SpECS Vibrating String Experiment User`s Manual
Contents
1. because it allows each computer experiment to represent the behavior of a whole infinite class of similar experiments Each vibrating string experiment thus elucidates physical behavior that is in this sense generic In this way computers are used to discover modes of physical behavior of complex systems that are general and are not tied to only a single highly specific situation In a sense this is related to the difference between computational science and computational engineering SpECS Vibrating String Experiment User s Manual University of Minnesota Laboratory for Computational Science amp Engineering Nov 11 1999 Back to the initialization of the experiment with two point masses The on screen display was shown above as it appears after clicking on the select button for the second point mass Note that default values have resulted in our setting the initial values of the velocities of both point masses to zero This corresponds to a physical experiment where we pull the two masses to their initial positions shown in the plot labeled Displacement hold them stationary there until at time O in the time system of this experiment we let them both go The on screen density plot shows that the density associated with each point mass is unity This may not be evident without putting a ruler up against the screen display The text boxes at the left and right under each of the plots on the display give the values of the minimum2. consists of two segments of equal length This physical model may seem needlessly complex However it allows the program to associate with each segment of the continuous string that is with each grid cell not only a mass and a tension but also a density The density is of course the mass per unit length For a continuous string it is the density that together with the tension determines the vibrational properties Anyone who has looked inside a piano will understand this fact Thus the concept of string density which goes naturally with the concept of a grid cell applies when we are trying to approximate a continuous string When we are computing the vibrations of individual masses as we do when the selected number of such masses or grid cells is small then the concept of string density gets a bit confusing The vibrating string experiment has this duality and although it can lead to some confusion it is an inherent property of computational models which are all by their fundamental nature discretized The consequences of our discretization that is of our model for the continuous string as a series of point masses joined together by springs is clearest when the number of these masses is small For a small number N of point masses we can see most clearly why we subdivide the string into N 7 rather than just N grid cells The reason is that using springs of equal length we need N 1 springs to join N point masses to each other and also t
3. active form and then press the Alt and PrntScrn keys on the keyboard simultaneously The display of our experiment will be captured on the clipboard from which it may be pasted into a Microsoft Word document or otherwise be packaged for output to a printer or whatever This is how the screen displays have been inserted into this user s manual Setting the display time interval and the number of the difference scheme Before we begin our experiment we should set the value of a parameter that will control the speed at which the experiment will run In the text box to the left of the msec per step button we may enter the number of milliseconds that we wish to pass between successive computations and accompanying plots on the screen The default value of 5 msec as shown in the screen as it appears below will let the experiment run at pretty much full speed A value of 50 msec will not slow it down very much but a value of 500 msec will give only two display updates per second a rather graceful pace of change To freeze the display and interrupt the computation we can simply click on the pause button To resume the computation we can click on continue While the experiment is paused we can change some of its parameters like the stop time and output interval and even the difference scheme we are using As is explained in detail in the notes accompanying this experiment a number of different numerical s
4. connected by points on walls Paul Woodward Noy 11 1999 Saving and or restoring a setup file We are now just about ready to begin our experiment However it might be useful at this point before any of our settings change as a result of running the experiment to save our settings in a disk file We can do this by clicking on the Write Setup button This allows us to specify a file name into which all our settings will be written At a later point we could read this file in by clicking on the Read Setup button and selecting this same file for input This would cause all our present settings to be restored In this fashion we can construct experiments that we may urge our friends to perform so that they can see and perhaps usefully comment upon some unexpected phenomenon we might have discovered For the curious a listing of our setup file is as follows NDifferenceScheme 2 NGridCells 2 WaveSpeedo 1 CourantNumber 5 YMax 201 05 YMin 0105 RhoMax 105 RhoMin lt 25 SpECS Vibrating String Experiment User s Manual University of Minnesota Laboratory for Computational Science amp Engineering Nov 11 1999 10 VYMax 01 VY Man 01 Periods 10 NPlotsPerPeriod 40 NSelected 2 u MSelected F y Rho WW yy 0 Oy l 01 2 01 34 0 PRPRR oo CO w SpECS Vibrating String Experiment EA Reset Mic OC f a B ey eee 0 EE Vibration of
