Class Wiki - User contributions [en]

Robert's HW

2010-11-01T23:51:37Z

Silent protagonist:

I decided to model the natural response of a boat given a small initial list on flat water.

Assume a boat of arbitrary geometry, with a given displacement(weight) D, mass moment of inertia I, and metacentric height GM (The metacenter is a theoretical point in a boat through which the buoyant force always passes for small angles of list.).

[[Image:Righting_arm.png|thumb|right|500px|]]

As the boat lists at angle φ, the centroid of the displaced volume of water shifts in the same direction, causing the buoyant force to be offset, resulting in a moment acting to right the boat. This righting moment is equal to the displacement of the boat times the righting arm GZ, where:

GZ = GM*sin φ

Which at small angles of φ can be approximated:

GZ = GM*φ

Thus, summing moments about the center of gravity of the boat:

<math>\sum M_G = (D) (GM) \varphi = I \ddot{\varphi}</math>

Rearranging:

<math> \ddot{\varphi} - \dfrac{ (D) (GM) }{I} \varphi = 0</math>

This is a simple ODE that may be solved using Laplace transforms.

<math>\displaystyle\mathcal{L} \left\{ \ddot{\varphi} - \dfrac{ (D) (GM) }{I} \varphi \right\} = \left ( s^2 \Phi(s) - s\varphi_0 - \dot{\varphi_0} \right ) - \left( \dfrac{ (D) (GM) }{I} \Phi(s) \right) </math>

Solving for <math>\Phi</math>:

<math> \Phi(s)= \dfrac{ s\varphi_0 + \dot{\varphi_0} }{s^2- \dfrac{ (D) (GM) }{I} } = \varphi_0 \left( \dfrac {s}{s^2- \dfrac{ (D) (GM) }{I}} \right) + \dot{\varphi_0} \left( \dfrac{1}{s^2- \dfrac{ (D) (GM) }{I}} \right) </math>

Assuming no initial angular velocity (<math> \dot{\varphi_0}=0 </math>) and converting back to time domain:

<math> \varphi(t)= \mathcal{L}^{-1} \{ \Phi(s) \} = \varphi_0 cos \left( - \sqrt{\dfrac{ (D) (GM) }{I}} t \right)u(t) </math>

This implies that, like a pendulum, a boat has a natural period of oscilation determined only by it's own physical properties. This period is important when considering more dynamic problems such as the motion of a boat in a storm.

Robert's HW

2010-11-01T23:49:37Z

Silent protagonist:

I decided to model the natural response of a boat given a small initial list on flat water.

Assume a boat of arbitrary geometry, with a given displacement(weight) D, mass moment of inertia I, and metacentric height GM (The metacenter is a theoretical point in a boat through which the buoyant force always passes for small angles of list.).

[[Image:Righting_arm.png|thumb|right|500px|]]

As the boat lists at angle φ, the centroid of the displaced volume of water shifts in the same direction, causing the buoyant force to be offset, resulting in a moment acting to right the boat. This righting moment is equal to the displacement of the boat times the righting arm GZ, where:

GZ = GM*sin φ

Which at small angles of φ can be approximated:

GZ = GM*φ

Thus, summing moments about the center of gravity of the boat:

<math>\sum M_G = (D) (GM) \varphi = I \ddot{\varphi}</math>

Rearranging:

<math> \ddot{\varphi} - \dfrac{ (D) (GM) }{I} \varphi = 0</math>

This is a simple ODE that may be solved using Laplace transforms.

<math>\displaystyle\mathcal{L} \left\{ \ddot{\varphi} - \dfrac{ (D) (GM) }{I} \varphi \right\} = \left ( s^2 \Phi(s) - s\varphi_0 - \dot{\varphi_0} \right ) - \left( \dfrac{ (D) (GM) }{I} \Phi(s) \right) </math>

Solving for <math>\Phi</math>:

<math> \Phi(s)= \dfrac{ s\varphi_0 + \dot{\varphi_0} }{s^2- \dfrac{ (D) (GM) }{I} } = \varphi_0 \left( \dfrac {s}{s^2- \dfrac{ (D) (GM) }{I}} \right) + \dot{\varphi_0} \left( \dfrac{1}{s^2- \dfrac{ (D) (GM) }{I}} \right) </math>

Assuming no initial angular velocity (<math> \dot{\varphi_0}=0 </math>) and converting back to time domain:

<math> \varphi(t)= \mathcal{L}^{-1} \{ \Phi(s) \} = \varphi_0 cos \left( - \dfrac{ (D) (GM) }{I} t \right)u(t) </math>

This implies that, like a pendulum, a boat has a natural period of oscilation determined only by it's own physical properties. This period is important when considering more dynamic problems such as the motion of a boat in a storm.

