** **

**A
Guest posty by Ian Huitson (Hui**

**)**

Firstly, a thank you to Daniel for allowing

me to make this post on his Excelhero.com blog.

I have admired Daniels work since he first

started this blog and this post will demonstrate some of the Charting, Named

Formula and VBA techniques that I have picked up from studying the examples highlighted

throughout the site and from what he also teaches in the Excel Hero Academy.

**DANCING
PENDULUMS**

A few weeks ago I visited the Newton Excel

Bach web site where I spotted the Dynamically Defined Dancing Pendulums.

Having noticed that Doug had done a nice

animation in Strand7 (a Finite Element Analysis program), I thought “I can do

that in Excel” and so I did.

This post looks at the techniques used

within Excel to produce this model.

I am not going to describe the Physics

involved in Pendulum motion here but I am going to discuss the implementation

of this model in Excel.

The Pendulum Project is broken into 4 Main areas:

·

Pendulums

·

Control

·

Animation

Each is discussed below.

** **

** **

**PENDULUMS**

A pendulum consists of a Fixed End

(Fulcrum), Moving End (Bob) and Connector or Wire.

I am not going to describe the Physics

involved in Pendulum motion nor the mathematics

But we need to know a few simple things about

the pendulum.

One point will be fixed at the fulcrum and

assigned an arbitrary location of (x=0, y=0), we choose this to simplify the

maths, the other/moving end, the Bob, will move according to the equations of

pendulum motion.

For a single Pendulum we need to know the

location of the pendulum at any time, t.

The location is expressed in angular terms

as ø = ø_{max}.sin( sqrt(g/L ).t ), where ø is in

Radians and · is multiply, G is Gravitational Acceleration 981.24cm/s^{2}

and L is the Pendulums length.

Knowing the pendulums angular location ø

and its length, we can transform that to X & Y coordinates using simple

geometry.

** **

** **

**In
Excel**

In Excel a Pendulum can easily be

represented on an Excel Scatter Chart as a single series consisting of two points

with a joining line.

We can then add any number of new series to

represent different pendulums.

Each Pendulum is modelled as a single

series consisting of 2 coordinates, one being the fixed or Fulcrum end which

we will locate at (0,0) and the Bob of the Pendulum, which will move according

to the equations for the pendulums motion.

To setup the pendulum I have used Named Formulas for the Coordinates of the Pendulum.

The advantage of Named Formulas is several

but Named Formulas,

·

Calculate blindingly fast;

·

Are flexible enough to allow us

to represent our two point series; and

·

Can easily be controlled using

some simple VBA code to allow animation

·

Can easily be controlled using built

in Excel Controls.

** **

**NAMED
FORMULAS**

To setup a Pendulum in Excel we will use a

number of named formulas.

** **

**General
Named Formulas**

t =0 Run

time starts at 0 and increments every unit

g =981.24^{ }Gravitational

acceleration, cm/s^{2}

OMax =’1’!$B$4*PI()/180 Max angle of

Pendulum, radians.

** **

**Named
Formula for each Pendulum**

Each pendulum is represented by 4 Named

Formulas, starting with p1 for pendulum 1 and then a suffix for the various

measurement:

p1Len | =’1’!$B$9 | The Length is stored on the worksheet (Discussed later) |

p1o | =OMax*SIN(SQRT(g/p1Len)*t) | Current angular position of Pendulum 1 at time t |

p1x | =p1Len*SIN(p1o)*{0;1} | Current orthogonal X position of Pendulum 1 at time t |

p1y | =-p1Len*COS(p1o)*{0;1} | Current orthogonal Y position of Pendulum 1 at time t |

The formula for the X and Y locations are

worth examining. The Excel Scatter Chart needs a Range of

size 2 to retrieve the Start and End X locations of the Series and an

associated range for the Y values for that series.

In this model we will be using a Named Formula for the X and Y values. This is done in the Named Formula using an Array.

So for the X Series =p1Len*SIN(p1o)*{0;1}

The {0;1} are the two points of the Series

representing the pendulum.

In this case values of 0 and 1 have been

used.

The Fulcrum End will always return 0 as it

is multiplied by 0

The Bob end of the Pendulum will return the

value p1Len*SIN(p1o)

And similarly for the Y values for the

series =-p1Len*COS(p1o)*{0;1}

Noting once again the use of the Array {0;1}

for the series and the** ‘-‘** so that

all Y values are below 0.

It should be noted that these 4 formulas could have been combined into 2 formula for the X & Y position as:

p1x =’1’!$B$9*SIN(OMax*SIN(SQRT(g/’1’!$B$9)*t))*{0;1}

p1y =-‘1’!$B$9*COS(OMax*SIN(SQRT(g/’1’!$B$9)*t))*{0;1}

This would have meant that there would only

be 32 equations for the 16 Pendulum rather than the 64 used.

It is felt that for clarity of discussion

and readability of the example this method would be clearer.

Side project

The above 4 named fromulas need to be entered

for each of the other 15 pendulums.

During this project I instigated two small

side projects.

The first was a small piece of VBA code to

import the named formulas into Excel from a range of worksheet formulas.

The second was a small VBA code to add the

16 series to the Chart with all the details once again sourced from a worksheet

Formulas.

The construction of these two projects is

covered in detail at Chandoo.org.

** **

**THE
CHART**

Once the named formulas are established, a

Chart can be added to the sheet.

A new series is added to the chart for each

pendulum

Series Name: 1

Series X Values: =’1’!p1x

Series Y Values: =’1’!p1y

These were repeated for each of the 16

Series. Once again this process was automated and is described in detail at Chandoo.org.

