Dynamics of Mechanical Systems 2009 Part 2 potx

Example 2.7.1: Vector Product Computation and Geometric Properties of the Vector ProductLet vectors A and B be expressed in terms of mutually perpendicular dextral unit vectors b.. That

Returning to the product A × D, let A and D be depicted as in Figure 2.7.5a, where θ

is the angle between the vectors As before, let nA be a unit vector parallel to A Then,

nA× D is a vector perpendicular to nA and with magnitude Dsinθ From Figure

2.7.5b, we see that:

(2.7.18)

By similar reasoning we have:

(2.7.19)Therefore, by comparing Eqs (2.7.17) and (2.7.19), we have:

(2.7.20)

This establishes the distributive law.

Finally, suppose that n1, n2, and n3 are mutually perpendicular unit vectors, and suppose

that vectors A and B are expressed in the forms:

1 3

1 3

1 3

Example 2.7.1: Vector Product Computation and Geometric Properties of the Vector Product

Let vectors A and B be expressed in terms of mutually perpendicular dextral unit vectors

b If C is perpendicular to A, with the angle θ between C and A being 90˚, C • A

is zero because cosθ is zero Conversely, if C • A is zero, and neither C nor A is

zero, then cosθ is zero, making C perpendicular to A From Eq (2.6.21), C • A is:

which is seen to be from Eq (2.7.25)

On many occasions it is necessary to consider the product of three vectors Such products are called “triple” products Two triple products that will be helpful to use are the scalar triple product and the vector triple product.

Given three vectors A, B, and C, the scalar triple product has the form A · (B × C) The

result is a scalar The scalar triple product is seen to be a combination of a vector productand a scalar product

Recall from Eqs (2.6.21) and (2.7.23) that if n1, n2, and n3 are mutually perpendicularunit vectors, the algorithms for evaluating the scalar and vector products are:

(2.8.1)and

(2.8.2)

where the A i and B i (i = 1, 2, 3) are the n i components of A and B.

By comparing Eqs (2.8.1) and (2.8.2), we see that the scalar triple product A · (B × C) may be obtained by replacing n1, n2, and n3 in:

(2.8.3)

Recall from the elementary rules of evaluating determinants that the rows and columnsmay be interchanged without changing the value of the determinant Also, the rows andcolumns may be cyclically permuted without changing the determinant value If the rows

or columns are anticyclically permuted, the value of the determinant changes sign Hence,

we can rewrite Eq (2.8.3) in the forms:

(2.8.4)

By examining Eq (2.8.4), we see that the dot and the cross may be interchanged in theproduct Also, the parentheses are unnecessary Finally, the vectors may be cyclicallypermuted without changing the value of the product An anticyclic permutation changesthe sign of the result Specifically,

If the vectors A, B, and C coincide with the edges of a parallelepiped as in Figure 2.8.1,

the scalar triple product of A, B, and C is seen to be equal to the volume of the

parallel-epiped That is, the volume V is:

Verify the equalities of Eq (2.8.5)

Solution: From Eq (2.8.2), the vector products A × B and B × C are:

(2.8.8)

(2.8.9)

Then, from Eq (2.6.22), the triple scalar products A × B · C and B × C · A are:

(2.8.10)and

(2.8.11)

The other equalities are verified similarly (see Problem P2.8.1)

Next, the vector triple product has one of the two forms: A × (B × C) or (A × B) × C.

