Appendix A: Linear Algebra

(1)

Appendix A: Linear Algebra

In this appendix, we summarize some important results from vector and matrix algebra that are useful in our development in this book. Most of the vector and matrix properties presented in the following sections are elementary and can be found in standard texts on linear algebra.

A.1 Matrices

An m n matrix A is an ordered rectangular array that has m n elements. The matrix A can be written in the form

A ¼ a

ij

¼

a

11

a

12

a

1n

a

₂₁

a

₂₂

a

2n

⋮ ⋮ ⋱ ⋮

a

_m1

a

_m2

a

mn

2 6 6 6 6 6 4

3 7 7 7 7

7 5 ðA:1Þ

The matrix A is called an m n matrix since it has m rows and n columns. The scalar element a

ij

lies in the ith row and jth column of the matrix A. Therefore, the index i, which takes the values 1, 2, . . ., m, denotes the row number, while the index j, which takes the values 1, 2, . . ., n, denotes the column number.

A matrix A is said to be square if m ¼ n. An example of a square matrix is

A ¼

3 :0 2:0 0:95 6 :3 0 :0 10 :0 9 :0 3 :5 1 :25 2

6 4

3 7 5

In this example m ¼ n ¼ 3, consequently, A is a 3 3 matrix.

The transpose of an m n matrix A is an n m matrix denoted as A

^T

and de ﬁned as

# Springer Nature Switzerland AG 2019

A. Shabana, Vibration of Discrete and Continuous Systems,

Mechanical Engineering Series,https://doi.org/10.1007/978-3-030-04348-3

379

(2)

A

^T

¼ a

ji

¼

a

₁₁

a

₂₁

a

m1

a

₁₂

a

₂₂

a

m2

⋮ ⋮ ⋱ ⋮

a

_1n

a

_2n

a

mn

2 6 6 4

3 7 7

5 ðA:2Þ

For example, let A be the matrix

A ¼ 2 :0 4:0 7:5 23:5 0 :0 8 :5 10 :0 0 :0

The transpose of A is given by

A ¼

2 :0 0:0

4:0 8:5

7:5 10:0 23 :5 0:0 2

6 6 6 4

3 7 7 7 5

That is, the transpose of the matrix A is obtained by interchanging the rows and columns.

Deﬁnitions A square matrix A is said to be symmetric if a

ij

¼ a

ji

, that is, if the elements on the upper right half can be obtained by ﬂipping the matrix about the diagonal. For example,

A ¼

3 :0 2:0 1:5

2:0 0 :0 2:3 1 :5 2 :3 1:5 2

6 4

3 7 5

is a symmetric matrix. Note that if A is symmetric, then A is the same as its transpose, that is, A ¼ A

^T

.

A square matrix is said to be an upper triangular matrix if a

_ij

¼ 0 for i > j. That is, every element below each diagonal element of an upper triangular matrix is zero. An example of an upper triangular matrix is

A ¼

6 :0 2:5 10:2 11:0 0 :0 8:0 5:5 6 :0 0 :0 0:0 3:2 4:0 0 :0 0:0 0:0 2:2 2

6 6 6 4

3 7 7 7 5

A square matrix is said to be a lower triangular matrix if a

_ij

¼ 0 for i < j. That is,

every element above the diagonal elements of a lower triangular matrix is zero. An

example of a lower triangular matrix is

(3)

A ¼

6 :0 0 :0 0 :0 0 :0 2 :5 8 :0 0 :0 0 :0 10 :2 5 :5 3 :2 0 :0

11:0 6:0 4:0 2:2 2

6 6 6 4

3 7 7 7 5

The diagonal matrix is a square matrix such that a

_ij

¼ 0 if i 6¼ j; that is, a diagonal matrix has element a

_ij

along the diagonal with all other elements equal to zero. For example,

A ¼

5 :0 0:0 0:0 0 :0 1:0 0:0 0 :0 0:0 7:0 2

6 4

3 7 5

is a diagonal matrix.

The null matrix or the zero matrix is de ﬁned to be the matrix in which all the elements are equal to zero. The unit matrix or the identity matrix is a diagonal matrix whose diagonal elements are nonzero and equal to one.

A skew-symmetric matrix is a matrix such that a

_ij

¼ a

ji

. Note that since a

_ij

¼ a

ji

for all i and j values, the diagonal elements should be equal to zero. An example of a skew-symmetric matrix ~ Ais

~A ¼

0 :0 3:0 5:0 3 :0 0 :0 2 :5 5 :0 2:5 0 :0 2

6 4

3 7 5

Observe that ~ A

^T

¼ ~A.

The trace of an n n square matrix A, denoted by tr A is the sum of the diagonal elements of A. The trace of A can thus be written as

tr A ¼ X

ⁿ

i¼1

a

ii

ðA:3Þ

Note that the trace of an n n identity matrix is n, while the trace of a skew- symmetric matrix is zero.

A.2 Matrix Operations

In this section, we discuss some of the basic matrix operations which are used throughout this book.

Matrix Addition The sum of two matrices A and B, denoted by AþB is given by A þ B ¼ a

ij

þ b

ij

ðA:4Þ

A.2 Matrix Operations 381

(4)

where b

_ij

are the elements of B. In order to add two matrices A and B, it is necessary that A and B have the same dimensions, that is, the same number of rows and the same number of columns. It is clear from Eq.

A.4

that matrix addition is commuta- tive, that is,

A þ B ¼ B þ A ðA:5Þ

Example A.1

The two matrices A and B are deﬁned as

A ¼ 3 :0 1:0 5:0 2 :0 0:0 2 :0

, B ¼ 2 :0 3:0 6 :0

3:0 0:0 5:0

The sum AþB is given by

A þ B ¼ 3 :0 1:0 5:0 2 :0 0:0 2 :0

þ 2 :0 3:0 6 :0

3:0 0:0 5:0

¼ 5 :0 4:0 1 :0

1:0 0:0 3:0

while A B is given by

A B ¼ 3 :0 1:0 5:0 2 :0 0:0 2 :0

2 :0 3:0 6 :0

3:0 0:0 5:0

¼ 1 :0 2:0 11:0 5 :0 0 :0 7 :0

Matrix Multiplication The product of two matrices A and B is another matrix C deﬁned as

C ¼ AB ðA:6Þ

The element c

ij

of the matrix C is deﬁned by multiplying the elements of the ith row in A by the elements of the jth column in B according to the rule

c

_ij

¼ a

i1

b

_{1 j}

þ a

i2

b

_{2 j}

þ þ a

in

b

_nj

¼ X

k

a

_ik

b

_kj

ðA:7Þ

Therefore, the number of columns in A must be equal to the number of rows in B.

