lecture #32 higher order equations harmonic motionpolking/slides/fall02/lecture32p.pdf · 1 math...
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![Page 1: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/1.jpg)
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Math 211Math 211
Lecture #32
Higher Order Equations
Harmonic Motion
November 11, 2002
![Page 2: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/2.jpg)
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Higher Order EquationsHigher Order Equations
![Page 3: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/3.jpg)
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Higher Order EquationsHigher Order Equations
• Linear homogenous equation of order n.
![Page 4: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/4.jpg)
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Higher Order EquationsHigher Order Equations
• Linear homogenous equation of order n.
y(n) + a1y(n−1) + · · · + an−1y
′ + any = 0
![Page 5: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/5.jpg)
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Higher Order EquationsHigher Order Equations
• Linear homogenous equation of order n.
y(n) + a1y(n−1) + · · · + an−1y
′ + any = 0
• Linear homogenous equation of second order.
y′′ + py′ + qy = 0
![Page 6: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/6.jpg)
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Higher Order EquationsHigher Order Equations
• Linear homogenous equation of order n.
y(n) + a1y(n−1) + · · · + an−1y
′ + any = 0
• Linear homogenous equation of second order.
y′′ + py′ + qy = 0
• Equivalent system: x′ = Ax, where
x =(
y
y′
)and A =
(0 1−q −p
).
![Page 7: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/7.jpg)
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Linear IndependenceLinear Independence
![Page 8: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/8.jpg)
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Linear IndependenceLinear Independence
• A fundamental set of solutions for the system consists
of two linearly independent solutions.
![Page 9: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/9.jpg)
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Linear IndependenceLinear Independence
• A fundamental set of solutions for the system consists
of two linearly independent solutions.
Definition: Two functions u(t) and v(t) are linearly
independent if neither is a constant multiple of the other.
![Page 10: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/10.jpg)
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Linear IndependenceLinear Independence
• A fundamental set of solutions for the system consists
of two linearly independent solutions.
Definition: Two functions u(t) and v(t) are linearly
independent if neither is a constant multiple of the other.
• u(t) and v(t) are linearly independent solutions to
y′′ + py′ + qy = 0 ⇔(
u
u′
)&
(v
v′
)are linearly
independent solutions to the equivalent system.
![Page 11: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/11.jpg)
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General SolutionGeneral Solution
![Page 12: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/12.jpg)
Return LI System
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General SolutionGeneral Solution
Theorem: Suppose that y1(t) & y2(t) are linearly
independent solutions to the equation
y′′ + py′ + qy = 0.
Then the general solution is
y(t) = C1y1(t) + C2y2(t).
![Page 13: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/13.jpg)
Return LI System
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General SolutionGeneral Solution
Theorem: Suppose that y1(t) & y2(t) are linearly
independent solutions to the equation
y′′ + py′ + qy = 0.
Then the general solution is
y(t) = C1y1(t) + C2y2(t).
Definition: A set of two linearly independent solutions
is called a fundamental set of solutions.
![Page 14: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/14.jpg)
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Solutions to y′′ + py′ + qy = 0.Solutions to y′′ + py′ + qy = 0.
![Page 15: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/15.jpg)
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Solutions to y′′ + py′ + qy = 0.Solutions to y′′ + py′ + qy = 0.
• The equivalent system has exponential solutions.
![Page 16: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/16.jpg)
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Solutions to y′′ + py′ + qy = 0.Solutions to y′′ + py′ + qy = 0.
• The equivalent system has exponential solutions.
• Look for exponential solutions to the 2nd order
equation of the form y(t) = eλt.
![Page 17: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/17.jpg)
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Solutions to y′′ + py′ + qy = 0.Solutions to y′′ + py′ + qy = 0.
• The equivalent system has exponential solutions.
• Look for exponential solutions to the 2nd order
equation of the form y(t) = eλt.
• Characteristic equation: λ2 + pλ + q = 0.
![Page 18: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/18.jpg)
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Solutions to y′′ + py′ + qy = 0.Solutions to y′′ + py′ + qy = 0.
• The equivalent system has exponential solutions.
• Look for exponential solutions to the 2nd order
equation of the form y(t) = eλt.
• Characteristic equation: λ2 + pλ + q = 0.
� Characteristic polynomial: λ2 + pλ + q.
