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Problem 1 - Derive EOM (SC-N1-ODE)Edit

From Meeting 13, p. 13-1 ~ p. 13-2

GivenEdit

Figure shows the Trajectory of a projectile (ex:Rocket):

 

FindEdit

  Drive Equation of Motion (EOM)
  Particular case   : Verify   is parabolla
  Consider  ,  ,
  Find  ,   for   = constant
  Find  ,   if  

SolveEdit

Part 1

Consider the trajectory of a projectile (ex. Rocket)

Various forces acting on the projectile at time 't' are:
1) Weight of the projectile
 
2) Inertia force
  for particle with constant mass
3) Air resistance which is proportional to the velocity of particle
 

Now consider the force equilibrium in both horizontal and vertical direction

a) Force Equilibrium in horizontal direction:
 
 ,
where   horizontal component of velocity

 

 

b) Force Equilibrium in the vertical direction:
 

 ,

where   vertical component of velocity

 

 

Part 2

Particular case: When  

Eq.1.1 reduces to

 

Integrating the above equation gives:

 

 

Apply 'initial condition' to determine integration constant, 

 

Now Eq.1.3 becomes:

 

Integrate the above equation to obtain , 

 

Then 'Initial condition' is applied to determine , 

 

Therefore,

 

Now  , can be expressed in terms of  ,

 

 

Similarly When  , Eq.1.2 reduces to

 

Integrate the above equation to evaluate,  

 

 

Apply 'initial condition' to obtain  

 

Now Eq.1.5 becomes,

 

Then integrate the above equation to determine,  

 

  is determined using 'initial condition' as:

 

Therefore,

 

Now Substitute Eq.1.4 for   in the above equation;

 

 

Eq.1.6 is in the form of a parabolic equation. Therefore   is parabola.

Part 3

With  , and   (from the geometry)

 

(1.13)

Which presents an interesting fact, once   is equal to zero it remains zero as its derivative is also zero (except when   and  ). Thus, if   the   velocity remains zero and the equations of motion reduce to equation 1.2.

Part 3.1Edit

Solving

 

(1.2)

To show exactness, we first put into a form that shows the first condition of exactness is met

 

(1.14)

so that

 

(1.15)

 

(1.16)

The second condition of exactness

 

(1.17)

Which is met if   is not a function of time

 

(1.18)

Thus, the equation is non-exact.

The equation can be made exact through the integrating factor method

 

(1.19)

Then expressing as a total derivative and testing for exactness

 

(1.20)

 

(1.21)

 

(1.22)

Letting  

 

(1.23)

 

(1.24)

 

(1.25)

 

(1.26)

Then

 

(1.27)

 

(1.28)

 

(1.29)

 

(1.30)

Combining

 

(1.30)

Though the equation is not integrable, by making it exact though the integrating factor method an expression was found.

Part 3.2Edit

If   the integrating factor method is complicated in equation 1.22 as the partial of   with respect to time remains, complicating the expression for  

Author and Proof-readerEdit

[Author]

[Proof-reader]

Problem 2 - Derive EOM (SC-L1-ODE)Edit

From Meeting 13, p. 13-3

GivenEdit

 

 

 

 

  : mass of the each pendulum

  : the angle from the vertical to the each pendulum

  : applied forces to the each pendulum

  : length of the pendulum

  : force constant(or spring constant)

  : acceleration of gravity

FindEdit

1. Derive (2-1) and (2-2)

2. Write (2.1) and (2-2) in form of (2-3)

 

 

where,

 

 

Dimension of matrix

 

 

 

 

 

SolveEdit

Step 1. Derivation

Background Knowledge

1. Torque [1]

2. Hooke's Law [2]

3. Pendulum [3]

4. Moment [4]

5. Moment of inertia [5]

6. Angular acceleration [6]

Derive Using above background,

  Torque of the spring force + Torque of the gravity force + Torque of the applied force )

 

where,

  : torque

  : moment of inertia

  : angular acceleration

Therefore, left hand side is  

Torque of the spring force

From the backgroud (Hooke's Law(wikipedia)),

 

 

where,
  : restoring force
  : spring constant
  : displacement from the equilibrium position (in this case, x = a)

Therefore, Torque of the spring force is,

 

 

Torque of the gravity force

 

 

Torque of the applied force

 

 

Using (2-4) ~ (2-8)

 

(2-2) can be verified with same procedure.

 

Step 2. Find A, B and U

Let's make a equation of a matrix with the information that we already have.

