Vector Analysis

Vector Algebra

• Three types of vectors
  • Free vector: can be displaced parallel to itself
  • Sliding vecotr: can be displaced along its line of action
  • Bound vector: has the origin and destination
• a quantity characterized by a magnitude and a direction is not necessaryly a vector
  • The quantity must obey the laws of vector algebra
• Vector addition
  • commutative:A+B=B+A
  • associative: (A+B)+C=A+(B+C)=A+B+C
• Vector subtraction
  • the same magnitude but with the opposite direction
• Projection of a vector onto an axis $u_0$
  • A projectino of a vector: A_u=A\cos\phi=A\cos(A,u_0)
• Multiplication of a vector by a scalar
  • multiply the magnitude of the vector

Bases and basis vectors

• a three-dimentional space can be defined by a set of three linearly independent vectors
  • oblique coordinate system((x^1,x^2,x^3)) vs orthonormal coordinate((x_1,x_2,x_3))
• The position of a point M is uniquely determined by its radius vector relative to the fixed origin in the coordinate system
  • However, before making any calculations, we must introduce a coordinate sysytem to denote the coordinates of the point in the coorinate system and in a certain unit
• Orthonormal systems: a coordinate system whose basis vectors intersect at right angles
• Curvilinear coordinate: a coordinate system whose coordinate curves are not straint lines(<-> rectangular, oblique coordinates)
  • Polar coordinate is a curvilinear orthonormal coordinate except when elliptic

Products of two vectors

• Scalar product(dot product): A\cdot B=|A||B|\cos(A,B)=A_1B_1+A_2B_2+A_3B_3
  • commutative: A\cdot B=B\cdot A
  • distributive: A\cdot (B+C)=A\cdot B+A\cdot C
  • A\perp B \leftrightarrow A\cdot B=0
• Vector product(cross product)
  • magnitude: |A||B|\sin(A,B)
  • orientation: perpendicular to the plane of A and B, direction from which rotation from A->B appears to be counter-clockwise
  • non-commutative: A\times B=-B\times A
  • A||B \leftrightarrow A\times B=0
  • C=A\times B=\begin{vmatrix} i_1 & i_2 & i_3\\ A_1 & A_2 & A_3\\ B_1 & B_2 & B_3\\ \end{vmatrix}
• Scalar triple product: V=(A\times B)\cdot C=|A\times B|C_{A\times B}=|A\times B|h
  • the volume of the parallelepiped taken with the sign depending on whether the angle between C and A\times B is acute or obtuse
  • (A\times B)\cdot C=\begin{vmatrix} i_1 & i_2 & i_3\\ A_1 & A_2 & A_3\\ B_1 & B_2 & B_3\\ \end{vmatrix}\cdot(C_1i_1+C_2i_2+C_3i_3)= \begin{vmatrix} C_1 & C_2 & C_3\\ A_1 & A_2 & A_3\\ B_1 & B_2 & B_3\\ \end{vmatrix}
  • commutative: (A\times B)\cdot C=(B\times C)\cdot A=(C\times A)\cdot B
  • (A\times B)\cdot C=0 if three vectors are coplanar(linearly dependent)
• Vector triple product: A\times (B\times C)
  • perpendicular to A and lies on the B,C plane

4.3: Scalar Fields

Level Surfaces

Gradient and directional derivative

Gradient vector: grad \phi=i_i\frac{\partial \phi}{\partial x_1}+i_2\frac{\partial \phi}{\partial x_2}+i_3\frac{\partial \phi}{\partial x_3}=i_k\frac{\partial \phi}{\partial x_k} - describes the inhomogeneity of the field Directional derivative: \frac{d\phi}{dl}=l\cdot grad \phi where l is a unit vector

• Properties of the gradient \frac{d\phi}{dl}=|grad\phi|cos(l,grad\phi)
  • The rate of change is the greatest in the direction of $grad \phi$: \bigg(\frac{d\phi}{dl}\bigg)_{max}=|grad\phi|=\sqrt{\bigg(\frac{\partial \phi}{\partial x_1}\bigg)^2+\bigg(\frac{\partial \phi}{\partial x_2}\bigg)^2+\bigg(\frac{\partial \phi}{\partial x_3}\bigg)^2}
  • vecotr grad $\phi$ points at the direction that’s normal to the level surface
• $\nabla$ (‘del’ or ‘nabla’) symbol is used

4.4 Vector Fields

Trajectory

Flux

A\cdot n dS=A_n dS or by integrating through the surface \int \int_SA\cdot n dS=\int\int_SA_ndS

• let dS be an element of smooth surface S, Flux is the amount flowing through dS per unit time
  • note $n$ is the unit exterior normal to S
• dS defines an elementary tube of flow whose surface is formed by the trajectories of the vector field
• the flow amount can be calculated by the height of the tubeh=|v|dt\cos(v,n)=|v\cdot n|dt
  • the flow amount is then: dQ=|v|cos(v,n)dS=v\cdot ndS
  • by integrating over the whole surface: Q=\int \int _Sv\cdot n dS
• if the fluid is incompressible, the mass m of the flow through the surface S per unit time is m=\rho Q
• if the flux of a field of a volume V, enclosed by a surface S, is positive, then flow out > flow in: decrease in density

