Analytic Mechanics
Table of Contents
1. Orbital Mechanics
1.1. Tidal Force
A body with a significant volume under a gravitational field (or any other inhomogeneous field for that matter), each part of it will experience different force, in an accelerating frame with the center of mass as the reference point each part appears to be experiencing the difference between the local force and the force on the center of mass.
2. Wave Theory
Wave is an organized transfer of energy.
2.1. Wave Equation
3D wave \[ \laplacian u = \frac{1}{c^2} \pdv[2]{u}{t}. \]
1D wave \[ \pdv[2]{u}{x} = \frac{1}{c^2} \pdv[2]{u}{t}. \] Hyperbolic differential equation that admits any solution of the form \[ u(x,t) = u_0(x \pm ct). \]
Wave equation can be solved by linear combination of sinusoidal waves. Sinusoidal wave can be trigonometric function \[ u(\mathbf{x},t) = A \sin(\mathbf{k}\vdot \mathbf{x} - \omega t + \phi), \] or the real part of the complex exponential \[ u(\mathbf{x},t) = \Re{A e^{i(\mathbf{k}\vdot \mathbf{x} - \omega t + \phi)}}. \] They are characterized by few parameters:
- \(\mathbf{k}\) wave vector
- \(Ae^{i\phi}\) phasor
By the completeness of sinusoidal waves in function space, any solution can be expressed as linear combination of such waves: \[ u(\mathbf{x}, t) = \int \frac{\dd{\mathbf{k}}}{(2\pi)^3} A(\mathbf{k}) e^{i(\mathbf{k}\vdot \mathbf{x} - \omega t)}. \] This is exactly the
2.2. Transverse Wave
2.2.1. In a String
\[ \pdv[2]{y}{x} = \frac{\lambda}{\tau} \pdv[2]{y}{t}. \] where \(y(x,t)\) is the vertical displacement, \(\lambda\) is the mass per unit length, \(\tau\) is the tension in the string.
2.2.1.1. Real String
The wave in a plucked string is angular, which is a superposition of harmonics.
They settle down to lower harmonics after enough period.
3. Beam Theory
3.1. Load
Why \(M, V, w\)? They don't sound appealing.
3.1.1. Distributed Force
\(w(x)\)
3.1.2. Shear Force
The shear force on the cross-section of the beam. \[ V(x) = -\int_0^x w\dd{x} \]
3.1.3. Bending Moment
The moment with respect to the fixed origin. \[ M(x) = \int_0^x V \dd{x} \]
They are produced by tensile stress.
Notice that the the integration is justified by the fact that the second term of infinitesimal moment is zero for perpendicular load: \[ \dd{\vb{M}} = \dd{\vb{r}}\cross \vb{F} + \vb{r} \times \dd{\vb{F}}. \]
3.2. Static Beam Equation
\[ \dv[2]{x} \left( EI \dv[2]{w}{x}\right) = q \] where \( w(x) \) represent the deflection in perpendicular direction along the beam, \(q\) is distributed load in force per unit length, \(E\) is elastic modulus, \(I\) is second moment of area.
\[ Q = -\dv{x} \left( EI \dv[2]{w}{x} \right) \] is the shear force, and \[ M = -EI \dv[2]{w}{x} \] is the bending moment in the beam.
3.3. Dynamic Beam Equation
\[ \frac{\partial^2}{\partial x^2}\left(EI\frac{\partial^2 w}{\partial x^2}\right) = -\mu\frac{\partial^2 w}{\partial x^2} + q(x). \]