A and C ae obseved by tanslating obseve moving with B. Z Then absolute position, velocity and acceleation of, e.g., A ae given by: RA = RB + A B, VA = VB + va B, AA = AB + aa B R C R B X C C/B Y B A/C R A A/B x A y We could just as well get to move with A and obseve B: RB = RA + BA, VB = VA + vba, AB = AA + aba Clealy, then: Thus, elativity of obsevations of tanslating obseves is staightfowad (because ates of change they see ae not affected by otations) BA = AB, vba = vab, aba = aab If moves with C: VA = VC + va C (1) whee VC = VB + vc B () and VA = VB + va B (3) (1), () VA = VB + va C + vc B (4) (3), (4) v A B = v A C + v C B L4:7

Example: Helicopte landing on ship in coss wind Ship has fowad speed of 0 knots, due Noth Ocean is stationay Wind is fom the West at 0 knots Contolle wants helicopte to descend vetically at 10 knots

8 Example: Helicopte landing on ship in coss wind Ship has fowad speed of 0 knots, due Noth Ocean is stationay Wind is fom the West at 0 knots Contolle wants helicopte to descend vetically at 10 knots What helicopte velocity elative to the ai is equied? I K L4:8 J (Noth) Absolute velocity of ship: V = S 0 J Absolute velocity of ai: V = A 0 I Velocity of helicopte elative to ship: v = H S 10 K Absolute velocity of helicopte: VH = VS + vh S = VA + vh A Hence: vh A = VS + vh S VA = 0 J 10 K 0 I (knots) Helicopte speed elative to ai: v = H A 0 = 30 knots An obseve v H A

9 K O 1 Motion elative to otating obseve L4:9 Inetial position of paticle P J I k R x j y i P R = RO 3 + i.e., position of P elative to O 1 = position of P elative to O 3 plus position of O 3 elative to O 1 dr d R O d 3 Inetial velocity of paticle P V dt dt dt = = + whee d/dt denotes time ate of change seen by inetial obseve Obseve may choose to descibe using Catesian coodinate system with unit vectos i j k, which he/she sees as constants: R O 3 O 3 = xi+ yj+ k Velocity seen by is v = = x i+ y j + k t 3 whee / t denotes time ate of change seen by moving obseve

10 d e dt We have: V d R O 3 d = +, dt dt = xi+ yj+ k Howeve, because i j k ae now otating, sees these unit vectos changing with time; i.e., v = t 3 d d d d = xi+ yj+ k+ x i + y j + k dt dt dt dt, v = = x i+ y j + k t 3 L4:10 Recall the ates of change of unit vectos in cylindical coodinates: d e = θ e θ = ω e e = ω e y θ = dt θ e d e = d e θ e θ θ = θ e = ω e e θ = ω e θ e dt dt θ 1 e θ Note that the angula velocity of the x igid tiad e e θ e is a vecto quantity: ω = θ ω = ω θ e = ω e magnitude is instantaneous ate of change of oientation diection is instantaneous axis of otation ight-hand scew convention e θ e ω e

11 We have: d di dj dk = + x + y + dt t dt dt dt 3 Think of unit vecto i as position vecto of the point x = 1 on the x-axis; then di/dt is the velocity of this point k = i j O 3 α ω v = sinθ ω n = ω i y 1 sinθ x Thus d i dt = ω i dj dk Similaly = ω j, = ω k dt dt Hence Rate of change seen by d = + xω i + yω j + ω k dt t 3 = + ω ( xi+ yj+ k) t 3 d = + ω dt t 3 L4:11 i.e., Coiolis Theoem (tue fo any vecto) Rate of change seen by Absolute angula velocity of

12 Motion elative to otating obseve L4:1 Inetial position of paticle P R = RO 3 + dr d R O d 3 Inetial velocity of paticle P V = = + dt dt dt i.e., V = VO 3 + v+ ω Absolute vel. of P Absolute Vel. of P vel. of O 3 seen by Coection fo obseve s otation K O 1 O 1 J k R x j y i P R O 3 O 3 I

13 Motion elative to otating obseve We have: V = VO 3 + v+ ω Inetial acceleation of paticle P dv d V O 3 dv dω d A ω dt dt dt dt dt = = v ω A = AO ω v + + ω ω + ω + ω t t t Absolute acc. of P Acc. of P seen by Coiolis component 3 ( ) A = A + a+ ω + ω v + ω ω O Absolute acc. of O 3 Tangential component Centipetal component L4:13

14 Application to spial stunt manoeuve Assume that the aicaft x-axis is aligned with the tangent unit vecto t Plane must oll about t though an angle ϕ to maintain a coodinated tun V Lsin ϕ = m ρ Lcos ϕ = mgcos β V tan ϕ = cos β ρ g ϕ = 17 j ϕ n b mg cosβ Lift L k View along x-axis Coodinate tansfomation i t 0 cosϕ sinϕ j = n 0 sinϕ cosϕ k b i.e., { } { } A xy Tϕ Atnb = s ρ Z n b e θ Y X θ G e L4:14 β t e

15 Application to spial stunt manoeuve Absolute acc. of P ( ) A= A + a+ ω + ω v + ω ω G Absolute acc. of G Let's assemble the components we need y G x L4:15 Acc. of P seen by. component is vibation we want to measue! Abs. acceleation of G: Acceleomete P { } = { } = = 4.55 m/s Axy Tϕ Atnb Components in aicaft axes xy of absolute acceleation of G

