17: Vibrating Systems
- Page ID
- 7043
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 17.1: Introduction
- This page covers the dynamics of mass-spring systems, starting with a single mass and Hooke's law, leading to simple harmonic motion. It then explores a dual-mass setup akin to a diatomic molecule, highlighting the advantages of Lagrangian methods over traditional Newtonian approaches for analyzing multiple masses and springs. The content prepares readers for more complex examples ahead.
- 17.2: The Diatomic Molecule
- This page explores the dynamics of two particles connected by an elastic spring, emphasizing the calculation of their oscillation period. It introduces the spring's extension as the internal coordinate \( q \) and derives kinetic and potential energies. Through Lagrange’s equations, it concludes that the motion is simple harmonic, establishing the period as \( 2\pi\sqrt{\frac{m}{k}} \) with \( m \) as the reduced mass.
- 17.3: Two Masses, Two Springs and a Brick Wall
- This page explores the dynamics of a two-mass system connected by springs, using Lagrangian formalism to derive equations of motion and analyze equilibrium and non-equilibrium states. It identifies two angular frequencies for oscillation: a slow in-phase mode and a fast out-of-phase mode.
- 17.4: Double Torsion Pendulum
- This page examines the dynamics of two cylinders with different rotational inertias and torsion constants, detailing their suspended rotational motion. It presents the kinetic and potential energies involved, drawing parallels to a prior example with linear parameters. The discussion includes the analysis of in-phase and out-of-phase motion modes, highlighting the analogous nature of their equations of motion.
- 17.5: Double Pendulum
- This page explores a two-mass system connected by rods, utilizing small angle approximations and Lagrangian mechanics to derive equations of motion. It outlines kinetic and potential energy, and applies Lagrange's equations, leading to solutions for angular displacements and frequencies of normal modes—both slow in-phase and fast out-of-phase. The study emphasizes how initial conditions influence the oscillation ratios between the masses based on specific mass and length values.
- 17.6: Linear Triatomic Molecule
- Now we are going to discuss three masses held together by springs, of force constants k1 and k2 . We are going to allow it to vibrate, but not to rotate. Also, for the time being, I don’t want the molecule to bend, so we’ll put it inside a drinking straw to that all the vibrations are linear.
- 17.7: Two Masses, Three Springs, Two brick Walls
- This page explores the oscillatory motion of a system comprising three equal masses linked by springs. It defines the kinetic and potential energies upon displacing the first two masses and derives the equations of motion using Lagrange’s equations. The analysis reveals two normal modes of oscillation, characterized by in-phase and out-of-phase movements. The overall motion is a combination of these modes, showing time-varying amplitudes and a regular alternation in the movement of the masses.
- 17.8: Transverse Oscillations of Masses on a Taut String
- This page covers the oscillation of a light string with three equal masses under tension, detailing the calculation of kinetic and potential energy. By applying Lagrange's equations, it derives the normal modes of oscillation and their frequencies. The separation of normal coordinates is discussed, showing how each leads to independent equations akin to simple harmonic motion, which is crucial for analyzing vibrating systems and simplifying energy expressions.
- 17.9: Vibrating String
- This page explores the analogy between vibrational modes of masses on a string and harmonic vibrations of a stretched string. It explains how more masses create normal modes similar to harmonics and derives an equation for vertical motion based on angles and tension. The wave equation is introduced, detailing the speed of disturbances along the string, thereby laying groundwork for understanding wave motion.
- 17.10: Water
- This page explores the physical model of a water molecule, emphasizing the vibrational motions of its oxygen and hydrogen atoms connected by springs. It analyzes the system's kinetic and potential energies through internal coordinates, providing formulas for both energies related to spring extensions and angle changes. Additionally, it presents a method for deriving normal mode frequencies using mass and spring constants.
- 17.11: A General Vibrating System
- This page covers the vibrational dynamics of molecules, including the calculation of vibrational freedom for nonlinear and linear molecules, and explores the relationships between potential and kinetic energy using matrix notation. It also delves into dimensional analysis in Lagrangian mechanics, leveraging Lagrange's equation to derive motion equations.
- 17.12: A Driven System
- This page explores a mechanical system with two masses under a periodic force, employing Lagrange's equations without damping. It defines generalized forces and derives equations of motion, presenting solutions for mass displacements. The findings highlight the variation of amplitude with forcing frequency, noting infinite amplitudes at specific frequencies and antiresonance, where one mass's motion becomes zero at certain angular frequencies.
- 17.13: A Damped Driven System
- This page analyzes a system of two coupled masses under periodic forces and damping, focusing on the equations of motion that include damping forces and spring tensions. It seeks steady-state solutions to express the amplitudes of each mass's motion relative to the input force. Parameters \( k_1, k_2, m, \) and the damping constant \( b \) simplify the analysis, while a coupling coefficient \(\alpha\) evaluates the middle spring's influence on amplitude responses for varying \(\alpha\).
Thumbnail: A double pendulum consists of two pendulums attached end to end. (CC BY-SA 3.0; 100Miezekatzen).