Mixing
Minimising a complex equation
$begingroup$
I'm trying to find a way to find a value(s) of $omega$ which minimise
$$mathbb{R} = 1 - frac{1+R_0^4-2R_0^2}{1+R_0^4 -2R_0^2cos(2omegabeta_1 L)}.$$
The terms in the equation depend on $omega$, namely
$$R_0 = frac{barrho_0sqrt{frac{bar rho_1}{c_{44}}}+barrho_1 sqrt{frac{barrho_1}{mu_0}}}{bar rho_1 sqrt{frac{bar rho_0}{mu_0}}-bar rho_0 sqrt{frac{bar rho_1}{c_{44}}}}$$
where
$$beta_1 = sqrt{frac{bar rho_1}{c_{44}}},$$
$$bar rho_0 = rho_0 - frac{k_1^2mu_0}{omega^2},$$
$$bar rho_1 = rho_1 - frac{k_1^2c_{66}}{omega^2}-frac{k_1^2c_{64}^2}{omega^2 c_{44}},$$
$$bar rho_2 = rho_2 - frac{k_1^2mu_2}{omega^2}.$$
The constants in this equation are
$$mu_{0,2}, rho_{0,1,2}, c_{44}, c_{64},c_{66},k_1,L$$
Am I in over my head here? Is there anything I could do to make analytical headway?
calculus optimization classical-mechanics
edited Jan 31 at 21:44
gt6989b
35.6k22557
asked Jan 31 at 21:38
rodger_kicksrodger_kicks
11010
$endgroup$
add a comment |
$begingroup$
I'm trying to find a way to find a value(s) of $omega$ which minimise
$$mathbb{R} = 1 - frac{1+R_0^4-2R_0^2}{1+R_0^4 -2R_0^2cos(2omegabeta_1 L)}.$$
The terms in the equation depend on $omega$, namely
$$R_0 = frac{barrho_0sqrt{frac{bar rho_1}{c_{44}}}+barrho_1 sqrt{frac{barrho_1}{mu_0}}}{bar rho_1 sqrt{frac{bar rho_0}{mu_0}}-bar rho_0 sqrt{frac{bar rho_1}{c_{44}}}}$$
where
$$beta_1 = sqrt{frac{bar rho_1}{c_{44}}},$$
$$bar rho_0 = rho_0 - frac{k_1^2mu_0}{omega^2},$$
$$bar rho_1 = rho_1 - frac{k_1^2c_{66}}{omega^2}-frac{k_1^2c_{64}^2}{omega^2 c_{44}},$$
$$bar rho_2 = rho_2 - frac{k_1^2mu_2}{omega^2}.$$
The constants in this equation are
$$mu_{0,2}, rho_{0,1,2}, c_{44}, c_{64},c_{66},k_1,L$$
Am I in over my head here? Is there anything I could do to make analytical headway?
calculus optimization classical-mechanics
edited Jan 31 at 21:44
gt6989b
35.6k22557
asked Jan 31 at 21:38
rodger_kicksrodger_kicks
11010
$endgroup$
$begingroup$
It looks really ugly. Where did it come from? Continuum mechanics?
$endgroup$
– Botond
Jan 31 at 21:41
$begingroup$
Yes it's a result from a wave problem. I'm trying to find a way to minimise this to take fourier transform to look at waves in time domain...
$endgroup$
– rodger_kicks
Jan 31 at 21:45
add a comment |
$begingroup$
I'm trying to find a way to find a value(s) of $omega$ which minimise
$$mathbb{R} = 1 - frac{1+R_0^4-2R_0^2}{1+R_0^4 -2R_0^2cos(2omegabeta_1 L)}.$$
The terms in the equation depend on $omega$, namely
$$R_0 = frac{barrho_0sqrt{frac{bar rho_1}{c_{44}}}+barrho_1 sqrt{frac{barrho_1}{mu_0}}}{bar rho_1 sqrt{frac{bar rho_0}{mu_0}}-bar rho_0 sqrt{frac{bar rho_1}{c_{44}}}}$$
where
$$beta_1 = sqrt{frac{bar rho_1}{c_{44}}},$$
$$bar rho_0 = rho_0 - frac{k_1^2mu_0}{omega^2},$$
$$bar rho_1 = rho_1 - frac{k_1^2c_{66}}{omega^2}-frac{k_1^2c_{64}^2}{omega^2 c_{44}},$$
$$bar rho_2 = rho_2 - frac{k_1^2mu_2}{omega^2}.$$
The constants in this equation are
$$mu_{0,2}, rho_{0,1,2}, c_{44}, c_{64},c_{66},k_1,L$$
Am I in over my head here? Is there anything I could do to make analytical headway?
