3D Rotational Analysis

Transcription

1 3D Rotatioal Aalysis by Jesús Dapea Departmet of Kiesiology Idiaa Uiversity Bloomigto, IN USA Notes for a tutorial give at the 19 th Cogress of the Iteratioal Society of Biomechaics, Duedi, New Zealad, Tel dapea@idiaa.edu Web:

2 1 2D traslatio = easy. 3D traslatio = easy to make the trasitio, just add oe more dimesio. 2D rotatio = harder, probably because (1) momet of iertia is variable; (2) vectors are less ituitive tha i traslatio. 3D rotatio = the hardest, because it s ot such a simple trasitio from 2D as it is i traslatio.

3 2 Most iterestig parameters: Traslatio: 0 th derivative level = liear locatio 1 st derivative level = liear velocity, liear mometum 2 d derivative level = liear acceleratio, force Rotatio: 0 th derivative level = agular locatio 1 st derivative level = agular velocity, agular mometum 2 d derivative level = agular acceleratio, torque

4 3 Agular mometum: The equatio H = I ω is ot always valid. It s oly good for a rigid object rotatig i 2D. Let s cosider a very geeral situatio: a amoeba. For a sigle particle i, the agular mometum H Oi about a referece poit O ca always be computed correctly by: H Oi = m i (r i " v i ) H Oi = m i (r i " dr i dt ) H Oi = m i (r i " dr i ) dt

5 4 Graphical cocept of cross product Cross product of two vectors: a b = c Magitude of the cross product: c = a b si θ The magitude of the cross product is equal to two times the area of the triagle formed by the two vectors: This holds whether the two vectors are draw with their tails together or with oe vector startig at the tip of the other:

6 5 Back to agular mometum... Goig back to our amoeba, we ca ow see that the agular mometum of the particle of material is directly proportioal to the area swept per uit of time by the particle relative to the referece poit: H Oi = 2 m i amout of area swept per secod This allows us to compare graphically the amouts of agular mometum of differet particles:

7 6 To calculate the agular mometum of the whole system (the whole amoeba!), we would eed to use the equatio: # H O = m i (r i " v i ) A problem with usig this equatio is that it would require us to compute the agular mometum values of a ifiite umber of idividual particles. So we are lucky that we do t study amoebas! The huma body ca be cosidered a series of rigid segmets that ca rotate relative to each other at joits. So the huma body is somethig itermediate betwee a sigle rigid block ad a amoeba. We will deal with it later.

8 7 For ow, we will cotiue developig our equatios, to obtai a expressio for the agular mometum of a rigid object. If all the poits of the amoeba were oly able to move i circles about poit O, we would have: H O = # # m i r i " v i = m i r i " i " r i ad thus, # H O = m i r i 2" i If all particles also have the same agular velocity vector, we have that: # H O = ( m i r i 2)" H O = I ω This is the traditioal formula for agular mometum. We have see that it ca oly be applied to a rigid object rotatig i 2D.

9 8 Agular mometum eeds to be calculated about a give poit Agular mometum eeds to be calculated about either: (a) a iertial poit, i.e., a poit that is static or that traslates at costat liear velocity or (b) the ceter of mass of the system. This is valid eve if the ceter of mass does ot have costat liear velocity. Usually, (b) is the best choice.

10 9 Calculatio of the agular mometum of the huma body about its ow ceter of mass The huma body is cosidered to be composed of rigid segmets that rotate relative to each other. Each segmet has two agular mometum elemets: remote agular mometum: associated with the motio of the segmet s c.m. about the c.m. of the whole body local agular mometum: associated with the motio of the segmet about its ow c.m. remote agular mometum = shak mass 2 area swept per secod (i.e. like amoeba particle) local agular mometum = shak momet of iertia about ow c.m. shak agular velocity (i.e. like rigid object)

11 10 The local terms of agular mometum are usually much smaller tha the remote terms. So if you igore the local terms, you will have some error but ot very much. This is very useful for qualitative aalysis, because the remote terms ca be used to make estimates of the agular mometum of each body segmet: I the tutorial, we will look at the agular mometum of each segmet durig the airbore motio of a hitch-kick style ( bicyclig ) log jump. I 2D aalysis, clockwise areas idicate a agular mometum vector poitig ito the blackboard; couterclockwise areas idicate a agular mometum vector poitig out from the blackboard toward us.

