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# Biomechanics BIOEN 5201

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This 6 page Class Notes was uploaded by Kaylee Schowalter on Monday October 26, 2015. The Class Notes belongs to BIOEN 5201 at University of Utah taught by J. Weiss in Fall. Since its upload, it has received 50 views. For similar materials see /class/229915/bioen-5201-university-of-utah in Bioengineering at University of Utah.

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Date Created: 10/26/15

Homework 1 HW1 Assigned 82511 BIOEN 5201 Fall 2011 Due 9 8 1 1 BEGINNING OF CLASS PLEASE BE SURE TO FOLLOW THE GUIDELLTNES FOR PREPARING HOMEWORK Estimated time to complete 46 hours 1 Background 5 points What assumptions are made in the application of continuum mechanics Why can we apply continuum mechanics as a mathematical model for biomechanics They could mention that continuum mechanics is concerned with materials that are homogenous on a macroscopic scale They should also discuss the concept of a continuum in which density displacement velocity etc are continuous and de ned at every point Continuum mechanics is appropriate for biomechanics because the characteristic scales at which most experiments are performed greatly exceed the dimensions of the tissue microstructure 2 Background 7 points I Listed below are some common measurements utilized by engineers For each measurement state whether that quantity is expressed as a scalar or a vector 1 point each a Position vector b Acceleration vector c Pressure scalar d Temperature scalar e Force vector II In your own words describe how a vector is different than a scalar 2 points This could be phrased several ways but essentially a physical quantity that can be described as a real number alone is a scalar while a physical quantity that can is described by both a number and a direction is a vector 3 Linear algebra 24 points 3 points each Linear algebra is a fundamental component of continuum mechanics and most engineering disciplines Solve the following matrix and vector problems showing all steps As it is always a good idea to be able to find and correct your own mistakes you are encouraged to use MATLAB Maple Mathematica etc to check your answers but you still need to show all steps 2 7 a Determine ifthe vectors u 3 and v 2 2 10 are perpendicular to each other To determine if the two vectors are perpendicular use the vector alot product 11 I V 27322 10 14 6 20 0 Since the dot product of the two vectors is 0 the vectors are indeed perpendicular Fquot 4 5 A plane is described by the vectors u 6 and v Find a vector that is normal to this plane 2 9 When you take the vector cross product of two vectors that are not parallel the result is a vector that is normal to both of the original vectors This is frequently used to define orthogonal coordinate systems O F D E E 6 u x v 41 26 23 69 1mm 009 50352 001 tax 6mg 5 1 9 56 11 X v 26 34 Please note that if you took the cross product of v X u you would still nd a vector normal to the plane still a correct answer as the problem is written but it wouldjust be pointing the opposite direction 56 26 vgtltu 34 find the outertensor product 8 4 With the following vectors u 2 and v 0 1 1 11 X v Just as in the case of the cross product the order of the multiplication makes all the difference For example if you took the outer product of V 8 u the result would be the transpose of the outer product requested in the problem statement That would not be a correct answer in this case Anyway moving on 8 8X4 80 81 32 0 8 u vH 4 0 1 2x4 20 2 1 8 0 2 1 DH 10 11 4 0 1 Say that you want to rotate a vector by 15 in this case you are simply de ning the same vector in a different coordinate system You can do this by multiplying the vector by a special matrix aptly named the rotation matrix Use the rotation matrix M to find X39 by rotating the vector X by 639 15 Note The prime on the X39 indicates a different coordinate system not the Matlab transpose 1 c056 sin6 0 X 3 M sing c056 0 X MX 0 0 0 1 c056 sin9 0 1 0966 0259 0 1 X sinQ c056 0 3 0259 0966 0 3 0 0 1 0 0 0 1 0 09661 02593 0189 x 02591 09663 3157 0 0 Prove that I am not lying to you about the rotation matrix Use the innerscalaralot product to confirm that the angle between X and X39 inpart dis 15 X X XX c056 10189 33157 00 J12 32 02 J01892 31572 02 cos 9 966 316316 cos 9 c056 0967 96 3162 9 cosquot1 0967 147 z 15 f What is the trace of the matrix B Give the de nition of the trace operation in index notation 311 312 313 321 322 323 B31 B32 B33 13 To determine the trace just sum the elements along the diagonal To receive full credit students must give the operation in index notation If they don t take 2 points off ME 5ijBij 2 Bit 2 311 322 333 g Is the matrix K invertible 4 4 3 6 K 1 2 9 3 14 2 7 For a matrix to be invertible the determinant of the matrix must be nonzero detK 439 I124 29739 439 29739 36 12439 4 4 36 1 2 9 14 27 4227 149 4127 3936114 32 472 4X0 368 0 The matrix is not invertible because the determinant is zero h Find the product of these two matrices 1 3 2 8 1 2 4 7 5 6 3 7 0 10 6 5 2 10 1 3 2 8 1 2 183625 113322 12372 10 4 7 5 6 3 7 487655 417352 42775 10 0 10 6 5 2 10 0810665 0110362 021076 10 36 14 3 99 35 7 90 42 10 4 Index notation 7 30 points 9 points each except Part b is only 3 points Show all of your work a Index notation also