Wave Forcing of Small Bodies - Revision history

Adi Kurniawan: /* The Marchenko Wave Drift Model */

2009-12-11T12:11:12Z

The Marchenko Wave Drift Model

← Older revision		Revision as of 12:11, 11 December 2009
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	Therefore		Therefore
	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	\frac{\mathrm{d}\bar{V}_{\tau}}{\mathrm{d}t} & =\frac{\mathrm{d}}{\mathrm{d}t}\left( \bar{V}_{\bar{x}}		\frac{\mathrm{d}\bar{V}_{\tau}}{\mathrm{d}t} &=\frac{\mathrm{d}}{\mathrm{d}t}\left( \bar{V}_{\bar{x}}
	\sqrt{1+\bar{\eta}^{\prime}{}^{2}}\right) \\		\sqrt{1+\bar{\eta}^{\prime}{}^{2}}\right) \\
	& =\frac{\mathrm{d}\bar{V}_{\bar{x}}}{\mathrm{d}t}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}+\bar		&=\frac{\mathrm{d}\bar{V}_{\bar{x}}}{\mathrm{d}t}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}+\bar
	{V}_{\bar{x}}\frac{\mathrm{d}}{\mathrm{d}t}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}\\		{V}_{\bar{x}}\frac{\mathrm{d}}{\mathrm{d}t}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}\\
	& =\frac{\mathrm{d}\bar{V}_{\bar{x}}}{\mathrm{d}t}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}+\bar		&=\frac{\mathrm{d}\bar{V}_{\bar{x}}}{\mathrm{d}t}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}+\bar
	{V}_{\bar{x}}\frac{1}{\sqrt{1+\bar{\eta}^{\prime}{}^{2}}}\bar{\eta}^{\prime		{V}_{\bar{x}}\frac{1}{\sqrt{1+\bar{\eta}^{\prime}{}^{2}}}\bar{\eta}^{\prime
	}\bar{\eta}^{\prime\prime}\frac{\mathrm{d}\bar{x}}{\mathrm{d}t}.		}\bar{\eta}^{\prime\prime}\frac{\mathrm{d}\bar{x}}{\mathrm{d}t}.

Adi Kurniawan: /* The Marchenko Wave Drift Model */

2009-12-11T12:10:12Z

The Marchenko Wave Drift Model

← Older revision		Revision as of 12:10, 11 December 2009
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	\sqrt{1+\bar{\eta}^{\prime}{}^{2}}\right) \\		\sqrt{1+\bar{\eta}^{\prime}{}^{2}}\right) \\
	& =\frac{\mathrm{d}\bar{V}_{\bar{x}}}{\mathrm{d}t}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}+\bar		& =\frac{\mathrm{d}\bar{V}_{\bar{x}}}{\mathrm{d}t}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}+\bar
	{V}_{\bar{x}}\frac{d}{dt}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}\\		{V}_{\bar{x}}\frac{\mathrm{d}}{\mathrm{d}t}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}\\
	& =\frac{\mathrm{d}\bar{V}_{\bar{x}}}{\mathrm{d}t}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}+\bar		& =\frac{\mathrm{d}\bar{V}_{\bar{x}}}{\mathrm{d}t}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}+\bar
	{V}_{\bar{x}}\frac{1}{\sqrt{1+\bar{\eta}^{\prime}{}^{2}}}\bar{\eta}^{\prime		{V}_{\bar{x}}\frac{1}{\sqrt{1+\bar{\eta}^{\prime}{}^{2}}}\bar{\eta}^{\prime