Observe that if A is an m n matrix and B is an n p matrix, then C is an m p matrix. Observe also that, in general, AB 6¼ BA. That is, matrix multiplication is not commutative. Matrix multiplication, however, is distributive, that is, if A and B are m p matrices and C is a p n matrix, then

A þ B

ð ÞC ¼ AC þ BC ðA:8Þ

(5)

Example A.2

Let

A ¼

0 4 1

2 1 1

3 2 1

2 6 4

3 7 5, B ¼

0 1 0 0 5 2 2 6 4

3 7 5

Then

AB ¼

0 4 1

2 1 1

3 2 1

2 6 4

3 7 5

0 1 0 0 5 2 2 6 4

3 7 5 ¼

5 2 5 4 5 5 2 6 4

3 7 5

Observe that the product BA is not deﬁned in this example since the number of columns in B is not equal to the number of rows in A.

The associative law is valid for matrix multiplications. That is, if A is an m p matrix, B is a p q matrix, and C is a q n matrix, then (AB)C ¼ A(BC) ¼ ABC.

Determinant The determinant of an n n square matrix A, denoted as jAj, is a scalar de ﬁned as

A j j ¼

a

₁₁

a

₁₂

a

1n

a

₂₁

a

₂₂

a

2n

⋮ ⋮ ⋱ ⋮

a

n1

a

n2

a

nn

2 6 6 6 6 6 4

3 7 7 7 7

7 5 ðA:9Þ

In order to be able to evaluate the unique value of the determinant of A, some basic de ﬁnitions have to be made ﬁrst. The minor M

ij

corresponding to the element a

_ij

is the determinant formed by deleting the ith row and jth column from the original determinant jAj. The cofactor C

ij

of the element a

_ij

is de ﬁned as

C

ij

¼ 1 ð Þ

ⁱ^þj

M

ij

ðA:10Þ Using this de ﬁnition of the cofactors C

ij

, which are determinants of order n 1, the value of the determinant in Eq.

A.9

can be obtained in terms of the cofactors of the elements of an arbitrary row i as follows:

A j j ¼ X

ⁿ

j¼1

a

ij

C

ij

ðA:11Þ

(6)

If A is a 2 2 matrix deﬁned as

A ¼ a

11

a

12

a

₂₁

a

₂₂

,

the cofactors C

_ij

associated with the elements of the ﬁrst row are

C

₁₁

¼ 1 ð Þ

²

a

₂₂

¼ a

22

, C

₁₂

¼ 1 ð Þ

³

a

₂₁

¼ a

21

ðA:12Þ According to the de ﬁnition of Eq.

A.11, the determinant of the 2

2 matrix A can be determined using the cofactors of the elements of the ﬁrst row as

A

j j ¼ a

11

C

11

þ a

12

C

12

¼ a

11

a

22

a

12

a

21

If A is a 3 3 matrix deﬁned as

A ¼

a

11

a

12

a

13

a

21

a

22

a

23

a

31

a

32

a

33

2 6 4

3 7 5,

the determinant of A in terms of the cofactors of the ﬁrst row is given by

A j j ¼ X

³

j¼1

a

1 j

C

1 j

¼ a

11

C

11

þ a

12

C

12

þ a

13

C

13

where

C

11

¼ a

22

a

23

a

32

a

33

, C

¹²

¼ a

21

a

23

a

31

a

33

, C

¹³

¼ a

21

a

22

a

31

a

32

That is, the determinant of A is

A j j ¼ a

11

a

22

a

23

a

32

a

33

a

¹²

a

21

a

23

a

31

a

33

þ a

¹³

a

21

a

22

a

31

a

32

¼ a

11

ð a

22

a

33

a

23

a

32

Þ a

12

ð a

21

a

33

a

23

a

31

Þ þ a

13

ð a

21

a

32

a

22

a

31

Þ

ðA:13Þ

One can show that the determinant of a square matrix is equal to the determinant of

its transpose, that is, | A| ¼ |A

^T

|, and the determinant of a diagonal matrix is equal to

the product of the diagonal elements. Furthermore, the interchange of any two

columns or rows changes only the sign of the determinant. One can also show that

if a matrix has two identical rows or two identical columns, the determinant of this

matrix is equal to zero. This can be demonstrated by the example of Eq.

A.13, for

example, if the second and third rows are identical, a

₂₁

¼ a

31

, a

₂₂

¼ a

32

, and

(7)

a

₂₃

¼ a

33

. Using these equalities in Eq.

A.13, one can show that the determinant of

the matrix A in this special case is equal to zero. A matrix whose determinant is equal to zero is said to be a singular matrix. For an arbitrary square matrix, singular or nonsingular, it can be shown that the value of the determinant does not change if any row or column is added to or subtracted from another.

Inverse of a Matrix A square matrix A

¹

that satis ﬁes the relationship

A

¹

A ¼ AA

¹

¼ I ðA:14Þ

where I is the identity matrix, is called the inverse of the matrix A. The inverse of the matrix A is deﬁned as

A

¹

¼ C

t

A

j j ðA:15Þ

where C

t

is the adjoint of the matrix A. The adjoint matrix C

t

is the transposed matrix of the cofactors C

_ij

of the matrix A.

Example A.3

Determine the inverse of the matrix

A ¼

1 1 1

0 1 1

0 0 1

2 6 4

3 7 5

Solution. The determinant of the matrix A is equal to one, that is, |A| ¼ 1. The cofactors of the elements of the matrix A are

C

₁₁

¼ 1, C

12

¼ 0, C

13

¼ 0, C

21

¼ 1 C

22

¼ 1, C

23

¼ 0, C

31

¼ 0, C

32

¼ 1 C

33

¼ 1

The adjoint matrix, which is the transpose of the matrix of the cofactor elements, is given by

C

t

¼

C

₁₁

C

₂₁

C

₃₁

C

₁₂

C

₂₂

C

₃₂

C

₁₃

C

₂₃

C

₃₃

2

4

3 5 ¼ 1 1 0

0 1 1

0 0 1

2 4

3 5

Therefore,

(continued)

(8)

A

¹

¼ C

t

A j j ¼

1 1 0

0 1 1

0 0 1

2 4

3 5

It follows that

A

¹

A ¼

1 1 0

0 1 1

0 0 1

2 4

3 5 1 1 1

0 1 1

0 0 1

2 4

3 5 ¼ 1 0 0

0 1 0

0 0 1

2 4

3 5 ¼ AA

¹

Note that if A is the 2 2 matrix

A ¼ a

₁₁

a

₁₂

a

₂₁

a

₂₂

, the inverse of A can be simply written as

A

¹

¼ 1 A j j

a

22

a

12

a

21

a

₁₁

where | A| ¼ (a

11

a

12

a

12

a

21

). If the determinant of A is equal to zero, the inverse of A does not exist. This is the case of a singular matrix.

