University of Florida/Egm4313/s12.team14.report2

Problem 1

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Given

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Find

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Find the non-homogeneous L2-ODE-CC in standard form and the solution in terms of the initial conditions and the general excitation  

Consider no excitation:


 

 

Plot the solution.

Solution

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Characteristic equation:

 

 


Substituting   into  :

 

 


 

 


Non-homogeneous solution:

 

 


Homogeneous solution:

 

 


Overall solution:

 

 


No excitation:


 

 


From intitial conditions:

 

 


 

 


Solving for   and  :

 

and

 


So, the final solution is:

 


 

Problem 2

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Given

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Find

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Find and plot the solution for  

Solution

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Due to no excitation,   becomes:

 

 

Substituting   into  :

 

 

Factoring, and solving for  :

 

 


 

Since   is a double root, the general solution:

 

 

From intitial conditions:

 

 


 

 


Solving for   and  :

 

and

 


So, the final solution is:

 


 

Problem 3

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Given

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(a)

 

(b)

 

Find

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General solution to the differential equation

Solution

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(a)

Characteristic equation:

 

Using the quadratic equation:

 

where:

 

Place values into quadratic equation:

 
 
 
 
 
 

(b)

Characteristic equation:

 

Using the quadratic equation:

 

where:

 

Place values into quadratic equation:

 
 
 
 
 

Problem 4

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Given

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(5)

 

 


(6)

 

 

h

Find

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For both (5) and (6), find a general solution. Then, check answers by substitution.

Solution

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(5)

To obtain the general solution for (Eq. 1), we let

 

This yields:

 

Therefore, we have:

 

Where   is a solution to the characteristic equation

 

We can see that

  and
 .

Next, we must examine the quadratic formula to determine whether the system is over-, under-, or critically-damped. The discriminant is:

 
 
 
Therefore, the equation is critically-damped.

The general solution is therefore represented by:

 

Where  .

We can determine the value of   by again using the quadratic equation, this time in full:

 

for which we find that  .

Therefore, the general solution to the equation is:

 

Verification by substitution:

 
 

Plugging into Eq. 1 and combining terms, it is found:

 

All the terms cancel, so the statement is true, and the solution is verified.


(6)

To obtain the general solution for (Eq. 2), let

 

So then, we have

 

as a solution to the characteristic equation. All terms must be divided by 10, to put the equation in standard form:

 

Examining the discriminant, we find that

 
Therefore, the equation is critically-damped.

The general solution is represented by:

 

and  .

To determine the value of  , we set:

 

for which we find that  .

Therefore, the general solution to the equation is:

 

Verification by substitution:

 
 

Plugging into Eq. 1 and combining terms, it is found:

 

All the terms cancel, so the statement is true, and the solution is verified.

Problem 5

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Given

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Given the basis:

(a)

 


(b)

 


Find

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For the given Information above, Find an ODE for both (a) and (b)

Solution

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(a)The general solution can be written as:

 


The characteristic equation can be written in the following way:

 


Giving the ODE:
 



(b) The general solution can be written as:

 


The characteristic equation can be written in the following way:

 


Giving the ODE:
 


Problem 6

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Given

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Mass Spring Dashpot system FBD's

Spring-dashpot equation of motion from sec 1-5

 


Find

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Spring-dashpot-mass system in series. Find the values for the parameters k,c,m with a double real root of  

Solution

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Consider the double real root

 

Characteristic equation is

 

 

Recall

 

 

Thus

 
 
 

Solve for c

 

Therefore

 

Problem 7

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Given

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Taylor Series at t=0

Find

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Develop the MacLaurin Series for  

Solution

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Taylor series is defined as

 

 

The MacLaurin series occurs when t=0

 

 

Development of MacLaurin series for  

 

 

Final Answer

 

Explicit form can be written as
 


Development of MacLaurin series for  

 

 

Final Answer

 

Explicit form can be written as
 


Development of MacLaurin series for  

 

 

Final Answer

 

Explicit form can be written as
 

Problem 8

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Given

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(8)

 

 



(15)

 

 

Find

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Solution

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(8)

Assume that the solution is of the form

 

 


 

 


And

 

 



Substituting equations 2, 3, and 4 into equation 1a yields

 
 
Since
 
 

We must use the quadratic formula to obtain values for lambda
 
 
 
 
 
Where  

 

In order to verify solution, we must evaluate it's first and second derivative, then plug them into the equation
Let  
 
 
 
 

Plugging   into Eq. 1 and collecting terms gives
 
 
 

Therefore the solution holds for any values of  




(15)

Assume that the solution is of the form

 

 


 

 


And

 

 



Substituting equations 2, 3, and 4 into equation 1b yields

 
 
Since
 
 

We must use the quadratic formula to obtain values for lambda
 
 
 
 
Where  

 

In order to verify solution, we must evaluate it's first and second derivative, then plug them into the equation
Let  
 
 
 
 

Plugging   into Eq. 1 and collecting terms gives
 
 
 

Therefore the solution holds for any values of  

Problem 9

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Given

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Find

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Solution

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Problem 10

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Given

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Differential equation, initial conditions, and forcing function as shown:

 

 

 

 



Find

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Find

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The solution to Eq. 1

Solution

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Substituting Equations 2, 3, and 4 into Equation 1 yields

 

 

To solve for the constants:

 

 

 

 

 

 

 

 

Solving Equations 6-9 yields

 

The homogeneous solution:

 

 

The total solution:

 

 

Setting   yields

 

 

Solving for   yields

 

 

Finding   yields

 

 

Setting   yields

 

 

Solving for   yields

 

 

Therefore, the total solution is

 

 


Team Member Tasks

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Name Responsibilities Checked by
Bo Turano Problem R2. --
David Parsons Problem R2.4 --
Dean Pickett Problem R2.8 --
Giacomo Savardi Problem R2. --
Isaac Kimiagarov Problem R2.3 --
Kyle Steiner Problem R2. --
Tony Han Problem R2.6, R2.7 --

All team members contributed to the coding of this page.