Dynamics of the Rigid Solid with General Constraints by a Multibody Approach (eBook)

Nicolae Pandrea, NICOLAE-DORU STANESCU (Autoren)

eBook Download: EPUB

2016
John Wiley & Sons (Verlag)
978-1-118-95440-9 (ISBN)

Infos

Covers both holonomic and non-holonomic constraints in a study of the mechanics of the constrained rigid body. Covers all types of general constraints applicable to the solid rigid Performs calculations in matrix form Provides algorithms for the numerical calculations for each type of constraint Includes solved numerical examples Accompanied by a website hosting programs

Autor

Nicolae Pandrea, Mechanical engineer, PhD. eng., Professor at the University of Pite?ti. Member of the Academy of Technical Sciences in Romania, has published 250 papers in Romania, U.S.A. and Europe and 7 books. Member of the Romanian Society of Acoustics. Winner of the "Traian Vuia" prize of the Romanian Academy. Co-author of the book Numerical Analysis with Applications in Mechanics and Engineering (Wiley, 2013). Nicolae-Doru St?nescu, Mechanical engineer, Mathematician, PhD. eng., PhD. math., Professor at the University of Pite?ti, has published 200 papers in Romania, Europe and U.S.A. and 11 books. Member of the International Institute of Acoustics and Vibrations, U.S.A, Member of the Société des Ingénieurs des l'Automobile, France. Winner of the "Traian Vuia" prize of the Romanian Academy. Co-author of the book Numerical Analysis with Applications in Mechanics and Engineering (Wiley, 2013).

Textauszug

1
Elements of Mathematical Calculation

This chapter is an introduction presenting the elements of mathematical calculation that will be used in the book.

1.1 Vectors: Vector Operations

A vector (denoted by a) is defined by its numerical magnitude or modulus |a|, by the direction Δ, and by sense. The vector is represented (Fig. 1.1) by an orientated segment of straight line.

Figure 1.1 Representation of a vector.

The sum of two vectors a, b is the vector c (Fig. 1.2) represented by the diagonal of the parallelogram constructed on the two vectors; it reads

(1.1)

Figure 1.2 The sum of two vectors.

The unit vector u of the vector a (or of the direction Δ) is defined by the relation

(1.2)

If one denotes by i, j, k the unit vectors of the axes of dextrorsum orthogonal reference system Oxyz, and by ax, ay, az the projections of vector a onto the axes, then one may write the analytical expression

(1.3)

The scalar (dot) product of two vectors is defined by the expression

(1.4)

where α is the angle between the two vectors.

We obtain the equalities

(1.5)

and, consequently, one deduces the analytical expressions

(1.6)

(1.7)

(1.8)

The vector (cross) product of two vectors, denoted by c,

(1.9)

is the vector perpendicular onto the plan of the vectors a and b, while the sense is given by the rule of the right screw when the vector a rotates over the vector b (making the smallest angle); the modulus has the expression

(1.10)

α being the smallest angle between the vectors a and b.

One obtains the equalities

(1.11)

and the analytical expression

(1.12)

The mixed product of three vectors, defined by the relation and denoted by (a, b, c), leads to the successive equalities

(1.13)

The mixed product (a, b, c) is equal to the volume with sign of the parallelepiped constructed having the three vectors as edges (Fig. 1.3). It is equal to zero if and only if the three vectors are coplanar.

Figure 1.3 The geometric interpretation of the mixed product of three vectors.

The double vector product satisfies the equality

(1.14)

The reciprocal vectors of the (non-coplanar) vectors a, b, c are defined by the expressions

(1.15)

and satisfy the equality

(1.16)

An arbitrary vector v may be written in the form

(1.17)

or as

(1.18)

1.2 Real Rectangular Matrix

By real rectangular matrix we understand a table with m rows and n columns ( )

(1.19)

where the elements aij are real numbers.

Sometimes, we use the abridged notation

(1.20)

The multiplication between a matrix and a scalar is defined by the relation

(1.21)

while the sum of two matrices of the same type (with the same number of rows and the same number of columns) is defined by

(1.22)

The zero matrix or the null matrix is the matrix denoted by [0], which has all its elements equal to zero.