![Page 19: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/19.jpg)
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Solutions to y′′ + py′ + qy = 0.Solutions to y′′ + py′ + qy = 0.
• The equivalent system has exponential solutions.
• Look for exponential solutions to the 2nd order
equation of the form y(t) = eλt.
• Characteristic equation: λ2 + pλ + q = 0.
� Characteristic polynomial: λ2 + pλ + q.
� Same for the 2nd order equation and the system.
![Page 20: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/20.jpg)
Return General solution
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Real RootsReal Roots
![Page 21: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/21.jpg)
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Real RootsReal Roots
• If λ is a root to the characteristic polynomial then
y(t) = eλt is a solution.
![Page 22: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/22.jpg)
Return General solution
6
Real RootsReal Roots
• If λ is a root to the characteristic polynomial then
y(t) = eλt is a solution.
� If the characteristic polynomial has two distinct real
roots λ1 and λ2, then y(1t) = eλ1t and y2(t) = eλ2t
are a fundamental set of solutions.
![Page 23: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/23.jpg)
Return General solution
6
Real RootsReal Roots
• If λ is a root to the characteristic polynomial then
y(t) = eλt is a solution.
� If the characteristic polynomial has two distinct real
roots λ1 and λ2, then y(1t) = eλ1t and y2(t) = eλ2t
are a fundamental set of solutions.
• If λ is a root to the characteristic polynomial of
multiplicity 2, then y1(t) = eλt and y2(t) = teλt are a
fundamental set of solutions.
![Page 24: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/24.jpg)
Return General solution Two roots
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Complex RootsComplex Roots
![Page 25: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/25.jpg)
Return General solution Two roots
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Complex RootsComplex Roots
• If λ = α + iβ is a complex root of the characteristic
equation, then so is λ = α − iβ.
![Page 26: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/26.jpg)
Return General solution Two roots
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Complex RootsComplex Roots
• If λ = α + iβ is a complex root of the characteristic
equation, then so is λ = α − iβ.
• A complex valued fundamental set of solutions is
z(t) = eλt and z(t) = eλt.
![Page 27: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/27.jpg)
Return General solution Two roots
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Complex RootsComplex Roots
• If λ = α + iβ is a complex root of the characteristic
equation, then so is λ = α − iβ.
• A complex valued fundamental set of solutions is
z(t) = eλt and z(t) = eλt.
• A real valued fundamental set of solutions is
x(t) = eαt cos βt and y(t) = eαt sin βt.
![Page 28: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/28.jpg)
Return General solution Real roots Complex roots
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ExamplesExamples
![Page 29: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/29.jpg)
Return General solution Real roots Complex roots
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ExamplesExamples
• y′′ − 5y′ + 6y = 0
![Page 30: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/30.jpg)
Return General solution Real roots Complex roots
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ExamplesExamples
• y′′ − 5y′ + 6y = 0, with y(0) = 0 and y′(0) = 1.
![Page 31: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/31.jpg)
Return General solution Real roots Complex roots
8
ExamplesExamples
• y′′ − 5y′ + 6y = 0, with y(0) = 0 and y′(0) = 1.
• y′′ + 4y′ + 13y = 0
![Page 32: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/32.jpg)
Return General solution Real roots Complex roots
8
ExamplesExamples
• y′′ − 5y′ + 6y = 0, with y(0) = 0 and y′(0) = 1.
• y′′ + 4y′ + 13y = 0, with y(0) = −1 and y′(0) = 14.
![Page 33: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/33.jpg)
Return General solution Real roots Complex roots
8
ExamplesExamples
• y′′ − 5y′ + 6y = 0, with y(0) = 0 and y′(0) = 1.
• y′′ + 4y′ + 13y = 0, with y(0) = −1 and y′(0) = 14.
• y′′ + 4y′ + 4y = 0
![Page 34: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/34.jpg)
Return General solution Real roots Complex roots
8
ExamplesExamples
• y′′ − 5y′ + 6y = 0, with y(0) = 0 and y′(0) = 1.
• y′′ + 4y′ + 13y = 0, with y(0) = −1 and y′(0) = 14.
• y′′ + 4y′ + 4y = 0, with y(0) = 2 and y′(0) = 0.