known as summation convention or Einstein summation convention is just a shorthand notation to simplify the writing of systems of equations Perform the following i Write the following system of equations in matrix form 8 4b2 6 X1 2 20 8 4122 6 X1 22 8 0 0 g s 0 1 cos9 0 x3 4 0 1 c0509 0 X3 4 ii Write the system of equations in index notation Mijxj bi J i Mijxj yi respectively iii How many free indices does the system have in index notation How many dummy indices What is the range of each of the indices There is one free index in the system 139 There is one dummy index in the system j Each of the indices range from 1 to 3 Obviously the actual letters they use for the indices don t matter but the answer should be something like this 8261 4492262 6x3 2 20 8261 4x3 2 361 cos6x2 4 0 b How many equations does the following index notation equality represent Label the free and dummy indices What is the range or numerical values of the indices aij nkCka There are 9 equations i and j are the free indices k is the dummy index The numerical range is 13 c The Kronecker Delta symbol 5g aids in simplifying systems of equations Expand the following equality into a system of nine equations Sij PRU 5iijm 511 PRii 17111 T22 T33 512 PRiz 513 PR13 522 PRzz 17111 T22 T33 521 PR21 523 PR23 533 PR33 17111 T22 T33 531 PR31 532 PR32 d What linear algebra operation does the following equation represent Expand the equation using the nonzero components of the permutation symbol EUk Ci sijkujvk This operation is the vector cross product Cl 5123u2v3 5132713 2 0 2 3 3v2 CZ 5231u3v1 5213u1U3 0 31 1U3 C3 5312u1v2 532171215 0 1 U2 u2v1 2v3 3U2 c u3v1 u1v3 ivz u2v1 5 Expansion and reduction in index notation Kronecker delta permutation symbol 7 16 points Showing all steps show that 4 points each a 6iijnuin mn39LL39Uj The only permutations of the Kronecer delta 6 that are nonzero are when the indices are equal ie 13939 Therefore you can see SHu uj Remember since it is index notation the order the vectorstensors are multiplied in does not matter Fquot Ameqn6mk6qk Amemn This is the same process as in part a but it is simply applied twice 6mk6qk3qn 61111131111 317111 OR Apm mk Apk amp Bqn6qk Bkn Remember the choice of individual letters does not matter so ApkBkn Amemn c 6mksmnk 0 Kronecker delta 6mk is only nonzero when mk Permutation symbol smnk is only nonzero when m at n at k Thus all permutations of this expression would result in a zero product 1 6ii6mm 9 Remember when the indices are the same it indicates the trace of the matrix Since the Kronecker is essentially a matrix of 0 1 0 the result for each term is 0 0 1 Thus3x39 6 Tensors 7 6 points What is the definition of a tensor Give one example each of quantities measured as zero order tensor lSL order tensor 2quotd order tensor A tensor is a mathematical object that is independent of any particular coordinate system This can be expressed mathematically as a linear combination of unit dyads that is to say the result of the outer product of two unit vectors where each term has been multiplied by a coefficient gives a 2quotd order tensor This expression is shown below but the students do not need to state it to receive full credit They only need to state what is essentially expressed in the first sentence of this paragraph A Aijei ej A zero order tensor is also known as a scalar in a Cartesian coordinate system Examples include but are not limited to mass pressure temperature etc This value is obviously completely independent of the coordinate system A 15 order tensor is also called a vector Examples include but are not limited to velocity temperature gradient force etc A 2ml order tensor in a Cartesian coordinate system would be referred to as a 2 dimensional matrix in linear algebra Examples include but are not limited to stress strain etc lst and 2nd order tensors are also independent of coordinate system but the actual value of the numbers they contain can change when switching from one coordinate system to another 7 Direct notation 7 16 points 4 points each 2 for the expression in indem direct notation 2 for the order of the result 5 Rewrite b and c in direct notation and rewrite a and d in index notation State whether the result of evaluation of the equation represents a 2nd order tensor component a lst order tensor component or a scalar a A u b Ajivaljum C uinAmm d m v u v Choice of index does not matter as long as dummies appear only twice and free appear only once While in index notation terms can be moved around as long as the order of indices within a term is not changed Also there may be alternative ways of representing these quantities It is up to the student to competently support any alternative formulations if points are to be argued a A I u lt2 Aijuj lSt order tensor vector b Ajivaljum ltgt A BTv u Scalar c uivjAmm ltgt trAu v 2quotd order tensor d m v u v lt gt ujmivkvj 2quotd order tensor 9 Coordinate transformation 7 10 points Coordinate system ex is transformed into coordinate system e by a 320 counterclockwise rotation Write the rotation matrix M that maps every point in coordinate system e1 to its new position inel39 e 3 1 0 0 1 0 0 e1 0 e2 1 e3 0 e 1 0 e 1 cos 32 e 1 sin 32 0 0 1 0 sin 32 cos 32 2 1 61 2 1 62 8 1 e3 1 0 0 Mi 2 e l I e 2 9392 I el e z I ez 9392 I e3 0 COS SlIl e 3 61 e 3 e2 e 3 e3 0 sin 32 cos 32

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