Adi Kurniawan at 12:09, 11 December 2009

2009-12-11T12:09:33Z

← Older revision		Revision as of 12:09, 11 December 2009
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	system is		system is
	<center><math>		<center><math>
	m\frac{d\bar{V}_{\tau}}{dt}=-mg\frac{\bar{\eta}^{\prime}}{\sqrt{1+\bar{\eta		m\frac{\mathrm{d}\bar{V}_{\tau}}{\mathrm{d}t}=-mg\frac{\bar{\eta}^{\prime}}{\sqrt{1+\bar{\eta
	}^{\prime}{}^{2}}} + F_d\,\,\,(1)		}^{\prime}{}^{2}}} + F_d\,\,\,(1)
	</math></center>		</math></center>
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	Therefore		Therefore
	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	\frac{d\bar{V}_{\tau}}{dt} & =\frac{d}{dt}\left( \bar{V}_{\bar{x}}		\frac{\mathrm{d}\bar{V}_{\tau}}{\mathrm{d}t} & =\frac{\mathrm{d}}{\mathrm{d}t}\left( \bar{V}_{\bar{x}}
	\sqrt{1+\bar{\eta}^{\prime}{}^{2}}\right) \\		\sqrt{1+\bar{\eta}^{\prime}{}^{2}}\right) \\
	& =\frac{d\bar{V}_{\bar{x}}}{dt}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}+\bar		& =\frac{\mathrm{d}\bar{V}_{\bar{x}}}{\mathrm{d}t}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}+\bar
	{V}_{\bar{x}}\frac{d}{dt}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}\\		{V}_{\bar{x}}\frac{d}{dt}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}\\
	& =\frac{d\bar{V}_{\bar{x}}}{dt}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}+\bar		& =\frac{\mathrm{d}\bar{V}_{\bar{x}}}{\mathrm{d}t}\sqrt{1+\bar{\eta}^{\prime}{}^{2}}+\bar
	{V}_{\bar{x}}\frac{1}{\sqrt{1+\bar{\eta}^{\prime}{}^{2}}}\bar{\eta}^{\prime		{V}_{\bar{x}}\frac{1}{\sqrt{1+\bar{\eta}^{\prime}{}^{2}}}\bar{\eta}^{\prime
	}\bar{\eta}^{\prime\prime}\frac{d\bar{x}}{dt}.		}\bar{\eta}^{\prime\prime}\frac{\mathrm{d}\bar{x}}{\mathrm{d}t}.
	\end{matrix}</math></center>		\end{matrix}</math></center>
	Substituting this in equation (1) we obtain		Substituting this in equation (1) we obtain
	<center><math>		<center><math>
	m\frac{d\bar{V}_{\bar{x}}}{dt}=-m(g+\left( \bar{V}_{\bar{x}}\right) ^{2}		m\frac{\mathrm{d}\bar{V}_{\bar{x}}}{\mathrm{d}t}=-m(g+\left( \bar{V}_{\bar{x}}\right) ^{2}
	\bar{\eta}^{\prime\prime})\frac{\bar{\eta}^{\prime}}{1+\bar{\eta}^{\prime}		\bar{\eta}^{\prime\prime})\frac{\bar{\eta}^{\prime}}{1+\bar{\eta}^{\prime}
	{}^{2}} + F_d.\,\,\,(2)		{}^{2}} + F_d.\,\,\,(2)
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	[[Rumer et. al. 1979]] included added mass and this makes the model		[[Rumer et. al. 1979]] included added mass and this makes the model
	<center><math>		<center><math>
	m\left( 1+C_{m}\right) \frac{~~dV_~~{x}}{dt}=-m(g+\left( V_{x}\right) ^{2}		m\left( 1+C_{m}\right) \frac{\mathrm{d}V_{x}}{\mathrm{d}t}=-m(g+\left( V_{x}\right) ^{2}
	\eta^{\prime\prime})\frac{\eta^{\prime}}{1+\eta^{\prime}{}^{2}} + F_d,\,\,\,(3)		\eta^{\prime\prime})\frac{\eta^{\prime}}{1+\eta^{\prime}{}^{2}} + F_d,\,\,\,(3)
	</math></center>		</math></center>
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	<center><math>		<center><math>
	m\left( 1+C_{m}\right) \frac{dV}{dt}=-mg\eta^{\prime}+\rho		m\left( 1+C_{m}\right) \frac{\mathrm{d}V}{\mathrm{d}t}=-mg\eta^{\prime}+\rho
	_{w}C_{w}A_{i}\,\,\bigg\|V_{w}-V\bigg\|\bigg(V_{w}-V\bigg)\,\,\,(4)		_{w}C_{w}A_{i}\,\,\bigg\|V_{w}-V\bigg\|\bigg(V_{w}-V\bigg)\,\,\,(4)
	</math></center>		</math></center>
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	(4) into (5) gives		(4) into (5) gives
	<center><math>		<center><math>
	\left( 1+C_{m}\right) m\frac{dV}{dt}\,=-m\,g\,\frac{kA}{2}\cos(kx-\omega		\left( 1+C_{m}\right) m\frac{\mathrm{d}V}{\mathrm{d}t}\,=-m\,g\,\frac{kA}{2}\cos(kx-\omega
	t)		t)
	+ \,\rho_{w}AC_{w}\|kcA\sin(k(x-ct))-V\|(kcA\sin(k(x-ct))-V). \,\,\,(6)		+ \,\rho_{w}AC_{w}\|kcA\sin(k(x-ct))-V\|(kcA\sin(k(x-ct))-V). \,\,\,(6)
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	about the behaviour of the equations using ideas from dynamical systems theory.		about the behaviour of the equations using ideas from dynamical systems theory.