File:Righting arm.png

2010-11-01T23:46:13Z

Silent protagonist: Source: http://en.wikipedia.org/wiki/File:Righting_arm.png

Source:
http://en.wikipedia.org/wiki/File:Righting_arm.png

Robert's HW

2010-11-01T23:42:46Z

Silent protagonist:

I decided to model the natural response of a boat given a small initial list on flat water.

Assume a boat of arbitrary geometry, with a given displacement(weight) D, mass moment of inertia I, and metacentric height GM (The metacenter is a theoretical point in a boat through which the buoyant force always passes for small angles of list.).

[[Image:Listing Boat|thumb|left|220px|]]

As the boat lists at angle φ, the centroid of the displaced volume of water shifts in the same direction, causing the buoyant force to be offset, resulting in a moment acting to right the boat. This righting moment is equal to the displacement of the boat times the righting arm GZ, where:

GZ = GM*sin φ

Which at small angles of φ can be approximated:

GZ = GM*φ

Thus, summing moments about the center of gravity of the boat:

<math>\sum M_G = (D) (GM) \varphi = I \ddot{\varphi}</math>

Rearranging:

<math> \ddot{\varphi} - \dfrac{ (D) (GM) }{I} \varphi = 0</math>

This is a simple ODE that may be solved using Laplace transforms.

<math>\displaystyle\mathcal{L} \left\{ \ddot{\varphi} - \dfrac{ (D) (GM) }{I} \varphi \right\} = \left ( s^2 \Phi(s) - s\varphi_0 - \dot{\varphi_0} \right ) - \left( \dfrac{ (D) (GM) }{I} \Phi(s) \right) </math>

Solving for <math>\Phi</math>:

<math> \Phi(s)= \dfrac{ s\varphi_0 + \dot{\varphi_0} }{s^2- \dfrac{ (D) (GM) }{I} } = \varphi_0 \left( \dfrac {s}{s^2- \dfrac{ (D) (GM) }{I}} \right) + \dot{\varphi_0} \left( \dfrac{1}{s^2- \dfrac{ (D) (GM) }{I}} \right) </math>

Assuming no initial angular velocity (<math> \dot{\varphi_0}=0 </math>) and converting back to time domain:

<math> \varphi(t)= \mathcal{L}^{-1} \{ \Phi(s) \} = \varphi_0 cos \left( - \dfrac{ (D) (GM) }{I} t \right)u(t) </math>

This implies that, like a pendulum, a boat has a natural period of oscilation determined only by it's own physical properties. This period is important when considering more dynamic problems such as the motion of a boat in a storm.

Robert's HW

2010-11-01T21:45:22Z

Silent protagonist:

I decided to model the natural response of a boat given a small initial list on flat water.

Assume a boat of arbitrary geometry, with a given displacement(weight) D, mass moment of inertia I, and metacentric height GM (The metacenter is a theoretical point in a boat through which the buoyant force always passes for small angles of list.).

[[Image:Listing Boat|thumb|left|220px|]]

As the boat lists at angle φ, the centroid of the displaced volume of water shifts in the same direction, causing the buoyant force to be offset, resulting in a moment acting to right the boat. This righting moment is equal to the displacement of the boat times the righting arm GZ, where:

GZ = GM*sin φ

Which at small angles of φ can be approximated:

GZ = GM*φ

Thus, summing moments about the center of gravity of the boat:

<math>\sum M_G = (D) (GM) \varphi = I \ddot{\varphi}</math>

Rearranging:

<math> \ddot{\varphi} - \dfrac{ (D) (GM) }{I} \varphi = 0</math>

This is a simple ODE that may be solved using Laplace transforms.

