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math problem (sinusoidal motion)

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.

RE: math problem (sinusoidal motion)

The answer is that it depends on A and B.  It's great that you say A and B are constant, but their values must be known before you solve this one.  There are no general t values that will work for all selections of A and B.



If you "heard" it on the internet, it's guilty until proven innocent. - DCS

RE: math problem (sinusoidal motion)

Ever heard of algebra? Trig identities?

(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)

from there it's even simpler ...
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)

Whoops, good catch, rb.  blush

RE: math problem (sinusoidal motion)

That reduces it, guys, but you still have to know A and B to solve for t, which is apparently what he wants.

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)

i suspect that an "answer" in terms of A and B would meet the needs of the "problem"

RE: math problem (sinusoidal motion)

Don, the complete analytical solution as completed by rb1957 yields your answer in the "y=0" solution that is independent of A and B, along with the other two answers in terms of A and B.  It is implicitly obvious that to find actual values for the other two solutions one must know both A and B.  However, the way the original equation was posted leaves me with doubts as to whether the OP is capable of following along and understanding the solution anyway.

RE: math problem (sinusoidal motion)

my two cents: assuming 't' is real valued, then cos(pi*t/9) is real valued. If so, then Sqrt(3A-B/3) has to be real, and this is true if and only if the argument in the Sqrt is positive or zero, that is, 3A>=B/3, or A>=B/9 (that is, A is greater than or equal to B/9). This tells you not what A and B are, only that A has a constraint, it must be greater than or equal to one-ninth of B.

RE: math problem (sinusoidal motion)

Hi guys,

Not to be a pill... but why are we answering something that is obviously a homework problem?

Wes C.
------------------------------
No trees were killed in the sending of this message, but a large number of electrons were terribly inconvenienced.

RE: math problem (sinusoidal motion)

(OP)
Its not a homework problem and I am perfectly capable of following the solution along.  I don't deal with these sort of equations on a regular basis as 99% of my work involves statics.  I was hoping for a little help to save me the time of digging through text books I haven't looked at in 10 years.  I know the values of A and B & posted the equation that way in hopes of getting an answer in terms of A and B so that they could be put in a spread sheet & get new values of t by changing the inputs.  Yes it is obvious that t = 4.5 is one solution & t = 13.5 is another.  I am interested in the other times when the cos of A cos(PI*t/6)=-B cos(pi*t/18).  If A=B t=2.25,4.5 6.75 etc. IF B>A the only solution is t = 4.5 & 13.5 .  But if A>B there will be 6 solutions t=4.5, t=13.5 & 4 others.  The 4 others are the ones I'm interersted in.  My lack of familiarity with trig identity functions prevented me from solving this equation in a timely manner.

Thanks for the help.

I'll chock up the condescending comments to not putting enough detail in my question.

RE: math problem (sinusoidal motion)

rb1957 made a little mistake above, and it turns out that cos(πt/9)=±√(0.75-B/12A), so, when A>B/9 so the radical is real and is also <1, there are 4 more solutions besides the two obvious t2=4.5 and t5=13.5
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
http://www.megamag.it : Magnetic brakes for fun rides
http://www.levitans.com : Air bearing pads

RE: math problem (sinusoidal motion)

I would have taken the easy way out.

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)

prex made a little mistake in his factoring (at least the way i read his expression).  inside the sqrt i started with 3A-B/3 = 3A/4-B/12 when you bring the "2" inside the sqrt; ok.

this = A*(3/4-B/12A), not (3/4-B/12A)

RE: math problem (sinusoidal motion)

Well rb1957, your expression
y*(-4y^2+(3A-B/3)) = 0
should be
y*(-4Ay^2+(3A-B/3)) = 0
so ...wink
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)

touche, another "small" mistake !

RE: math problem (sinusoidal motion)

scully44,

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

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