The zero matrix verifies the relations

(1.23)

The transpose matrix [A]T is the matrix obtained transforming the rows of the matrix [A] into columns, that is

(1.24)

The transposing operation has the following properties

(1.25)

where we assumed that the sum can be performed.

The matrix with one column bears the name column matrix or column vector and it is denoted by {A}, that is

(1.26)

while the matrix with one row is called row matrix or row vector and is denoted as

(1.27)

or

(1.28)

where

(1.29)

If the matrix [A] has m rows and n columns, and the matrix [B] has n rows and p columns, then the two matrices can be multiplied and the result is a matrix [C] with m rows and p columns

(1.30)

where the elements cij, , , of the matrix [C] satisfy the equality

(1.31)

that is, the elements of the product matrix are obtained by multiplying the rows of matrix [A] by the columns of matrix [B].

The transpose of the product matrix is given by the relation

(1.32)

In some cases, there may exist matrices of matrices and the multiplication is performed as in the following example

(1.33)

where we assumed that the operations of multiplication and addition of matrices can be performed for each separate case.

1.3 Square Matrix

The matrix [A] is a square matrix if the number of rows is equal to the number of columns; hence

(1.34)

where the number n is the dimension or the order of the matrix.

The determinant associated to the matrix [A] is denoted by det[A].

If [Aij] is the matrix obtained from the matrix [A] by the suppression of the row i and the column j, then the algebraic complement is given by the expression

(1.35)

and the following relation holds true

(1.36)

The determinants of the matrices satisfy the equalities

(1.37)

(1.38)

where we assumed that the matrices [A] and [B] have the same order.

In general, the multiplication of matrices is not commutative,

(1.39)

but it is associative and distributive, that is

(1.40)

(1.41)

where the matrices [A], [B] and [C] have the same order.

The trace of a matrix, denoted by Tr[A] is equal to the sum of the elements situated on the principal diagonal

(1.42)

The diagonal matrix is the matrix with all the elements equal to zero, except some elements situated on the principal diagonal.

The unity matrix, generally denoted by [I], is the diagonal matrix that has all the elements of the principal diagonal equal to unity,

(1.43)

The unity matrix verifies the relations

(1.44)

The adjunct matrix is defined by the relation

(1.45)

The matrix [A] is called singular if ; it is called a non-singular one if .

The non-singular matrices [A] admit inverse matrices ; the inverse matrices fulfill the conditions

(1.46)

(1.47)

(1.48)

The matrix [A] is called symmetric if

(1.49)

it is called anti-symmetric or skew if

(1.50)

The matrix [A] is called orthogonal if it fulfills the condition

(1.51)

The orthogonal matrix [A] satisfies the equalities

(1.52)

The equation of nth degree

(1.53)

is the characteristic equation of the matrix [A]; its roots λ1, λ2, …, λn are called the eigenvalues of the matrix [A].

The vectors which are obtained from the equality

(1.54)

are called eigenvectors and, if the matrix [A] is a symmetric one, then its eigenvectors are orthogonal

(1.55)

Using the notation

(1.56)

one obtains the characteristic equation

(1.57)

where the coefficients cj are given by the iterative relations

(1.58)

Observation 1.3.1.

1. The eigenvalues of the matrix [A] of order n can be real or complex, distinct or not.
2. One or more eigenvectors correspond to an eigenvalue λm, depending on the order of multiplicity for that eigenvalue.
3. No matter if the eigenvalue is real or not, keeping into account that the matrix [A] has real components, the...

Produkt-Details

• Erscheint lt. Verlag: 3.5.2016
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: algorithms • Computational / Numerical Methods • constraints • Curves • Differential Equations • Dynamics • Equilibrium • Examples • Festkörpermechanik • Festkörpermechanik • Forces • General Constraints • holonomic • Kinematics • Maschinenbau • Mathematical • Mathematics • Mathematik • matrix form • mechanical engineering • Mechanics • Motion • Multibody • Multibody Approach • Non-holonomic • Numerical • Numerical Methods • numerische Methoden • Points • Rechnergestützte / Numerische Verfahren im Maschinenbau • Rechnergestützte / Numerische Verfahren im Maschinenbau • rigid • rigid body • solid • solid mechanics • solid rigid • Solutions • Surface
• ISBN-10: 1-118-95440-8 / 1118954408
• ISBN-13: 978-1-118-95440-9 / 9781118954409