me 563 mechanical vibrations lecture #1 - purdue …deadams/me563/lecture110.pdf · me 563...
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![Page 1: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/1.jpg)
ME 563 Mechanical Vibrations
Lecture #1 Derivation of equations of motion
(Newton-Euler Laws)
![Page 2: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/2.jpg)
Derivation of Equation of Motion Define the vibrations of interest - Degrees of freedom (translational, rotational, etc.) - Frequency range (<5 Hz, >15 Hz, etc.) - Amplitude range (<2 g, >10 g, linear or nonlinear, etc.) Develop a model representation -Discrete/lumped elements (springs, dampers, etc.) -Continuous elements (beams, rods, membranes, plates, etc.) -Excitation function (ground motion, wind, machinery, etc.) Define motions (kinematics) -u, v, w, θ, etc. -Undeformed or deformed datum, direction w/r/t gravitational field -Constraints on/between the variables and #DOFs (base motion, gears) Derive equations of motion -Newton-Euler laws -Energy/power methods Calculate system parameters -Strength of materials or experimentation -Catalogues from vendors (bushings, mounts, couplings, etc.)
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![Page 3: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/3.jpg)
Define the vibrations of interest
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![Page 4: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/4.jpg)
Derivation of Equation of Motion Define the vibrations of interest - Degrees of freedom (translational, rotational, etc.) - Frequency range (<5 Hz, >15 Hz, etc.) - Amplitude range (<2 g, >10 g, linear or nonlinear, etc.) Develop a model representation -Discrete/lumped elements (springs, dampers, etc.) -Continuous elements (beams, rods, membranes, plates, etc.) -Excitation function (ground motion, wind, machinery, etc.) Define motions (kinematics) -u, v, w, θ, etc. -Undeformed or deformed datum, direction w/r/t gravitational field -Constraints on/between the variables and #DOFs (base motion, gears) Derive equations of motion -Newton-Euler laws -Energy/power methods Calculate system parameters -Strength of materials or experimentation -Catalogues from vendors (bushings, mounts, couplings, etc.)
Kee
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of a
ssum
ptio
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![Page 5: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/5.jpg)
Define the vibrations of interest (Degrees of freedom)
Panel rigid body degrees of freedom
Panel flexible body degrees of freedom
4
![Page 6: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/6.jpg)
Derivation of Equation of Motion Define the vibrations of interest - Degrees of freedom (translational, rotational, etc.) - Frequency range (<5 Hz, >15 Hz, etc.) - Amplitude range (<2 g, >10 g, linear or nonlinear, etc.) Develop a model representation -Discrete/lumped elements (springs, dampers, etc.) -Continuous elements (beams, rods, membranes, plates, etc.) -Excitation function (ground motion, wind, machinery, etc.) Define motions (kinematics) -u, v, w, θ, etc. -Undeformed or deformed datum, direction w/r/t gravitational field -Constraints on/between the variables and #DOFs (base motion, gears) Derive equations of motion -Newton-Euler laws -Energy/power methods Calculate system parameters -Strength of materials or experimentation -Catalogues from vendors (bushings, mounts, couplings, etc.)
Kee
p tr
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of a
ssum
ptio
ns
5
![Page 7: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/7.jpg)
Define the vibrations of interest (Frequency range)
Low-frequency (<50 Hz) response
High-frequency (>100 Hz) response
6
![Page 8: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/8.jpg)
Derivation of Equation of Motion Define the vibrations of interest - Degrees of freedom (translational, rotational, etc.) - Frequency range (<5 Hz, >15 Hz, etc.) - Amplitude range (<2 g, >10 g, linear or nonlinear, etc.) Develop a model representation -Discrete/lumped elements (springs, dampers, etc.) -Continuous elements (beams, rods, membranes, plates, etc.) -Excitation function (ground motion, wind, machinery, etc.) Define motions (kinematics) -u, v, w, θ, etc. -Undeformed or deformed datum, direction w/r/t gravitational field -Constraints on/between the variables and #DOFs (base motion, gears) Derive equations of motion -Newton-Euler laws -Energy/power methods Calculate system parameters -Strength of materials or experimentation -Catalogues from vendors (bushings, mounts, couplings, etc.)