Each series was then selected and a Built

in Circle Marker put at one end of Size 15 and a small marker put at the other

end.

The Axis need to be scaled so that the

scale of each axis is equal or the pendulums will swing in a non circular path.

This is easily done manually by adding a Square Shape to the Worksheet and changing the chart’s size so that the two axis are the same length.

Once they are correctly scaled the Axes can

be removed along with the Titles and Grid Lines, by selecting them and pressing

**Delete**.

The Bob was then changed to a Circle, The

fill color was set and it was resized to 15 Pts. The Pendulum’s wire, the

series line, was also set to a Black Color and resized to 0.5 Pt.

The above is then repeated for each

Pendulum.

In the Controls section below, I will discuss

how to change the colors of the Bob and Wires of the pendulum during the

animation.

** **

**ANIMATION**

In the above section we saw how to setup a

Pendulum as an Excel scatter chart based on a number of Named Formulas. The Named

Formulas ultimately rely on a single named constant, t, which holds the time value.

To animate the chart all we need to do is

increment the time named formula and the chart will adjust itself to the new

position based on the revised time.

To do this I have used a simple piece of

code which is shown below:

The animation code:

Sub Pendulum() Dim ti As Double Dim i As Double ' Stop people breaking execution Application.EnableCancelKey = xlDisabled ' Set a default Mouse Cursor while animating Application.Cursor = xlNorthwestArrow ti = [tinc] Do While [D7] = "True" For i = 0 To 2000 Step ti ' Add a Name to the Name Manager ' called t with a new Time Value i Application.Names.Add "t", i DoEvents ' Exit if Start/Stop Pressed If [D7] = "False" Then End Next Loop ' Reset Mouse Cursor Application.Cursor = xlDefault End Sub

The code loops a variable i from 0 to 2000

in steps obtained from B3 via the ti variable.

For every loop of the variable i, the code:

·

Stores a new named constant ‘t’

with value = i

·

Executes a DoEvents command

which:

o

allows the Chart to Update

based on the new X & Y values as a result of the new time ‘t’

o

Checks any keyboard/mouse input

·

It then checks the value from

D7 and if False exits, or if True continues

The use of minimal and simple VBA code and named

formulas results in a system that can update quick enough to result in a very

smooth animation.

** **

** **

**CONTROLS**

A number of user definable controls have

been added to the Pendulum Project

The most obvious of these are the Check

Boxes and Option Buttons on the Chart, but there is also a number of numerical

inputs and a slider.** **

**Time
Inc, Swing (Max) and Gravity **

The 3 cells B3:B5 hold values for the Time

Inc, Swing (Max) and Gravity respectively.

These can be changed either before execution

of the pendulum or during execution. The Gravity is also controlled by the

slider to its right. The slider is linked to the cell D5 under the slider and

the Cell B5 divides the sliders value by 10.

The other two cells which allow manual

input are C9 & C10 and these will be discussed in the Pendulum lengths

section.

The Main Controls are Form Control

Controls.

The Start/Stop Check Box is linked to the cell

D7 as well as executing the **Sheet1!Pendulum_Animate**

subroutine which controls the animation.

The Pendulum Color Option Buttons are

grouped and linked to E9 and pressing any option button executes the **Set_Pendulum_Colors** subroutine. Which

reads the value of Cell E9 and calls the **Color_Markers**

subroutine with the Color type (1=Colored, 2=Grey & 3=Zebra) as an option.

The Wire Color Option Buttons are grouped

and linked to E10 and pressing any option button executes the **SetWires** subroutine. Which reads the

value of Cell E10 and calls the **Color_Wires**

subroutine with the with the Wire type (1=Black, 2=Grey & 3=None) as

options.

** **

**Reset
Parameters**

A subroutine, **Reset**, has

been provided which allows either user defined reset of all parameters as well

as on being executed automatically on startup.

Sub Reset() With Worksheets("1") .Range("D5").Value = 981.42 'Gravity .Range("D7").Value = "False" 'Starting position .Range("E9").Value = 1 'Bob Color .Range("E10").Value = 1 'Wire Color .Range("B3").Value = 0.015 'Initial Time Inc .Range("B4").Value = 15 'Initial Swing Angle SetColors SetWires End Sub

**PENDULUM
LENGTHS**

Earlier in the post I mentioned that the

use of named formulas was much faster than cell references, which it is. So why

are the pendulum lengths stored on the worksheet?

The calculation of the length of the

pendulums is important to ensure the harmonics are correctly modelled.

To get the nice synchronous effects seen

during a long run of the animation the series must be in harmony with each

other. This simply means that after a certain time that the pendulums will

re-align themselves.

The simplest way to do this is to choose

lengths of pendulums which are suited to integer repetitions within the same

time frame.

That is, choose a pendulum length so that

its natural frequency is evenly divided into say 60 Seconds.

The cell formulae from B8 to C24 do just

that.

Column C sets up a number of Integer

Frequencies based on the values in C8 and C9.

The lengths in Column B are then pro-rated

to a standard 60cm pendulum based on the required Cycles/T.

You can change C9 & C10 during

execution to see what effect it has.

** **

**LINKS**

**CLOSING**

You can see by the animated GIF at the top

of the post and the attached workbook that smooth animation of 16 Pendulum is

possible within Excel.

** **

Here is the workbook:

Huis_Excel_Hero_Pendulum.xlsm.

** **

For best results, run the dancing pendulums with no other workbooks open…

** **

**Hui…**

– Mutually Interfering Shapes (Optical Illusion)