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R1.1: Spring-dash pot system in parallel with a mass and applied force f(t)
[edit]Initial Information
[edit]From lecture slide 1-4
Variables:
Methods
[edit]Kinematics:
Derived from (Eq.1)
Kinetics:
Solution
[edit]Final Equation:
R1.2: Spring-mass-dashpot with applied force r(t) on the ball(Fig. 53, p.85, K2011)
[edit]Initial Information
[edit]Variables:
Methods
[edit]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
[edit]From lecture slide 2-2, capacitance is defined as,
Solution
[edit]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
[edit]Initial Information
[edit]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
[edit]Solution
[edit]R1.6
[edit]Initial Information
[edit]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
[edit]References
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Solution
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