5. in the Periods to run text box and clicking on the continue button or if we want to restart the run by clicking on the Restart button and then on the Begin button perhaps to better observe something we may have missed However we don t have to run the experiment over again if we have saved data periodically during the run To save data in this way we need to enter a value in the text box to the left of the Plots per Period label The default is 20 plots per period and we can change this to any desired integer value If we choose the default the results of our experiment are saved in the computer s memory initially and after every time interval equal to 1 20 of a period From this saved data the program can generate plots of all the information from SpECS Vibrating String Experiment User s Manual University of Minnesota Laboratory for Computational Science amp Engineering Nov 11 1999 the experiment in its three plotting windows Thus if we run the experiment for 10 periods saving 40 plots per period we will have 401 stored states of the system of the vibrating string to peruse dynamically at the end of the experiment We may access these stored states by manipulating the vertical scroll bar in the center of the on screen display We may then click in any of the three windows at a desired horizontal location or we may manipulate the horizontal M Selected scroll bar in order to view detaile
6. left and maximum right values for the y axis ranges in each plot The x axis is of course always just as long as the string namely one length unit This sort of display is fully adequate to give a visual impression of the physical configuration but it is not as quantitative as one might desire To get full quantitative detail the user has only to manipulate the horizontal scroll bar next to the label M Selected By moving the scroll bar or equivalently by entering the number of any desired point mass in the text box labeled M Selected the user can read the precise values of the position point mass number density velocity and transverse displacement at that point along the string For convenience the user can simply click in any of the three plot windows at the horizontal location where he or she desires to obtain detailed numerical values of these quantities and they will appear in the appropriate text boxes This can also be done while the experiment is running Setting the time to stop the experiment and to save and display results The experiment will begin by definition at time zero We need to specify how long we want it to run and at what regular interval we would like its results to be saved for later visual or quanti tative perusal We have discussed earlier the time units for our experiment that is the units in which we are to interpret the times that will be displayed in the window at the bottom center of the on
7. note that the current problem time is displayed in the box just above this title SpECS Vibrating String Experiment User s Manual University of Minnesota Laboratory for Computational Science amp Engineering Nov 11 1999 The Vibrations of Two Point Masses Connected by Springs to Each Other and to Walls Enough about boundary conditions To perform our experiment with two point masses and three springs we execute the program StringGUl exe by clicking on it This action brings up the on screen display On the previous page the screen display for the experiment is shown where the user has already entered a new value of 2 in the text window labeled N Pt Masses This value of 2 has replaced the program s default value of 501 We will set up one of the two normal modes of oscillation for these two point masses from which all other oscillatory modes may be obtained by superposition as explained in the notes for this experiment By default the point mass on the left has an initial displacement of 0 01 a value that is significant yet still small compared to the length unity separating our two walls i e the length of our string The point mass on the right however has a default disolacement of O while we would like to assign to it an initial displacement of 0 01 To make this adjustment we can enter the value 2 in the text box labeled M Selected Equivalently we can adjust the horizontal scroll bar to
8. screen display To aid the user in choosing time intervals for saving output and for choosing when to stop the experiment a period is computed by the program This period is the time that it takes a small disturbance a small transverse displacement of the string to propagate along the entire length of the string This is at least the definition of the period in the case where we set the density constant along the entire length of the string In cases where the string density varies along its length this period is the time that such a disturbance would require to propagate along the entire string if its density were everywhere equal to its density in the first grid cell that is at the left most end of the string We will get a better understanding of this time interval that we call one period presently In the case of a string everywhere of density unity this time interval is just unity i e one unit of time in whatever system we are using in the experiment We can tell the program when to stop the experiment by entering the number of periods we want it to run in the text box to the left of the label Periods to run The default value of 2 appears in this window followed by a decimal point to emphasize that we may enter any positive number here not just an integral number of periods When the program reaches this point in the experiment it will pause and ask us if we want to continue which would require entering a new larger value