	If the velocity of the body, <math>V,</math> is large, <math>dV/dt</math> can therefore be		If the velocity of the body, <math>V,</math> is large, <math>\mathrm{d}V/\mathrm{d}t</math> can therefore be
	approximated by		approximated by
	<center><math>		<center><math>
	\frac{dV}{dt}\approx-\sigma\tau\,\|V\|V,		\frac{\mathrm{d}V}{\mathrm{d}t}\approx-\sigma\tau\,\|V\|V,
	</math></center>		</math></center>
	which means that if <math>V</math> is large and positive the <math>V</math> will decrease, if <math>V</math> is		which means that if <math>V</math> is large and positive the <math>V</math> will decrease, if <math>V</math> is
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	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	\iint_{S}\left( \frac{\partial P}{\partial V}+\frac{\partial Q}		\iint_{S}\left( \frac{\partial P}{\partial V}+\frac{\partial Q}
	{\partial\theta}\right) \ ~~dVd~~\theta & =\oint_{\Gamma}\left( Pd\theta		{\partial\theta}\right) \ \mathrm{d}V\mathrm{d}\theta & =\oint_{\Gamma}\left( P\mathrm{d}\theta
	-~~QdV~~\right) \\		-Q\mathrm{d}V\right) \\
	& =\int_{0}^{T}\left( P\frac{d\theta}{dt}-Q\frac{dV}{dt}\right) dt\\		& =\int_{0}^{T}\left( P\frac{\mathrm{d}\theta}{\mathrm{d}t}-Q\frac{\mathrm{d}V}{\mathrm{d}t}\right) \mathrm{d}t\\
	& =\int_{0}^{T}\left( PQ-QP\right) dt=0,		& =\int_{0}^{T}\left( PQ-QP\right) \mathrm{d}t=0,
	\end{matrix}</math></center>		\end{matrix}</math></center>
	which is known as Bendixon's criterion. However,		which is known as Bendixon's criterion. However,
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	<center><math>		<center><math>
	\iint_{S}\frac{\partial P}{\partial V}+\frac{\partial Q}{\partial\theta		\iint_{S}\frac{\partial P}{\partial V}+\frac{\partial Q}{\partial\theta
	}\ ~~dVd~~\theta>0		}\ \mathrm{d}V\mathrm{d}\theta>0
	</math></center>		</math></center>
	and a closed limit cycle cannot exist. Futhermore, we can use a similar		and a closed limit cycle cannot exist. Futhermore, we can use a similar
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	valid. At an equilibrium point <math>(\theta_{j},V_{j})</math> the conditions		valid. At an equilibrium point <math>(\theta_{j},V_{j})</math> the conditions
	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	\frac{dV}{dt} & =0,\\		\frac{\mathrm{d}V}{\mathrm{d}t} & =0,\\
	\frac{d\theta}{dt} & =0,		\frac{\mathrm{d}\theta}{\mathrm{d}t} & =0,
	\end{matrix}</math></center>		\end{matrix}</math></center>
	are satisfied. The second equation implies <math>V_{j}=1/\omega,</math> (which means that		are satisfied. The second equation implies <math>V_{j}=1/\omega,</math> (which means that

Adi Kurniawan at 12:05, 11 December 2009

2009-12-11T12:05:33Z

← Older revision		Revision as of 12:05, 11 December 2009
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	=Introduction=		==Introduction==

	While large bodies on the water surface will reflect and		While large bodies on the water surface will reflect and
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	[[Shen and Zhong 2001]]).		[[Shen and Zhong 2001]]).