If A and B are nonsingular square matrices, then AB

ð Þ

¹

¼ B

¹

A

¹

ðA:16Þ

It can also be veri ﬁed that

A

¹

T

¼ A

^T ₁

ðA:17Þ That is, the transpose of the inverse of a matrix is equal to the inverse of its transpose.

A square matrix A is said to be orthogonal if

A

^T

A ¼ AA

^T

¼ I ðA:18Þ

In this case, A

^T

¼ A

¹

. That is, the inverse of an orthogonal matrix is equal to its transpose. An example of orthogonal matrices is

A ¼ cos θ sin θ sin θ cos θ

ðA:19Þ

For this matrix, one has

(9)

A

^T

A ¼ cos θ sin θ

sin θ cos θ

cos θ sin θ sin θ cos θ

¼ cos

²

θ þ sin

²

θ 0 0 sin

²

θ þ cos

²

θ

Using the trigonometric identity cos

²

θ + sin

²

θ ¼ 1, one obtains A

^T

A ¼ I, and the matrix A deﬁned by Eq.

A.19

is indeed an orthogonal matrix.

A.3 Vectors

An n-dimensional vector a is an ordered set

a ¼ a ð

1

; a

2

; . . . ; a

n

Þ ðA:20Þ of n scalars. The scalar a

_i

,i ¼ 1,2,. . .,n, is called the ith component of a.

An n-dimensional vector can be considered as an n 1 matrix that consists of only one column. Therefore, the vector a can be written in the following column form

a ¼ a

1

a

2

⋮ a

n

2 6 6 4

3 7 7

5 ðA:21Þ

The transpose of this column vector de ﬁnes the n-dimensional row vector a

^T

¼ [a

1

a

₂

. . . a

n

]. The vector a of Eq.

A.21

can also be written as

a ¼ a ½

1

a

2

. . . a

n

^T

ðA:22Þ By considering the vector as special case of a matrix with only one column or one row, the rules of matrix addition and multiplication apply also to vectors. For example, if a and b are two n-dimensional vectors, deﬁned, respectively, as

a ¼ a ½

1

a

2

. . . a

n

^T

, b ¼ b ½

1

b

2

. . . b

n

^T

, then aþb is deﬁned as

a þ b ¼ a ½

1

þ b

1

a

₂

þ b

2

a

n

þ b

n

^T

Two vectors a and b are equal if and only if a

i

¼ b

i

for i ¼ 1,2,. . ., n.

The product of a vector a and scalar α is the vector

αa ¼ αa ½

1

αa

2

. . . αa

n

^T

ðA:23Þ

A.3 Vectors 387

(10)

The dot, inner, or scalar product of two vectors a ¼ a ½

1

a

2

. . . a

n

^T

and b ¼ b ½

1

b

₂

. . . b

n

^T

is de ﬁned by the following scalar quantity

ab ¼ a

^T

b ¼ a ½

1

a

2

. . . a

n

b

1

b

2

⋮ b

n

2 6 6 4

3 7 7 5

¼ a

1

b

1

þ a

2

b

2

þ þ a

n

b

n

ðA:24Þ

which can be written as

ab ¼ a

^T

b ¼ X

ⁿ

i¼1

a

i

b

i

ðA:25Þ

It follows that a b ¼ b a.

The length of a vector a, denoted as jaj, is deﬁned as the square root of the dot product of a with itself, that is,

a

j j ¼ ffiffiffiffiffiffiffiffi a

^T

a

p ¼ a

²1

þ a

²2

þ þ a

²n

1=2

ðA:26Þ

The terms modulus, magnitude, norm, and absolute value of a vector are also used to denote the length of a vector.

Example A.4

Let a and b be the two vectors

a ¼ 0 1 3 2 ½

^T

, b ¼ 1 0 2 3 ½

^T

then

a þ b ¼ 0 1 3 2 ½

^T

þ 1 0 2 3 ½

^T

¼ 1 1 5 5 ½

^T

The dot product of a and b is

a b ¼ a

^T

b ¼ 0 1 3 2 ½

1 0 2 3 2 6 6 6 6 4

3 7 7 7 7 5

¼ 0 þ 0 þ 6 þ 6 ¼ 12 The length of the vectors a and b is deﬁned as

(continued)

(11)

a

j j ¼ ffiffiffiffiffiffiffiffi a

^T

a

p ¼ 0 h ð Þ

²

þ 1 ð Þ

²

þ 3 ð Þ

²

þ 2 ð Þ

²

i

1=2

¼ ffiffiffiffiffi p 14

3:742 b

j j ¼ ffiffiffiffiffiffiffiffi b

^T

b

p ¼ 1 h ð Þ

²

þ 0 ð Þ

²

þ 2 ð Þ

²

þ 3 ð Þ

²

i

1=2

¼ ffiffiffiffiffi p 14

3:742

Differentiation In many applications in mechanics, scalar and vector functions that depend on one or more variables are encountered. An example of a scalar function that depends on the system velocities and possibly the system coordinates is the kinetic energy. Examples of vector functions are the coordinates, velocities, and accelerations that depend on time.

Let us ﬁrst consider a scalar function f that depends on several variables q

1

, q

₂

, . . ., q

_n

and the parameter t, that is,

f ¼ f q ð

1

; q

2

; . . . q

n

; t Þ ðA:27Þ where q

₁

, q

₂

, . . ., q

n

are functions of t, that is, q

_i

¼ q

i

(t).

The derivative of f with respect to t is df

dt ¼ ∂f

∂q

1

dq

₁

dt þ ∂f

∂q

2

dq

₂

dt þ þ ∂f

∂q

n

dq

_n

dt þ ∂f

∂t ðA:28Þ

which can be written using vector notation as

df dt ¼ ∂f

∂q

1

∂f

∂q

2

∂f

∂q

n

dq

₁

dt dq

₂

dt

⋮ dq

_n

dt 2 6 6 6 6 6 6 6 6 6 4

3 7 7 7 7 7 7 7 7 7 5

þ ∂f

∂t ðA:29Þ

This equation can be written compactly as df dt ¼ ∂f

∂q d q

dt þ ∂f

∂t ðA:30Þ

in which ∂f/∂t is the partial derivative of f with respect to t and q ¼ q ½

1

q

₂

q

n

^T

∂f

∂q ¼ f

_q

¼ ∂f

∂q

1

∂f

∂q

2

∂f

∂q

n

9 >

=

> ; ðA:31Þ

The second equation in this equation de ﬁnes the partial derivative of a scalar function with respect to a vector as a row vector. Note that if f is not an explicit function of time ∂f/∂t ¼ 0.

