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R1.1: Spring-dash pot system in parallel with a mass and applied force f(t)

Initial Information

From lecture slide 1-4

Variables:$k,$ $c,$ $m,$ $f(t),$ $y(t),$ $y_{k},$ $y_{c},$ $f_{i},$ $f_{k},$ $f_{c}$

Methods

Kinematics:

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

Derived from (Eq.1)

Kinetics:

$\displaystyle f(t)=my''_{k}+cy'_{k}+ky_{k}$

Solution

Final Equation:

$\displaystyle f(t)=my''_{k}+cy'_{k}+ky_{k}$

R1.2: Spring-mass-dashpot with applied force r(t) on the ball(Fig. 53, p.85, K2011)

Initial Information

Variables:$k,$ $c,$ $m,$ $f(t),$ $y_{k},$ $f_{I},$ $f_{k},$ $f_{c}$

Methods

Kinematics:

$\displaystyle y=y_{k}$

Kinetics:

$\displaystyle C{\frac {d^{2}v_{c}}{dt^{2}}}={\frac {di}{dt}}$

Failed to parse (syntax error): {\displaystyle \displaystyle (Eq.{1}') </p> |} ==Solution== It is blank here. =R1.3 Spring-dashpot-mass system FBD and Equation of Motion= <br /> Problem found on slide 1-6 ==Initial Information== From lecture slide 1-4, the spring-dashpot-mass system:<br /> [image, or link to image on earlier] ==Methods== ==Solution== =R1.4: RLC Circuit Modeling= ==Initial Information== From lecture slide 2-2, a general RLC circuit Kirchhoff's Voltage Law (KVL) equation, and two alternative formulations, are given: :{| style="width:100%" border="0" align="left" |- |<math>\displaystyle V = LC \frac{d^{2}v_{c}}{dt^{2}} + RC \frac{dv_{c}}{dt}+ v_{c}}

$\displaystyle (Eq.2)$

$\displaystyle {V}'=L{I}''+R{I}'+{\frac {1}{C}}I$

$\displaystyle (Eq.3)$

$\displaystyle V=L{Q}''+R{Q}'+{\frac {1}{C}}Q$

$\displaystyle (Eq.4)$

We are being asked to derive (3) and (4) from (2).

Methods

From lecture slide 2-2, capacitance is defined as,

$\displaystyle Q=Cv_{c}\Rightarrow \int idt=Cv_{c}\Rightarrow i=C{\frac {dv_{c}}{dt}}$

$\displaystyle (Eq.1)$

Solution

Deriving (1), we get:

$\displaystyle C{\frac {d^{2}v_{c}}{dt^{2}}}={\frac {di}{dt}}$

$\displaystyle (Eq.{1}')$

Also, by solving (1) for $v_{c}$, we obtain:

$\displaystyle v_{c}={\frac {1}{C}}\int idt$

$\displaystyle (Eq.{1}'')$

Substituting equations (1), (1'), and (1") into (2)

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

$\displaystyle (Eq.{2}')$

Which is an "integro-differential equation." Therefore, to eliminate the integral we differentiate (2') with respect to t, to get:

$\displaystyle {V}'=L{I}''+R{I}'+{\frac {1}{C}}I$

$\displaystyle (Eq.3)$

Since $Q=\int idt$ from (1), substituting this into (2') yields:

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

$\displaystyle (Eq.4)$

R1.5: General Solution of ODE

Initial Information

From pg. 59 problem 4,

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

$\displaystyle (Eq.4)$

And from pg. 59 problem 5,

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

$\displaystyle (Eq.5)$

Find a general solution for Equations (4) and (5) and check the answer by substitution.

Methods

Solution

R1.6

Initial Information

We are asked to determine the order, linearity and whether the principle of superposition can be applied to the following examples.
The order of a differential equation is found by looking at the highest occurring derivative of the dependent variable.
A differential equation is linear if the dependent variable and all of its derivatives occur linearly throughout the equation.

Falling Stone

Governing Equation:

$\displaystyle {y}''=g=constant$

Order: 2

Linearity: Yes

Parachutist

Governing Equation:

$\displaystyle {mv'}=mg-bv^{2}$

Order: 1

Linearity: No

Outflowing water from a tank

Govering Equation:

$\displaystyle {h'}=-k{\sqrt {h}}$

Order: 1

Linearity: No

Vibrating mass on a spring

Governing Equation:

$\displaystyle {my''+ky}=0$

Order: 2

Linearity: Yes

Beats of a vibrating system

Governing Equation:

$\displaystyle {y''+\omega ^{2}y}=cos\omega {t}$

Order: 2

Linearity: Yes

Current I in an RLC Circuit

Governing Equation:

$\displaystyle {V}'=L{I}''+R{I}'+{\frac {1}{C}}I$

Order: 2

Linearity: Yes

Beam Deformation

Governing Equation:

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

Order: 0

Linearity: No

Pendulum

Governing Equation:

$\displaystyle {L\theta ''+g*sin(\theta )}=0$

Order: 2

Linearity: Yes

Solution

References

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Solution

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