	=The Marchenko Wave Drift Model=		==The Marchenko Wave Drift Model==

	We present the models for wave forcing of small floating bodies		We present the models for wave forcing of small floating bodies
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	==Including added mass==		===Including added mass===

	[[Rumer et. al. 1979]] included added mass and this makes the model		[[Rumer et. al. 1979]] included added mass and this makes the model
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	</math></center>		</math></center>

	==Drag==		===Drag===

	So far we have not considered the drag force although in practice the drag		So far we have not considered the drag force although in practice the drag
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	co-ordiante system.		co-ordiante system.

	=The Drift Velocity for a Linear Sinusoidal Wave=		==The Drift Velocity for a Linear Sinusoidal Wave==

	If we are going to use the slope sliding models in the context of linear wave		If we are going to use the slope sliding models in the context of linear wave
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	==Non-dimensionalisation==		===Non-dimensionalisation===

	We will now non-dimensionalise equation ((finalsystemxb)). This will		We will now non-dimensionalise equation ((finalsystemxb)). This will
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	It should also be noted that all the constants must be positive.		It should also be noted that all the constants must be positive.

	==The behaviour of the solutions==		===The behaviour of the solutions===

	Analytic solutions for system (4) cannot be obtained easily.		Analytic solutions for system (4) cannot be obtained easily.
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	is false (as long as the surface friction is not assumed to be zero).		is false (as long as the surface friction is not assumed to be zero).

	==Equilibrium points==		===Equilibrium points===

	We will now investigate the existence of the equilibrium points. For this		We will now investigate the existence of the equilibrium points. For this

Meylan: /* Equilibrium points */

2006-06-15T23:35:00Z

Equilibrium points

← Older revision		Revision as of 23:35, 15 June 2006
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	-\omega^{2}\,\cos{\theta}+\sigma\left\vert \omega\sin{\theta}-1/\omega		-\omega^{2}\,\cos{\theta}+\sigma\left\vert \omega\sin{\theta}-1/\omega
	\right\vert \,\left( \omega\sin{\theta}-1/\omega\right) =0\,\,\,(8)		\right\vert \,\left( \omega\sin{\theta}-1/\omega\right) =0\,\,\,(8)
	~~(equipoint)~~
	</math></center>		</math></center>
	If <math>0\leq\theta\leq{\pi}/{2}</math> or <math>{3\pi}/{2}\leq\theta\leq2\pi</math>, this equation		If <math>0\leq\theta\leq{\pi}/{2}</math> or <math>{3\pi}/{2}\leq\theta\leq2\pi</math>, this equation
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	\frac{\partial P}{\partial\theta}(\theta_{1},0) > 0,\quad \mathrm{and}\quad		\frac{\partial P}{\partial\theta}(\theta_{1},0) > 0,\quad \mathrm{and}\quad
	\frac{\partial P}{\partial\theta}(\theta_{2},0) < 0.\,\,\,(9)		\frac{\partial P}{\partial\theta}(\theta_{2},0) < 0.\,\,\,(9)
	~~(equiclassi)~~
	</math></center>		</math></center>
	We can determine the type of the equilibrium points by considering the		We can determine the type of the equilibrium points by considering the

