IntroductionCourse Goals

Review Topics

MCE371: Vibrations

Prof. Richter

Department of Mechanical Engineering

Handout 1Fall 2017

IntroductionCourse Goals

Review Topics

Outline

1 IntroductionVibration Problems in Engineering

2 Course Goals

3 Review TopicsHarmonic FunctionsExponentially-Decaying Oscillations

IntroductionCourse Goals

Review TopicsVibration Problems in Engineering

Vibration Analysis in Mechanical Engineering

Vibrations established intentionally: pneumatic jackhammer,atomic force microscope, musical instruments ...

Vibrations that should be supressed: aircraft wing flutter,tractor cab seats, gas turbine shafts ....

Give 2 more examples of intentional and undesirable vibrationin engineering systems.

In both cases, the ME must be prepared to handle vibrationproblems at three levels: analysis, design and field work(measurement and instrumentation)

IntroductionCourse Goals

Review TopicsVibration Problems in Engineering

Intentional Vibrations: Non-Contact Mode AFM

Image from Park Systems(http://www.parkafm.com/AFM_technology/cross_e_xe.php)

Interatomic forces modulate the cantilever vibrations. Data can beused to obtain an image as the sample is scanned in 3D.

Key idea: use feedback control to compensate for changes to thevibration amplitude caused by sample-tip distances.

More details:http://en.wikipedia.org/wiki/Atomic_force_microscopy

IntroductionCourse Goals

Review TopicsVibration Problems in Engineering

Unintentional Vibrations: Aircraft Wing Flutter

Phenomenon where interacting aerodynamic and elastic forcesresult in a self-induced oscillation.

Example: NASA test on twin-engine plane:http://www.youtube.com/watch?v=iTFZNrTYp3k&feature=relate

Tacoma Narrows 1940 bridge collapse:http://www.youtube.com/watch?v=3mclp9QmCGs

Both are examples of the same destructive self-inducedvibration phenomenon.

IntroductionCourse Goals

Review TopicsVibration Problems in Engineering

Valve Floating Problem

Problem: valves begin to stay open (float) at high rotational speeds

How would you begin to address this problem? (aiming at valvetrainredesign for a faster engine)

At the end of this course, you should be able to carry basic analysis

and prepare computer simulations to study problems like this

IntroductionCourse Goals

Review TopicsVibration Problems in Engineering

Vibration Control: Active/Semiactive Suspension

You need a course in control systems following vibrations to work on this!

IntroductionCourse Goals

Review Topics

Course Goals

1 Establish a solid foundation to describe dynamic mechanical systemsin terms of differential equations

2 Develop proficiency in understanding and using the properties of thesolutions of forced and unforced one-degree-of-freedom systems.

3 Develop proficiency in deriving the differential equations of motionfor systems with multiple degrees-of-freedom.

4 Develop proficiency in solving one-degree-of-freedom differentialequations using Laplace methods.

5 Develop working-level proficiency with computer simulation tools:Matlab and Simulink

6 Introduce matrix methods for systems with multipledegrees-of-freedom.

7 Introduce vibration suppression methods and vibration measurementinstrumentation. A laboratory session on accelerometers will beincluded.

IntroductionCourse Goals

Review Topics

Harmonic FunctionsExponentially-Decaying Oscillations

Using sinusoids to describe harmonic motion

Reference: Palm Sect. 1.2, Inman Sects. 1.1, 1.2If you measure real vibrations using a pickup, do you expect to see a

perfect sinewave?

An experimental vibration trace can be decomposed into contributing

sinusoids of various frequencies. This is why we need to be very familiarwith harmonic functions, their parameters and their graphicalrepresentations.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−10

−5

0

5

10Example: Fourier Decomposition

Vib

ratio

n A

mpl

itude

, mill

i−in

ches

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−4

−2

0

2

4

Time,ms

Vib

ratio

n C

ompo

nent

s, m

illi−

inch

es

IntroductionCourse Goals

Review Topics

Harmonic FunctionsExponentially-Decaying Oscillations

Sinewave Parameters

y(t) = A sin(wt + φ) = A sin(2πf + φ)

1 A is the amplitude (half of the distance from low peak to high peak)

2 w is the radian frequency measured in rad/s

3 f is the number of cycles per second (Hertz): w = 2πf .

4 φ is the phase in radians

5 T = 1/f is the period in sec.

IntroductionCourse Goals

Review Topics

Harmonic FunctionsExponentially-Decaying Oscillations

Useful Identities

cos(x + φ) = sin(x + φ+π

2)

− sin(x + φ) = sin(x + φ+ π)

Exercise: If y(t) = A sin(wt +φ) is the position, obtain the velocityand the acceleration in terms of sin and sketch the three functions.

IntroductionCourse Goals

Review Topics

Harmonic FunctionsExponentially-Decaying Oscillations

Plotting Harmonic Functions in Matlab

1 Create a time vector: >> t=[0:0.1:10]. Use a reasonable∆t. Rule-of-thumb: T/∆t > 8.

2 Type the expression for the sinewave: >> y=3*sin(2*t+1.5)

3 Produce a basic plot: >> plot(t,y)

4 Label the plot (A MUST): >> title(’Sine

Wave’);xlabel(’Time, sec.’);ylabel(’Position,

mm’)

IntroductionCourse Goals

Review Topics

Harmonic FunctionsExponentially-Decaying Oscillations

Some Plot Refinements

1 Suppose we want to add another trace on the same plot: >>y2=sin(4*t)

2 Plot with a yellow asterisk only: >> hold on;

plot(t,y2,’*y’)

3 Add a legend: >>legend(’y’,’y2’,’Location’,’NorthEast’)

4 Type >> help plot and carefully study the options

IntroductionCourse Goals

Review Topics

Harmonic FunctionsExponentially-Decaying Oscillations

Assigned Exercise

1 Download the data from the course website

2 Plot and label using the given time and position vectors. Usedata point markers instead of a solid line

3 Find the approximate amplitude, frequency and phase fromthe plot

4 Calculate a fitted sinewave using the above and superimpose asolid-line plot. Include a legend.

IntroductionCourse Goals

Review Topics

Harmonic FunctionsExponentially-Decaying Oscillations

Decaying and Increasing Oscillations

1 A damped vibratory system (auto suspension) has a decreasingamplitude.

2 Some unstable oscillations have increasing amplitudes.

3 The amplitude envelope can be fitted to an exponential function inmany practical cases.

4 Decreasing amplitude: y(t) = Ae−t/τ sin(wt + φ), τ > 0.

5 Increasing amplitude: y(t) = Aert sin(wt + φ), r > 0.

IntroductionCourse Goals

Review Topics

Harmonic FunctionsExponentially-Decaying Oscillations

Exponentially-Decaying Oscillation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time, sec.

y(t)

Exponentially−Decaying Oscillation

Ae−tτ

y(t)=Ae−t/τsin(wt+φ)

A=2τ=2w=10φ=π/4

Logarithmic decrement: δ = ln y(t)y(t+T ) , T = 2π/w is the period.

For non-consecutive peaks: δ = 1nln B1

Bn+1(see Palm, p.150)

IntroductionCourse Goals

Review Topics

Harmonic FunctionsExponentially-Decaying Oscillations

Some Properties of ye(t) = Ae−t/tau

τ is called the time constant.

ye(0) = A. At t = τ , ye has decreased to 37% of the initial value.

At t = 4τ , ye has decreased to 2% of its initial value.

A time equal to 4τ constants is usually designated as settling time.

These facts can be used to estimate τ from an experimental trace.