A.3 Vectors 389

(12)

Example A.5

Consider the function

f q ð

₁

; q

2

; t Þ ¼ q

²1

þ 3q

³2

t

²

where q

₁

and q

₂

are functions of the parameter t. The total derivative of f with respect to the parameter t is

df dt ¼ ∂f

∂q

1

dq

₁

dt þ ∂f

∂q

2

dq

₂

dt þ ∂f

∂t where

∂f

∂q

1

¼ 2q

1

, ∂f

∂q

2

¼ 9q

²2

, ∂f

∂t ¼ 2t Hence,

df dt ¼ 2q

1

dq

₁

dt þ 9q

²₂

dq

₂

dt 2t

¼ 2q

1

9q

²₂

dq

₁

dt dq

₂

dt 2 6 6 4

3 7 7 5 2t

where ∂f/∂q can be recognized as the row vector

∂f

∂q ¼ f

_q

¼ 2q

1

9q

²₂

Consider the case of vector functions that depend on several variables. These vector functions can be written as

f

₁

¼ f

1

q

₁

; q

2

; . . . ; q

n,

t f

₂

¼ f

2

q

₁

; q

2

; . . . ; q

n,

t

⋮

f

_m

¼ f

m

ð q

₁

; q

2

; . . . ; q

n

; t Þ 9 >

> >

=

> >

> ;

ðA:32Þ

where q

_i

¼ q

i

(t),i ¼ 1,2,. . ., n. Using the procedure previously outlined in this section,

the total derivative of an arbitrary function f

j

can be written as

(13)

df

_j

dt ¼ ∂f

j

∂q d q

dt þ ∂f

j

∂t , j ¼ 1,2, . . . , m ðA:33Þ in which ∂f

j

/ ∂q is the row vector

∂f

j

∂q ¼ ∂f

j

∂q

1

∂f

j

∂q

2

∂f

j

∂q

n

ðA:34Þ

Consequently,

d f dt ¼

df

₁

dt df

₂

dt

⋮ df

_m

dt 2 6 6 6 6 6 6 6 6 6 4

3 7 7 7 7 7 7 7 7 7 5

¼

∂f

1

∂q

1

∂f

1

∂q

2

∂f

1

∂q

n

∂f

2

∂q

1

∂f

2

∂q

2

∂f

2

∂q

n

⋮ ⋮ ⋱ ⋮

∂f

m

∂q

1

∂f

m

∂q

2

∂f

m

∂q

n

2 6 6 6 6 6 6 6 6 6 6 4

3 7 7 7 7 7 7 7 7 7 7 5

dq

₁

dt dq

₂

dt

⋮ dq

_n

dt 2 6 6 6 6 6 6 6 6 6 4

3 7 7 7 7 7 7 7 7 7 5

þ

∂f

1

∂t

∂f

2

∂t

⋮

∂f

m

∂t 2 6 6 6 6 6 6 6 6 6 4

3 7 7 7 7 7 7 7 7 7 5

ðA:35Þ

where

f ¼ f ½

1

f

₂

f

m

^T

ðA:36Þ Equation

A.35

can be written compactly as

d f dt ¼ ∂f

∂q d q

dt þ ∂f

∂t ðA:37Þ

where the m n matrix ∂f/∂q, the n-dimensional vector dq/dt, and the m-dimen- sional vector ∂f/∂t can be recognized, respectively, as

∂f

∂q ¼ f

q

¼

∂f

1

∂q

1

∂f

1

∂q

2

∂f

1

∂q

n

∂f

2

∂q

1

∂f

2

∂q

2

∂f

2

∂q

n

⋮ ⋮ ⋱ ⋮

∂f

m

∂q

1

∂f

m

∂q

2

∂f

m

∂q

n

2 6 6 6 6 6 6 6 6 6 6 4

3 7 7 7 7 7 7 7 7 7 7 5

, d q dt ¼ q

t

¼

dq

₁

dt dq

₂

dt

⋮ dq

_n

dt 2 6 6 6 6 6 6 6 6 4

3 7 7 7 7 7 7 7 7 5

, ∂f

∂t ¼ f

t

¼

∂f

1

∂t

∂f

2

∂t ⋮

∂f

m

∂t 2 6 6 6 6 6 6 6 6 6 4

3 7 7 7 7 7 7 7 7 7 5

ðA:38Þ

If f

_j

is not an explicit function of the parameter t, then ∂f

j

/ ∂t is equal to zero.

Note also that the partial derivative of an m-dimensional vector function f with respect to an n-dimensional vector q is the m n matrix f

q

de ﬁned in the preceding equation.

A.3 Vectors 391

(14)

Example A.6

Consider the vector function f deﬁned as

f ¼ f

₁

f

₂

f

₃

2 6 4

3 7 5 ¼

q

²₁

þ 3q

³2

t

²

8q

²₁

3t 2q

²₁

6q

1

q

₂

þ q

²2

2 6 4

3 7 5

The total derivative of the vector function f is

d f dt ¼

df

₁

dt df

₂

dt df

₃

dt 2 6 6 6 6 6 6 4

3 7 7 7 7 7 7 5

¼

2q

₁

9q

²₂

16q

₁

0 4q

₁

6q

2

ð Þ ð 2q

₂

6q

1

Þ 2

6 4

3 7 5

dq

₁

dt dq

₂

dt 2 6 4

3 7 5 þ

2t

3 0 2 6 4

3 7 5

where the matrix f

q

can be recognized as

f

_q

¼

2q

₁

9q

²₂

16q

₁

0 4q

₁

6q

2

ð Þ ð 2q

₂

6q

1

Þ 2

6 4

3 7 5

and the vector f

t

is

∂f

∂t ¼ f

t

¼ 2t 3 0 ½

^T

In the analysis of mechanical systems, one may also encounter scalar functions in the form Q ¼ q

^T

Aq. Following a similar procedure to the one previously outlined in this section, one can show that ∂Q/∂q ¼ q

^T

( AþA

^T

). If A is a symmetric matrix, that is, A ¼ A

^T

, one has ∂Q/∂q ¼ 2q

^T

A. Linear Independence The concepts to be introduced here are of fundamental importance in the development presented in this book. Their use is crucial in formulating many of the dynamic relationships presented in several chapters of this text.

