# University of Florida/Egm4313/IEA-f13-team10/R2

## Report 2[edit | edit source]

### Problem 1[edit | edit source]

#### Problem Statement[edit | edit source]

Show the equivalence between the two statements that define the linearity of differential operators.

**Statement (1):**

The definition of a linear operator, , is

for any two constants and , and for any two functions and

**Statement (2):**

The operator is linear iff

for any constant and for any function ,

and

for any two functions and .

#### Solution[edit | edit source]

We saw on piazza how to prove statement (1) from statement (2). To prove statement (2) from statement (1) we choose clever values for the constants and and for the functions and .

##### Proof for the Scaling Property[edit | edit source]

Since and can be any constant, and and can be any function, let us set

and

Then (1) becomes

Now let us say *2c* is equal to some constant . The above equation becomes

which is the scaling property of a linear operator found in statement (2).

##### Proof for the Addition Property[edit | edit source]

Since and can be any constant, let us set

Then (1) becomes

which is the addition property of a linear operator found in statement (2).

#### Honor Pledge[edit | edit source]

On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

### Problem 2[edit | edit source]

#### Problem Statement[edit | edit source]

Build on my note in the same piazza post on differential operators on the notation

for general differential operators, and the notation

as defined in Eq.(1) in K 2011, sec.2.3, to explain the similarities and differences between these two notations.

#### Solution[edit | edit source]

The similarity with both and is that they look like they're both upper case D's. Both of these D's also symbolized the differential operator.

The difference between both of these D's is that

is a generalized differential operator, which can be seen as

and refers to the first differential operator:

#### Honor Pledge[edit | edit source]

On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

### Problem 3[edit | edit source]

#### Problem Statement[edit | edit source]

Identify the polynomial as defined in K 2011, sec.2.3, for this problem.

Then rewrite the above problem in terms of the linear differential operator .

Use its linear property to demonstrate the above problem into a homogeneous part and a particular part.

How would the initial conditions be written in terms of the homogeneous solution and the particular solution?

#### Solution[edit | edit source]

Let

Splitting into homogeneous and particular,

Inputting the initial conditions,

#### Honor Pledge[edit | edit source]

On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

### Problem 4[edit | edit source]

#### Problem Statement[edit | edit source]

Given the two roots an the initial conditions:

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[edit | edit source]

Therefore,

Solve for constants and :

so,

#### Honor Pledge[edit | edit source]

### Problem 5[edit | edit source]

R 2.5 K 2011, sec 2.2, p. 59, pb.5, with initial conditions

#### Problem Statement[edit | edit source]

Find the general solution to the ODE

With the initial values y(0) = 1 and y'(0) = .5

#### Solution[edit | edit source]

This is a linear, first order ODE with constant coefficients.

To find the general solution to this ODE set

so that and

Substituting in y to the ODE and factoring out we get:

Using the quadratic formula to solve for r we get

where and

Solving to get

Since we have a repeated root, we need to find v(x) so that y2(x) = v(x)y1(x)

Taking the first and second derivative of y2(x) we get:

solving for so

So y2(x) = x y1(x)

We get the general solution

Now with the initial values y(0) = 1 and y'(0) = .5

,

#### Honor Pledge[edit | edit source]

### Problem 6[edit | edit source]

#### Problem Statement[edit | edit source]

Discuss the similarities and differences between Coulomb friction / damping (piazza post) with the damper used in the SDOF spring-mass-damper system with 2 ends fixed.

#### Solution[edit | edit source]

##### Similarities[edit | edit source]

With both ends fixed for SDOF system, we can say that
(1)

In the theory of Coulomb friction/damping, the friction changes the direction of the oscillating object by "damping" the force of motion.

In the case of the SDOF system, the damper fixed to one of the ends is friction force shown in the theory.

##### Differences[edit | edit source]

The equation of motion for the SDOF system with two fixed ends is

The equation of motion for a typical coulomb damping system is

or

As seen above, the typical system involves a vague force to change the direction of an oscillating object while the SDOF system has a specific damper as function of .