![Page 35: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/35.jpg)
Return General solution Real roots Complex roots
8
ExamplesExamples
• y′′ − 5y′ + 6y = 0, with y(0) = 0 and y′(0) = 1.
• y′′ + 4y′ + 13y = 0, with y(0) = −1 and y′(0) = 14.
• y′′ + 4y′ + 4y = 0, with y(0) = 2 and y′(0) = 0.
• y′′ + 25y = 0
![Page 36: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/36.jpg)
Return General solution Real roots Complex roots
8
ExamplesExamples
• y′′ − 5y′ + 6y = 0, with y(0) = 0 and y′(0) = 1.
• y′′ + 4y′ + 13y = 0, with y(0) = −1 and y′(0) = 14.
• y′′ + 4y′ + 4y = 0, with y(0) = 2 and y′(0) = 0.
• y′′ + 25y = 0, with y(0) = 3 and y′(0) = −2.
![Page 37: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/37.jpg)
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The Vibrating SpringThe Vibrating Spring
![Page 38: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/38.jpg)
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The Vibrating SpringThe Vibrating Spring
Newton’s second law: ma = total force.
![Page 39: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/39.jpg)
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The Vibrating SpringThe Vibrating Spring
Newton’s second law: ma = total force.
• Forces acting:
![Page 40: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/40.jpg)
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The Vibrating SpringThe Vibrating Spring
Newton’s second law: ma = total force.
• Forces acting:
� Gravity
![Page 41: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/41.jpg)
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The Vibrating SpringThe Vibrating Spring
Newton’s second law: ma = total force.
• Forces acting:
� Gravity mg.
![Page 42: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/42.jpg)
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The Vibrating SpringThe Vibrating Spring
Newton’s second law: ma = total force.
• Forces acting:
� Gravity mg.
� Restoring force R(x).
![Page 43: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/43.jpg)
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The Vibrating SpringThe Vibrating Spring
Newton’s second law: ma = total force.
• Forces acting:
� Gravity mg.
� Restoring force R(x).
� Damping force D(v).
![Page 44: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/44.jpg)
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The Vibrating SpringThe Vibrating Spring
Newton’s second law: ma = total force.
• Forces acting:
� Gravity mg.
� Restoring force R(x).
� Damping force D(v).
� External force F (t).
![Page 45: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/45.jpg)
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• Including all of the forces, Newton’s law becomes
ma = mg + R(x) + D(v) + F (t)
![Page 46: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/46.jpg)
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• Including all of the forces, Newton’s law becomes
ma = mg + R(x) + D(v) + F (t)
• Hooke’s law: R(x) = −kx.
![Page 47: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/47.jpg)
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• Including all of the forces, Newton’s law becomes
ma = mg + R(x) + D(v) + F (t)
• Hooke’s law: R(x) = −kx.
� k > 0 is the spring constant.
![Page 48: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/48.jpg)
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• Including all of the forces, Newton’s law becomes
ma = mg + R(x) + D(v) + F (t)
• Hooke’s law: R(x) = −kx.
� k > 0 is the spring constant.
� Spring-mass equilibrium
![Page 49: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/49.jpg)
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• Including all of the forces, Newton’s law becomes
ma = mg + R(x) + D(v) + F (t)
• Hooke’s law: R(x) = −kx.
� k > 0 is the spring constant.
� Spring-mass equilibrium x0 = mg/k.
![Page 50: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/50.jpg)
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10
• Including all of the forces, Newton’s law becomes
ma = mg + R(x) + D(v) + F (t)
• Hooke’s law: R(x) = −kx.
� k > 0 is the spring constant.
� Spring-mass equilibrium x0 = mg/k.
� Set y = x − x0.
![Page 51: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/51.jpg)
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10
• Including all of the forces, Newton’s law becomes
ma = mg + R(x) + D(v) + F (t)
• Hooke’s law: R(x) = −kx.
� k > 0 is the spring constant.
� Spring-mass equilibrium x0 = mg/k.
� Set y = x − x0. Newton’s law becomes
my′′ = −ky + D(y′) + F (t).
![Page 52: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/52.jpg)
Return Vibrating spring
11
• Damping force D(y′)
![Page 53: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/53.jpg)
Return Vibrating spring
11
• Damping force D(y′) = −µy′.