12 11 3D Rotatio Qualitative aalysis of agular mometum There are two possible ways to make a qualitative aalysis of agular mometum i a 3D motio: (1) Try to make the aalysis directly i 3D. This ivolves visualizig 3D vectors i a multitude of directios. It s difficult to do, ad geerally ot a good idea. (2) Look at the 3D motio as three separate projected 2D motios, by viewig i successio from three differet directios perpedicular to each other. Each of these orthogoal directios will give us iformatio about the agular mometum vector compoet i the directio alog the lie of sight.

13 12 Quatitative aalysis of agular mometum Those are the methods that we would use for a ituitive, qualitative approach. To make accurate calculatios with a computer, we would use other methods, as we will see ext. Calculatig the remote terms of agular mometum is very simple, both i 2D ad 3D aalysis. The formula: # H O = m i (r i " v i ) would be applied to the liear locatio ad liear velocity of the ceter of mass of each segmet relative to the whole body ceter of mass. I a true 2D motio, calculatig the local terms of agular mometum would also be quite simple. We would just eed to multiply each segmet s cetroidal momet of iertia by the segmet s agular velocity about its ow ceter of mass. This is the simple formula H = I ω. However, i 3D motio there is a problem for the calculatio of the local terms of agular mometum. The formula H = I ω does ot work i 3D motio. We will see why.

14 13 Both objects i the drawig above are rotatig about a vertical axis, ad the agular velocity vector ω is the same for both. I the view from the positive Z directio, the particles that costitute the two objects sweep the same amouts of couterclockwise areas, ad i the view alog the X axis o areas are swept at all i either case at the istat show i the drawig. However, i the view alog the Y axis the object o the left is sweepig o area, while the object o the right is sweepig couterclockwise areas whe viewed from the egative Y directio: This implies that the object o the left has oly a positive Z compoet of agular mometum, while the object o the right has a positive Z compoet ad a egative Y compoet of agular mometum: Therefore, i the object o the right the agular velocity ad agular mometum vectors do ot poit i the same directio. Cosequetly, H I ω.

15 14 It is clear that the problem stems from the fact that the mass o oe side of the agular velocity vector is higher up alog the vector tha the mass o the other side: This asymmetry is what makes the agular mometum be misaliged with the agular velocity. The agular mometum ad agular velocity vectors would be aliged if the masses o either side were set more evely: Physicists have foud a way to quatify this asymmetry, ad to use it to calculate correct values for the agular mometum of a rigid object i 3D.

16 15 Momets of iertia: 2 I XX = " dm(y i + Z 2 2 i ) I YY = " dm(x i + Z 2 i ) 2 I ZZ = " dm(x i + Y 2 i ) Products of iertia: I XY = " dm(x i # Y i ) I XZ = " dm(x i # Z i ) I YX = " dm(y i # X i ) I YZ = " dm(y i # Z i ) I ZX = " dm(z i # X i ) I ZY = " dm(z i # Y i ) (NOTE: I some books there are o mius sigs before the sigmas i the defiitios of the products of iertia. Such a defiitio is valid too, but it requires chages also i other equatios. Be sure ot to mix the two alterative covetios!)

17 16 Let s look at the I XZ product of iertia: We see that the products of iertia give us good iformatio about the asymmetry of the object.

18 17 Physicists like to show the momets of iertia ad the products of iertia together i a ordered array: " I XX I XY I XZ & # I YX I YY I YZ ' % I ZX I ZY I ZZ ( This is called the tesor of iertia. It s ofte desigated simply as { I}. The formula to calculate the agular mometum of a rigid object usig the momets ad products of iertia is the followig: H = I XX ω X + I XY ω Y + I XZ ω Z + I YX ω X + I YY ω Y + I YZ ω Z + I ZX ω X + I ZY ω Y + I ZZ ω Z where ω X, ω Y ad ω Z are the three compoets of the ω vector. This ca be expressed as the product of two matrices: " I XX I XY I XZ & " H = # I YX I YY I YZ ' ) # % I ZX I ZY I ZZ ( % * X * Y * Z & ' ( where # % % & " X " Y " Z ' % ( % ) is the agular velocity vector, which ca be desigated simply as {"}.

19 18 So i the ed we have that H = {I} {ω}, where {I} ad {ω} are actually matrices. Note that this is ot the same as H = I ω, which would ot produce the correct results, because it is missig the terms ivolvig the products of iertia. Havig said all this, it IS possible to calculate the agular mometum of a rigid object i 3D without usig the products of iertia! But before we lear how to do this we eed to defie the cocept of pricipal axes.

