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

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Initial Information

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From lecture slide 1-4

Variables:

Methods

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Kinematics:

Derived from (Eq.1)


Kinetics:


Solution

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Final Equation:


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

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Initial Information

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Variables:

Methods

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Kinematics:

Kinetics:

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}}

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

Methods

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

Solution

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Deriving (1), we get:

Also, by solving (1) for , we obtain:

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

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

Since from (1), substituting this into (2') yields:

R1.5: General Solution of ODE

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Initial Information

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From[1] pg. 59 problem 4,




And from[1] pg. 59 problem 5,

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

Methods

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Solution

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R1.6

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Initial Information

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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:


Order: 2


Linearity: Yes

Parachutist

Governing Equation:


Order: 1


Linearity: No

Outflowing water from a tank

Govering Equation:


Order: 1


Linearity: No

Vibrating mass on a spring

Governing Equation:


Order: 2


Linearity: Yes

Beats of a vibrating system

Governing Equation:


Order: 2


Linearity: Yes

Current I in an RLC Circuit

Governing Equation:


Order: 2


Linearity: Yes

Beam Deformation

Governing Equation:


Order: 0


Linearity: No

Pendulum

Governing Equation:


Order: 2


Linearity: Yes

Solution

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References

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  1. ^ a b Kreyszig, "Advanced Engineering Mathematics," John Wiley & Sons, 2011.

r

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sol

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a

Solution

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a

s

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a