The result is a vector The position of the parentheses is important, as the two formsgenerally produce different results

To explore this, let the vectors A, B, and C be expressed in terms of mutually dicular unit vectors n with scalar components A , B , and C (i = 1, 2, 3) Then, by using

perpen-FIGURE 2.8.1

A parallelepiped with vectors A, B,

and C along the edges.

the algorithms of Eqs (2.8.1) and (2.8.2), we see that the vector triple products may beexpressed as:

(2.8.12)and

(2.8.13)Observe that the last terms in these expressions are different

Example 2.8.2: Validity of Eqs (2.8.6) and (2.8.7) and the Necessity of Parentheses

Verify Eqs (2.8.6) and (2.8.7) using the vectors of Eq (2.8.7)

Solution: From Eqs (2.8.2) and (2.8.9), A × (B × C) is:

Observe that the results in Eqs (2.8.14) and (2.8.15) are identical and thus consistentwith Eq (2.8.12) Similarly, the results of Eqs (2.8.16) and (2.8.17) are the same, thusverifying Eq (2.8.13) Finally, observe that the results of Eqs (2.8.14) and (2.8.16) aredifferent, thus demonstrating the necessity for parentheses on the left sides of Eqs (2.8.12)and (2.8.13).

Recall from Eq (2.6.10) that the projection A of a vector A along a line L is:

(2.8.18)

where n is a unit vector parallel to L, as in Figure 2.8.2 The vector triple product may be

used to obtain A⊥, the component of perpendicular to L That is,

(2.8.19)

To see this, use Eqs (2.8.6) and (2.8.8) to expand the product That is,

(2.8.20)

Observe in the previous sections that expressing a vector in terms of mutually ular unit vectors results in a sum of products of the scalar components and the unit vectors

perpendic-Specifically, if v is any vector and if n1, n2, and n3 are mutually perpendicular unit vectors,

we can express v in the form:

(2.9.1)

Because these sums occur so frequently, and because the pattern of the indices is similarfor sums of products, it is convenient to introduce the “summation convention”: if anindex is repeated, there is a sum over the range of the index (usually 1 to 3) This means,

for example, in Eq (2.9.1), that the summation sign may be deleted That is, v may be

V= vn

In using this convention, several rules are useful First, in an equation or expression, arepeated index may not be repeated more than one time Second, any letter may be usedfor a repeated index For example, in Eq (2.9.2) we may write:

(2.9.3)

Finally, if an index is not repeated it is a “free” index In an equation, there must be acorrespondence of free indices on both sides of the equation and in each term of theequation

In using the summation convention, we can express the scalar and vector products as

follows: If n1, n2, and n3 are mutually perpendicular unit vectors and if vectors a and b

are expressed as:

(2.9.4)

then the products a · b and a × b are:

(2.9.5)

where, as before, e ijk is the permutation symbol (see Eq (2.7.7))

With a little practice, the summation convention becomes natural in analysis procedures

We will employ it when it is convenient

Example 2.9.1: Kronecker Delta Function Interpreted

The Kronecker delta function may then be interpreted as an index operator, substituting

an i for the j, thus the name substitution symbol.

2.10 Review of Matrix Procedures

In continuing our review of vector algebra, it is helpful to recall the elementary procedures

in matrix algebra For illustration purposes, we will focus our attention primarily on square

matrices and on row and column arrays Recall that a matrix A is simply an array of

numbers a ij (i = 1,…, m i ; j = 1,…, n) arranged in m rows and n columns as:

(2.10.1)

The entries a ij are usually called the elements of the matrix The first subscript indicates

the row position, and the second subscript designates the column position

Two matrices A and B are said to be equal if they have equal elements That is,

(2.10.2)

If all the elements of matrix are zero, it is called a zero matrix If a matrix has only one row, it is called a row matrix or row array If a matrix has only one column, it is called a column matrix or column array A matrix with an equal number of rows and columns is a square matrix If all the elements of a square matrix are zero except for the diagonal elements, the matrix is called a diagonal matrix If all the diagonal elements of a diagonal matrix have the value 1, the matrix is called an identity matrix If a square matrix has a zero determinant, it is said to be a singular matrix The transpose of a matrix A (written AT)

is the matrix obtained by interchanging the rows and columns of A If a matrix and its

transpose are equal, the matrix is said to be symmetric If a matrix is equal to the negative

of its transpose, it is said to be antisymmetric.