The vectors a

1

, a

2

, . . ., a

n

are said to be linearly dependent if there exist scalars e

₁

, e

₂

, . . . , e

n

, which are not all zeros, such that

e

1

a

1

þ e

2

a

2

þ þ e

n

a

n

¼ 0 ðA:39Þ

Otherwise, the vectors a

1

, a

2

, . . ., a

n

are said to be linearly independent. In the case

of linearly independent vectors, none of these vectors can be expressed in terms of

(15)

the others. On the other hand, if Eq.

A.39

holds, and not all the scalars e

₁

, e

₂

, . . ., e

n

are equal to zero, one or more of the vectors a

1

, a

2

, . . ., a

n

can be expressed in terms of the other vectors.

Equation

A.39

can be written in matrix form as

a

1

a

2

a

n

½

e

1

e

₂

⋮ e

n

2 6 6 6 4

3 7 7

7 5 ¼ 0 ðA:40Þ

which can be written compactly as Ae ¼ 0, where e ¼ e ½

1

e

₂

e

n

^T

and the columns of the coef ﬁcient matrix A are the vectors a

1

, a

2

, . . ., a

n

, that is, A ¼ a ½

1

a

2

a

n

. If the vectors a

1

, a

2

, . . ., a

n

are linearly dependent, the system of homogeneous algebraic equations Ae ¼ 0 has a nontrivial solution. On the other hand, if the vectors a

1

, a

2

, . . ., a

n

are linearly independent vectors, then A must be a nonsingular matrix since the system of homogeneous algebraic equations Ae ¼ 0 has only the trivial solution e ¼ A

¹

0 ¼ 0. Consequently, in the case where the vectors a

1

, a

2

, . . ., a

n

are linearly dependent, the square matrix A must be singular. The number of linearly independent columns in a matrix is called the column rank of the matrix, while the number of independent rows is called the row rank of the matrix. It can be shown that for any matrix, the row rank is equal to the column rank and is equal to the rank of the matrix. Therefore, a square matrix which has a full rank is a matrix which has linearly independent rows and linearly independent columns. One concludes, therefore, that a matrix which has a full rank is a nonsingular matrix. Consequently, if a

1

, a

2

, . . ., a

n

are n-dimensional linearly independent vectors, any other n-dimensional vector can be expressed as a linear combination of these vectors. For instance, let b be another n-dimensional vector.

We show that this vector has a unique representation in terms of the linearly independent vectors a

1

, a

2

, . . ., a

n

. To this end, we write b as

b ¼ x

1

a

1

þ x

2

a

2

þ þ x

n

a

n

ðA:41Þ where x

₁

, x

₂

, . . ., x

n

are scalars. In order to show that x

₁

, x

₂

, . . ., x

n

are unique, Eq.

A.41

can be written as b ¼ Ax, where A ¼ a ½

1

a

2

. . . a

n

is a square matrix and x ¼ x ½

1

x

2

x

n

^T

. Since the vectors a

1

, a

2

, . . ., a

n

are assumed to be linearly independent, the coef ﬁcient matrix A has a full row rank and, therefore, it is nonsingular. The system of algebraic equations Ax ¼ b has a unique solution x which can be written as x ¼ A

¹

b. That is, an arbitrary n-dimensional vector b has a unique representation in terms of the linearly independent vectors a

1

, a

2

, . . ., a

n

. A familiar and important special case is the case of three-dimensional vectors.

One can show that the three vectors

a

1

¼ 1 0 0 2 4

3 5, a

2

¼

0 1 0 2 4

3 5, a

3

¼ 0 0 1 2 4

3 5,

A.3 Vectors 393

(16)

are linearly independent. Any other three-dimensional vector b ¼ b ½

1

b

2

b

3

^T

can be written in terms of the linearly independent vectors a

1

, a

2

, and a

3

as b ¼ b

1

a

1

+ b

₂

a

2

+ b

₃

a

3

, where the coef ﬁcients x

1

, x

₂

, and x

₃

can be recognized in this special case as x

₁

¼ b

1

, x

₂

¼ b

2

, and x

₃

¼ b

3

. The coef ﬁcients x

1

, x

₂

, and x

₃

are called the coordinates of the vector b in the basis deﬁned by the vectors a

1

, a

2

, and a

3

.

Example A.7

Show that the vectors

a

1

¼ 1 0 0 2 6 4

3 7 5, a

²

¼

1 1 0 2 6 4

3 7 5, a

³

¼ 1 1 1 2 6 6 4

3 7 7 5

are linearly independent. Find also the representation of the vector b ¼ 1 3 0 ½

^T

in terms of the vectors a

1

, a

2

, and a

3

.

Solution. In order to show that the vectors a

1

, a

2

, and a

3

are linearly indepen- dent, we must show that the relationship

e

₁

a

1

þ e

2

a

2

þ e

3

a

3

¼ 0 holds only when e

₁

¼ e

2

¼ e

3

¼ 0. To this end, we write

e

₁

1 0 0 2 6 4

3 7 5 þ e

2

1 1 0 2 6 4

3 7 5 þ e

3

1 1 1 2 6 4

3 7 5 ¼ 0

which leads to

e

1

þ e

2

þ e

3

¼ 0 e

2

þ e

3

¼ 0 e

3

¼ 0 Back substitution shows that

e

₃

¼ e

2

¼ e

1

¼ 0

That is, the vectors a

1

, a

2

, and a

3

are linearly independent.

In order to ﬁnd the unique representation of the vector b in terms of these linearly independent vectors, we write

(continued)

(17)

b ¼ x

1

a

1

þ x

2

a

2

þ x

3

a

3

which can be written in a matrix form as b ¼ Ax, where

A ¼

1 1 1

0 1 1

0 0 1

2 6 4

3 7 5, b ¼

1 3 0 2 6 4

3 7 5

Therefore, the vector of coordinates x can be obtained as

x ¼ x

1

x

₂

x

3

2 6 4

3 7 5 ¼ A

¹

b ¼

1 1 1

0 1 1

0 0 1

2 6 4

3 7 5

1 3 0 2 6 4

3 7 5 ¼

4 3 0 2 6 4

3 7 5

A.4 Eigenvalue Problem

In the analysis of structural systems, we often encounter a system of homogeneous algebraic equations in the form

Ay ¼ λy ðA:42Þ

where A is a square matrix, y is an unknown vector, and λ is an unknown scalar.

Equation

A.42

can be written in the form A λI

ð Þy ¼ 0 ðA:43Þ

where I is the identity matrix. Equation

A.43

represents an algebraic system of homogeneous equations which have a nontrivial solution if and only if the coef ﬁ- cient matrix ( A λI) is singular. That is, the determinant of this matrix is equal to zero. This leads to

A λI

j j ¼ 0 ðA:44Þ

This is called the characteristic equation of the matrix A. If A is an n n matrix, Eq.

