From f04625d539c1d1477a68fe62f092eb2dab9064d3 Mon Sep 17 00:00:00 2001 From: Alvie Rahman Date: Mon, 6 Mar 2023 22:25:45 +0000 Subject: [PATCH] vibrations lecture --- approximate methods --- .../approximate_methods.md | 68 ++++++++++++++++++ ...HEET 6 - Approximate Methods Solutions.pdf | Bin 0 -> 665314 bytes ...bratioon SHEET 6 - Approximate Methods.pdf | Bin 0 -> 221209 bytes .../lecture_notes/1-dof Handout 07_v2.pdf | Bin 0 -> 438484 bytes ...Approximate Methods - No Dunkerley - 6.pdf | Bin 0 -> 473694 bytes .../lecture_notes/Beam Vibrations 07_v2.pdf | Bin 0 -> 411601 bytes .../Free vibration response 07_v2.pdf | Bin 0 -> 479088 bytes .../Harmonic response 08_v2 - 5.pdf | Bin 0 -> 661920 bytes .../lecture_notes/Isolation 7.pdf | Bin 0 -> 503067 bytes .../lecture_notes/N-dof Systems 07_v2.pdf | Bin 0 -> 560871 bytes .../2021 Control Lecture 5.pptx | Bin 0 -> 1021117 bytes .../Control Consolidation 3.pptx | Bin 0 -> 950530 bytes .../Vibrations - Approximate Methods.pdf | Bin 0 -> 506467 bytes 13 files changed, 68 insertions(+) create mode 100755 uni/mmme/2046_dynamics_and_control/approximate_methods.md create mode 100644 uni/mmme/2046_dynamics_and_control/exercise_sheets/Vibratioon SHEET 6 - Approximate Methods Solutions.pdf create mode 100644 uni/mmme/2046_dynamics_and_control/exercise_sheets/Vibratioon SHEET 6 - Approximate Methods.pdf create mode 100644 uni/mmme/2046_dynamics_and_control/lecture_notes/1-dof Handout 07_v2.pdf create mode 100644 uni/mmme/2046_dynamics_and_control/lecture_notes/Approximate Methods - No Dunkerley - 6.pdf create mode 100644 uni/mmme/2046_dynamics_and_control/lecture_notes/Beam Vibrations 07_v2.pdf create mode 100644 uni/mmme/2046_dynamics_and_control/lecture_notes/Free vibration response 07_v2.pdf create mode 100644 uni/mmme/2046_dynamics_and_control/lecture_notes/Harmonic response 08_v2 - 5.pdf create mode 100644 uni/mmme/2046_dynamics_and_control/lecture_notes/Isolation 7.pdf create mode 100644 uni/mmme/2046_dynamics_and_control/lecture_notes/N-dof Systems 07_v2.pdf create mode 100644 uni/mmme/2046_dynamics_and_control/lecture_slides/2021 Control Lecture 5.pptx create mode 100644 uni/mmme/2046_dynamics_and_control/lecture_slides/Control Consolidation 3.pptx create mode 100644 uni/mmme/2046_dynamics_and_control/lecture_slides/Vibrations - Approximate Methods.pdf diff --git a/uni/mmme/2046_dynamics_and_control/approximate_methods.md b/uni/mmme/2046_dynamics_and_control/approximate_methods.md new file mode 100755 index 0000000..f4143b8 --- /dev/null +++ b/uni/mmme/2046_dynamics_and_control/approximate_methods.md @@ -0,0 +1,68 @@ +--- +author: Akbar Rahman +date: \today +title: MMME2046 // Approximate Methods +tags: [ vibrations, approximate_methods, rayleighs_method ] +uuid: 7cd5b86f-74df-4ec6-b3c6-9204cf949093 +lecture_slides: [ ./lecture_slides/Vibrations - Approximate Methods.pdf ] +lecture_notes: [ ./lecture_notes/Approximate Methods - No Dunkerley - 6.pdf ] +exercise_sheets: [ ./exercise_sheets/Vibratioon SHEET 6 - Approximate Methods.pdf, ./exercise_sheets/Vibratioon SHEET 6 - Approximate Methods Solutions.pdf ] +--- + +branson says 3rd year dynamics is way harder + +This module will focus exclusively on Rayleigh's method. +In general: + +$$\omega_{n,\text{Rayleigh}} \geq \omega_{n,\text{exact}}$$ + +This means that it is best to try several mode shapes and the lowest frequency of +those shapes will be the most accurate. + +The most important frequency is the first natural frequency. +If the operating frequency never achieves this frequency, then there won't be any issues. + +# Instantaneous Strain Energy + +$$E = 0.5k(\Delta L)^2$$ + +where $k$ is the stiffness, $\Delta L$ is the length + +# Rayleigh's Method for Shafts and Beams + +$$T_\text{max} = 0.5\omega^2 \int^L_0 \rho A [Y(x)]^2 \mathrm{d} x$$ + +$$U_\text{max} = 0.5\int^L_0 EI \left(\frac{\mathrm{d}^2Y}{\mathrm dx^2}\right)^2 \mathrm{d} x$$ + +where $Y(x)$ is the mode shape function, which defines the amplitude of vibration of the shaft or +beam along its length. This function needs to be guessed. + +## $Y(x) = Cx^2$ + +![](./images/vimscrot-2023-03-06T16:38:11,596006361+00:00.png) + +This guess is not that close, but the only one expected to be used in exams. + +It gives: + +$$T_\text{max} = \omega^2\frac{\rho A C^2L^5}{10}$$ + +$$U_\text{max} = 2EIC^2L$$ + +Equating them gives + +$$\omega_n = \frac{4.47}{L^2}\sqrt{\frac{EI}{\rho A}}$$ + +... which is an **error of 27%** (not close). + +## Other choices of $Y(x)$ + +Are detailed in lecture slides (p12-14). + +# Dynamically Equivalent Systems (slides p22-37) + +![](./images/vimscrot-2023-03-06T17:07:02,646825881+00:00.png) + + + + diff --git a/uni/mmme/2046_dynamics_and_control/exercise_sheets/Vibratioon SHEET 6 - Approximate Methods Solutions.pdf b/uni/mmme/2046_dynamics_and_control/exercise_sheets/Vibratioon SHEET 6 - Approximate Methods Solutions.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3cd6c7f71e5b3d70a05004a4a82039f65b87ce4f GIT binary patch literal 665314 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