math problem (sinusoidal motion)
math problem (sinusoidal motion)
(OP)
Does anyone know how to solve the following equation:
-A*2*PI/6*COS(2*PI*t/6)-B*2*PI/18*COS(2*PI*t/18)=0
Where: A is a constant
B is a constant
t = time
Find: all values of t between 0 & 18 such that the above statment is true.
the function is velocity of a point subjected to sinusoidal motion of 2 frequencies & 2 amplitudes. Times of Max. acceleration to design some hydraulic actuators to impart said motion.
-A*2*PI/6*COS(2*PI*t/6)-B*2*PI/18*COS(2*PI*t/18)=0
Where: A is a constant
B is a constant
t = time
Find: all values of t between 0 & 18 such that the above statment is true.
the function is velocity of a point subjected to sinusoidal motion of 2 frequencies & 2 amplitudes. Times of Max. acceleration to design some hydraulic actuators to impart said motion.





RE: math problem (sinusoidal motion)
If you "heard" it on the internet, it's guilty until proven innocent. - DCS
RE: math problem (sinusoidal motion)
(PI/3)*(-Acos(PI*t/3) - B/3*cos(PI*t/9))=0
PI/3 term drops out. Let x=PI*t/9
-Acos(3x)-B/3cos(x)=0
use triple angle trig identity:
-4A(cos(x)^3)+3Acos(x)-B/3cos(x)=0
let y = cos(x)
-4Ay^3+(3A-B/3)y=0
Problem is reduced to a simple cubic.
RE: math problem (sinusoidal motion)
factor ...
y*(-4y^2+(3A-B/3)) = 0
y = 0, sqrt(3A-B/3)/2 (+ve and -ve)
and y = cos(pi/9*t) ...
RE: math problem (sinusoidal motion)
RE: math problem (sinusoidal motion)
You can find at least *some* of the values of t without knowing A and B though, since the equation is looking for zero-crossings.
If both the cosines are zero, then the equation is satisfied regardless of A and B, since they are multiplied by 0. Cosines are zero at pi/2, 3*pi/2, 5*pi/2, etc. If you set the terms inside the cosines to n*pi/2, where n is an odd integer, then pi*t/3 solves to
t = 1.5 n
and pi*t/9 solves to
t = 4.5 n
where n is an odd integer.
The only time t<18 that both these equations are true is when
t = 4.5
This may not be the only solution, but it is a solution regardless of the values of A and B.
Don
Kansas City
RE: math problem (sinusoidal motion)
RE: math problem (sinusoidal motion)
RE: math problem (sinusoidal motion)
RE: math problem (sinusoidal motion)
Not to be a pill... but why are we answering something that is obviously a homework problem?
Wes C.
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RE: math problem (sinusoidal motion)
Thanks for the help.
I'll chock up the condescending comments to not putting enough detail in my question.
RE: math problem (sinusoidal motion)
The solutions are:
t1=(9/π)cos-1(√(0.75-B/12A))
t3=(9/π)cos-1(-√(0.75-B/12A))
t4=t1+9
t6=t3+9
Unfortunately scully44, if you looked for simpler formulae, that's it. However it turns out that t1 tends to 1.5 when A/B goes towards infinite, that it is of course equal to 4.5 when A=B/9 and that it is already =~1.76 when A=B. Similarly t3 tends to 7.5 (starting again at 4.5) and is already =~7.24 when A=B.
prex
http://www.xcalcs.com : Online tools for structural design
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RE: math problem (sinusoidal motion)
I would have graphed it over 0<t<18 and see where it hits 0. This might have been tough if all you have available is a spreadsheet but most math programs could do this. Took a minute or two to perform this in Maple and then you can manipulate the numbers to better model the system instead of just looking a few numbers.
RE: math problem (sinusoidal motion)
this = A*(3/4-B/12A), not (3/4-B/12A)
RE: math problem (sinusoidal motion)
y*(-4y^2+(3A-B/3)) = 0
should be
y*(-4Ay^2+(3A-B/3)) = 0
so ...
It is instructive to follow this other way of reasoning: the cos() function results necessarily in a non dimensional number, so its argument must also be non dimensional. Now A and B may be dimensional (e.g. an amplitude, hence a length), therefore the quantity under the radical may only depend on the ratio A/B to be non dimensional...
It is BTW an evidence from the first equation presented by scully44 that the roots may only depend on A/B, not on their separate values.
prex
http://www.xcalcs.com : Online tools for structural design
http://www.megamag.it : Magnetic brakes for fun rides
http://www.levitans.com : Air bearing pads
RE: math problem (sinusoidal motion)
RE: math problem (sinusoidal motion)
hopefully the attached file will help you. In this file, A=B=1. So, you can change the values of A and B in cell B7.
You can use the Solver feature or maybe even the Goal Seek (doubt it) feature in Excel to further assist you.
Good Luck!
-pmover