User talk:Egm6321.f10.team5.oh/hw3

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Problem 1 - Derive EOM (SC-N1-ODE)[edit source]

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

Given[edit source]

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

Sudheesh problem2.jpg

Find[edit source]

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

Solve[edit source]

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 3.1[edit source]

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 3.2[edit source]

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-reader[edit source]

[Author]

[Proof-reader]

Problem 2 - Derive EOM (SC-L1-ODE)[edit source]

From Meeting 13, p. 13-3

Given[edit source]

 : 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

Find[edit source]

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

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

where,

Dimension of matrix

Solve[edit source]

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)


Given[edit source]

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

Find[edit source]

Derive equation of motion:

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

Given

and

Solution[edit source]

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:
Egm6321 f10 team3 Sudheesh fig3.jpg
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 Members[edit source]

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

Author and Proof-reader[edit source]

[Author]

[Proof-reader]

Problem 3 - Derive (L1-ODE-CC)[edit source]

From Meeting 14, p. 14-1

Given[edit source]

Find[edit source]

Derive (3-2)

Solve[edit source]

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-reader[edit source]

[Author]

[Proof-reader]

Problem 4 - Expand Taylor series(exponential and exponential matrix)[edit source]

From Meeting 14, p. 14-2

Given[edit source]

Find[edit source]

1) Derive (4-1)

2) Derive (4-2)

Solve[edit source]

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-reader[edit source]

[Author] Oh, Sang Min

[Proof-reader]

Problem 5 - Generalized to SC-L1-ODE-VC[edit source]

From Meeting 14, p. 14-2

Given[edit source]

L1-ODE-CC :

L1-ODE-VC :

SC-L1-ODE-CC :

Dimension of matrix

Find[edit source]

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

Solve[edit source]

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-reader[edit source]

[Author] Oh, Sang Min

[Proof-reader]

Problem 6 - Obtaining SC-L1-ODE-CC with int. factor method[edit source]

From Meeting 15, p. 15-1

Given[edit source]

Find[edit source]

Solve[edit source]

Author and Proof-reader[edit source]

[Author]

[Proof-reader]

Problem 7 - Application SC-L1-ODE-CC about rolling control of rocket[edit source]

From Meeting 15, p. 15-1

Given[edit source]

= aileron angle(deflection)

= roll angle

= roll angular velocity

= aileron efflectiveness

= roll time constant

Find[edit source]

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

Solve[edit source]

Failed to parse (syntax error): {\displaystyle \underline{A}= \begin{pmatrix} 0 & 1 & 0 \\ 0 & \frac{-1}{\tau} & \frac{Q}{\tau}\\ 0 & 0 & 0 \end{pmatrix} , \ \underline{B}= \begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix} }

Author and Proof-reader[edit source]

[Author] Oh, Sang Min

[Proof-reader]

Problem 8[edit source]

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 9[edit source]

Problem 10[edit source]

References[edit source]

  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)