Recall that the algebra of matrices is based upon a few simple rules: First, the

multipli-cation of a matrix A by a scalar s produces a matrix B whose elements are equal to the elements of A multiplied by s That is,

(2.10.3)

Next, the sum of two matrices A and B is a matrix C whose elements are equal to thesum of the respective elements of A and B That is,

(2.10.4)

Matrix subtraction is defined similarly

The product of matrices is defined through the “row–column” product algorithm Theproduct of two matrices A and B (written AB) is possible only if the number of columns

in the first matrix A is equal to the number of rows of the second matrix B When this

occurs, the matrices are said to be conformable If C is the product AB, then the element c ij

is the sum of products of the elements of the ith row of A with the corresponding elements

of the jth column of B Specifically,

Matrix products are distributive over addition and subtraction That is,

(2.10.6)and

(2.10.7)

Matrix products are also associative That is, for conformable matrices A, B, and C, wehave:

(2.10.8)

Hence, the parentheses are unnecessary

Next, it is readily seen that the transpose of a product is the product of the transposes

in reverse order That is:

(2.10.11) Then A–1 is the matrix with elements:

then A is said to be orthogonal In this case, the rows and columns of A may be considered

as components of mutually perpendicular unit vectors

2.11 Reference Frames and Unit Vector Sets

In the analysis of dynamical systems, it is frequently useful to express kinematical anddynamical quantities in several different reference frames Orthogonal transformationmatrices (as discussed in this section) are useful in obtaining relationships between therepresentations of the quantities in the different reference frames

To explore these ideas consider two unit vector sets ni and i and an arbitrary V as in

Figure 2.11.1 Let the sets be inclined relative to each other as shown Recall from

Eq (2.6.19) that V may be expressed in terms of the ni as:

(2.11.1)

where the Vi are the scalar components of V Similarly, V may be expressed in terms of

the i as:

(2.11.2)

Given the relative inclination of the unit vector sets, our objective is to obtain relations

between the Vi and the i To that end, let S be a matrix with elements S ij defined as:

Similarly, if we express 1, 2, and 3 in terms of the i, we have:

(2.11.7)

Observe the difference between Eqs (2.11.6) and (2.11.7): in Eq (2.11.6), the sum is taken

over the second index of S ij , whereas in Eq (2.11.7) it is taken over the first index This is

consistent with Eq (2.11.3), where we see that the first index is associated with the ni andthe second with the i Observe the same pattern in Eqs (2.11.6) and (2.11.7)

By substituting from Eqs (2.11.6) and (2.11.7) into Eqs (2.11.1) and (2.11.2), we obtain:

(2.11.8)and

(2.11.9)Hence, we have:

(2.11.10)

Observe that Eq (2.11.10) has the same form as Eqs (2.11.6) and (2.11.7)

By substituting from Eqs (2.11.6) and (2.11.7), we obtain the expression:

To illustrate these ideas, imagine the unit vector sets ni and i to be aligned with each

other such that ni and i are parallel (i = 1, 2, 3) Next, let the i be rotated relative to the

ni and i so that the angle between 2 and n2 (and also between 3 and n3) is α, as shown

in Figure 2.11.2 Then, by inspection of the figure, ni and i are related by the expressions:

ˆ ˆ

ˆˆˆ

where sα and cα represent sinα and cosα Hence, from Eq (2.11.3), the matrix S has the

(In this context, the transformation matrix of Eq (2.11.15) might be called A.)

Finally, let the i have a general inclination relative to the ni as shown in Figure 2.11.1.The may be brought into this configuration by initially aligning them with the n and

00

00

00

and

c

ˆn ˆn

then by performing three successive dextral rotations of the i about 1, 2, and 3 throughthe angles α, β, and γ The transformation matrix S between the ni and the i is, then:

(2.11.21)

2.12 Closure

The foregoing discussion is intended primarily as a review of basic concepts of vector andmatrix algebra Readers who are either unfamiliar with these concepts or who want toexplore the concepts in greater depth may want to consult a mathematics or vectormechanics text as provided in the References We will freely employ these conceptsthroughout this text

2.4 Shields, P C., Elementary Linear Algebra, Worth Publishers, New York, 1968.

2.5 Ayers, F., Theory and Problems of Matrices, Schawn Outline Series, McGraw-Hill, New York,

1962.

