CALCGATE SOFTWARE LICENESE AGREEMENT

A grant of the licenese permits you to use one copy of the software ‘calcgate’, i.e., licensed as a single product. The product calcgate is not for sale, distribution, rent, or lease. The product may be used on the purchaser’s own computer and loaded into temporary memory i.e, RAM, or installed into permanent memory such as a hard disk or other storage media as a backup. For use on a network, one copy must be purchased for each user. Although the purchaser may modify the program for calcgate for use on his or her computer, modification for the purposes of sale as a commercial product are strictly prohibited.You may transfer your rights under this license agreement to another party provided you transfer the software, associated printed material, retain no copies and the recipient agrees to the terms of the license agreement.

To run ‘calcgate’ from within the Matlab computing environment insert the cd rom into the compact disk drive and set the current directory to d:\. At the comand prompt type calcgate. For best viewing, please set the command window screen width to about 80 percent of maximum or until the left edge of the command window touches the right edge of the calcgate main graphical user interface menu.

For technical assitance for any reason please contact calcgate immediately at:

e-mail: support@calcgate.com or

Tel: (818)571-1960.

Introduction:

Calcgate is designed to run within the Matlab computing
environment, and uses a mathematical transform that enables the user to perform
integral calculus on a variety of equations with incredible speed and accuarcy.
Calcgate stands for: **CALC**ulus **G**eometric **A**rea **T**ransform **E**quation, hence ‘Calcgate’.
Calcgate uses a mathematical transform method to calculate work, energy, and
integral area for any polynominal series with an accuracy of sixteen decimal places,
instantly. Calcgate can also perform integral calculus with a fourier series as
input. This allows work, energy and also co-variant integral equations to be
determined. The fourier series program, intended primarily as a simulation tool
for determining work, energy, and integral area, is now accurate to sixteen
decimal places with a given harmonic series input. Another version, that is
also much faster (including the equivalent of more harmonics) using discrete
values for the force and integral area vector component(s), along with linear
and exponential interpolation, with accuracies of sixteen or more decimal
places for any given equivalent harmonic series input, will be available in the
future sometime. Using polynominal input, Calcgate can calculate work and
energy, with the Energy program, polynominal equations with powers up to 21
instantly and accurate to sixteen decimal places. Calcgate can also perform
integral calculus on non-invariant equations, using polynominal input, with the
same speed and accuracy using the Area program, however, the differential can
be a power of up to 23. The Energy program also includes an initial velocity
input, and select initial velocity from last calculation feature allowing
different polynominal functions to calculated in series. This would enable say,
the work and energy to be determined for a triangle or sawtooth force function
by performing two separate calculation(s) in series then adding the two energy
values, using the display total energy menu option and again the results
accurate to within sixteen decimal places. For example, using a force f(t) = t,
and f(t) = -t + 1 (and using an initial velocity v(o) or select initial
velocity from last calculation feature on the integration time input GUI), on
intervals of 0 to 1 seconds calculates the work and energy of a triangle force
on the limits of integration of 0 to 2 seconds. Or say, for example f(t) = t^2,
and f(t) = -t + 1 (and using an initial velocity v(o) or select initial
velocity from last calculation on the integration time input GUI), calculates
the work and energy of a parabola and ramp, on the limits of integration 0 to 2
seconds, accurate to within sixteen decimal places by simply doing the
calculation(s) in series and adding the results using the display total energy
option on the main menu. Other interesting combinations using algebraic
equations with polynomial series as input, and using the initial velocity v(o)
can be used to determine work and energy for the vector components to within
sixteen decimal places instantly. Applications for calculation could include
heat energy due to friction in mechanical systems, maxwell’s equations,
simulations of high energy physics, astrophysical forces, forces between the
earth and moon, co-variant integral equations, and much more,

Calcgate.

Calcgate Users Guide:

Main menu-

input ft: input force coefficients for t^0 through t^21

input f(x): input force vector component coefficients for t^0 through t^21

input mass: input the mass of an object or particle

get files: load f(t), f(x), or mass from mat files. F(t) must have matrix dimensions (1,28), f(x) must have matrix dimensions (7,28), and mass must have matrix dimensions (1,1)

store files: save f(t), f(x), and mass in mat files

Calc/graph: calculate work and energy and graph results

Calc Area: go to ‘Area’ program main menu

powersolve: go to power solve main menu

fourier: go to fourier series main menu

display total energy: displays total energy, distance, and time after completing series calculations

clear total energy: clears total energy, distance, and time.

exit: exit Calcgate

input f(t)-

inputs the coefficient for each power of t (0-21)

clear: clears all f(t) and f(x) coefficients

done: returns to main menu

input f(x)-

inputs the coefficient for each power of t (0-21)

row: selects the row number (one of 1-7) for each f(x) vector component

done: returns to main menu

input mass-

inputs mass of object or particle

done: returns to main menu

get files-

load f(t): get f(t) from file (must have matrix dimensions (1,28))

load f(x): get f(x) from file (must have matrix dimensions (7,28))

load mass: get mass from file (must have matrix dimensions (1,1)

store files-

save f(t): store f(t) in mat file

save f(x): store f(x) in mat file

save mass: store mass in mat file

save all: store f(t), f(x), and mass in mat file

done: returns to main menu

calculate and graph-

distance: selects distance as the limits of integration

time: selects time as the limits of integration

integration distance-

xi: inputs the initial distance xi for the limits of integration

xf: inputs the final distance xf for the limits of integration.

