# University of Florida/Egm4313/s12.team6.reports/R1

Report 1

## Problem 1: Spring-dashpot system in parallel with a mass and applied force

Derive the equation of motion of a spring-dashpot system in parallel, with a mass and applied force ${\displaystyle f(t)}$

### Given

Spring-dashpot system in parallel

### Solution

The kinematics of the system can be described as,

 ${\displaystyle \displaystyle x=x_{k}=x_{c}}$ ${\displaystyle \longrightarrow (1)}$

The kinetics of the system can be described as,

 ${\displaystyle \displaystyle m{\ddot {x}}+f_{I}=f(t)}$ ${\displaystyle \longrightarrow (2)}$

and,

 ${\displaystyle \displaystyle f_{I}=f_{k}+f_{c}}$ ${\displaystyle \longrightarrow (3)}$

Given that,

 ${\displaystyle \displaystyle f_{k}=kx_{k}}$ ${\displaystyle \longrightarrow (3a)}$
 ${\displaystyle \displaystyle f_{c}=c{\dot {x_{c}}}}$ ${\displaystyle \longrightarrow (3b)}$

From (1), it can be found that,

${\displaystyle \displaystyle {\dot {x}}={\dot {x_{k}}}={\dot {x_{c}}}}$

and,

${\displaystyle \displaystyle {\ddot {x}}={\ddot {x_{k}}}={\ddot {x_{c}}}}$

From (3), it can be found that,

${\displaystyle \displaystyle f_{I}=c{\dot {x_{c}}}+kx_{k}}$

Finally, it can be found that

 ${\displaystyle m{\ddot {x}}+c{\dot {x}}+kx=f(t)}$

### Author

Solved by: EGM4313.s12.team6.hill 19:05, 1 February 2012 (UTC)

## Problem 2: Spring-dashpot system in parallel with an applied force

#### Question

Derive the equation of motion of the spring - mass - dashpot in Fig. 53, in K 2011 p. 85, with an applied force r(t) on the ball.

Figure 53, page 85 K 2011

#### Solution

There are 2 possible cases that can occur in this problem, depending on the direction of the applied force.
In both cases,

${\displaystyle F_{applied}=r(t)}$

${\displaystyle F_{inertia}=my''}$

${\displaystyle F_{damping}=cy'}$

${\displaystyle F_{spring}=ky}$

##### Case 1
Case 1: Positive Force, Positive Displacement

The applied force is in the positive direction, and therefore the displacement is in the positive direction.

From the Free Body Diagram, we get the equation

${\displaystyle F_{applied}+F_{inertia}-F_{damping}-F_{spring}=0}$

Rearranging the equation, we get

${\displaystyle F_{applied}=-F_{inertia}+F_{spring}+F_{damping}}$

Replacing Force variables, we get

${\displaystyle r(t)=my''+cy'+ky}$

##### Case 2
Case 2: Negative Force, Negative Displacement

The applied force is in the negative direction, and therefore the displacement is in the negative direction.

From the Free Body Diagram, we get the equation

${\displaystyle -F_{applied}-F_{inertia}+F_{damping}+F_{spring}=0}$

Rearranging the equation, we get

${\displaystyle F_{applied}=-F_{inertia}+F_{spring}+F_{damping}}$

Replacing Force variables, we get

${\displaystyle r(t)=my''+cy'+ky}$

#### Conclusion

Since both cases return the same solution, the equation of motion is derived as:

 ${\displaystyle r(t)=my''+cy'+ky}$

### Author

Solved by: Egm4313.s12.team6.hickey 20:04, 1 February 2012 (UTC)

## Problem 3: Spring-dashpot-mass system

For the spring-dashpot-mass system on p.1-4, draw the FBDs and derive the equation of motion (2) p.1-4

### Given

spring-dashpot-mass system

### Solution

Problem 3

The equation of motion

${\displaystyle my''+f_{i}=f(t)}$

where:

${\displaystyle my''}$

is the inertial force

${\displaystyle f_{i}}$

is the internal force

${\displaystyle f(t)}$

is the applied force
this analysis assumes:

### Assumptions:

Motion in the horizontal direction
Massless spring
Massless dashpot
massless connections

### Solution:

To analyze the system we look at the kinematics and kinetics of the system
Kinematics: This involves the displacement which affects the mass. The displacement, represented by:

${\displaystyle y}$: This is the total displacement of the spring plus the displacement of the dashpot.
${\displaystyle y_{k}}$: represents the displacement of the spring
${\displaystyle y_{c}}$: represents the displacement of the dashpot
${\displaystyle y=y_{k}+y_{c}}$

Kinetics: this involves the forces associated with the displacements. The spring and dashpot are in series, therefore at any section in the series the internal for will be the same. This is denoted as:${\displaystyle y_{i}}$. The force in the dashpot is proportional to the first time derivative of displacement (velocity)

${\displaystyle f_{k}=f_{c}=f_{i}=ky_{k}=cy'_{c}}$

Where :${\displaystyle k}$ is the spring constant.