9. the linear ramp in string density indicated in the density plot The first several waveforms following the experiment through the first wave transmission and reflection event are shown on the next page This screen capture indicates the kind of analysis that can be done after the computation is completed by clearing the plotting windows clicking on the Multi button and displaying several successive stored states SpECS Vibrating String Experiment User s Manual University of Minnesota Laboratory for Computational Science amp Engineering Nov 11 1999 16 ia SpECS Vibrating String Experiment Reflection and transmission of a triangular pulse at a linear ramp in string density from 1 to 16 Paul Woodward Noy 11 1999 SpECS Vibrating String Experiment User s Manual University of Minnesota Laboratory for Computational Science amp Engineering Nov 11 1999
10. the right of this label so that 2 appears in the M Selected window After resetting the displacement to 0 01 the on screen display looks as shown below ia SpECS Vibrating String Experiment pe ce I pe ED T 7 Periods to Run oe ni coon b gl msec per step oo oo 00 Your name and your title for this experiment go here To provide this information just click on the get title button at the top middle of this form Please note that the current problem time is displayed in the box just above this title The change we have requested in the transverse displacement of the second point mass will not be reflected in the graph until we click on the Select button The user interface works by letting us choose values for all the parameters associated with the selected point mass namely its transverse displacement its transverse velocity and its density before we apply all these choices by clicking on the Select button If we wish to revise our choices we can re enter them and have SpECS Vibrating String Experiment User s Manual University of Minnesota Laboratory for Computational Science amp Engineering Nov 11 1999 the new values take effect by clicking again on Select The meaning of the density has already been explained The mass of our selected point is just the selected density multiplied by the length of its grid cell In the present case with unit density selected thi
11. SpECS Vibrating String Experiment User s Manual Paul R Woodward and B Kevin Edgar University of Minnesota Laboratory for Computational Science amp Engineering Introduction The purpose of this experiment is to provide a computational laboratory for studying wave propagation in one dimension The computer program implements a discrete model of a vibrating string by approximating the continuous elastic system as a string of point masses joined together in a straight line by massless springs The concept of a point mass that is of a mass concentrated entirely at a single point is obtained as the limit of a series of objects all of the same mass which are denser and denser and smaller and smaller Similarly the massless spring is the limit of a conceptual series of springs all of the same length and all exerting the same force when stretched which have increasingly less and less mass One can build in the laboratory excellent representa tions of the strings of masses joined by springs that come very close to the conceptual ideal of point masses and massless springs i e the masses are small and dense while the springs are strong but light Such realizable physical models are the inspiration for the discretization of the continuous string used by the computer simulation in this experiment They should behave very much in accordance with the results of the computations performed by this program Checking that this is indeed correct is left
12. The point mass at the wall thus never moves and the spring joining it to the first point mass in the series representing our string might just as well be attached to an immovable wall as to this fictitious point mass In our program we will have to set the positions and velocities of the fictitious point masses before each time we update the state of our model vibrating string but in this update computation we need make no allowance for special treatment of any of the point masses real or fictitious This construction is called a reflecting boundary condition because we obtain the position and velocity of the fictitious point mass behind the wall by reflecting the position and velocity of the point mass near the wall just as in a mirror We will see in our experiments that wave signals that approach the wall will also be reflected by the wall The behavior of these waves at the wall is equivalent to the result we would obtain by letting the waves propagate right through the wall while the corresponding mirrored waves propagating toward the wall from the other side moving along the reflected fictitious point masses and springs emerge from the wall just as the original waves disappear into it ia SpECS Vibrating String Experiment 2 aame E a e ee m a m E F om am oo oo Eo Your name and your title for this experiment go here To provide this information just click on the get title button at the top middle of this form Please
13. am by a common factor of 10 multiply all the times produced in the program output by this same scale factor and multiply all the masses by this factor as well Because the velocities are all measured in lengths traversed per unit of time these would not be changed The tension in the string since it is a force would also be unchanged the masses are now ten times larger and their separations are ten times greater but the time intervals are ten times greater as well and these enter into the force raised to the second power in the denominator Thus the same numerical results of one string vibration experiment hold when lengths are interpreted as centimeters times as seconds and masses as grams and also when lengths are interpreted as meters times as measured in numbers of 100 second intervals and masses as measured in numbers of 100 gram increments In this case velocities would be interpreted as numbers of meters that would be traversed in 100 second intervals or equivalently in centimeters per second The new unit of force would be that required to accelerate a mass of 100 grams or 0 1 kg by a velocity increment of one meter per 100 seconds or 0 01 m sec every 100 seconds That is this unit of force would still be one dyne It is left as an exercise for the reader to figure out how many Newtons the unit of force in the mks system this force of one dyne is This discussion may strike the reader as confusing but it is worth thinking through
