Learning Objective Dump

Just a place to dump excess, redundant learning objectives in case we need them.

model a physical vibrating system and understand it's essential motion characteristics
numerically integrate non-linear systems
understand how the eigenvalue problem can model a vibrational system
design a mechanical system with vibration in mind
measure vibrations and interpret the meaning of the data
to analyze a vibrational system using computation

1 Introduction [100 mins]

The goal here is to lead the students through solving a vibration engineering problem while simultaneously introducing the foundation Python commands they will need to build on. This shouldn't be overwhelming with too many new Python things but should ease them in but show them that they can solve something useful the first day.

Students will be able to use the core Python computing commands in the class.
Students Will be able solve an introductory vibration engineering problem.

TODO : What problem to open with?

2 Modeling Vibrating Systems [50 min]

The key thing here is that we want students to be able to look at real physical objects and visualize what the essential motion is. They should be able to sketch out free body diagrams that indicate:

an appropriate number of degrees of freedom
appropriate generalized coordinate definitions
appropriate lumped elements that describe the rigid bodies and lumped elements (springs, dampers) and external loads acting on the system

I'd also like them to come away with an appreciation of why it may be important to try to create the simplest model that is capable of explaining the phenomena of interest.

3 Formulating Equations of Motion [150 min]

recognize that the relationship between mass/inertia, acceleration, and the loads acting on a system are second order ordinary differential equations
be able to express the linear and angular velocity magnitude expressions of important points and rotating reference frames
be able to write the system's kinetic energy in terms of generalized coordinates
be able to write the system's potential energy in terms of the generalized coordinates
be able to form the Lagrangian
be able write the Lagrange's equation of the first kind and evaluate it
add non-conservative forces with Rayleigh's principle
recognize advantages over a Newton-Euler formulation
be able to convert between first and second order form
be able to convert from canonical form to state space
single and multi dof
linearizing eoms

https://en.wikipedia.org/wiki/Lagrangian_mechanics

4 Free harmonic motion with and without viscous damping

determine the natural frequency
find the solution to the ODE
compare different ways to express ODE solution
write the SDoF system in terms of nat. freq and damping ratio
underdamped, critically damped, overdamped

Estimating system parameters from vibrations (live experiment where we give them data)
Forced harmonic motion with and without viscous damping
Non-linear vibration (Coulomb) + simulation of non-linear systems
Impulse response (heaviside)
Stability: book balance
Base excitation: car on bumpy road
Mass imbalance
Arbitrary forcing (convolution integral)
Arbitrary periodic forcing (Fourier series)
Modal analysis of decomposable systems: building
Modal analysis of non-decomposable: bicycle modeshapes
Isolator design
Vibration absorbers

Other:

Stiffness
Equivalency in stiffness, damping, mass, etc
Free response to two dof
Transform methods
Response random inputs
Analogy to electrical circuits or other energy domains
More in-depth non-linear vibratory systems
Relationship to FEA of structures
Beams and membranes: continuous systems (Euler’s beam equation)

Analyzing Vibrating Systems

After finishing this section students will be able to:

state the three fundamental characteristics that make a system vibrate
describe different methods of measuring vibrations
choose appropriate sensors and a placement
visualize the vibrational measurements
visualize a system’s free response
identify critically damped, underdamped, and overdamped behavior
excite a system with different input signals
use time domain techniques to characterize a system's behavior
use frequency domain techniques to characterize a system's behavior
identify a MDoF system and see effects of coupling through time and frequency domain
interactively adjust core system parameters to affect response

The story arc:

Start with a conversation about real things that vibrate. Show vibrating systems and ask for examples. Have class discussion on commonalities with the instructor goal of identifying mass/inertia, flexibility/stiffness, damping as core common attributes of vibratory systems.
How might we measure the motion of these things? What are important measurements? Talk about position, velocity, and acceleration measurement techniques. Mass, stiffness, damping measurement?
Introduce a 1 DoF system (block on an ice rink) (mass and damping?). Show how to simulate an initial value problem and have the students inspect the measurements (position, velocity, acceleration, .. jerk). What do you observe with initial velocity? (velocity decreases over time) Have them look at short sim, so velocity doesn't change, and then a long sim where velicity does change (goes to zero)
Introduce a rotational 1 DOF system (mass and damping) and it's measurements (angle, angular velocity, angular acceleration)
Now introduce a 1 DoF ?bungee jumper? system (or something without gravity). Explain that this sysem has an elastic element that resists the motion of mass. Let the students simulate this and talk about the characteristics of the measurements? What does position functino look like? What is the relationship among the three measurements? Can they determine the dreivative relationship?
Show how to adjust the mass, and stiffness values of the system. Let students explore to discover effects of the parameters on the free response. Can they find out that m and k primarily affect the frequency of oscillation? Introduce the concept of natural frequency.
Introduce autocorrleation to see period of signal?
Call the signal frequency spectrum method (FFT of signal) to look at the frequency vs amplitude of the signal. Introduce the frequency domain view of the signal.
Introduce the effects of damping on the free response: over, under, critical
Introduce multi dof system (two bungee jumpers?), talk about dof, coordinates, etc
Simulate initial value problem and look at signals in time and frequency domain.
Now let's poke at the systems with an input. Starting with a sinusoidal input. What does the output look like? Should identify that it is also sinusoidal but diff amplitude. Try different frequencies.
Introduce the frequency response. Use a sweep frequency input, call a underlying sys id method that produces the frequency response plot? Talk about amplitude and phase relationships.
Explore how the mass, stifneess, and damping affect the response plot.
Introduce resonance.
Look at frequency response of mdof systems.

Modeling Vibrating Systems

After finishing this section students will be able to:

identify the essential aspects to model (# degrees of freedom, coordinates, modeling assumptions)
draw a vibration free body diagram of a real system
write the non-linear equations of motion of a vibrating system
write a linear equation of motion for a vibrating system
transform a mathematical model into a computational model
validate that the model behaves like the real system in a way that is optimal for answering your questions about the system

Designing Vibrating Systems

After finishing this section students will be able to:

realize that mass, stiffness, damping, and geometry affect behavior in a complex coupled way. Understand the tradeoffs among adjusting model parameters
use common methods like isolators, mass balance, mass abosrber to control behavior

moorepants / resonance