![Page 54: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/54.jpg)
Return Vibrating spring
11
• Damping force D(y′) = −µy′.
� µ ≥ 0 is the damping constant.
![Page 55: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/55.jpg)
Return Vibrating spring
11
• Damping force D(y′) = −µy′.
� µ ≥ 0 is the damping constant.
� Newton’s law becomes
![Page 56: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/56.jpg)
Return Vibrating spring
11
• Damping force D(y′) = −µy′.
� µ ≥ 0 is the damping constant.
� Newton’s law becomes
my′′ = −ky − µy′ + F (t)
![Page 57: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/57.jpg)
Return Vibrating spring
11
• Damping force D(y′) = −µy′.
� µ ≥ 0 is the damping constant.
� Newton’s law becomes
my′′ = −ky − µy′ + F (t), or
my′′ + µy′ + ky = F (t)
![Page 58: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/58.jpg)
Return Vibrating spring
11
• Damping force D(y′) = −µy′.
� µ ≥ 0 is the damping constant.
� Newton’s law becomes
my′′ = −ky − µy′ + F (t), or
my′′ + µy′ + ky = F (t), or
y′′ +µ
my′ +
k
my =
1m
F (t).
![Page 59: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/59.jpg)
Return Vibrating spring
11
• Damping force D(y′) = −µy′.
� µ ≥ 0 is the damping constant.
� Newton’s law becomes
my′′ = −ky − µy′ + F (t), or
my′′ + µy′ + ky = F (t), or
y′′ +µ
my′ +
k
my =
1m
F (t).
• This is the equation of the vibrating spring.
![Page 60: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/60.jpg)
Return Vibrating spring equation
12
RLC CircuitRLC CircuitL
C
R
E
+
−
I
I
![Page 61: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/61.jpg)
Return Vibrating spring equation
12
RLC CircuitRLC CircuitL
C
R
E
+
−
I
I
LI ′′ + RI ′ +1C
I = E′(t)
![Page 62: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/62.jpg)
Return Vibrating spring equation
12
RLC CircuitRLC CircuitL
C
R
E
+
−
I
I
LI ′′ + RI ′ +1C
I = E′(t), or
I ′′ +R
LI ′ +
1LC
I =1L
E′(t).
![Page 63: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/63.jpg)
Return Vibrating spring equation
12
RLC CircuitRLC CircuitL
C
R
E
+
−
I
I
LI ′′ + RI ′ +1C
I = E′(t), or
I ′′ +R
LI ′ +
1LC
I =1L
E′(t).
• This is the equation of the RLC circuit.
![Page 64: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/64.jpg)
Return
13
Harmonic MotionHarmonic Motion
![Page 65: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/65.jpg)
Return
13
Harmonic MotionHarmonic Motion
• Spring: y′′ + µmy′ + k
my = 1mF (t).
![Page 66: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/66.jpg)
Return
13
Harmonic MotionHarmonic Motion
• Spring: y′′ + µmy′ + k
my = 1mF (t).
• Circuit: I ′′ + RL I ′ + 1
LC I = 1LE′(t).
![Page 67: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/67.jpg)
Return
13
Harmonic MotionHarmonic Motion
• Spring: y′′ + µmy′ + k
my = 1mF (t).
• Circuit: I ′′ + RL I ′ + 1
LC I = 1LE′(t).
• Essentially the same equation.
![Page 68: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/68.jpg)
Return
13
Harmonic MotionHarmonic Motion
• Spring: y′′ + µmy′ + k
my = 1mF (t).
• Circuit: I ′′ + RL I ′ + 1
LC I = 1LE′(t).
• Essentially the same equation. Use
x′′ + 2cx′ + ω20x = f(t).
![Page 69: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/69.jpg)
Return
13
Harmonic MotionHarmonic Motion
• Spring: y′′ + µmy′ + k
my = 1mF (t).
• Circuit: I ′′ + RL I ′ + 1
LC I = 1LE′(t).
• Essentially the same equation. Use
x′′ + 2cx′ + ω20x = f(t).
� We call this the equation for harmonic motion.
![Page 70: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/70.jpg)
Return
13
Harmonic MotionHarmonic Motion
• Spring: y′′ + µmy′ + k
my = 1mF (t).
• Circuit: I ′′ + RL I ′ + 1
LC I = 1LE′(t).