14. as an exercise for the ambitious user of this program The program assumes that the string has unit length i e one inch one foot one meter one mile or whatever and that the string is fixed at its two end points The tension along the string is also assumed to be unity see discussion of units below The transverse displacement of the string at any point along its length is computed by the program as a function of time and is displayed dynamically as the computation proceeds It is assumed that this transverse displacement y which is set initially by the user is always small The force law used by the program and explained in the accompanying notes does not make sense if this assumption is violated The user may enter transverse displacements that violate this restriction but if this is done the results of the computational experiment will be in error Consequently when the user enters large displace ments a message box pops up to give an appropriate warning This program can be used to investigate the behavior of either actual strings of masses joined by springs or of continuous strings In the first instance the user selects a small number of point masses while in the second the user chooses a very large number of point masses The behavior to be expected when the number of point masses is small is explained in the notes for this experi ment Below we will explain how to set up such an experiment to investigate the vibrations of 2 poi
15. boratory for Computational Science amp Engineering Nov 11 1999 14 Selected F y Rho I yy 1 1 1 OOO O 1002 QO0002Z5 1400 009975 1401 lt 01 1402 009975 1800 000025 PRPPRPPRRPRPRPHRE BRB H L03 Losa IG 16 16 L6 O00O 0O0OO0OO0OO0O0OO0O0OO0O0OO0O0OO0OCO0OOOOOOO O N O O N OOOO 00000 O w SpECS Vibrating String Experiment mor M setccr hosie f reso Wien i ME E Booo uoo ap Reflection and transmission of a triangular pulse at a linear ramp in string density from 1 to 16 Paul Woodward Noy 11 1999 The display captured on the next page shows the waveform at the end of the experiment revealing the direct descendant of the original isolated single triangular signal shown in the setup screen above as well as several reflected signals The space time display at the bottom right shows SpECS Vibrating String Experiment User s Manual University of Minnesota Laboratory for Computational Science amp Engineering Nov 11 1999 15 w SpECS Vibrating String Experiment oor mmm H a j ao E E a a 0 01821 0 01821 EA Reflection and transmission of a triangular pulse at a linear ramp in string density from 1 to 16 Paul Woodward Nov 11 1999 the propagation of all these signals as they are transmitted and reflected again and again at
16. chemes are provided for use in this experiment The default is to use scheme number 3 This scheme is the most robust and in some respects it is also the least accurate Scheme number 4 is much more accurate in producing the time history of the transverse displacement but it achieves this accuracy at the cost of introducing very high frequency oscillations for strings with very many point masses in the velocity distribution Schemes 3 and 4 both keep high frequency velocity oscillations at bay by introducing amplitude damping of the string vibration In the case of just 2 point masses that we are concerned with at present this amplitude damping means that either of these schemes will cause the point masses to stop vibrating after only several periods This behavior is easily observed by performing such an experiment However we will select SpECS Vibrating String Experiment User s Manual University of Minnesota Laboratory for Computational Science amp Engineering Nov 11 1999 scheme 2 which has the special property that it has no dissipation so our point masses will oscillate forever as physical theory says that they should under ideal circumstances i e no friction no losses involved with stretching and relaxation of the springs etc We select scheme 2 by entering a 2 in the Scheme number text box as shown ia SpECS Vibrating String Experiment So a a ya a oo C Se Vibration of two point masses
17. d quantitative information in the various text windows on the display We may even generate a text file with long columns of numbers by clicking on the Write Plotfile button Such a text file can be used as input to a standard plotting package or it can be perused for detailed analysis by some user generated computer program In normal practice we would not expect to be generating such large text files since the graphics of the on screen display should give us a very good feel for what is going on in our experiments Putting a title onto the on screen display describing our experiment and printing the display Especially if we intend to capture and print the on screen display for communicating the results of our experiment to others it is useful to generate a label for the experiment Ample space is provided for this purpose at the bottom of the display We need only to click on the Get Title button and to enter our title into the pop up input box It is a good idea to include in this title our name the date and the purpose of the experiment Of course the entire display will be captured so there is no need to include redundant information To get instructions on how to perform this display capture we can click on the button labeled PRNT Because of Microsoft bugs in Visual Basic there is no proper way to simply print this display Instead we must click somewhere on the form not on a button to make sure that it is the