A.44

is a polynomial of order n in λ. This equation can be written in the following form:

a

n

λ

ⁿ

þ a

n1

λ

ⁿ¹

þ þ a

0

¼ 0 ðA:45Þ where a

_i

, i ¼ 0, 1, 2, . . ., n, are the scalar coefﬁcients of the polynomial. The solution of Eq.

A.45

de ﬁnes n roots λ

1

, λ

2

, . . ., λ

n

. These roots are called the characteristic values or the eigenvalues of the matrix A. Associated with each of these eigenvalues,

A.4 Eigenvalue Problem 395

(18)

there is an eigenvector y

i

which can be determined to within an arbitrary constant by solving the system of equations

A λ

i

I

ð Þy

i

¼ 0 ðA:46Þ

If A is a real-symmetric matrix, one can show that the eigenvectors associated with distinctive eigenvalues are orthogonal, that is,

y

i^T

y

j

¼ 0 if i 6¼ j y

_i^T

y

j

6¼ 0 if i ¼ j

)

ðA:47Þ

Example A.8

Find the eigenvalues and eigenvectors of the matrix

A ¼ 4 1 2

1 0 0

2 0 0

2 4

3 5

Solution. The characteristic equation of this matrix is

A λI

j j ¼

4 λ 1 2

1 λ 0

2 0 λ

¼ 4 λ ð Þλ

²

þ λ þ 4λ ¼ 0 This equation can be rewritten as

λ λ 5 ð Þ λ þ 1 ð Þ ¼ 0 which has the roots

λ

1

¼ 0, λ

2

¼ 5, λ

3

¼ 1

The ith eigenvector associated with the eigenvalue λ

i

can be obtained using the equation

A λ

i

I ð Þy

i

¼ 0 The solution of this equation yields

y

1

¼ 0 2

1 2 4

3 5, y

2

¼

5 1 2 2 4

3 5, y

3

¼ 1

1 2 2 4

3

5

(19)

Problems

A.1. Find the sum of the following two matrices

A ¼ 3:0 8 :0 20:5 5 :0 11:0 13 :0 7 :0 20:0 0 2

4

3 5, B ¼ 0 3 :2 0

17:5 5:7 0

12 :0 6:8 10:0 2

4 3 5

Evaluate also the determinant and the trace of A and B.

A.2. Find the product AB and BA, where A and B are given in Problem 1.

A.3. Find the inverse of the following matrices:

A ¼ 1 2 1

2 1 0

0 1 1

2 4

3 5, B ¼ 0 3 5

2 2 3

6 2 0

2 4

3 5

A.4. Show that an arbitrary square matrix A can be written as A ¼ A

1

þ A

2

where A

1

is a symmetric matrix and A

2

is a skew-symmetric matrix.

A.5. Show that the interchange of any two rows or columns of a square matrix changes only the sign of the determinant.

A.6. Show that if a matrix has two identical rows or two identical columns, the determinant of this matrix is equal to zero.

A.7. Let

A ¼ A

¹¹

A

12

A

21

A

22

be a nonsingular matrix. If A

11

is square and nonsingular, show by direct matrix multiplications that

A

¹

¼ A

¹11

þ B

1

H

¹

B

2

B

1

H

¹

H

¹

B

2

H

¹

where

B

1

¼ A

¹11

A

12

, B

2

¼ A

21

A

¹11

H ¼ A

22

B

2

A

12

¼ A

22

A

21

B

1

¼ A

22

A

21

A

¹11

A

12

Problems 397

(20)

A.8. Let a and b be the two vectors

a ¼ 1 0 3 2 5 ½

^T

, b ¼ 0 1 2 3 8 ½

^T

Find aþb, a b, |a| and |b|.

A.9. Find the total derivative of the function

f q ð

₁

; q

2

; q

3

; t Þ ¼ q

1

q

₃

3q

²2

þ 5t

⁵

with respect to the parameter t. De ﬁne also the partial derivative of the function f with respect to the vector q(t), where q t ð Þ ¼ q ½

1

ð Þ q t

2

ð Þ q t

3

ð Þ t

^T

. A.10. Find the total derivative of the vector function

f ¼ f

₁

f

₂

f

₃

2 4

3 5 ¼ q

²₁

þ 3q

²2

5q

³4

þ t

³

q

²₂

q

²3

q

₁

q

₄

þ q

2

q

₃

þ t 2

4 3 5

with respect to the parameter t. De ﬁne also the partial derivative of the function f with respect to the vector q ¼ q ½

1

q

₂

q

₃

q

₄

^T

.

A.11. Let Q ¼ q

^T

Aq, where A is an n n square matrix and q is an n-dimensional vector. Show that

∂Q

∂q ¼ q

^T

A þ A

^T

A.12. Show that the vectors

a

1

¼ 0 0 1 2 4

3 5, a

²

¼

0 1 1 2 4

3 5, a

³

¼ 1 1 1 2 4

3 5

are linearly independent. Determine also the coordinates of the vector b ¼ 1 5 3 ½

^T

in the basis a

1

, a

2

, and a

3

.

A.13. Find the rank of the following matrices

A ¼

2 5 1 6 9 3 4 0 2 2

4

3 5, B ¼ 3 5 1 0

2 0 1 3

7 1 2 9

2 4

3

5

(21)

A.14. Find the eigenvalues and eigenvectors of the following two matrices

A ¼

2 1 0

1 2 1

0 1 1

2 4

3 5, B ¼ 6 2 0

2 2 3

4 4 3

2 4

3 5

A.15. Show that if A is a real-symmetric matrix, then the eigenvectors associated with distinctive eigenvalues are orthogonal.

Problems 399

(22)

Aktan, A. E., Farhey, D. N., Helmicki, A. J., Brown, D. L., Hunt, V. J., Lee, K.-L., & Levi, A. (1997). Structural identiﬁcation for condition assessment: Experimental arts. Journal of Structural Engineering, 123(12), 1674–1684.

Atkinson, K. E. (1978). An introduction to numerical analysis. New York: Wiley.

Baruch, M. (1997). Modal data are insufﬁcient for identiﬁcation of both mass and stiffness matrices.

AIAA Journal, 35(11), 1797–1798.

Bathe, K. J. (1996). Finite element procedures. Englewood Cliffs: Prentice Hall, Inc.

Beer, E. P., & Johnston, E. R. (1977). Vector mechanics for engineering: Statics and dynamics (3rd ed.). New York: McGraw-Hill.