2.6 Pettofrezzo, A J., Elements of Linear Algebra, Prentice Hall, Englewood Cliffs, NJ, 1970 2.7 Usamani, R A., Applied Linear Algebra, Marcel Dekker, New York, 1987.

2.8 Borisenko, A I., and Tarapov, I E., Vector and Tensor Analysis with Applications (translated by

R A Silverman), Prentice Hall, Englewood Cliffs, NJ, 1968.

0

0 1 00

00

Section 2.3 Vector Addition

P2.3.1: Given the vectors A, B, C, and D as shown in Figure P2.3.1, let the magnitudes of

these vectors be:

Section 2.4 Vector Components

P2.4.1: Given the unit vectors i and j and the vectors A and B as shown in Figure P2.4.1, let the magnitudes of A and B be 10 N and 15 N, respectively Express A and B in terms

of i and j.

P2.4.2: See Problem P2.4.1 Let C be the resultant (or sum) of A and B.

a Express C in terms of the unit vectors i and j.

b Find the magnitude of C.

P2.4.3: Given the unit vectors i and j and the vectors A and B as shown in Figure P2.4.3, let the magnitudes of A and B be 15 lb and 26 lb, respectively Express A and B in terms

of i and j.

P2.4.4: See Problem 2.4.3 Let C be the resultant of A and B.

a Express C in terms of i and j.

b Find the magnitude of C.

P2.4.5: Given the Cartesian coordinate system as shown in Figure P2.4.5, let A, B, and C

be points in space as shown Let nx, ny, and nz be unit vectors parallel to X, Y, and Z.

a Express the position vectors AB, BC, and CA in terms of nx, ny, and nz

b Find unit vectors parallel to AB, BC, and CA.

P2.4.6: See Problem P2.4.5 Let R be the resultant of AB, BC, and CA Express R in terms

P2.4.7: See Figure P2.4.7 Forces F1,…, F6 act along the edges and diagonals of a

parallel-epiped (or box) as shown Let i, j, and k be mutually perpendicular unit vectors parallel

to the edges of the box Let the box be 12 inches long, 3 inches high, and 4 inches deep

Let the magnitudes of F1,…, F6 be:

a Find unit vectors n1,…, n6 parallel to the forces F1,…, F6 Express these unit vectors

in terms of i, j, and k.

b Express F1,…, F6 in terms of unit vectors n1,…, n6

c Express F1,…, F6 in terms of i, j, and k.

d Find the resultant R of F1 Express R in terms of i, j, and k.

e Find the magnitude of R.

P2.4.8: Let forces be given as:

where n1, n2, and n3 are mutually perpendicular unit vectors

a Find a force F such that the resultant of F and the forces F1,…, F6 is zero Express

F in terms of n1, n2, and n3

b Determine the magnitude of F.

P2.4.9: See Figure P2.4.9 Let P and Q be points of a Cartesian coordinate system with

coordinates as shown Let nx, ny, and nz be unit vectors parallel to the coordinate axes asshown

a Construct the position vectors OP, OQ, and PQ and express the results in terms

of nx, ny, and nz

b Determine the distance between P and Q if the coordinates are expressed in

meters

c Find the angles between OP and the X-, Y-, and Z-axes.

d Find the angles between OQ and the X-, Y-, and Z-axes.

e Find the angles between PQ and the X-, Y-, and Z-axes.

Section 2.5 Angle Between Two Vectors

P2.5.1: From the definition in Section 2.5, what is the angle between two parallel vectorswith the same sense? What is the angle if the vectors have opposite sense? What is theangle between a vector and itself?