Kee
p tr
ack
of a
ssum
ptio
ns
7
![Page 9: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/9.jpg)
Define the vibrations of interest (Amplitude range)
Ceramic tile attached to orbiter using adhesive bond Strain isolation
pad stress/strain
8
![Page 10: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/10.jpg)
Derivation of Equation of Motion Define the vibrations of interest - Degrees of freedom (translational, rotational, etc.) - Frequency range (<5 Hz, >15 Hz, etc.) - Amplitude range (<2 g, >10 g, linear or nonlinear, etc.) Develop a model representation -Discrete/lumped elements (springs, dampers, etc.) -Continuous elements (beams, rods, membranes, plates, etc.) -Excitation function (ground motion, wind, machinery, etc.) Define motions (kinematics) -u, v, w, θ, etc. -Undeformed or deformed datum, direction w/r/t gravitational field -Constraints on/between the variables and #DOFs (base motion, gears) Derive equations of motion -Newton-Euler laws -Energy/power methods Calculate system parameters -Strength of materials or experimentation -Catalogues from vendors (bushings, mounts, couplings, etc.)
Kee
p tr
ack
of a
ssum
ptio
ns
9
![Page 11: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/11.jpg)
Develop model representation
(Lumped/discrete elements)
≅ M
K1
K2
K3
K4 A
irfra
me
Airf
ram
e
Parallel elements – same motion Series elements – same force
10
![Page 12: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/12.jpg)
Derivation of Equation of Motion Define the vibrations of interest - Degrees of freedom (translational, rotational, etc.) - Frequency range (<5 Hz, >15 Hz, etc.) - Amplitude range (<2 g, >10 g, linear or nonlinear, etc.) Develop a model representation -Discrete/lumped elements (springs, dampers, etc.) -Continuous elements (beams, rods, membranes, plates, etc.) -Excitation function (ground motion, wind, machinery, etc.) Define motions (kinematics) -u, v, w, θ, etc. -Undeformed or deformed datum, direction w/r/t gravitational field -Constraints on/between the variables and #DOFs (base motion, gears) Derive equations of motion -Newton-Euler laws -Energy/power methods Calculate system parameters -Strength of materials or experimentation -Catalogues from vendors (bushings, mounts, couplings, etc.)
Kee
p tr
ack
of a
ssum
ptio
ns
11
![Page 13: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/13.jpg)
Develop model representation
(Continuous elements)
≅ M
Airframe
Airframe
12
![Page 14: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/14.jpg)
Derivation of Equation of Motion Define the vibrations of interest - Degrees of freedom (translational, rotational, etc.) - Frequency range (<5 Hz, >15 Hz, etc.) - Amplitude range (<2 g, >10 g, linear or nonlinear, etc.) Develop a model representation -Discrete/lumped elements (springs, dampers, etc.) -Continuous elements (beams, rods, membranes, plates, etc.) -Excitation function (ground motion, wind, machinery, etc.) Define motions (kinematics) -u, v, w, θ, etc. -Undeformed or deformed datum, direction w/r/t gravitational field -Constraints on/between the variables and #DOFs (base motion, gears) Derive equations of motion -Newton-Euler laws -Energy/power methods Calculate system parameters -Strength of materials or experimentation -Catalogues from vendors (bushings, mounts, couplings, etc.)
Kee
p tr
ack
of a
ssum
ptio
ns
13
![Page 15: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/15.jpg)
Develop model representation
(Excitation function)
M
K1
K2
K3
K4
Airf
ram
e
Airf
ram
e
Base motion
M
K1
K2
K3
K4 A
irfra
me
Airf
ram
e
Applied force (impact)
14
![Page 16: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/16.jpg)
Develop model representation
(Excitation function)
15
![Page 17: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/17.jpg)
Derivation of Equation of Motion Define the vibrations of interest - Degrees of freedom (translational, rotational, etc.) - Frequency range (<5 Hz, >15 Hz, etc.) - Amplitude range (<2 g, >10 g, linear or nonlinear, etc.) Develop a model representation -Discrete/lumped elements (springs, dampers, etc.) -Continuous elements (beams, rods, membranes, plates, etc.) -Excitation function (ground motion, wind, machinery, etc.) Define motions (kinematics) -u, v, w, θ, etc. -Undeformed or deformed datum, direction w/r/t gravitational field -Constraints on/between the variables and #DOFs (base motion, gears) Derive equations of motion -Newton-Euler laws -Energy/power methods Calculate system parameters -Strength of materials or experimentation -Catalogues from vendors (bushings, mounts, couplings, etc.)