calculus optimization classical-mechanics
edited Jan 31 at 21:44
gt6989b
35.6k22557
asked Jan 31 at 21:38
rodger_kicksrodger_kicks
11010
$endgroup$
I'm trying to find a way to find a value(s) of $omega$ which minimise
$$mathbb{R} = 1 - frac{1+R_0^4-2R_0^2}{1+R_0^4 -2R_0^2cos(2omegabeta_1 L)}.$$
The terms in the equation depend on $omega$, namely
$$R_0 = frac{barrho_0sqrt{frac{bar rho_1}{c_{44}}}+barrho_1 sqrt{frac{barrho_1}{mu_0}}}{bar rho_1 sqrt{frac{bar rho_0}{mu_0}}-bar rho_0 sqrt{frac{bar rho_1}{c_{44}}}}$$
where
$$beta_1 = sqrt{frac{bar rho_1}{c_{44}}},$$
$$bar rho_0 = rho_0 - frac{k_1^2mu_0}{omega^2},$$
$$bar rho_1 = rho_1 - frac{k_1^2c_{66}}{omega^2}-frac{k_1^2c_{64}^2}{omega^2 c_{44}},$$
$$bar rho_2 = rho_2 - frac{k_1^2mu_2}{omega^2}.$$
The constants in this equation are
$$mu_{0,2}, rho_{0,1,2}, c_{44}, c_{64},c_{66},k_1,L$$
Am I in over my head here? Is there anything I could do to make analytical headway?
calculus optimization classical-mechanics
calculus optimization classical-mechanics
edited Jan 31 at 21:44
gt6989b
35.6k22557
asked Jan 31 at 21:38
rodger_kicksrodger_kicks
11010
edited Jan 31 at 21:44
gt6989b
35.6k22557
asked Jan 31 at 21:38
rodger_kicksrodger_kicks
11010
edited Jan 31 at 21:44
gt6989b
35.6k22557
edited Jan 31 at 21:44
gt6989b
35.6k22557
edited Jan 31 at 21:44
gt6989b
35.6k22557
35.6k22557
asked Jan 31 at 21:38
rodger_kicksrodger_kicks
11010
asked Jan 31 at 21:38
rodger_kicksrodger_kicks
11010
asked Jan 31 at 21:38
rodger_kicksrodger_kicks
11010
11010
$begingroup$
It looks really ugly. Where did it come from? Continuum mechanics?
$endgroup$
– Botond
Jan 31 at 21:41
$begingroup$
Yes it's a result from a wave problem. I'm trying to find a way to minimise this to take fourier transform to look at waves in time domain...
$endgroup$
– rodger_kicks
Jan 31 at 21:45
add a comment |
$begingroup$
It looks really ugly. Where did it come from? Continuum mechanics?
$endgroup$
– Botond
Jan 31 at 21:41
$begingroup$
Yes it's a result from a wave problem. I'm trying to find a way to minimise this to take fourier transform to look at waves in time domain...
$endgroup$
– rodger_kicks
Jan 31 at 21:45
$begingroup$
It looks really ugly. Where did it come from? Continuum mechanics?
$endgroup$
– Botond
Jan 31 at 21:41
$begingroup$
It looks really ugly. Where did it come from? Continuum mechanics?
$endgroup$
– Botond
Jan 31 at 21:41
$begingroup$
Yes it's a result from a wave problem. I'm trying to find a way to minimise this to take fourier transform to look at waves in time domain...
$endgroup$
– rodger_kicks
Jan 31 at 21:45
$begingroup$
Yes it's a result from a wave problem. I'm trying to find a way to minimise this to take fourier transform to look at waves in time domain...
$endgroup$
– rodger_kicks
Jan 31 at 21:45
add a comment |
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$begingroup$
It looks really ugly. Where did it come from? Continuum mechanics?
$endgroup$
– Botond
Jan 31 at 21:41
$begingroup$
Yes it's a result from a wave problem. I'm trying to find a way to minimise this to take fourier transform to look at waves in time domain...
$endgroup$
– rodger_kicks
Jan 31 at 21:45