  Divergence

  \mathrm{div}A=\lim_{V\rightarrow0}\frac{1}{V}\int\int_SA\cdot ndS=\frac{1}{V}\int\int_V\bigg(\frac{\partial A_1}{\partial x_1}+\frac{\partial A_2}{\partial x_2}+\frac{\partial A_3}{\partial x_3}\bigg)dV=\frac{\partial A_1}{\partial x_1}+\frac{\partial A_2}{\partial x_2}+\frac{\partial A_3}{\partial x_3}

• the flux of A through S divided by V is the average strength of the sources and sinks inside V
  • its limit as the V and S shrinks to a point M: divergence
  • the definition of div A is independnet of the choice of coordinate system
• field div A does not exist for every field A, - but only exists at every point where the components and thier derivatives are continuous
• From Gauss’ theorem, the surface integration may be carried over to the volume integration
  • this volume integration of components yields the definition of divergence of A in a rectangular coordinates
• Also, by rewriting Gauss’ theorem with divergence, divergence theorem is obtained: \int\int\int_V\mathrm{div} A dV=\int\int_S A\cdot ndS
• \mathrm{div}A is a sca
• lar product of $\nabla$ and $A$: \mathrm{div} A=\frac{\partial A_k}{\partial x_k}

Mode

Compliance=Displacement/force mechanical impedance=1/mobility=force/velocity accelarance=acceleration/force

Shell Physics

free vector bound vector

Vector Fields “Vector and tensor analysis with applications”

Trajectories: A curve whose tangent at every point has the same direction as a vector field A=A(r) Flux: Let S be a two-sided piecewise-smooth surface in a vector field A(r), whose elment is called dS. n is a unit vector normal to dS. Flux is the A\cdot n dS=A_ndSf

Divergence

• Given any point M in a vector field A=A(r), let S be an arbitrary closed surface S surrounding M, and enclosing a volume V
  • \frac{1}{V}
• the limit of V and S is called the divergence of the field A

4th order PDE: vibrating beam(Faculty of Kahn)

Equation of vibration: \frac{\partial^2 u}{\partial t^2}=-\alpha^2\frac{\partial^4u}{\partial x^4} \alpha-\frac{EI}{\mu} :flextual rigidity

• E:Young modulus, I:area moment of inertia w.r.t x-axis, $\mu$: mass per length Boundary condition: u=0 -> Bending moment of the beam:\frac{\partial^2 u}{\partial x^2}=0 (from structural mechanics point of view)

Initial conditions in the beam:

• $u(x,0)=u_0(x)$

• $\frac{\partial u }{\partial t}

  _{t=0}=v_o(x)$

PDE: 4th order in x–4 initial conditions in x - $u(x,t)=u(L,t)=0$ - \frac{\partial^2 u}{\partial x^2}|_{x=0}=\frac{\partial^2 u}{\partial x^2}|_{x=L}=0 PDE: 2th order in t–2 initial conditions in t - u(x.0)=u_o(x), \frac{\partial u}{\partial t}|_{t=0}=v_0(x)

• PDE and boundary conditions are homogeneous: use separation of variables–let $u(x,t)=X(x)T(t)$
  • then PDE is: $X\frac{d^2T}{d^2t}=-\alpha^2T\frac{d^4 X}{dx^4}$
  • or $\frac{1}{T}\frac{d^2 T}{dt^2}=\frac{-\alpha^2}{X}\frac{d^4X}{dx^4}$, which has to be constant
  • 

Formulation of equations of motion: “Introduciton to finite element vibration analysis”

Weak form for CPT:..? D\int_\Omega\nabla ^4wvd\Omega+\rho h\ddot w=f D\int_\Omega\nabla^2w\nabla^2vd\Omega+D\bigg[\int_S \bigg\{ v(\nabla(\nabla w)n) - (\nabla w)(\nabla v n)\bigg\}\bigg] + \rho h\ddot w=f

Strong and weak forms

Strong forms: governs the system of equations, requires strong continuity on the dependent field variables–differentiable up to the order of the partial differential equations(=strong form)

• Obtaining exact solution for a strong form is usually very difficult
  • The finite difference method(FDM): anapproximated solution to strong form, but only works for simple and regular geometry and boundary conditions

Weak form: usually created using one of the following methods, a weaker continuity on the field variables, produces a set of discretized system equations that give much more stable and accurate results especially for complex geometry

• Energy principles: special form of variational principle, suited to mechanics of solids and structures
• Weighted residual methods: general mathematical tool for all kinds of PDE