• Essentially the same equation. Use
x′′ + 2cx′ + ω20x = f(t).
� We call this the equation for harmonic motion.
� It includes both the vibrating spring and the RLC
circuit.
![Page 71: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/71.jpg)
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14
The Equation for Harmonic MotionThe Equation for Harmonic Motion
x′′ + 2cx′ + ω20x = f(t).
![Page 72: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/72.jpg)
Return
14
The Equation for Harmonic MotionThe Equation for Harmonic Motion
x′′ + 2cx′ + ω20x = f(t).
• ω0 is the natural frequency.
![Page 73: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/73.jpg)
Return
14
The Equation for Harmonic MotionThe Equation for Harmonic Motion
x′′ + 2cx′ + ω20x = f(t).
• ω0 is the natural frequency.
� Spring: ω0 =√
k/m.
![Page 74: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/74.jpg)
Return
14
The Equation for Harmonic MotionThe Equation for Harmonic Motion
x′′ + 2cx′ + ω20x = f(t).
• ω0 is the natural frequency.
� Spring: ω0 =√
k/m.
� Circuit: ω0 =√
1/LC.
![Page 75: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/75.jpg)
Return
14
The Equation for Harmonic MotionThe Equation for Harmonic Motion
x′′ + 2cx′ + ω20x = f(t).
• ω0 is the natural frequency.
� Spring: ω0 =√
k/m.
� Circuit: ω0 =√
1/LC.
• c is the damping constant.
![Page 76: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/76.jpg)
Return
14
The Equation for Harmonic MotionThe Equation for Harmonic Motion
x′′ + 2cx′ + ω20x = f(t).
• ω0 is the natural frequency.
� Spring: ω0 =√
k/m.
� Circuit: ω0 =√
1/LC.
• c is the damping constant.
� Spring: 2c = µ/m.
![Page 77: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/77.jpg)
Return
14
The Equation for Harmonic MotionThe Equation for Harmonic Motion
x′′ + 2cx′ + ω20x = f(t).
• ω0 is the natural frequency.
� Spring: ω0 =√
k/m.
� Circuit: ω0 =√
1/LC.
• c is the damping constant.
� Spring: 2c = µ/m.
� Circuit: 2c = R/L.
![Page 78: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/78.jpg)
Return
14
The Equation for Harmonic MotionThe Equation for Harmonic Motion
x′′ + 2cx′ + ω20x = f(t).
• ω0 is the natural frequency.
� Spring: ω0 =√
k/m.
� Circuit: ω0 =√
1/LC.
• c is the damping constant.
� Spring: 2c = µ/m.
� Circuit: 2c = R/L.
• f(t) is the forcing term.
![Page 79: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/79.jpg)
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15
Simple Harmonic MotionSimple Harmonic Motion
![Page 80: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/80.jpg)
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15
Simple Harmonic MotionSimple Harmonic Motion
No forcing
![Page 81: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/81.jpg)
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15
Simple Harmonic MotionSimple Harmonic Motion
No forcing , and no damping.
![Page 82: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/82.jpg)
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15
Simple Harmonic MotionSimple Harmonic Motion
No forcing , and no damping.
x′′ + ω20x = 0
![Page 83: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/83.jpg)
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15
Simple Harmonic MotionSimple Harmonic Motion
No forcing , and no damping.
x′′ + ω20x = 0
• p(λ) = λ2 + ω20
![Page 84: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/84.jpg)
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15
Simple Harmonic MotionSimple Harmonic Motion
No forcing , and no damping.
x′′ + ω20x = 0
• p(λ) = λ2 + ω20 , λ = ±iω0.
![Page 85: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/85.jpg)
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15
Simple Harmonic MotionSimple Harmonic Motion
No forcing , and no damping.
x′′ + ω20x = 0
• p(λ) = λ2 + ω20 , λ = ±iω0.
• Fundamental set of solutions:
![Page 86: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/86.jpg)
Return
15
Simple Harmonic MotionSimple Harmonic Motion
No forcing , and no damping.
x′′ + ω20x = 0
• p(λ) = λ2 + ω20 , λ = ±iω0.
• Fundamental set of solutions: x1(t) = cos ω0t &
x2(t) = sinω0t.