If multiple roots available, user can select which root to use for calculation.

integration time-

ti: inputs the initial time for limits of integration

tf: inputs the final time for the limits of integration

v(0): inputs the initial velocity v(0)

graph menu-

add: choose f(x) vector component (one of 1-7) to graph

remove: choose f(x) component to remove from graph

graph all: choose all f(x) 1-7 to graph work and energy

graph: start graph of work and energy ∫ f(x) dx

cancel: return to main menu

add graph-

inputs f(x) (one of 1-7) to add to graph output

done: returns to graph menu

remove graph-

inputs f(x) (one of 1-7) to remove from graph output

done: returns to main menu

graph precision-

user defined: selects user defined graphing precision

default: selects default graphing precision of .01 seconds

user defined-

inputs user defined graphing precision

cancel: returns to main menu

NOTE: after graphing, user is prompted to close and or save figures to continue

(Version 6.5)

powersolve-

add: selects f(x) vector component (one of 1-7) to calculate time to average power.

remove: selects f(x) vector component (one of 1-7) to remove from time to average power calculation

calculate: inputs average power value in units of joules/sec

cancel: return to main menu

fourier-

get f(t): loads f(t) from mat file (must have matrix dimensions of at least (2,2))

get f(x): loads f(x) from mat file (must have matrix dimensions of at least (2,2))

get mass: loads mass from mat file (must have matrix dimensions of (1,1))

Calculate:

displays integration time menu-

ti: inputs initial time ti for limits of integration (ti must be positive and less than or equal to tf )

tf: inputs final time tf for limits of integration (tf must be a positive value, i.e., greater than zero and less than or equal to the period of the fundamental frequency)

calculate: calculate work and energy ∫ f(x) dx

cancel: returns to fourier series menu

select f(x)1: switches f(x)1 off and on to add or remove from calculation

select f(x)2: switches f(x)2 off and on to add or remove from calculaton

done: returns to main menu

Area main menu-

input f(t): input f(t) coefficients for t^1 through t^23

input f(x): input f(x) vector component coefficients for t^0 through t^21

get files: load f(t), f(x). F(t) must have matrix dimensions (1,28), f(x) must have matrix dimensions (7,28)

store files: save f(t), f(x)

Calc/graph: calculate area and graph results

Calc Energy: go to ‘Energy’ program main menu

fourier: go to fourier series main menu

display total area: displays total area, change in differential, and time after completing series calculations

clear total area: clears total area, change in differential, and time

exit: exit Calcgate

input f(t)-

inputs the coefficient for each power of t (1-23)

clear: clears all f(t) and f(x) coefficients

done: returns to main menu

input f(x)-

inputs the coefficient for each power of t (0-21)

row: selects the row number (one of 1-7) for each f(x) vector component

done: returns to main menu

get files-

load f(t): get f(t) from mat file (must have matrix dimensions (1,28))

load f(x): get f(x) from mat file (must have matrix dimensions (7,28))

load mass: get mass from mat file (must have matrix dimensions (1,1))

store files-

save f(t): store f(t) in mat file

save f(x): store f(x) in mat file

save all: store f(t), f(x) in mat file

done: returns to main menu

calculate and graph-

differential: selects differential as the limits of integration

time: selects time as the limits of integration

cancel: returns to main menu

integration differential-

xi: inputs the initial differential xi for the limits of integration

xf: inputs the final differential xf for the limits of integration.

If multiple roots available, user can select which root to use for calculation.

integration time-

ti: inputs the initial time ti for the limits of integration

tf: inputs the final time tf for the limits of integration

graph menu-

add: choose f(x) vector component (one of 1-7) to graph

remove: choose f(x) component to remove from graph

graph all: choose all f(x) 1-7 to graph area

graph: start graph of area ∫ f(x) dx

cancel: return to main menu

add graph-

inputs f(x) (one of 1-7) to add to graph output

done: returns to graph menu

remove graph-

inputs f(x) (one of 1-7) to remove from graph output

done: returns to main menu

graph precision-

user defined: selects user defined graphing precision

default: selects default graphing precision of .01 seconds

user defined:

inputs user defined graphing precision

cancel: returns to main menu

NOTE: after graphing, user is prompted to close and or save figures to continue

(Version 6.5)

fourier-

get f(t): loads f(t) from mat file (must have matrix dimensions of at least (2,2))

get f(x): loads f(x) from mat file (must have matrix dimensions of at least (2,2))

get mass: loads mass from mat file (must have matrix dimensions of (1,1))