Meylan: /* The Drift Velocity for a Linear Sinusoidal Wave */

2006-06-15T23:33:18Z

The Drift Velocity for a Linear Sinusoidal Wave

← Older revision		Revision as of 23:33, 15 June 2006
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	<center><math>		<center><math>
	m\left( 1+C_{m}\right) \frac{dV}{dt}=-mg\eta^{\prime}+\rho		m\left( 1+C_{m}\right) \frac{dV}{dt}=-mg\eta^{\prime}+\rho
	_{w}C_{w}A_{i}\,\,\bigg\|V_{w}-V\bigg\|\bigg(V_{w}-V\bigg)\,\,\,(4~~) (linear_force~~)		_{w}C_{w}A_{i}\,\,\bigg\|V_{w}-V\bigg\|\bigg(V_{w}-V\bigg)\,\,\,(4)
	</math></center>		</math></center>
	where <math>V</math> is the velocity in the tangential or <math>x</math> direction. This equation is		where <math>V</math> is the velocity in the tangential or <math>x</math> direction. This equation is
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	the wave profile is given by a single frequency linear wave		the wave profile is given by a single frequency linear wave
	<center><math>		<center><math>
	\eta(x,t)=\frac{H}{2}\sin(k(x-ct)),\,\,\,(5~~) (linear_wave~~)		\eta(x,t)=\frac{H}{2}\sin(k(x-ct)),\,\,\,(5)
	</math></center>		</math></center>
	where <math>H</math> is the wave height and <math>k</math> is the wave number. The		where <math>H</math> is the wave height and <math>k</math> is the wave number. The
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	</math></center>		</math></center>
	where we have assumed that the water depth in infinite. Substituting equation		where we have assumed that the water depth in infinite. Substituting equation
	(4~~)(linear_wave)~~) into (5~~)(linear_force)~~) gives		(4) into (5) gives
	<center><math>		<center><math>
	\left( 1+C_{m}\right) m\frac{dV}{dt}\,=-m\,g\,\frac{kA}{2}\cos(kx-\omega		\left( 1+C_{m}\right) m\frac{dV}{dt}\,=-m\,g\,\frac{kA}{2}\cos(kx-\omega
	t)		t)
	+ \,\rho_{w}AC_{w}\|kcA\sin(k(x-ct))-V\|(kcA\sin(k(x-ct))-V). \,\,\,(6~~) (finalsystemxb~~)		+ \,\rho_{w}AC_{w}\|kcA\sin(k(x-ct))-V\|(kcA\sin(k(x-ct))-V). \,\,\,(6)
	</math></center>		</math></center>

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	\frac{d\theta}{d\tilde{t}}\,=\,\omega^{2}\tilde{V}-\omega		\frac{d\theta}{d\tilde{t}}\,=\,\omega^{2}\tilde{V}-\omega
	\end{matrix}\,\,\,(7)		\end{matrix}\,\,\,(7)
	~~(autonomous)~~
	</math></center>		</math></center>
	where the variables are now non-dimensional and where		where the variables are now non-dimensional and where

Meylan: /* Including added mass */

2006-06-15T23:31:59Z

Including added mass

← Older revision		Revision as of 23:31, 15 June 2006
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	m\left( 1+C_{m}\right) \frac{dV_{x}}{dt}=-m(g+\left( V_{x}\right) ^{2}		m\left( 1+C_{m}\right) \frac{dV_{x}}{dt}=-m(g+\left( V_{x}\right) ^{2}
	\eta^{\prime\prime})\frac{\eta^{\prime}}{1+\eta^{\prime}{}^{2}} + F_d,\,\,\,(3)		\eta^{\prime\prime})\frac{\eta^{\prime}}{1+\eta^{\prime}{}^{2}} + F_d,\,\,\,(3)
	~~(rumer_correct_added_mass)~~
	</math></center>		</math></center>

Meylan: /* The Marchenko Wave Drift Model */

2006-06-15T23:31:36Z

The Marchenko Wave Drift Model

← Older revision		Revision as of 23:31, 15 June 2006
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	<center><math>		<center><math>
	m\frac{d\bar{V}_{\tau}}{dt}=-mg\frac{\bar{\eta}^{\prime}}{\sqrt{1+\bar{\eta		m\frac{d\bar{V}_{\tau}}{dt}=-mg\frac{\bar{\eta}^{\prime}}{\sqrt{1+\bar{\eta
	}^{\prime}{}^{2}}} + F_d\,\,\,(1~~) (marchenko_original~~)		}^{\prime}{}^{2}}} + F_d\,\,\,(1)
	</math></center>		</math></center>
	where <math>\bar{V}_{\tau}</math> is the tangential velocity and		where <math>\bar{V}_{\tau}</math> is the tangential velocity and
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	m\frac{d\bar{V}_{\bar{x}}}{dt}=-m(g+\left( \bar{V}_{\bar{x}}\right) ^{2}		m\frac{d\bar{V}_{\bar{x}}}{dt}=-m(g+\left( \bar{V}_{\bar{x}}\right) ^{2}
	\bar{\eta}^{\prime\prime})\frac{\bar{\eta}^{\prime}}{1+\bar{\eta}^{\prime}		\bar{\eta}^{\prime\prime})\frac{\bar{\eta}^{\prime}}{1+\bar{\eta}^{\prime}
	{}^{2}} + F_d.\,\,\,(2~~) (marchenko_travelling~~)		{}^{2}} + F_d.\,\,\,(2)
	</math></center>		</math></center>
	which is the Marchenko model in the <math>\bar{x}</math> direction.		which is the Marchenko model in the <math>\bar{x}</math> direction.