Belytschko, T., Liu, W. K., & Moran, B. (2000). Nonlinear ﬁnite elements for continua and structures. New York: Wiley.

Berman, A. (2000). Inherently incompleteﬁnite element model and its effects on model updating.

AIAA Journal, 38(11), 2142–2146.

Bonet, J., & Wood, R. D. (1997). Nonlinear continuum mechanics forﬁnite element analysis.

Cambridge, UK: Cambridge University Press.

Carnahan, B., Luther, H. A., & Wilkes, J. O. (1969). Applied numerical methods. New York: Wiley.

Cha, P. D., & Tuck-Lee, J. P. (2000). Updating structural system parameters using frequency response data. Journal of Engineering Mechanics, 126(12), 1240–1246.

Clough, R. W., & Penzien, J. (1975). Dynamics of structures. New York: McGraw-Hill.

Cook, R. D. (1981). Concepts and applications ofﬁnite element analysis. New York: Wiley.

Craig, R. R., Jr. (1983). Structural dynamics: An introduction to computer methods. New York:

Wiley.

Craig, R. R., Jr., & Bampton, M. C. C. (1968). Coupling of substructures for dynamic analyses.

AIAA Journal, 6(7), 1313–1319.

Crisﬁeld, M. A. (1991). Nonlinear ﬁnite element analysis of solids and structures, Vol. 1:

Essentials. Chichester: Wiley.

Dym, C. L., & Shames, I. H. (1973). Solid mechanics: A variational approach. New York:

McGraw-Hill.

Fischer, M., & Eberhard, P. (2014). Linear model reduction of large scale industrial models in elastic multibody dynamics. Multibody System Dynamics, 31, 27–46.

Friswell, M. I., & Mottershead, J. E. (1995). Finite element model updating in structural dynamics.

Dordrecht: Kluwer Academic Publishers.

Goldenweizer, A. (1961). Theory of thin elastic shells. Oxford, UK: Pergamon Press.

Goldstein, H. (1950). Classical mechanics. Cambridge, MA: Addison-Wesley.

Gould, P. L. (1988). Analysis of shells and plates. New York: Springer-Verlag.

Greenberg, M. D. (1978). Foundations of applied mathematics. Englewood Cliffs: Prentice Hall.

Greenwood, D. T. (1988). Principles of dynamics (2nd ed.). Englewood Cliffs: Prentice Hall.

Grossi, E., & Shabana, A. A. (2018A). ANCF analysis of the crude oil sloshing in railroad vehicle systems. Sound and Vibration, 433, pp. 493–516.

401

(23)

Grossi, E., & Shabana, A. A. (2018B). Verification of a total Lagrangian ANCF solution procedure forfluid-structure interaction problems. ASME Journal of Verification, Validation, and Uncer- tainty Quantification, 2(4).https://doi.org/10.1115/1.4038904.

Hara, K., & Watanabe, M. (2018). Development of an efﬁcient calculation procedure for elastic forces in the ANCF beam element by using a constrained formulation. Multibody System Dynamics, 43, 369–386.

Hartog, D. (1968). Mechanical vibration. New York: McGraw-Hill.

Holzapfel, G. A. (2000). Nonlinear solid mechanics: A continuum approach for engineering.

Chichester: Wiley.

Hubert, J. S., & Palencia, E. S. (1989). Vibration and coupling of continuous systems: Asymptotic methods. Berlin: Springer-Verlag.

Hutton, D. V. (1981). Applied mechanical vibrations. New York: McGraw Hill.

Inman, D. J. (1994). Engineering vibration. Englewood Cliffs: Prentice Hall.

Johnson, W. (1980). Helicopter theory. Princeton: Princeton University Press.

Kane, T. R., Ryan, R. R., & Banerjee, A. K. (1987). Dynamics of a cantilever beam attached to a moving base. AIAA Journal of Guidance, Control, and Dynamics, 10(2), 139–151.

Kobayashi, N., Wago, T., & Sugawara, Y. (2011). Reduction of system matrices of planar beam in ANCF by component mode synthesis method. Multibody System Dynamics, 26, 265–281.

Liang, Y., & McPhee, J. (2018). Symbolic integration of multibody system dynamics with theﬁnite element method. Multibody System Dynamics, 43, 387–405.

Meirovitch, L. (1986). Elements of vibration analysis. New York: McGraw-Hill.

Miller, R. K., & Michel, A. N. (1982). Ordinary differential equations. New York: Academic Press.

Mottershead, J. E., & Friswell, M. I. (1993). Model updating in structural dynamics: A survey.

Journal of Sound and Vibration, 167(2), 347–375.

Muller, P. C., & Schiehlen, W. O. (1985). Linear vibrations. Dordrecht: Martinus Nijhoff Publishers.

Naghdi, P. M. (1972). The theory of shells and plates. Handbuch der Physik, 6(a/2), 425–640, Springer-Verlag, Berlin.

Nayfeh, A. H., & Mook, D. T. (1979). Nonlinear oscillations. New York: Wiley.

Nicolsen, B., Wang, L., & Shabana, A. (2017). Nonlinearﬁnite element analysis of liquid sloshing in complex vehicle motion scenarios. Journal of Sound and Vibration, 405, 208–233.

Noor, A. K. (1990). Bibliography of monographs and surveys on shells. Applied Mechanics Review, 43, 223–234.

O’Shea, J. J., Jayakumar, P., Mechergui, D., Shabana, A. A., & Wang, L. (2018). Reference conditions and sub-structuring techniques in ﬂexible multibody system dynamics. ASME Journal of Computational and Nonlinear Dynamics, 13(2018), 041007-1–041007-12.

Ogden, R. W. (1984). Non-linear elastic deformations. Mineola: Dover Publications.

Pappalardo, C. M., Wang, T., & Shabana, A. A. (2017). On the formulation of the planar ANCF triangularﬁnite elements. Nonlinear Dynamics, 89(2), 1019–1045.

Pappalardo, C. M., Zhang, Z., & Shabana, A. A. (2018). Use of independent volume parameters in the development of new large displacement ANCF triangular plate/shell elements. Nonlinear Dynamics, 91(4), 2171–2202.

Patel, M. D., Pappalardo, C. M., Wang, G., & Shabana, A. A. (2018). Integration of geometry and small and large deformation analysis for vehicle modelling: Chassis, and airless and pneumatic Tireﬂexibility. International Journal of Vehicle Performance, Accepted for publication.

Pipes, L. A., & Harvill, L. R. (1970). Applied mathematics for engineers and physicists. New York:

McGraw-Hill.

Rao, S. S. (1986). Mechanical vibrations. Reading: Addison-Wesley.