18. e point masses at the walls account for a third of the mass of the string in this case see below The attachment of the first and last spring to immovable walls is what is called a boundary condition We can choose to treat this boundary condition specially performing different computations at the ends of our string of point masses than we do in its middle However we can also treat this boundary condition by determining a configuration of point masses and springs at and beyond each end of our string which will generate precisely the desired behavior at the ends of SpECS Vibrating String Experiment User s Manual University of Minnesota Laboratory for Computational Science amp Engineering Nov 11 1999 the string when we treat all point masses both real and fictitious in exactly the same way in our program This way of treating boundary conditions is encountered again and again in computa tional science We can meet all necessary conditions by placing additional point masses on the other sides of each of the walls If the transverse displacement and velocity of such a fictitious point mass just behind a wall is always set precisely equal to and of opposite sign from the corresponding real point mass adjacent to the wall then a moment s thought leads to the conclusion that the point mass located right at the wall will never experience any net force The two springs connected to it always pull equally and in opposite directions
19. ed state in the text window above this scroll bar The graphic in the lower right hand plotting window that has now appeared at the completion of the experiment takes a bit of explanation This is a grayscale space time plot showing at a single glance the entire time development of the distribution of transverse displacement of the string For each of the 400 stored states a thin horizontal slice of this picture has been created In that thin slice the area corresponding to each grid cell or to each point mass has been shaded from black to white according to the small or large value of the transverse displacement of that point mass at that time in the experiment Note that the half grid cells associated with the point masses that our computational method places at the walls to implement the boundary conditions are shaded a neutral gray at all times since they always have transverse displacements of zero Had we used difference scheme 3 or 4 this space time display would have revealed the damping of the oscillation because as we go up in this picture in that case the color would go over to a nearly uniform shade of middle level gray To understand the real usefulness of the space time plot and also to understand the function of the Wave button we must graduate to an experiment with many point masses representing a continuous vibrating string The Vibration of a Continuous String Plucked at the Center We will now consider an experiment i
20. een above but it makes it difficult to set up continuous smooth waveforms like sine waves The only SpECS Vibrating String Experiment User s Manual University of Minnesota Laboratory for Computational Science amp Engineering Nov 11 1999 13 way in which such waves could be set up is for the user to generate with some externally provided program a setup file containing an entry for every grid cell This would not be terribly difficult but it would not be easy either Nevertheless there is nothing sacred about the sine wave and one can perform fascinating wave propagation experiments without them As an example consider the case shown on the next page which comes from initializing a rightward traveling triangular wave which propagates from a region of unit density into one of density 16 w SpECS Vibrating String Experiment _ O x for oo so eat B 0 M i so 1 0000 F FE 3 Em lt 0o mi hme AAA u Vibration of a string resulting from initiation of a rightward traveling triangular wave Paul Woodward Novy 11 1999 The setup file for this experiment is as follows NDifferenceScheme 3 NGridCells 4001 WaveSpeedo 1 CourantNumber 25 YMax 0105 YMin 0105 RhoMax 16 7525 RhoMin 95 VYMax 110330125000003 VYMin 1055525 Periods 6 NPlotsPerPeriod 40 NSelected 17 SpECS Vibrating String Experiment User s Manual University of Minnesota La
21. he string returns almost exactly to its original position lie almost on top of each other in pairs To the extent that they do not lie directly on top of each other these velocity distributions indicate the dissipation of the numerical scheme Even with the selected huge number of point masses 4001 the dissipation is visibly rounding the corners of the sudden jumps in the velocity For an ideal string with no dissipation these jumps would be perfectly sharp But of course neither this numerical model nor any real string could possibly do that The vibration of this plucked string is a standing wave mode This is clear from the space time plot in the panel at the bottom right in the on screen display This sort of standing wave vibration should be familiar to anyone who has closely observed the strings of a guitar as the SpECS Vibrating String Experiment User s Manual University of Minnesota Laboratory for Computational Science amp Engineering Nov 11 1999 12 instrument is played To see wave motion we must not simply pluck the string we must assign it non zero initial velocities as well as transverse displacements To create the initial conditions for a rightward traveling wave we have only to set up our desired initial transverse displacements and then to click on the Wave button Using the same set of initial displacements as above the result of running this new experiment is shown on the next page ia SpECS Vibrat