 

 

rearrange the derived equations (2-1) and (2-2),

 

 

Let's put them to (2-9)

 

GivenEdit

Shown in figure are the two Pendulums connected by a spring:

 

FindEdit

  Derive equation of motion:
 

 

 

 

  Write Eq.2.1 and Eq.2.2 in the form of Mtg 13 (c),page2 , of:
 

 

Given

  and  

SolutionEdit

  Derive equation of motion:
(a) Consider Free Body Diagram of left pendulum:
For small angle:
  and  
 
 
 

Now using D'Alembert's_principle, sum of the moments about pivot(A)is equal to zero

 

 

 

(b) Consider Free Body Diagram of right pendulum:
For small angle:
  and  
 
 
 

Using D'Alembert's_principle, sum of the moments about pivot(B)is equal to zero

 

 

 

  Write Eq.2.1 and Eq.2.2 in the form of Eq.2.3(system of coupled equation):
Eq.2.1 can be rearranged as,
 

 

 

 

Now Eq.2.4 and Eq.2.5 can be put in the form of Eq.2.3 as:

 

 

Where:
 
 

Contributing MembersEdit

Solved and posted by Egm6321.f10.team3.Sudheesh 15:39, 4 October 2010 (UTC)

Author and Proof-readerEdit

[Author]

[Proof-reader]

Problem 3 - Derive (L1-ODE-CC)Edit

From Meeting 14, p. 14-1

GivenEdit

 

 

 

 

FindEdit

Derive (3-2)

SolveEdit

We can rearrange the eqn(3.1). As it is not time variable problem, let  

 

(3.2)

Let's find the integrating factor first.

As the coefficient for the   is 1,

 

(3.3)

Multiply the integrating factor to eqn(3.2) on both side.

 

(3.4)

 

(3.5)

let's integrate for the interval  

 

(3.6)

 

(3.7)

rearrange eqn(3.7),

 

(3.8)

 

(3.9)

Author and Proof-readerEdit

[Author]

[Proof-reader]

Problem 4 - Expand Taylor series(exponential and exponential matrix)Edit

From Meeting 14, p. 14-2

GivenEdit

   

 

   

 

FindEdit

1) Derive (4-1)

2) Derive (4-2)

SolveEdit

Solution of 1)

Using Taylor series [7],

 

which can be written in the more compact sigma notation [8] as

 

In the particular case where a = 0, the series is also called a Maclaurin series [9]

   

 

 

 

 

 

 

 

 

Solution of 2)

Using Taylor series and Maclaurin series

 

 

<Background Knowledge> - Exponential Matrix, [10] Identity Matrix [11]

 

 

 

 

 

Author and Proof-readerEdit

[Author] Oh, Sang Min

[Proof-reader]

Problem 5 - Generalized to SC-L1-ODE-VCEdit

From Meeting 14, p. 14-2

GivenEdit

L1-ODE-CC :

 

 

L1-ODE-VC :

 

 

SC-L1-ODE-CC :

 

 

Dimension of matrix

 

 

 

 

FindEdit

Generalized (5-3) to SC-L1-ODE-VC

SolveEdit

SC-L1-ODE-CC can be generalized to SC-L1-ODE-CC as same as L1-ODE-CC is generalized to L1-ODE-VC

Using (5-1) ~ (5-3)

SC-L1-ODE-VC

 

Dimension of matrix

 

 

 

 

Author and Proof-readerEdit

[Author] Oh, Sang Min

[Proof-reader]

Problem 6 - Obtaining SC-L1-ODE-CC with int. factor methodEdit

From Meeting 15, p. 15-1

GivenEdit

FindEdit

SolveEdit

Author and Proof-readerEdit

[Author]

[Proof-reader]

Problem 7 - Application SC-L1-ODE-CC about rolling control of rocketEdit

From Meeting 15, p. 15-1

GivenEdit

 

 

 

 

 

 

 

 

  = aileron angle(deflection)

  = roll angle

  = roll angular velocity

  = aileron efflectiveness

  = roll time constant

FindEdit

Put (7-1) ~ (7-3) in form of (7-4)

SolveEdit

 

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Author and Proof-readerEdit

[Author] Oh, Sang Min

[Proof-reader]

Problem 8Edit

We can rewrite the eqn(8.1) as (8.2).

 

(8.3)

We are familiar with this equation, as we learned already. Total derivative - Egm6321.f10_HW1_prob#1_team6

 

(8.4)

As  , we know that   only.
It means  

Hence, eqn(8.2) becomes,

 

(8.5)

There are two possible solutions.

1)  

2)  

If 1) were satisfied, whole problems became zero, which is trivial. We can conclude that 2) is the solution.

As   and  ,

 

(8.6)

Problem 9Edit

Problem 10Edit

ReferencesEdit

  1. Torque(wikipedia)
  2. Hooke's Law(wikipedia)
  3. Pendulum(wikipedia)
  4. Moment(wikipedia)
  5. Moment of inertia(wikipedia)
  6. Angular acceleration(wikipedia)
  7. Taylor series(wikipedia)
  8. sigma notation(wikipedia)
  9. Maclaurin series(http://mathworld.wolfram.com)
  10. Exponential Matrix(wikipedia)
  11. Identity Matrix(wikipedia)
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