Roberson, R. E., & Schwertassek, R. (1988). Dynamics of multibody systems. Berlin: Springer- Verlag.

Ruzzene, M., & Baz, A. (2000). Finite element modeling of vibration and sound radiation from ﬂuid-loaded damped shells. Thin-Walled Structures, 36, 21–46.

402 References

(24)

Schilhans, M. J. (1958). Bending frequency of a rotating cantilever beam. ASME, Journal of Applied Mechanic, 25, 28–30.

Shabana, A. A. (1986). Dynamics of inertia-variantﬂexible systems using experimentally identiﬁed parameters. ASME Journal of Mechanisms, Transmissions, and Automation in Design, 108(3), 358–1366.

Shabana, A. A. (1996A). Finite element incremental approach and exact rigid body inertia. ASME Journal of Mechanical Design, 118, 171–178.

Shabana, A. A. (1996B). Deﬁnition of the slopes and the ﬁnite element absolute nodal coordinate formulation. Multibody System Dynamics, 1, 339–348.

Shabana, A. A. (2010A). Computational dynamics (3rd ed.). Chichester: Wiley.

Shabana, A. A. (2010B). On the deﬁnition of the natural frequency of oscillations in nonlinear large rotation problems. Journal of Sound and Vibration, 329, 3171–3181.

Shabana, A. A. (2013). Dynamics of multibody systems (4th ed.). New York: Cambridge University Press.

Shabana, A. A. (2018A). Computational continuum mechanics (3rd ed.). Chickester: Wiley.

Shabana, A. A. (2018B). Theory of vibration; An introduction (3rd ed.). New York: Springer- Verlag.

Shabana, A. A., Zaher, M. H., Recuero, A. M., & Rathod, C. (2011). Study of nonlinear system stability using eigenvalue analysis: Gyroscopic motion. Sound and Vibration, 330, 6006–6022.

Skrinjar, L., Slaviˇc, J., & Boltežar, M. (2017). Absolute nodal coordinate formulation in a pre-stressed large-displacements dynamical system. Strojniski Vestnik-Journal of Mechanical Engineering, 63, 417–425.

Steidel, R. F. (1989). An introduction to mechanical vibration. New York: Wiley.

Stolarski, H., Belytschko, T., & Lee, S. H. (1995). A review of shellﬁnite elements and corotational theories. Computational Mechanics Advances, 2(2), 125–212.

Strang, G. (1988). Linear algebra and its applications (3rd ed.). Philadelphia: Saunders College Publishing.

Thomson, W. T. (1988). Theory of vibration with applications. Englewood Cliffs: Prentice Hall.

Thorp, O., Ruzzene, M., & Baz, A. (2005). Attenuation of wave propagation inﬂuid-loaded shells with periodic shunted piezoelectric rings. Smart Materials and Structures, 14, 594.

Timoshenko, S. (1958). Strength of materials: Elementary theory and problems (3rd ed.).

Princeton: Van Nostrand-Reinhold.

Timoshenko, S., Young, D. H., & Weaver, W. (1974). Vibration problems in engineering.

New York: Wiley.

Ugural, A. C., & Fenster, K. (1979). Advanced strength and applied elasticity. Elsevier.

Wei, C., Wang, L., & Shabana, A. A. (2015). A total Lagrangian ANCF liquid sloshing approach for multibody system applications. ASME Journal of Computational and Nonlinear Dynamics, 10, 051014-1–051014-10.

Wylie, C. R., & Barrett, L. C. (1982). Advanced engineering mathematics. New York: McGraw- Hill.

Zienkiewicz, O. C. (1977). Theﬁnite element method (3rd ed.). New York: McGraw-Hill.

Zienkiewicz, O. C., & Taylor, R. L. (2000). Theﬁnite element method, Vol. 2: Solid mechanics (5th ed.). New York: Butterworth Heinemann.

(25)

Index

A

Absolute nodal coordinate formulation (ANCF), 98, 281, 329–333 Analysis of higher modes, 169, 232 Angular acceleration, 6, 9, 12, 101 Angular oscillations, 23, 51, 58, 112, 135,

187, 200

Approximation methods, 201, 255–264, 334 Arbitrary forcing function, 1, 39–42, 265 Assumed displacementﬁeld, 269, 279, 281,

286, 287, 289, 296, 313, 314, 323, 326 Assumed-modes method, 269–271

B

Beam element, 280, 285–287, 289, 291–293, 298, 307, 308, 316, 324, 330, 332, 337–342

Boolean matrix, 291, 296

Boundary conditions, 184, 188, 201, 205, 208–210, 212, 213, 216, 218–220, 226, 227, 229, 235, 238, 240, 241, 243, 245, 247, 248, 253, 256–260, 265, 266, 268, 269, 272–274, 313, 335

Brick element, see Solid element

C

Cayley–Hamilton theorem, 347, 348, 376 Characteristic equation, 17, 18, 20, 21, 119,

121, 137, 138, 159, 165, 184, 186, 189, 206, 264, 313, 314, 320, 355, 395 Characteristic matrix, 155, 159, 343, 346, 350,

351, 353, 354, 357, 360, 361, 376 Characteristic polynomial, 345–349, 356 Characteristic values, 119, 357, 395 Characteristic vector, 119

Coefﬁcient of sliding friction, 26 Completeness, 287

Concentrated loads, 242 Condition for similarity, 350 Connectivity conditions, 290, 291 Connectivity of theﬁnite elements, 279,

290–296

Conservation of energy, 55, 90, 107, 140–143, 253, 255, 256

Conservation theorems, 55, 88–93 Conservative systems, 55, 88, 89, 93 Consistent-mass formulation, 296 Constant strain triangular element, 283 Constitutive equations, 301, 303, 333 Constrained motion, 7

Continuous systems, 1, 11, 16, 42, 192, 201–271, 279–281, 300, 314 Convergence of theﬁnite-element solution,

281, 313–317 Convolution integral, 40

Coordinate reduction, 270, 334, 335 Coordinate transformation, 125, 290, 329,

330, 335

Coulomb damping, 1, 26, 28

Critical damping coefﬁcient, 18–24, 30, 35, 39, 50

Critically damped systems, 20, 21, 48

D

Damped natural frequency, 22, 38, 39, 48, 147

Damping

general viscous, 108, 152

proportional damping, 107, 146, 147, 161 viscous damping, 17, 24–26, 28, 108, 151 Damping factor, 18–24, 31, 33, 35, 38,

39, 48, 147, 148

Damping matrix, 146, 147, 151, 154, 310 Deformation modes, 134, 139, 176, 229, 239 Degree of freedom, 7

405

Appendix A: Linear Algebra