20 Pricipal axes 19

21 20 I objects that have symmetry, the pricipal axes are easy to fid. For istace, i the huma truk: We ca use pricipal axes to avoid havig to use the products of iertia.

22 21 Calculatio of local agular mometum i 3D without usig products of iertia Let s cosider a cylider with elliptical cross-sectio to represet the huma truk. The XYZ axes show here are ot pricipal axes. They are parallel to the axes of the referece frame attached to the groud. The stadard equatio to calculate agular mometum is: " I XX I XY I XZ & " H = # I YX I YY I YZ ' ) # % I ZX I ZY I ZZ ( % * X * Y * Z & ' ( We will ow show a temporary o-rotatig referece frame abc such that axes a, b ad c happe to coicide istataeously with the pricipal axes of the cylider:

23 22 The agular velocity of the cylider will first be calculated i terms of the XYZ referece frame attached to the groud, usig stadard film aalysis or video aalysis procedures. The, we ca choose to express the agular velocity of the cylider i terms of the abc referece frame,. If we do this, we will have: " I aa I ab I ac & " H = # I ba I bb I bc ' ) # % I ca I cb I cc ( % * a * b * c & ' ( But, sice axes a, b ad c are pricipal axes, the products of iertia are zero. Therefore: " I aa 0 0 & " H = # 0 I bb 0 ' ) # % 0 0 I cc ( % * a * b * c & ' ( = I aa ω a + I bb ω b + I cc ω c For istace, if: I aa = 1.4 I bb = 1.6 I cc = 0.5 ad if: ω a = 27 u a ω b = 31 u b ω c = 45 u c where u a, u b ad u c are the uit vectors of the abc referece frame, we would have: H = (1.4 27) u a + (1.6 31) u b + (0.5 45) u c = 37.8 u a u b u c Kg m 2 /s

24 23 So that would be the agular mometum value that we were lookig for. But there is a slight problem: This agular mometum vector is expressed i terms of the abc referece frame istead of the XYZ referece frame, which is icoveiet. But this problem is actually very easy to solve. Remember that the agular velocity vector was iitially expressed i terms of the XYZ referece frame. Whe we project this vector oto the a, b ad c axes, the ω a, ω b ad ω c vectors that we get will each be expressed still i terms of the XYZ referece frame. For istace, we could have: ω a = 27 u a = 19.2 u X u Y u Z ω b = 31 u b = u X u Y u Z ω c = 45 u c = u X 6.4 u Y u Z (for example!) (for example!) (for example!) where u X, u Y ad u Z are the uit vectors of the XYZ referece frame I such case, the value of the agular mometum vector would be: H = 1.4 (19.2 u X u Y u Z ) (-19.7 u X u Y u Z ) (-13.7 u X 6.4 u Y u Z ) = u X u Y u Z Kg m 2 /s So, i effect, we are usig H = I aa ω a + I bb ω b + I cc ω c, but with ω a, ω b ad ω c expressed i terms of the XYZ referece frame, which is a trivial thig to do, as we have see.

25 24 That is how we would calculate the local agular mometum of the truk about its ow c.m. The other segmets are usually cosidered cyliders of circular (istead of elliptical) cross-sectio. So for them the logitudial axis is clearly defied, but there are ifiite possible orietatios for the other two axes. Ay two axes will do, as log as they are both perpedicular to the logitudial axis ad perpedicular to each other. This ambiguity allows us to calculate the local terms of agular mometum of these segmets with eve greater ease tha for the truk. We will pick the a or the b axis to be i the plae defied by ω ad by the logitudial axis c. For istace, let s assume that we choose to make the a axis be i that plae: I this case, ω b will be zero. Therefore: H = I aa ω a + I cc ω c where ω a ad ω c will be expressed i terms of the XYZ referece frame.

26 25 Agular mometum of the whole body: H = 14 " m i (r i v i ) remote terms + 13 " I LONGi ω TWi + I TRANSVi ω SOMi o-truk local terms + I LONG14 ω TW14 + I ML14 ω SOM-ML14 + I AP14 ω SOM-AP14 truk local term

27 26 Refereces For the represetatio of agular mometum by areas swept, see: Hopper, Berard J. The mechaics of huma movemet. America Elsevier, This excellet book is out of prit, but you may fid it at a library. Or you ca purchase a used copy through Iteret. For istace, try at: or at: For the computatio of 3D agular mometum, see: Dapea, J. A method to determie the agular mometum of a huma body about three orthogoal axes passig through its ceter of gravity. Joural of Biomechaics 11: , 1978.