<math>\displaystyle\mathcal{L} \left\{ \ddot{\varphi} - \dfrac{ (D) (GM) }{I} \varphi \right\} = \left ( s^2 \Phi(s) - s\varphi_0 - \dot{\varphi_0} \right ) - \left( \dfrac{ (D) (GM) }{I} \Phi(s) \right) </math>

Solving for <math>\Phi</math>:

<math> \Phi(s)= \dfrac{ s\varphi_0 + \dot{\varphi_0} }{s^2- \dfrac{ (D) (GM) }{I} } = \varphi_0 \left( \dfrac {s}{s^2- \dfrac{ (D) (GM) }{I}} \right) + \dot{\varphi_0} \left( \dfrac{1}{s^2- \dfrac{ (D) (GM) }{I}} \right) </math>

Assuming no initial angular velocity (<math> \dot{\varphi_0}=0 </math>) and converting back to time domain:

<math> \varphi(t)= \mathcal{L}^{-1} \{ \Phi(s) \} = \varphi_0 cos \left( - \dfrac{ (D) (GM) }{I} t \right)u(t) </math>

Robert's HW

2010-11-01T21:37:24Z

Silent protagonist:

I decided to model the natural response of a boat given a small initial list on flat water.

Assume a boat of arbitrary geometry, with a given displacement(weight) D, mass moment of inertia I, and metacentric height GM (The metacenter is a theoretical point in a boat through which the buoyant force always passes for small angles of list.).

[[Image:Listing Boat|thumb|left|220px|]]

As the boat lists at angle φ, the centroid of the displaced volume of water shifts in the same direction, causing the buoyant force to be offset, resulting in a moment acting to right the boat. This righting moment is equal to the displacement of the boat times the righting arm GZ, where:

GZ = GM*sin φ

Which at small angles of φ can be approximated:

GZ = GM*φ

Thus, summing moments about the center of gravity of the boat:

<math>\sum M_G = (D) (GM) \varphi = I \ddot{\varphi}</math>

Rearranging:

<math> \ddot{\varphi} - \dfrac{ (D) (GM) }{I} \varphi = 0</math>

This is a simple ODE that may be solved using Laplace transforms.

<math>\displaystyle\mathcal{L} \left\{ \ddot{\varphi} - \dfrac{ (D) (GM) }{I} \varphi \right\} = \left ( s^2 \Phi(s) - s\varphi_0 - \dot{\varphi_0} \right ) - \left( \dfrac{ (D) (GM) }{I} \Phi(s) \right) </math>

Solving for <math>\Phi</math>:

<math> \Phi(s)= \dfrac{ s\varphi_0 + \dot{\varphi_0} }{s^2- \dfrac{ (D) (GM) }{I} } = \varphi_0 \left( \dfrac {s}{s^2- \dfrac{ (D) (GM) }{I}} \right) + \dfrac{\dot{\varphi_0}}{s^2- \dfrac{ (D) (GM) }{I}} </math>

Converting back to time domain:

<math> \varphi= \mathcal{L}^{-1} \{ \Phi(s) \}

Robert's HW

2010-11-01T17:48:06Z

Silent protagonist: Created page with 'I decided to model the natural response of a boat given a small initial list on flat water. Assume a boat of arbitrary geometry, with a given displacement(weight) D, mass moment…'

Fall 2010

2010-11-01T17:24:20Z

Silent protagonist:

Put links for your reports here.

Links to the posted reports are found under the publishing author's name.

Please number and sort Authors alphabetically.

0. [[Banton, Alex]]
*[[Alex's Octave Assignment]]
*[[Alex's Assignment #8]]

1. [[Bidwell, Kelvin]]
*[[Kelvin's Octave Assignment]]

2. [[Blaire, Matthew]]
*[[Matthew's Octave Assignment]]
*[[Matthew's Asgn #8]]

3. [[Boyd, Aaron]]
*[[ Aaron's Octave Assignment]]
*[[Aaron Boyd's Assignment 8]]

4. [[Bryson, David]]
*[[David's Octave Assignment]]

5. [[Colls, David]]
*[[Colls Octave Assignment]]
*[[Fourier Series Assignment]]

6. [[Fullerton, Colby]]
*[[Colby's Octave Assignment]]