Kee
p tr
ack
of a
ssum
ptio
ns
16
![Page 18: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/18.jpg)
Define motions (DOFs) (Generalized coordinates,
datum, gravity)
x is defined w/r/t equilibrium position
(under acceleration of shuttle)
Dynamic response is large compared to
gravitational response
Acceleration
Acceleration
17
x
M
K1
K2
K3
K4
Airf
ram
e
Airf
ram
e
![Page 19: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/19.jpg)
Derivation of Equation of Motion Define the vibrations of interest - Degrees of freedom (translational, rotational, etc.) - Frequency range (<5 Hz, >15 Hz, etc.) - Amplitude range (<2 g, >10 g, linear or nonlinear, etc.) Develop a model representation -Discrete/lumped elements (springs, dampers, etc.) -Continuous elements (beams, rods, membranes, plates, etc.) -Excitation function (ground motion, wind, machinery, etc.) Define motions (kinematics) -u, v, w, θ, etc. -Undeformed or deformed datum, direction w/r/t gravitational field -Constraints on/between the variables and #DOFs Derive equations of motion -Newton-Euler laws -Energy/power methods Calculate system parameters -Strength of materials or experimentation -Catalogues from vendors (bushings, mounts, couplings, etc.)
Kee
p tr
ack
of a
ssum
ptio
ns
18
![Page 20: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/20.jpg)
Define motions (DOFs) (Constraints on
coordinates) x
Acceleration xairframe(t)
# D.O.F. = # generalized coordinates - # constraints = 2 – 1 = 1
M
K1
K2
K3
K4
Airf
ram
e
Airf
ram
e
19
![Page 21: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/21.jpg)
Derivation of Equation of Motion Define the vibrations of interest - Degrees of freedom (translational, rotational, etc.) - Frequency range (<5 Hz, >15 Hz, etc.) - Amplitude range (<2 g, >10 g, linear or nonlinear, etc.) Develop a model representation -Discrete/lumped elements (springs, dampers, etc.) -Continuous elements (beams, rods, membranes, plates, etc.) -Excitation function (ground motion, wind, machinery, etc.) Define motions (kinematics) -u, v, w, θ, etc. -Undeformed or deformed datum, direction w/r/t gravitational field -Constraints on/between the variables and #DOFs (base motion, gears) Derive equations of motion -Newton-Euler laws -Energy/power methods Calculate system parameters -Strength of materials or experimentation -Catalogues from vendors (bushings, mounts, couplings, etc.)
Kee
p tr
ack
of a
ssum
ptio
ns
20
![Page 22: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/22.jpg)
Derive equations of motion (Newton-Euler, force)
M
K1
K2
K3
K4
Airf
ram
e
Airf
ram
e
x
f(t)
M
f(t)
K1⋅x
K2⋅x
K3⋅x
K4⋅x
FBD
21
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Derive equations of motion (Newton-Euler, base motion)
M
K1
K2
K3
K4
Airf
ram
e
Airf
ram
e
x
xaf(t) M
K1⋅(x-xaf)
K2⋅(x-xaf)
K3⋅(x-xaf)
K4⋅(x-xaf)
FBD
22
![Page 24: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/24.jpg)
Derivation of Equation of Motion Define the vibrations of interest - Degrees of freedom (translational, rotational, etc.) - Frequency range (<5 Hz, >15 Hz, etc.) - Amplitude range (<2 g, >10 g, linear or nonlinear, etc.) Develop a model representation -Discrete/lumped elements (springs, dampers, etc.) -Continuous elements (beams, rods, membranes, plates, etc.) -Excitation function (ground motion, wind, machinery, etc.) Define motions (kinematics) -u, v, w, θ, etc. -Undeformed or deformed datum, direction w/r/t gravitational field -Constraints on/between the variables and #DOFs (base motion, gears) Derive equations of motion -Newton-Euler laws -Energy/power methods Calculate system parameters -Strength of materials or experimentation -Catalogues from vendors (bushings, mounts, couplings, etc.)
Kee
p tr
ack
of a
ssum
ptio
ns
23
![Page 25: ME 563 Mechanical Vibrations Lecture #1 - Purdue …deadams/ME563/lecture110.pdf · ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)](https://reader031.vdocument.in/reader031/viewer/2022022502/5aacbcbf7f8b9a693f8d6e33/html5/thumbnails/25.jpg)
Calculate system parameters
(Strength of materials)
≅ K2
f
Δ
€
Force = Stiffness×Deflectionf = K2 ×Δ
fΔ
= K2
Δ =fL3
3EI
24
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What about the rotational motion?
≅ Icm
K1
K2
K3
K4
25
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What about the rotational motion?
Icm
K1
K2
K3
K4
Icm
K1⋅θ θ K3⋅θ
K2⋅θ K4⋅θ
What is K2?
26