Section 2.6 Vector Multiplication: Scalar Product

P.2.6.1: Consider the vectors A and B shown in Figure P2.6.1 Let the magnitudes of A and B be 8 and 5, respectively Evaluate the scalar product A · B.

P2.6.2: See Problem P2.6.1 and Figure P2.6.1 Express A and B in terms of the unit vectors

n1 and n2 as shown in the figure Use Eq (2.6.2) to evaluate A · B.

P2.6.3: See Problems P2.6.1 and P2.6.2 Let C be the resultant of A and B Find the magnitude of C.

P2.6.4: Let n1, n2, and n3 be mutually perpendicular unit vectors Let vectors A and B be expressed in terms of n1, n2, and n3 as:

2

1

A= −3n +5n +6n and B=4n −2n +7n

a Determine the magnitudes of A and B.

b Find the angle between A and B.

P2.6.5: Let vectors A and B be expressed in terms of mutually perpendicular unit vectors as:

Find B y so that the angle between A and B is 90˚.

P2.6.6: See Figure P2.6.6 Let the vector V be expressed in terms of mutually perpendicular unit vectors nx, ny, and nz, as:

Let the line L pass through points A and B where the coordinates of A and B are as shown.

Find the projection of V along L.

P2.6.7: A force F is expressed in terms of mutually perpendicular unit vectors nx, ny, and

nz as:

If F moves a particle P from point A (1, –2, 4) to point B (–3, 4, –5), find the projection of

F along the line AB, where the coordinates (in feet) of A and B are referred to an X, Y, Z

Cartesian system and where nx, ny, and nz are parallel to X, Y, and Z.

P2.6.8: See Problem P2.6.7 Let the work W done by F be defined as the product of the projection of F along AB and distance between A and B Find W.

Section 2.7 Vector Multiplication: Vector Product

P2.7.1: Consider the vectors A and B shown in Figure P2.7.1 Using Eq (2.7.1), determine the magnitude of the vector product A × B.

a If A and B are in the X–Y plane, what is the direction of A × B?

b What is the direction of B × A?

L A(2,1,2)

F= −4nx+2ny−7nzlb

P2.7.2: See Problem P2.7.1 What is the magnitude of 6A × 2B?

P2.7.3: Consider the vectors A, B, and C shown in Figure P2.7.3 Let i, j, and k be mutually

perpendicular unit vectors as shown, with k being along the Z-axis.

a Evaluate the sum B + C.

b Determine the vector products A × B and A × C.

c Use the results of (b) to determine the sum of A × B and A × C.

d Use the results of (a) to determine the product A × (B + C) Compare the result

with that of (c)

P.2.7.4: Let n1, n2, and n3 be mutually perpendicular (dextral) unit vectors, and let vectors

A , B, and C be expressed in terms of n1, n2, and n3 as:

a Evaluate the sum B + C.

b Use Eq (2.7.24) to determine the vector products A × B and A × C.

c Use the results of (b) to determine the sum of A × B and A × C.

d Use the results of (a) to determine the product A × (B + C) Compare the result

P2.7.5: Let A and B be expressed in terms of mutually perpendicular unit vectors n1, n2,

and n3 as:

Find a unit vector perpendicular to A and B.

P2.7.6: See Figure P2.7.6 Let X, Y, and Z be mutually perpendicular coordinate axes, and let A, B, and C be points with coordinates relative to X, Y, and Z as shown.

a Form the position vectors AB, BC, and CA, and express the results in terms of the unit vectors nx, ny, and nz

b Evaluate the vector products AB × BC, BC × CA, and CA × AB.

c Find a unit vector n perpendicular to the triangle ABC.

