ME 563 Mechanical Vibrations Lecture #1. Derivation of equations of motion (Newton-Euler Laws)

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1 ME 563 Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws)

2 Derivation of Equation of Motion 1 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.) Keep track of assumptions

3 Define the 2 vibrations of interest

4 Derivation of Equation of Motion 3 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.) Keep track of assumptions

5 Define the 4 vibrations of interest (Degrees of freedom) Panel rigid body degrees of freedom Panel flexible body degrees of freedom

6 Derivation of Equation of Motion 5 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.) Keep track of assumptions

7 Define the 6 vibrations of interest (Frequency range) Low-frequency (<50 Hz) response High-frequency (>100 Hz) response

8 Derivation of Equation of Motion 7 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.) Keep track of assumptions

9 Define the 8 vibrations of interest (Amplitude range) Ceramic tile attached to orbiter using adhesive bond Strain isolation pad stress/strain

10 Derivation of Equation of Motion 9 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.) Keep track of assumptions

11 Develop model 10 representation (Lumped/discrete elements) K1 K2 M K3 K4 Parallel elements same motion Series elements same force

12 Derivation of Equation of Motion 11 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.) Keep track of assumptions

13 Develop model 12 representation (Continuous elements) M

14 Derivation of Equation of Motion 13 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.) Keep track of assumptions

15 Develop model 14 representation (Excitation function) K1 K3 K1 K3 K2 M K4 K2 M K4 Base motion Applied force (impact)

16 Develop model 15 representation (Excitation function)

17 Derivation of Equation of Motion 16 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.) Keep track of assumptions

18 Define motions (DOFs) 17 (Generalized coordinates, datum, gravity) Acceleration K1 x K3 K2 M K4 Acceleration x is defined w/r/t equilibrium position (under acceleration of shuttle) Dynamic response is large compared to gravitational response

19 Derivation of Equation of Motion 18 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.) Keep track of assumptions

20 Define motions (DOFs) 19 (Constraints on coordinates) K1 K2 M x K3 K4 Acceleration x airframe (t) # D.O.F. = # generalized coordinates - # constraints = 2 1 = 1

21 Derivation of Equation of Motion 20 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.) Keep track of assumptions

22 Derive equations of motion 21 (Newton-Euler, force) K1 x K3 K1 x FBD K3 x K2 M f(t) K4 K2 x M f(t) K4 x

23 Derive equations of motion 22 (Newton-Euler, base motion) K1 K2 M x K3 K4 x af (t) K1 (x-x af ) K2 (x-x af ) FBD M K3 (x-x af ) K4 (x-x af )

24 Derivation of Equation of Motion 23 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.) Keep track of assumptions

25 Calculate system 24 parameters (Strength of materials) f Δ K2 Force = Stiffness Deflection f = K 2 Δ f Δ = K 2 Δ = fl3 3EI

26 What about the 25 rotational motion? K1 K3 I cm K2 K4

27 What about the 26 rotational motion? K1 θ K3 K1 θ K3 θ I cm I cm K2 K4 K2 θ K4 θ What is K2?

Introduction to Mechanical Vibration

2103433 Introduction to Mechanical Vibration Nopdanai Ajavakom (NAV) 1 Course Topics Introduction to Vibration What is vibration? Basic concepts of vibration Modeling Linearization Single-Degree-of-Freedom