${\displaystyle c}$ is the damping coefficient.

and:${\displaystyle y'_{c}}$ is the velocity of the dashpot. From this we get :${\displaystyle y'_{c}=(k/c)y_{k}}$ which presents :${\displaystyle y'_{c}}$ in terms of :${\displaystyle y_{k}}$

section cut
Free body diagram

The constitutive relations:
The spring force is equal to the spring constant times the displacement of the spring. The damping force from the dashpot is equal to the damping coefficient times the velocity.

${\displaystyle y=y_{k}+y_{c}}$

this presents two unknown dependent variables by using the relation

${\displaystyle y''=y_{k}''+(y_{c}')'}$

Becomes:

${\displaystyle y''=y_{k}''+{\frac {k}{k}}y_{k}'}$

now we can rewrite the equation of motion:
${\displaystyle my''+f_{i}=f(t)}$

as :${\displaystyle m[yk''+{\frac {k}{c}}yk']+fi=f(t)}$
However >:${\displaystyle f_{i}=f_{k}}$

so we get: ${\displaystyle m[yk''+{\frac {k}{c}}yk']+fk=f(t)}$

 ${\displaystyle m[y_{k}''+{\frac {k}{c}}y_{k}']+f_{k}=f(t)}$:

### Author

Solved by: Hopeton87 19:42, 1 February 2012 (UTC)

## Problem 4: RLC circuit

Derive (3) and (4) from (2) on pg. 2-2

### Given:

${\displaystyle V=LC{\frac {d^{2}V_{c}}{dt^{2}}}+RC{\frac {dV_{c}}{dt}}+V_{c}}$

### Find:

Derive the below two equations from the given:

A) ${\displaystyle LI''+RI'+{\frac {1}{C}}I=V'}$
and
B) ${\displaystyle LQ''+RQ'+{\frac {1}{C}}Q=V}$

### Solution:

#### Part A:

From Eq. 2-2(1): ${\displaystyle Q=CV_{c}=\int Idt}$

${\displaystyle \therefore {\frac {dI}{dt}}=C{\frac {d^{2}V_{c}}{dt}}}$

substituting ${\displaystyle Q=CV_{c}=\int Idt}$ into ${\displaystyle RC{\frac {dV_{c}}{dt}}}$ we get: ${\displaystyle RI}$

substituting ${\displaystyle {\frac {dI}{dt}}=C{\frac {d^{2}V_{c}}{dt}}}$ into ${\displaystyle LC{\frac {d^{2}V_{c}}{dt^{2}}}}$ we get: ${\displaystyle LI'}$

So we now have: ${\displaystyle V=LI'+RI+V_{c}}$

Deriving the whole equation we get:

${\displaystyle V'=LI''+RI'+{\frac {dV_{c}}{dt}}}$

by multiplying ${\displaystyle {\frac {dV_{c}}{dt}}}$ by ${\displaystyle /frac{C}{C}}$ we get:

${\displaystyle V'=LI''+RI'+{\frac {CdV_{c}}{Cdt}}}$

we can now sub ${\displaystyle Q=CV_{c}=\int Idt}$ for ${\displaystyle {\frac {CdV_{c}}{dt}}}$ giving us:

 ${\displaystyle LI''+RI'+{\frac {1}{C}}I=V'}$

#### Part B:

From Eq. 2-2(1): ${\displaystyle Q=CV_{c}=\int Idt}$

multiplying ${\displaystyle V_{c}}$ by ${\displaystyle {\frac {C}{C}}}$ we get ${\displaystyle {\frac {CV_{c}}{C}}}$

Subbing ${\displaystyle Q=CV_{c}=\int Idt}$ into ${\displaystyle {\frac {CV_{c}}{C}}}$ we get:

${\displaystyle V=LC{\frac {d^{2}V_{c}}{dt^{2}}}+RC{\frac {dV_{c}}{dt}}+{\frac {Q}{C}}}$

taking the first and second derivative of ${\displaystyle Q=CV_{c}}$ we get:

${\displaystyle Q'=C{\frac {dV_{c}}{dt}}}$ and ${\displaystyle Q''=C{\frac {d^{2}V_{c}}{dt^{2}}}}$

substitute ${\displaystyle Q'=C{\frac {dV_{c}}{dt}}}$ into ${\displaystyle RC{\frac {dV_{c}}{dt}}}$ we get: ${\displaystyle RQ'}$

substitute ${\displaystyle Q''=C{\frac {d^{2}V_{c}}{dt^{2}}}}$ into ${\displaystyle LC{\frac {d^{2}V_{c}}{dt^{2}}}}$ we get: ${\displaystyle LQ''}$

 ${\displaystyle \therefore V=LQ''+RQ'+{\frac {1}{C}}Q}$

### Author

Solved by:Egm4313.s12.team6.mcpherson 19:52, 1 February 2012 (UTC)

         Egm4313.s12.team6.jagolinzer 19:58, 1 February 2012 (UTC)



## Problem 5:

This problem consists of finding a general solution to two linear, second order differential equations with constant coefficients. The problems are taken from the tenth edition of Erwin Kreyszig's Advanced Engineering Mathematics and can be found as problems 4 and 5 of problem set 2.2 on page 59.

### Problem 4

#### Given:

${\displaystyle y''+6y'+(\pi ^{2}+4)y=0}$

#### Find:

The homogeneous solution to the differential equation.

#### Solution:

Use the method of undetermined coefficients, also known as the method of trial solution, to solve the differential equation.

Let ${\displaystyle y=e^{\lambda x}}$

Therefore:

${\displaystyle y'=\lambda e^{\lambda x}}$
and
${\displaystyle y''=\lambda ^{2}e^{\lambda x}}$

So,

${\displaystyle y''+6y'+(\pi ^{2}+4)y=0}$

can be written as

${\displaystyle e^{\lambda x}(\lambda ^{2}+a\lambda +b)=0}$ where ${\displaystyle a=6}$ and ${\displaystyle b=\pi ^{2}+4}$

Since the exponential ${\displaystyle e^{\lambda x}}$ can never equal zero,

${\displaystyle \lambda ^{2}+a\lambda +b=0}$

This equation can be solved for lambda using the quadratic equation :

${\displaystyle \lambda ={\frac {-a\pm {\sqrt[{}]{a^{2}-4b}}}{2}}}$

Substituting values:

${\displaystyle \lambda ={\frac {-(6)\pm {\sqrt[{}]{(6)^{2}-4(\pi ^{2}+4)}}}{2}}}$

Which evaluates to:

${\displaystyle \lambda =-3\pm {\sqrt[{}]{-19.478}}}$
or
${\displaystyle \lambda =-3\pm 2.207i}$

Since the discriminant is less than zero we let ${\displaystyle \omega =2.207}$ and the homogeneous solution will be of the form:

${\displaystyle y=e^{\frac {-ax}{2}}(Acos(\omega x)+Bsin(\omega x))}$

Substituting values we have the homogeneous solution:

 ${\displaystyle y=e^{-3x}(Acos(2.21x)+Bsin(2.21x))}$

### Problem 5

#### Given:

${\displaystyle y''+2\pi y'+\pi ^{2}y=0}$

#### Find:

The homogeneous solution to the differential equation.

#### Solution:

Use the method of undetermined coefficients, also known as the method of trial solution, to solve the differential equation.

Let ${\displaystyle y=e^{\lambda x}}$

Therefore:

${\displaystyle y'=\lambda e^{\lambda x}}$
and
${\displaystyle y''=\lambda ^{2}e^{\lambda x}}$

So,

${\displaystyle y''+2\pi y'+\pi ^{2}y=0}$

can be written as

${\displaystyle e^{\lambda x}(\lambda ^{2}+a\lambda +b)=0}$ where ${\displaystyle a=2\pi }$ and ${\displaystyle b=\pi ^{2}}$

Since the exponential ${\displaystyle e^{\lambda x}}$ can never equal zero,

${\displaystyle \lambda ^{2}+a\lambda +b=0}$

This equation can be solved for lambda using the quadratic equation :

${\displaystyle \lambda ={\frac {-a\pm {\sqrt[{}]{a^{2}-4b}}}{2}}}$

Substituting values we can see that the discriminant is zero:

${\displaystyle \lambda ={\frac {-2\pi \pm {\sqrt[{}]{4\pi ^{2}-4\pi ^{2}}}}{2}}}$

Therefore, only one solution can be found from this equation:

${\displaystyle y_{1}=e^{-\pi x}}$

For the second solution reduction of order must be used:

Let:

${\displaystyle y_{2}=uy_{1}}$

Then,

${\displaystyle y_{2}'=u'y_{1}+y_{1}'u}$

and

${\displaystyle y_{2}''=u''y_{1}+2u'y_{1}'+y_{1}''u}$

Substituting these into the original equation gives:

${\displaystyle (u''y_{1}+2u'y_{1}'+y_{1}''u)+a(u'y_{1}+y_{1}'u)+buy_{1}=0}$

Which simplifies to:

${\displaystyle u''y_{1}'+u'(2y_{1}'+ay_{1})+u(y_{1}''+ay_{1}'+by_{1})=0}$

Since,

${\displaystyle 2y_{1}'=-ae^{{\frac {-a}{2}}x}=-ay_{1}}$

and

${\displaystyle y_{1}''+ay_{1}'+by_{1}=0}$

What remains is:

${\displaystyle u''y_{1}=0}$

or

${\displaystyle u''=0}$

Therefore,

${\displaystyle u=C_{1}x+C_{2}}$

Now we let ${\displaystyle C_{1}=1}$ and ${\displaystyle C_{2}=0}$ so that:

${\displaystyle u=x}$

Now,

${\displaystyle y_{2}=xy_{1}=xe^{-\pi x}}$

Using ${\displaystyle y_{1}}$ and ${\displaystyle y_{2}}$ we have the general solution:

 ${\displaystyle y=(C_{1}+C_{2}x)e^{-\pi x}}$

### Author

Solved by:EGM4313.s12.team6.davis 02:50, 1 February 2012 (UTC)

## Problem 6

For each ODE in Fig.2 in K 2011 p.3 (except the last one involving a system of 2 ODEs), determine the order, linearity (or lack of), and show whether the principle of superposition can be applied.

### Given

${\displaystyle (1){y}''=g=constant}$

${\displaystyle (2)m{v}'=mg-bv^{2}}$

${\displaystyle (3){h}'=-k{\sqrt {h}}}$

${\displaystyle (4)m{y}''+ky=0}$

${\displaystyle (5)y''+\omega _{0}^{2}=cos(\omega t)}$, ${\displaystyle \omega _{0}=\omega }$

${\displaystyle (6)L{I}''+R{I}'+{\frac {1}{c}}I={E}'}$

${\displaystyle (7)EIy^{iv}={\mathit {f(x)}}}$

${\displaystyle (8){\mathit {L{\theta }''+gsin{\theta }=0}}}$

### Solution

The order of an ODE is found by looking for the highest derivative. If an ODE is linear, than the principle of superposition can be used by finding the solution to the homogeneous equation and finding the particular solution and then adding up the two. If an equation is not linear, then the principle of superposition cannot be applied.
${\displaystyle (1){y}''=g=constant}$
Order:2nd order. The highest derivative is a 2nd derivative on the y.
Linearity:Linear.
Superposition:Yes.

${\displaystyle (2)m{v}'=mg-bv^{2}}$
Order:1st order. The highest derivative is a 1st on the v.
Linearity:Non-linear.
Superposition:No.

${\displaystyle (3){h}'=-k{\sqrt {h}}}$
Order:1st order
Linearity:Non-linear.
Superposition:No.

${\displaystyle (4)m{y}''+ky=0}$
Order:2nd order.
Linearity:Linear.
Superposition:Yes.

${\displaystyle (5)y''+\omega _{0}^{2}=cos(\omega t)}$, ${\displaystyle \omega _{0}=\omega }$
Order:2nd order
Linearity:Linear.
Superposition:Yes.

${\displaystyle (6)L{I}''+R{I}'+{\frac {1}{c}}I={E}'}$
Order:2nd order
Linearity:Linear.
Superposition:Yes.

${\displaystyle (7)EIy^{iv}={\mathit {f(x)}}}$
Order:4th oder
Linearity:Linear.
Superposition:Yes.

${\displaystyle (8){\mathit {L{\theta }''+gsin{\theta }=0}}}$
Order:2nd order
Linearity:Non-linear.
Superposition:Yes.

### Author

Solved by:Egm4313.s12.team6.berthoumieux 02:52, 1 February 2012 (UTC)

## Contributing Members

 Team Contribution Table Problem Number Lecture Assigned To Solved By Typed By Proofread By 1.1 R1.1 in Sec 1 p. 1-5 Hill Hill Hill Davis 1.2 R1.2 in Sec 1 p. 1-4 Hickey Hickey Hickey Hill 1.3 R1.3 in Sec 1 p. 1-5 Nembhard Nembhard Nembhard Hill 1.4 R1.4 in Sec 2 p. 2-2 Jagolinzer/McPherson Jagolinzer/McPherson Jagolinzer/McPherson Hill 1.5 R1.5 in Sec 2 p. 2-5 Davis Davis Davis Hill 1.6 R1.6 in Sec 2 p. 2-7 Berthoumieux Berthoumieux Berthoumieux Hill