Calculate:

ti: inputs initial time ti for limits of integration (ti must be positive and less than or equal to tf)

tf: inputs final time tf for limits of integration (tf must be a positive value, i.e., greater than zero and less than or equal to the period of the fundamental frequency)

calculate: calculate area ∫ f(x) dx

cancel: returns to fourier series menu

select f(x)1: switches f(x)1 off and on to add or remove from calculation

select f(x)2: switches f(x)2 off and on to add or remove from calculation

done: returns to main menu

to create f(t), f(x), and mass variables –

if f(t) is such that:

f(t) = t^7+t^6+t^5+t^4+t^3+t^2+t^1+1

and f(x) is:

f(x)1 = t^1+1

f(x)2 = t^2

f(x)3 = t^3

f(x)4 = t^4

f(x)5 = t^5

f(x)6 = t^6

f(x)7 = t^7

then perform the following:

a = zeros(1,28) initializes f(t) matrix.

a(1,1:8) = 1 sets the coefficients equal to that of those for f(t) which has powers powers 0-7

b = zeros(7,28) initializes the f(x) force vector component matrix

b(: ,1:1) = 1 sets the coefficients equal to that of those for the powers of f(x) which has powers of 0-7, in rows 1 through 7

mass = 1, sets mass equal to a value of 1.

The preceeding procedures creates variables f(t) = a, f(x) = b, mass = 1.

Note how ∑f(x)n = f(t), i.e., the sum of the vector components equals the force function f(t).

Now just save the variables a, b, and mass to a mat file in order to load them from within Calcgate.

F(x) could also be initialized with all variables in one row, instead of separate rows, by performing the following:

b = zeros(7,28), and then b(1,1:8) = 1 creates f(x)1 = t^7+t^6+t^5+t^4+t^3+t^2+t^1+1 which is stored in row 1, instead of the vector components being stored in separate rows 1–7 as in the preceeding example.

If the coefficients for the powers of t are other than 1, then they can be set accordingly in each matrix for both force f(t) and f(x) vector components.

The maximum powers, in the Energy program for both f(t) and f(x) are each 21. The preceeding steps can be used to set up the f(t) and f(x) variables for the Area program as well. The maximum powers of t, in the Area program, for f(t) is 23, and for f(x) is 21.

The fourier series program requires sinusoidal input in magnitude and phase form.

For example to create a force function with two vector components f(x)1 and f(x)2, perform the following:

a = zeros(501,2)

a(1,1) = 4

a(1,2) = 1e8

a(2,1) = 1

a(2,2) = 45*pi/180

a(501,1) = .1

a(501,2) = 10*pi/180

creates a force function having a fundamental frequency of
1e8 with a constant term of 4 where the magnitude and phase of the fundamental
are 1 and 45 respectively. The last value is the 500^{th} harmonic with
magnitued and phase of .1 and 10 respectively.

To create f(x)1 vector component perform the following:

b = zeros(501,2)

b(1,1) = 4.

b(501,1) = .1

b(501,2) = 10*pi/180

creates a vector component of f(x)1 that has a constant term
of 4 and 500^{th} harmonic

with magnitude and phase of .1 and 10 degrees (expressed in terms of radians) respectively.

F(x)2 consisting of only the fundamental can be created as in following:

c = zeros(2,2)

c(2,1) = 1

c(2,2) = 45*pi/180

the dimensions of (2,2) are because only one term, i.e., the fundamental makes up the vector component and thus only a 2 by 2 matrix is needed. If the vector component were to have more harmonics, the row dimensions would be one greater than the largest harmonic number.

The mass variable can be created as in the following:

mass = 100.

All that has to be done now is to save variables a, b, c, and mass in a mat file and then load them from within Calcgate fourier series. The same procedures can be performed to initialize the f(t) and f(x) components for the Area program verison of the fourier series program, however, the sum of the f(x) vector components do not have to be equivalent to f(t).

The fourier series program can now calculate work, energy, and integral area of a fourier series, for several thousand harmonics in a reasonable amount of time because of the introduction of a new additional mathematical theorem which is equivalent to the transform function, however is much easier to use. Future versions of the fourier series program will be accurate to even more decimal places, for any given defined force or other co-variant f(t) function input, and allow even more harmonics to computed much faster (including the equivalent of more harmonics) by using discrete values for both the force and integral area vector component(s), along with linear and exponential interpolation, with accuracies of 8, 12, and 16 decimal places for any given equivalent harmonic series input. Purchasers of Calcgate will be offered an upgrade at reduced prices when the updated version becomes available.