22. ing String Experiment oor Sooo ZO a os oms ah OFTE TA P Se T enten E O mE 0 02205 0 02101 02101 t 2 00012 Vibration of a string of uniform density gpm at its two ends after itis plucked at its center Paul Woodward Nov 11 We cannot get any idea of the wave propagation from the composite plots of the transverse displacement the velocity and the density Even watching these plots as the experiment runs or reviewing the time development after the fact by moving the vertical scroll bar through the series of 80 stored states does not give the impression of recognizable wave motion This is because the wave is constantly being superposed on its reflection from the fixed point at the one wall or the other However the space time plot at the bottom right in the on screen display shown above makes the wave propagation and its reflection immediately evident Some remarks about the significance of selected point masses or grid cells In the section above we skimmed over the significance of the selected grid cells or point masses The user initializes the distributions of transverse displacement transverse velocity and density by setting these properties for a set of selected grid cells The program then interpolates linearly between each pair of neighboring selected grid cells in order to set these same properties for all the grid cells This procedure makes it easy to set up the simple waveforms we have s
23. m up to a total mass of 501 502 or very nearly the value of unity appropriate for the whole string In this case the point masses at the walls in the computational model would make up only 1 502 of the total string mass a contribution that is pretty close to negligible SpECS Vibrating String Experiment User s Manual University of Minnesota Laboratory for Computational Science amp Engineering Nov 11 1999 An aside on the issue of units It is worthwhile at this point to make a brief aside to discuss the issue of units The program does not use any inherent set of units That is to say that the program manipulates numbers according to values entered by the user and assumes that these numbers represent values in some consistent set of units The burden is on the user to keep track of the units and to make sure that all the values entered into the program are consistent in the units he or she is using Thus it is the user who decides whether the length of the string is one inch one centimeter one foot one meter or indeed one light year The density values entered by the user must represent values of mass per unit of length where that length is measured in the units used to interpret the unit length of the string If one uses the so called cgs units centimeters grams and seconds then the string is one centimeter long and a density of unity must then mean a string of one gram per centimeter of length In this case each of our two p
24. nt masses To compute a good approximation to the vibration of a continuous string a large number of point masses should be used There is a trade off here between the number of point masses selected and hence the quality of the result and the cost of the computation in time and in memory consumed on the computer A default value of 501 point masses is implemented in the program but users will find that the results they can obtain using say 2001 point masses are much cleaner The maximum number of point masses allowed by the program is 5001 Just a Bit about How the Computer Model Works In order to understand properly the setup and working of the program it is necessary to delve just a little into its construction The experiment program divides the approximated continuous string into N 17 grid cells Because the length of the string is unity each of these grid cells has a length of 7 N 17 The correspondence between these grid cells and the mechanical model of point masses interconnected by springs is as follows Each grid cell contains a single point mass located at its center Each cell also contains two massless springs one on each side of the point mass Each of these springs is connected to the point mass and on the other end each is connected to the spring from the appropriate neighboring grid cell Thus the mechanical model used by the program is actually that of a series of point masses joined by massless springs each of which
25. ntended to approximate the behavior of a continuous string We will therefore use a lot of point masses namely 4001 This may seem like overkill but we want to obtain a truly clean result We will study the vibration of a string that is plucked as in a harpsichord rather than one that is struck as in a piano To keep things simple in this example we will pluck the string precisely at its center That is why we choose to use 4001 point masses or grid cells rather than the more natural number 4000 This will allow us to set the transverse displacement of the point mass or string element numbered 2001 at a value of 0 01 leaving all the velocities at zero and leaving the density of the string constant at a value of unity To accomplish this we merely execute the vibrating string program by clicking on its name StringGUl exe and then by entering the value 4001 into the N Pt Masses text window Everything else comes automatically with the program s defaults The result of this experiment is shown on the next page We can observe rather unusual behavior here which is better seen by watching the experi ment as it runs The string snaps back not all at once but beginning only at the center The stretched portions of the string at the sides remain stretched until signals have time to propagate along the string from the center carrying the vital information that the string has been released The velocity plots for these 2 periods in which t
