<< Chapter < Page Chapter >> Page >

As the damping factor grows from 0 to 6 π , the eigenvalues shift further left in the complex plane. At factors of π , the two smallest magnitude eigenvalues become completely real, with one moving left and one moving right in the complex plane. The physical significance of this lies in the fact that the real part of the eigenvalue furthest in the right half plane is determines proportionally how quickly the displacement u decays. For large values of a , the string becomes overdamped, floating in midair, while for smaller values of a , the system oscillates before coming to rest. At a = π , the damping is optimal for bringing the string to rest most quickly. This behavior is shown in .

Displacement of the midpoint of a string for different damping terms. Notice for damping factor a = π , the displacement reaches a steady state fastest.

Networks of strings

Unlike our simple one dimensional case, it is much more difficult to determine the closed form eigenvalues and eigenfunctions of a network of strings. To this end, we apply the finite element method to numerically simulate the behavior of a network wave equation.

Network wave equation

Let the i th string in a network of strings be defined on an interval from [ 0 , i ] , where i is the length of that particular string. To generalize the wave equation to a network of strings in three dimensions, we reference Schmidt's system of equations for the planar displacement u i ( x i , t ) of the i th string, where x i [ 0 , i ] . We define the "stensor" matrix

P i = k i [ ( s i - 1 ) I + v i v i T ]

where k i is stiffness, s i > 1 is prestress (string tension), and v i is a unit vector specifying 3-dimensional orientation of the i th string. We characterize network movement by

ρ i I 2 u i t 2 = P i 2 u i x i 2

where ρ i is the i th strings density. I is the 3-by-3 identity matrix. Our boundary conditions are Dirichlet at endpoints (displacement is fixed at 0) and a condition enforcing force balance laws and connectivity of each leg at the joint. We define an end of the first string to have position 0, and for the other endpoints, we consider them to be at position k on their respective k th string. Our Dirichlet conditions can be written as

u 1 ( 0 , t ) = 0 , u k ( k , t ) = 0

If we define the set S i to be the set of integer indices of all strings incident to a joint at the end of the i th string, the force-balance joint conditions connecting strings in the set { i , S i } can be described by

P i u i x i ( i , t ) = j S i P j u j x j ( 0 , t )

This network wave equation matrix P i can also be mathematically derived from the nonlinear model of Antman; the linear, one dimensional wave equation is derived by taking the orientation vector v to be a standard basis vector.

An example of the notation for the simple tritar case.

The network wave equation is much more tractable for a concrete example. We begin by covering the network wave equation for the simplest net - a Y-shaped net called a“tritar", in honor of the guitar with Y-shaped strings (see http://www.tritare.com). For our simple case, then, we have the boundary conditions

u 1 ( 0 , t ) = 0 , u 2 ( 2 , t ) = 0 , u 3 ( 3 , t ) = 0

with the force balance equation

Questions & Answers

the diagram of the digestive system
Assiatu Reply
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
Aron Reply
Diversity of living thing
ISCONT
what is cell division
Aron Reply
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
Allison Reply
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'The art of the pfug' conversation and receive update notifications?

Ask