![Page 87: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/87.jpg)
Return
15
Simple Harmonic MotionSimple Harmonic Motion
No forcing , and no damping.
x′′ + ω20x = 0
• p(λ) = λ2 + ω20 , λ = ±iω0.
• Fundamental set of solutions: x1(t) = cos ω0t &
x2(t) = sinω0t.
• General solution: x(t) = C1 cos ω0t + C2 sin ω0t.
![Page 88: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/88.jpg)
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15
Simple Harmonic MotionSimple Harmonic Motion
No forcing , and no damping.
x′′ + ω20x = 0
• p(λ) = λ2 + ω20 , λ = ±iω0.
• Fundamental set of solutions: x1(t) = cos ω0t &
x2(t) = sinω0t.
• General solution: x(t) = C1 cos ω0t + C2 sin ω0t.
• Every solution is periodic with the natural frequency
ω0.
![Page 89: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/89.jpg)
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15
Simple Harmonic MotionSimple Harmonic Motion
No forcing , and no damping.
x′′ + ω20x = 0
• p(λ) = λ2 + ω20 , λ = ±iω0.
• Fundamental set of solutions: x1(t) = cos ω0t &
x2(t) = sinω0t.
• General solution: x(t) = C1 cos ω0t + C2 sin ω0t.
• Every solution is periodic with the natural frequency
ω0.
� The period is T = 2π/ω0.
![Page 90: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/90.jpg)
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16
Amplitude and PhaseAmplitude and Phase
![Page 91: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/91.jpg)
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Amplitude and PhaseAmplitude and Phase
• Put C1 and C2 in polar coordinates:
![Page 92: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/92.jpg)
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16
Amplitude and PhaseAmplitude and Phase
• Put C1 and C2 in polar coordinates:
C1 = A cos φ, & C2 = A sin φ.
![Page 93: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/93.jpg)
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16
Amplitude and PhaseAmplitude and Phase
• Put C1 and C2 in polar coordinates:
C1 = A cos φ, & C2 = A sin φ.
• Then x(t) = C1 cos ω0t + C2 sin ω0t
![Page 94: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/94.jpg)
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16
Amplitude and PhaseAmplitude and Phase
• Put C1 and C2 in polar coordinates:
C1 = A cos φ, & C2 = A sin φ.
• Then x(t) = C1 cos ω0t + C2 sin ω0t
= A cos(ω0t − φ).
![Page 95: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/95.jpg)
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16
Amplitude and PhaseAmplitude and Phase
• Put C1 and C2 in polar coordinates:
C1 = A cos φ, & C2 = A sin φ.
• Then x(t) = C1 cos ω0t + C2 sin ω0t
= A cos(ω0t − φ).
• A is the amplitude
![Page 96: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/96.jpg)
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16
Amplitude and PhaseAmplitude and Phase
• Put C1 and C2 in polar coordinates:
C1 = A cos φ, & C2 = A sin φ.
• Then x(t) = C1 cos ω0t + C2 sin ω0t
= A cos(ω0t − φ).
• A is the amplitude; A =√
C21 + C2
2 .
![Page 97: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/97.jpg)
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16
Amplitude and PhaseAmplitude and Phase
• Put C1 and C2 in polar coordinates:
C1 = A cos φ, & C2 = A sin φ.
• Then x(t) = C1 cos ω0t + C2 sin ω0t
= A cos(ω0t − φ).
• A is the amplitude; A =√
C21 + C2
2 .
• φ is the phase
![Page 98: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/98.jpg)
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16
Amplitude and PhaseAmplitude and Phase
• Put C1 and C2 in polar coordinates:
C1 = A cos φ, & C2 = A sin φ.
• Then x(t) = C1 cos ω0t + C2 sin ω0t
= A cos(ω0t − φ).
• A is the amplitude; A =√
C21 + C2
2 .
• φ is the phase; tanφ = C2/C1.
![Page 99: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/99.jpg)
Return Amplitude & phase
17
ExamplesExamples
![Page 100: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/100.jpg)
Return Amplitude & phase
17
ExamplesExamples
• C1 = 3, C2 = 4
![Page 101: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/101.jpg)
Return Amplitude & phase
17
ExamplesExamples
• C1 = 3, C2 = 4 ⇒ A = 5
![Page 102: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/102.jpg)
Return Amplitude & phase
17
ExamplesExamples
• C1 = 3, C2 = 4 ⇒ A = 5, φ = 0.9273.