26. o two fixed walls Consider the case of just two point masses For our string of length unity it is natural to locate the point masses a distance of 1 3 from each wall and also from each other Then we have two point masses each with mass 1 3 if the string density is unity if you think these masses should have the value 12 see the discussion later in this section and three springs of equal length The remaining string mass of 1 3 must be associated with point masses located in fixed positions on the walls The program represents this situation internally as follows Each point mass represents the mass of its grid cell of length 1 3 by concentrating all that mass at the center of the section of the string represented by the grid cell The effect of the fixed walls is represented by placing a fictitious point mass at the location of each wall and by demanding that the transverse displacements and velocities of these point masses must always vanish their positions remain fixed Each of these point masses is associated with a grid cell too That grid cell centered on the wall extends half way into the region in front of and half way into the region behind the wall Thus our string of length unity is represented by two point masses located 1 3 and 2 3 of the distance along its length and by three grid cells one for each of these two point masses and half a cell for each of the two fictitious immovable point masses at each wall the half cells for th
27. oint masses would have a mass of 1 3 gram The user cannot change the value of the tension along the string so it is important to note that if cgs units are being used then this tension is one dyne a dyne is the force required to accelerate one gram by a velocity increment of one centimeter per second every second Similarly the velocities entered by the user must be consistently interpreted that is interpreted by the user in the way they will be interpreted by the program If cgs units are being used then these velocities should be understood to be given in centimeters per second Note also that times are displayed by the program When cgs units are in use these must be interpreted as seconds As we will see presently the program can create velocities for the user which gets around the difficulty of some figuring on the user s part However the program will display velocity values and these are useless to the user unless he or she knows their units in order to interpret this program output It is possible to write a program like this one without assuming a particular system of units because one can view the physical system as having a natural system of units all by itself This is because the results of the experiment can be scaled to arrive at equally valid results for strings of different lengths For example if we wanted to obtain results for a string of 10 centimeters we would only have to multiply all the lengths produced by the progr
28. s mass is just 1 3 After clicking on Select the on screen display reflects the new position of the second point mass as shown below ia SpECS Vibrating String Experiment I Select fj Deselect f Reset f Clear aj CT os R m s E a o l T m a Begin oo oo 0 0 Your name and your title for this experiment go here To provide this information just click on the get title button at the top middle of this form Please note that the current problem time is displayed in the box just above this title Note that we are approximating the behavior of a string of density unity and length unity hence a string of mass unity by two point masses of mass 1 3 and three massless springs These masses do not seem to add up properly The missing mass 1 3 is associated with the half grid cells corresponding to the fictitious point masses that we are placing at the walls in order to implement our boundary conditions We want these fictitious point masses to be just like the real ones so that we do not have to treat them specially This means that they must also have mass 1 3 Thus our two real point masses constitute only 2 3 of the mass of the string we are using them to approximate This should not be alarming because we cannot expect a good approximation of a continuous string to result from just two point masses and three springs If we had used instead the program s default value of 501 point masses then these would su
29. two p oint masses conn Sp rin gs to each other and to fixed p oints on walls Paul Woodward Mov 11 1999 Running the experiment Running our experiment is now a breeze We have only to click on the button labeled Begin When we get to the end that we specified after 10 periods a text box pops up to ask us if we really want to plot all our stored states on top of each other on the display For an experiment involving lots of stored states and lots of point masses this summary plotting could take quite some time It may not prove to be terribly valuable and therefore the default response is to skip this function For our little experiment the result of this summary plotting is shown above Note that because we saved 40 states per period when they are all plotted the screen is blackened Clearly SpECS Vibrating String Experiment User s Manual University of Minnesota Laboratory for Computational Science amp Engineering Nov 11 1999 11 this is not helpful We can manually control the display of stored states as follows First we can clear the plot windows by clicking on the Clear button at the top center of the display Then we can choose to display one plot or multiple plots at a time by clicking on either the Single or the Multi button respectively The plots that are displayed are then selected by manipulating the vertical scroll bar at the center of the display or by entering the number of a desired stor
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