16 Angula velocity of intinsic tiad n V/ρ b V/σ t V V ω int = t+ b ρ σ Angula velocity of aicaft: ω= ωint +ϕ t 1 V 1 VV = tan cos, ρ g = 1+ tan ϕ ρg ϕ β ϕ { } ( ) A= A + a+ ω + ω v + ω ω G Recall that ρ and σ ae constant fo cicula helix = V + + V ρ ϕ t σ b L4:16 T ω tnb = ad/s Angula acceleation of aicaft: i.e., V V V V V V ω = + ϕ ϕ ρ t+ b+ σ + ρ n ρ + n σ σ V V V ϕ V V ω = + ϕ ρ t ρ ρ σ n b σ { } T ω tnb = ad/s ϕ = ( ) ( 1+ tan ) Vcos β V V ϕ tanϕ ρg ϕ

17 ( ) A= A + a+ ω + ω v + ω ω G L4:17 Coiolis acceleation: a = co ω v j We don t know v, but we can wite it v = k v 5 k Hence ω v is pependicula to k, so that the -component of a co is eo Tangential acceleation: a = ω tang We don t know, but vibation amplitude is small, so we can wite 5j Let ω = α x i + αyj + αk Then ω = 5α i+ 5α xk Hence, -component of a tang is 5α x T ωxy = T ϕ ω tnb = Hence, -component of a tang is = m/s 3 By coodinate tansfom { } { } Centipetal acceleation: a = ω centip ( ω ) ω = 5ω i+ 5ω xk ω ( ω ) = 5ω xωyi+ 5( ωx + ω ) j+ 5ωy ω k Hence, -component of a centip is 5ω y ω = m/s { ωxy } T ϕ { ωtnb } = = T

The acceleomete eading L4:18 m + Aindicated A pieoelectic acceleomete Wing tip suface The acceleomete esponds to the -component of the absolute acceleation A, and the -component of the gavitational

18 The acceleomete eading L4:18 m + Aindicated A pieoelectic acceleomete Wing tip suface The acceleomete esponds to the -component of the absolute acceleation A, and the -component of the gavitational acceleation g (which egistes as a negative acceleation) g cos β k cos cos g β ϕ Positive acceleation I.e., the 'indicated' acceleation is ( ) cos cos A = ki A g = kia g β ϕ indicated We can see this by inspection, but let's wok it out fomally using coodinate tansfomations

19 Component of gavity along sensitive axis of acceleomete 0 g = ge, i.e., { g θ } = 0 g 0 cos β sin β 0 g sin β β θ = = 0 = 0 0 sinβ cosβ g gcosβ { g } T { g } tnb g sinβ 0 cosϕ sinϕ = 0 = 0 sinϕ cosϕ g cosβ g sin β gxy = gcos βsin ϕ g cos β cos ϕ as expected! { g } T ϕ { g } i.e., xy tnb { } s ρ Z j n b e θ Y X g θ L4:19 ϕ G β t e e

Absolute acc. of P Putting it all togethe! ( ) A= A + a+ ω + ω v + ω ω G Absolute acc. of G y G = ka i = Aindicated + gcos β cos ϕ ki AG + ω + ω v + ω ω = A indicated + 9.81 0.953 0.956 [ 1.300 + 0.

20 Absolute acc. of P Putting it all togethe! ( ) A= A + a+ ω + ω v + ω ω G Absolute acc. of G y G = ka i = Aindicated + gcos β cos ϕ ki AG + ω + ω v + ω ω = A indicated [ ] = A indicated m/s Requied coection. Phew!! x L4:0 Acc. of P seen by. component is vibation we want to measue! Acceleomete P ( ) cos cos A = ki A g = kia g β ϕ indicated Hence, vibation acceleation is ( )

21 Aticles fom Meiam & Kaige (5th edn) elevant to this lectue /8 Relative motion (tanslating axes) 5/1-5/ Intoduction, Rotation concepts of tanslation and otation of igid body; angula velocity 5/4 Relative velocity simplification fo plane motion 5/6 Relative acceleation simplification fo plane motion 5/7 Motion elative to otating axes simplification fo plane motion 7/-7/6 Geneal motion genealisation to thee dimensions L4:1

22 Example poblem: Meiam & Kaige 4 th edn 5/164 5/170 in 5 th edn L4:

23 B is a otating obseve, with angula velocity ω, who sees position, velocity and acceleation of A as, v and a VA = VB + v+ ω ( ) A = A + a+ ω + ω v + ω ω A B L4: V = i = i A =13.33 (m/s) V j B V A = = 60 A j j A A i B (m/s).963 (m/s ) =.963 (m/s ) ( ) = i+ j (m) A A V B j A B V A i ω ( ) 60 1 cos 45 = V B ω k 60 = 0. (ad/s)

24 VA = VB + v+ ω v = VA VB ω v= i j 0. k 9.80( i+ j ) = i j 6.63( j i ) i.e., v = 6.71 i j (m/s) ( ) A = A + a+ ω + ω v + ω ω A B ( ) a= A A ω ω v ω ω A B A A V B j A B V A A A V B j A B V A ( ) ( ) ( ) = + a j i 0 k i j k i j a v i i ω ω L4:4 i.e., a= 4.43 i+ 7.4 j (m/s )

Kinematics in 2-D (II)

Kinematics in 2-D (II) Unifom cicula motion Tangential and adial components of Relative velocity and acceleation a Seway and Jewett 4.4 to 4.6 Pactice Poblems: Chapte 4, Objective Questions 5, 11 Chapte