![Page 103: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/103.jpg)
Return Amplitude & phase
17
ExamplesExamples
• C1 = 3, C2 = 4 ⇒ A = 5, φ = 0.9273.
• C1 = −3, C2 = 4
![Page 104: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/104.jpg)
Return Amplitude & phase
17
ExamplesExamples
• C1 = 3, C2 = 4 ⇒ A = 5, φ = 0.9273.
• C1 = −3, C2 = 4 ⇒ A = 5
![Page 105: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/105.jpg)
Return Amplitude & phase
17
ExamplesExamples
• C1 = 3, C2 = 4 ⇒ A = 5, φ = 0.9273.
• C1 = −3, C2 = 4 ⇒ A = 5, φ = 2.2143.
![Page 106: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/106.jpg)
Return Amplitude & phase
17
ExamplesExamples
• C1 = 3, C2 = 4 ⇒ A = 5, φ = 0.9273.
• C1 = −3, C2 = 4 ⇒ A = 5, φ = 2.2143.
• C1 = −3, C2 = −4
![Page 107: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/107.jpg)
Return Amplitude & phase
17
ExamplesExamples
• C1 = 3, C2 = 4 ⇒ A = 5, φ = 0.9273.
• C1 = −3, C2 = 4 ⇒ A = 5, φ = 2.2143.
• C1 = −3, C2 = −4 ⇒ A = 5
![Page 108: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/108.jpg)
Return Amplitude & phase
17
ExamplesExamples
• C1 = 3, C2 = 4 ⇒ A = 5, φ = 0.9273.
• C1 = −3, C2 = 4 ⇒ A = 5, φ = 2.2143.
• C1 = −3, C2 = −4 ⇒ A = 5, φ = −2.2143.
![Page 109: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/109.jpg)
Return Amplitude & phase
18
ExampleExample
x′′ + 16x = 0, x(0) = −2 & x′(0) = 4
![Page 110: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/110.jpg)
Return Amplitude & phase
18
ExampleExample
x′′ + 16x = 0, x(0) = −2 & x′(0) = 4
• Natural frequency: ω20 = 16
![Page 111: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/111.jpg)
Return Amplitude & phase
18
ExampleExample
x′′ + 16x = 0, x(0) = −2 & x′(0) = 4
• Natural frequency: ω20 = 16 ⇒ ω0 = 4.
![Page 112: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/112.jpg)
Return Amplitude & phase
18
ExampleExample
x′′ + 16x = 0, x(0) = −2 & x′(0) = 4
• Natural frequency: ω20 = 16 ⇒ ω0 = 4.
• General solution:
![Page 113: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/113.jpg)
Return Amplitude & phase
18
ExampleExample
x′′ + 16x = 0, x(0) = −2 & x′(0) = 4
• Natural frequency: ω20 = 16 ⇒ ω0 = 4.
• General solution: x(t) = C1 cos 4t + C2 sin 4t.
![Page 114: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/114.jpg)
Return Amplitude & phase
18
ExampleExample
x′′ + 16x = 0, x(0) = −2 & x′(0) = 4
• Natural frequency: ω20 = 16 ⇒ ω0 = 4.
• General solution: x(t) = C1 cos 4t + C2 sin 4t.
• IC: −2 = x(0) = C1, and 4 = x′(0) = 4C2.
![Page 115: Lecture #32 Higher Order Equations Harmonic Motionpolking/slides/fall02/lecture32p.pdf · 1 Math 211Math 211 Lecture #32 Higher Order Equations Harmonic Motion November 11, 2002](https://reader035.vdocuments.us/reader035/viewer/2022071103/5fdc934fb675f44de94f3c35/html5/thumbnails/115.jpg)
Return Amplitude & phase
18
ExampleExample
x′′ + 16x = 0, x(0) = −2 & x′(0) = 4
• Natural frequency: ω20 = 16 ⇒ ω0 = 4.
• General solution: x(t) = C1 cos 4t + C2 sin 4t.
• IC: −2 = x(0) = C1, and 4 = x′(0) = 4C2.
• Solutionx(t) = −2 cos 2t + sin 2t
=√
5 cos(2t − 2.6779).