\kappa hG\nabla^2w+\kappa h G\Phi- \rho h\frac{\partial ^2w}{\partial t^2}=-F OO \gamma_{bs}\big\{\big\}

\nabla^2_{\eta\xi}W+\bar\Phi_{\eta\xi}+\gamma_{bs}\Omega^2W=0 r_{o}^2\gamma_{bs}\bigg\{ \frac{1}{2(1+v)}\nabla^2_{\eta\xi}\Psi_x+ \frac{1}{2(1-v)}\frac{\partial\bar\Phi_{\eta\xi}}{\partial\eta}\bigg\}+ (r_o^2\gamma_{bs}\Omega^2-1)\Psi_x- \frac{\partial W}{\partial \eta} =0 r_{o}^2\gamma_{bs}\bigg\{ \frac{1}{2(1+v)}\nabla^2_{\eta\xi}\Psi_y+ \frac{1}{2(1-v)}\frac{\partial\bar\Phi_{\eta\xi}}{\partial\xi}\bigg\}+ (r_o^2\gamma_{bs}\Omega^2-1)\Psi_y- \frac{\partial W}{\partial \xi} =0 where\(\bar\Psi_{\eta\xi}=\frac{\partial\Psi_x}{\partial\eta}+\frac{\partial\Psi_y}{\partial\xi}=\frac{1}{1+v}(\bar M_x+\bar M_y)\)

##

The Finite Element Method

stress strain deformation displacement Solid mechanics Structural mechanics Static<Dynamic Elastic,Elasticity Plastic,Plasticity linear relationship linear elastic anisotropic<isotropic

deflection:w rotation about x:theta_x rotation about y:theta_y

hoge

Solid Mechanics Theory

Tensor notation

Generalized Plane Strain(P.269)

Vectors and tensors

$\epsilon$: Physical Euclidean space, R^3(set of three real numbers)

• O: origin of $\epsilon$

ξ1, ξ2 and ξ3 are also called the coordinates of M in the coordinate system defined by O and (⃗ı1,⃗ı2,⃗ı3).

Invariant Quantities: qualities that remain unchanged regardless of how the coordinate system is set(scalars) Covariant Quantities: qualities that follow the coordinate system:when the coordinate is stretched or rotated, the quality follows the same - spacial derivatives(gradients etc.) depend on the system scale and rotation - forces depends on the orientation Contravariant Qualities: qualities that oppose the change in the coordinate system - displacements and velocities shrinks when the coordinate is streched and also rotates against the coordinate

Contravariant basis:superscript Covariant basis:subscript

Components of a vector in the coontravariant basis: covariant components? Components of a vector in the covariant basis: contravariant components?

Metric tensor: dot product of covariant-covariant components, or contravariant-contravariant components

• Why “metric”–dot product calculates the length of vectors ie distances
• If covariant and contravariant are orthgonal, there’s one metric tensor between them

\[g_{mn}=i_m\dot i_n\]

Metric can be used to exchange between contravariant and covariant

• the same goes for component-wise exchange via components of the metric tensor

To distinguish surface tensors from 3D tensors, surface tensors are denoted with a number of underbars corresponding to their order first fundamental form of the surface: one under bar, metric tensor of the tangent plane, can be used to convert covariant components into contravariant ones, useful to express surface integrals, second fundamental form of the surface: two under bar

Dual vector: can be described by covariant component, takes contravariant vector, returns a scalar

The total rank of V is m=p+q: (p,q)-tensor, where p:contravariant rank q:variant rank Tensor operation such as Summation, Scalar Multiplication, Linear Combination –they do not change the rank of the tensorst

Cordinates

Tensor: an object that generalizes vector to a higher dimension, that’s invariant under a change of coordinate systems, with components that change according to a special set of mathematical formulae

Tensor product:

Tangent space:A vector space that tangentially touches a smooth surface M at point T, described as $T_{p}M$,represents physical directions of movement Dual space:The dual space of the tangent space at p($T^{*}{p}M$) is the set of all linear functionsthat take all tangent vectors v and return a scalar,$\omega:T{p}M->R$ Metric tensor is a bilinear map.represents the measurements applied to vectors

Tangent Space: The space of vectors that describe directions at a point. Dual Space: The space of covectors (linear maps) that “act on” tangent vectors to produce scalars. Metric Tensor: The tool that connects these spaces, allowing transformations between covariant and contravariant components.

Injective mapping: a function that maps elements from one space to another in a way that preserves uniqueness–no two discinct elements in the original space are mapped to the same element in the target space.

The metric tensor $g_{ij}$

Chart:A smooth injective mapping from one bounded open subset of R, $Omega$ into another $\epsilon$

Invariant quantity: a quantity that does not depend on a particular choice of coordinate system. Velocity doesn’t depend on the coordinate, but it’s components do depend on the coordinate.

(ref)[http://inspire.starfree.jp/analyticalMechanics/introduction.html]

https://physnotes.jp/math/par_tot/