Meylan: /* Equilibrium points */

2006-06-14T23:55:06Z

Equilibrium points

← Older revision		Revision as of 23:55, 14 June 2006
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	<math>0\leq\theta\leq{\pi}/{2}</math> or <math>{3\pi}/{2}\leq\theta\leq2\pi</math> it follows that		<math>0\leq\theta\leq{\pi}/{2}</math> or <math>{3\pi}/{2}\leq\theta\leq2\pi</math> it follows that
	<center><math>		<center><math>
			\frac{\partial P}{\partial\theta}(\theta_{1},0) > 0,\quad \mathrm{and}\quad
	\frac{\partial P}{\partial\theta}(\theta_{1},0)~~\ &~~ >\ 0,\\		\frac{\partial P}{\partial\theta}(\theta_{2},0) < 0.\,\,\,(9)
	\frac{\partial P}{\partial\theta}(\theta_{2},0)~~\ &~~ <\ 0.

	(equiclassi)		(equiclassi)
	</math></center>		</math></center>
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	\left(		\left(
	\begin{matrix}		\begin{matrix}
	~~[c]{cc}~~
	-2\sigma\left\vert \omega\sin\omega-1/\omega\right\vert & \frac{\partial		-2\sigma\left\vert \omega\sin\omega-1/\omega\right\vert & \frac{\partial
	P}{\partial\theta}\\		P}{\partial\theta}\\
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	^{2}\theta}}}.		^{2}\theta}}}.
	</math></center>		</math></center>
	It follows from (~~(equiclassi)~~) that at <math>\theta_{1}</math> there is one		It follows from (9) that at <math>\theta_{1}</math> there is one
	eigenvalue with positive real part and one with negative real part. At		eigenvalue with positive real part and one with negative real part. At
	<math>\theta_{2}</math> both eigenvalues have a negative real part. Applying the		<math>\theta_{2}</math> both eigenvalues have a negative real part. Applying the
	equilibrium point classification theorems ~~([[perko91]])~~, <math>\theta_{1}</math> is a		equilibrium point classification theorems, <math>\theta_{1}</math> is a
	saddle point and <math>\theta_{2}</math> is an attracting node or a spiral.		saddle point and <math>\theta_{2}</math> is an attracting node or a spiral.

	In summary we have shown that all solutions to equation (~~(autonomous)~~) and		In summary we have shown that all solutions to equation (7) and
	hence to equation (~~(finalsystemxb)~~) must tend to either an equilibrium		hence to equation (6) must tend to either an equilibrium
	point (at which the velocity of the body is given by the wave phase speed) or		point (at which the velocity of the body is given by the wave phase speed) or
	to an encircling limit cycle. There can exist at most one encircling limit		to an encircling limit cycle. There can exist at most one encircling limit

Meylan: /* Equilibrium points */

2006-06-14T23:52:30Z

Equilibrium points

← Older revision		Revision as of 23:52, 14 June 2006
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	<center><math>		<center><math>
	-\omega^{2}\,\cos{\theta}+\sigma\left\vert \omega\sin{\theta}-1/\omega		-\omega^{2}\,\cos{\theta}+\sigma\left\vert \omega\sin{\theta}-1/\omega
	\right\vert \,\left( \omega\sin{\theta}-1/\omega\right) =0		\right\vert \,\left( \omega\sin{\theta}-1/\omega\right) =0\,\,\,(8)
	(equipoint)		(equipoint)
	</math></center>		</math></center>