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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


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Variables:$$k,$$ $$c,$$  $$m,$$  $$f(t),$$  $$y(t),$$  $$y_k,$$  $$y_c,$$  $$f_i,$$  $$f_k,$$  $$f_c$$
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 * }
 * }

Methods
Kinematics:


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 * $$\displaystyle y = y_k = y_c$$ ||  Derived from (Eq.1)
 * }
 * }

Kinetics:


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 * $$\displaystyle f(t) = my''_k + cy'_k + ky_k$$
 * }
 * }
 * }

Solution
Final Equation:


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 * $$\displaystyle f(t) = my''_k + cy'_k + ky_k$$
 * }
 * }
 * }

=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:


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 * $$\displaystyle y=y_k $$
 * }
 * }
 * }

Kinetics:


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C\frac{d^{2}v_{c}}{dt^{2}} = \frac{di}{dt}$$
 * $$\displaystyle
 * $$\displaystyle
 * \displaystyle (Eq.{1}')
 * }

Solution
It is blank here.

=R1.3 Spring-dashpot-mass system FBD and Equation of Motion=

Problem found on slide 1-6

Initial Information
From lecture slide 1-4, the spring-dashpot-mass system:

[image, or link to image on earlier]

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:


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V = LC \frac{d^{2}v_{c}}{dt^{2}} + RC \frac{dv_{c}}{dt}+ v_{c}$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq.2)
 * }
 * }


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{V}' = L{I}'' + R{I}' + \frac{1}{C}I$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq.3)
 * }
 * }


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V = L{Q}'' + R{Q}' + \frac{1}{C}Q$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq.4)
 * }
 * }

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

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


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Q = Cv_{c} \Rightarrow \int idt = Cv_{c} \Rightarrow i = C\frac{dv_{c}}{dt}$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq.1)
 * }
 * }

Solution
Deriving (1), we get:


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C\frac{d^{2}v_{c}}{dt^{2}} = \frac{di}{dt}$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq.{1}')
 * }
 * }

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


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v_{c} = \frac{1}{C}\int idt$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq.{1}'')
 * }
 * }

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


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V = L{I}' + RI + \frac{1}{C}\int I$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq.{2}')
 * }
 * }

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


 * {| style="width:100%" border="0" align="left"

{V}' = L{I}'' + R{I}' + \frac{1}{C}I$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq.3)
 * }
 * }

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


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V = L{Q}'' + RQ + \frac{1}{C}Q$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq.4)
 * }
 * }

=R1.5: General Solution of ODE=

Initial Information
From pg. 59 problem 4,


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

And from pg. 59 problem 5,


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{y}''+2\pi {y}'+{{\pi }^{2}}y=0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 5)
 * }
 * }

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

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  <DT>Governing Equation: <DD>$$\displaystyle{y}''= g = constant$$ </DL>

Order: 2

Linearity: Yes

<H4>Parachutist</H4> <DL> <DT>Governing Equation: <DD>$$\displaystyle{mv'}=mg-bv^2$$ </DL>

Order: 1

Linearity: No

<H4>Outflowing water from a tank</H4> <DL> <DT>Govering Equation: <DD>$$\displaystyle{h'}=-k\sqrt{h}$$ </DL>

Order: 1

Linearity: No

<H4>Vibrating mass on a spring</H4> <DL> <DT>Governing Equation: <DD>$$\displaystyle{my''+ky}=0$$ </DL>

Order: 2

Linearity: Yes

<H4>Beats of a vibrating system</H4> <DL> <DT>Governing Equation: <DD>$$\displaystyle{y''+\omega^2y}=cos\omega{t}$$ </DL>

Order: 2

Linearity: Yes

<H4>Current I in an RLC Circuit</H4> <DL> <DT>Governing Equation: <DD>$$\displaystyle{V}' = L{I}'' + R{I}' + \frac{1}{C}I$$ </DL>

Order: 2

Linearity: Yes

<H4>Beam Deformation</H4> <DL> <DT>Governing Equation: <DD>$$\displaystyle{EIy^{iv}}=f(x)$$ </DL>

Order: 0

Linearity: No

<H4>Pendulum</H4> <DL> <DT>Governing Equation: <DD>$$\displaystyle{L\theta''+g*sin(\theta)}=0$$ </DL>

Order: 2

Linearity: Yes

Solution
=References=

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