7. [[Hildebrand, Kurt]]
*[[Kurt's Octave Assignment]]
*[[Kurt's Assignment #8]]

8. [[Martinez, Jonathan]]
*[[Martinez's Octave Assignment]]
*[[Martinez's Fourier Assignment]]

9. [[Morgan, David]]
*[[David Morgan's Octave Assignment]]
*[[David Morgan's Fourier Series assignment]]

10. [[Roth, Andrew]]
*[[Andrew Roth's Fourier Series/Laplace Transform Project]]
*[[Andrew's Octave Assignment]]
*[[Example: LaTex format]]
*[[Chapter 22--Fourier Series: Fundamental Period, Frequency, and Angular Frequency]]

11. [[Stirn, Jed]]
*[[Jed's Octave Assignment]]
*[[HW#8:Laplace Transforms/Fourier Series]]

12. [[Stringer, Robert]]
*[[Robert's Octave Assignment]]
*[[Robert's HW #8]]

13. [[Wooley, Andy]]
*[[Andy's Octave Assignment]]
*[[Andy's Fourier Series Project]]

14. [[Zimmerly, Brian]]
*[[Brian's Octave Assignment]]

Robert's Octave Assignment

2010-10-04T17:38:58Z

Silent protagonist: Created page with 'Solving a System of Non-Linear Equations: Octave can solve a system of non-linear equations of the form f(x)=0 using the function fsolve fsolve (fcn, x0, options) [x, fvec, i…'

Solving a System of Non-Linear Equations:
Octave can solve a system of non-linear equations of the form

f(x)=0

using the function fsolve
fsolve (fcn, x0, options)
[x, fvec, info, output, fjac]= fsolve (fcn,... )
Where fcn is a vector containing the system, and x0 is a vector containing initial guesses for each variable (nescessary for this particular algorithm).

To solve the system
<math>−2x^2+3xy+4sin(y)-6=0</math>
<math>3x^2-2xy^2+3cos(x)+4=0</math>

Enter the following:
%define a function for fsolve to use, containing the system
functiony=f(x)
y(1)=-2*x(1)^2+3*x(1)*x(2)+4*sin(x(2))-6;
y(2)=3*x(1)^2-2*x(1)*x(2)^2+3*cos(x(1))+4;
endfunction

%call fsolve, and place the results into a vector
[x,fval,info]=fsolve(@f,[1;2])

See pg 331 in the Octave Manual for more info.

Stringer, Robert

2010-10-04T17:02:14Z

Silent protagonist:

Contact Info:
Phone: (804)929-7007
E-mail: silent.protagonist.42@gmail.com

Stringer, Robert

2010-10-04T17:00:31Z

Silent protagonist: Created page with 'Contact Info: Phone: (804)929-7007 E-mail: silent.protagonist.42@gmail.com'

Contact Info:
Phone: (804)929-7007
E-mail: silent.protagonist.42@gmail.com

Fall 2010

2010-10-04T16:58:45Z

Silent protagonist:

Put links for your reports here.

Links to the posted reports are found under the publishing author's name.

Please number and sort Authors alphabetically.

1. [[Bidwell, Kelvin]]
*[[Kelvin's Octave Assignment]]

2. [[Blaire, Matthew]]
*[[Matthew's Octave Assignment]]

3. [[Bryson, David]]
*[[David's Octave Assignment]]

4. [[Colls, David]]
*[[Colls Octave Assignment]]

5. [[Fullerton, Colby]]
*[[Colby's Octave Assignment]]

6. [[Martinez, Jonathan]]
*[[Martinez's Octave Assignment]]

7. [[Morgan, David]]
*[[David Morgan's Octave Assignment]]

8. [[Roth, Andrew]]
*[[Andrew's Octave Assignment]]
*[[Example: LaTex format]]
*[[Chapter 22--Fourier Series: Fundamental Period, Frequency, and Angular Frequency]]

9. [[Stirn, Jed]]
*[[Jed's Octave Assignment]]

10. [[Stringer, Robert]]
*[[Robert's Octave Assignment]]

11. [[Wooley, Andy]]
*[[Andy's Octave Assignment]]

12. [[Zimmerly, Brian]]
*[[Brian's Octave Assignment]]