P2.7.7: See Figure P2.7.7 Let LP and LQ be lines passing through points P1, P2 and Q1, Q2

as shown Let the coordinates of P1, P2, Q1, and Q2 relative to the X, Y, Z system be as

shown Find a unit vector n perpendicular to LP and LQ, and express the results in terms

of the unit vectors nx, ny, and nz shown in Figure P2.7.7

P2.7.8: See Figure P2.7.8 Let points P and Q have the coordinates (in feet) as shown Let

L be a line passing through P and Q, and let F be a force acting along L Let the magnitude

a Find a unit vector n parallel to L, in the direction of P to Q Express the results

in terms of the unit vectors nx, ny, and nz shown in Figure P2.7.7

b Express F in terms of nx, ny, and nz

c Form the vectors OP and OQ, calculate OP × F and OQ × F, and express the results in terms of nx, ny, and nz Compare the results

P2.7.9: Let e ijk and δjk be the permutation symbol and the Kronecker delta symbol as inEqs (2.7.7) and (2.6.7) Evaluate the sums:

Section 2.8 Vector Multiplication: Triple Products

P2.8.1: See Example 2.8.1 Verify the remaining equalities of Eq (2.8.5) for the vectors A,

B , and C of Eq (2.8.7).

P2.8.2: Use Eq (2.8.6) to find the volume of the parallelepiped shown in Figure P2.8.2,where the coordinates are measured in meters

P2.8.3: The triple scalar product is useful in determining the distance d between two

non-parallel, non-intersecting lines Specifically, d is the projection of a vector P1P2, which

connects any point P1 on one of the lines with any point P2 on the other line, onto the

common perpendicular between the lines Thus, if n is a unit vector parallel to the common

perpendicular, d is given by:

FIGURE P2.7.8

Coordinate system X, Y, Z with points P

and Q, line L, and force F.

1 3

C(1,1,5)

B(2,8,0)

d=P P n⋅

Using these concepts, find the distance d between the lines L1 and L2 shown in Figure

P2.8.3, where the coordinates are expressed in feet (Observe that n may be obtained from the vector product (P1Q1× P2Q2)/P1Q1× P2Q2.)

P2.8.4: The triple vector product is useful in determining second-moment vectors Let P

be a particle with mass m located at a point P with coordinates (x, y, z) relative to a Cartesian coordinate frame X, Y, Z, as in Figure P2.8.4 (If a particle is small, it may be identified by

a point.) Let position vector p locate P relative to the origin O Let n a be an arbitrarily

directed unit vector and let nx, ny, and nz be unit vectors parallel to axes X, Y, and Z as

shown The second moment vector Ia of P for O for the direction n a is then defined as:

a Show that in this expression the parentheses are unnecessary; that is, unlike thegeneral triple vector products in Eqs (2.8.12) and (2.8.13), we have here:

b Observing that p may be expressed as:

find I , I , and I , the second moment vectors for the directions n , n , and n

mp×(na×p)=m(p×n a)×p

p=xnx+yny+znz

c Let unit vector na be expressed in terms of nx, ny, and nz as:

show that Ia may then be expressed as:

Section 2.9 Use of the Index Summation Convention

P2.9.1: Evaluate and/or expand the following terms:

a δkk

b e ijkδjk

c a ij b jk

Section 2.10 Review of Matrix Procedures

P2.10.1: Given the matrices A, B, and C:

a Compute the products AB and BC

b Compute the products (AB)C and A(BC) and compare the results (see Eq.(2.8.10))

c Find BT, AT, and the product BTAT

d Find (AB)T and compare with BTAT (see Eq (2.10.9))

e Find , the matrix of adjoints for A (see Eq (2.10.11))

f Compute A–1

g Compute the products A–1A and AA–1

P2.10.2: Given the matrix S (an orthogonal matrix):

Section 2.11 Reference Frames and Unit Vector Sets

P2.11.1: Determine the numerical values of the transformation matrix elements of

Eq (2.11.19) if α, β, and γ have the values:

P2.11.2: See Problem P2.11.1 Show that ST = S–1 and that detS = 1

α= °30 , β= °45 , γ= °60