Introduction Course Goals Review Topics. MCE371: Vibrations. Prof. Richter. Department of Mechanical Engineering. Handout 1 Fall 2017

Transcription

1 MCE371: Vibrations Prof. Richter Department of Mechanical Engineering Handout 1 Fall 2017

2 Outline Introduction 1 Introduction Vibration Problems in Engineering 2 3

3 Vibration 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 vibration in engineering systems. In both cases, the ME must be prepared to handle vibration problems at three levels: analysis, design and field work (measurement and instrumentation)

4 Vibration Problems in Engineering Intentional Vibrations: Non-Contact Mode AFM Image from Park Systems ( Interatomic forces modulate the cantilever vibrations. Data can be used to obtain an image as the sample is scanned in 3D. Key idea: use feedback control to compensate for changes to the vibration amplitude caused by sample-tip distances. More details:

5 Vibration Problems in Engineering Unintentional Vibrations: Aircraft Wing Flutter Phenomenon where interacting aerodynamic and elastic forces result in a self-induced oscillation. Example: NASA test on twin-engine plane: Tacoma Narrows 1940 bridge collapse: Both are examples of the same destructive self-induced vibration phenomenon.

6 Valve Floating Problem Introduction Vibration Problems in Engineering Problem: valves begin to stay open (float) at high rotational speeds How would you begin to address this problem? (aiming at valvetrain redesign 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

7 Vibration Problems in Engineering Vibration Control: Active/Semiactive Suspension You need a course in control systems following vibrations to work on this!

8 Introduction 1 Establish a solid foundation to describe dynamic mechanical systems in terms of differential equations 2 Develop proficiency in understanding and using the properties of the solutions of forced and unforced one-degree-of-freedom systems. 3 Develop proficiency in deriving the differential equations of motion for systems with multiple degrees-of-freedom. 4 Develop proficiency in solving one-degree-of-freedom differential equations using Laplace methods. 5 Develop working-level proficiency with computer simulation tools: Matlab and Simulink 6 Introduce matrix methods for systems with multiple degrees-of-freedom. 7 Introduce vibration suppression methods and vibration measurement instrumentation. A laboratory session on accelerometers will be included.

9 Using sinusoids to describe harmonic motion Reference: Palm Sect. 1.2, Inman Sects. 1.1, 1.2 If 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 familiar with harmonic functions, their parameters and their graphical representations. Vibration Amplitude, milli inches Example: Fourier Decomposition Vibration Components, milli inches Time,ms

10 Sinewave Parameters Introduction y(t) = Asin(wt +φ) = Asin(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.

11 Useful Identities Introduction cos(x +φ) = sin(x +φ+ π 2 ) sin(x +φ) = sin(x +φ+π) Exercise: If y(t) = Asin(wt+φ) is the position, obtain the velocity and the acceleration in terms of sin and sketch the three functions.

12 Plotting 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 )

13 Some Plot Refinements Introduction 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

14 Assigned Exercise Introduction 1 Download the data from the course website 2 Plot and label using the given time and position vectors. Use data point markers instead of a solid line 3 Find the approximate amplitude, frequency and phase from the plot 4 Calculate a fitted sinewave using the above and superimpose a solid-line plot. Include a legend.

15 Decaying and Increasing Oscillations 1 A damped vibratory system (auto suspension) has a decreasing amplitude. 2 Some unstable oscillations have increasing amplitudes. 3 The amplitude envelope can be fitted to an exponential function in many practical cases. 4 Decreasing amplitude: y(t) = Ae t/τ sin(wt +φ), τ > 0. 5 Increasing amplitude: y(t) = Ae rt sin(wt +φ), r > 0.

16 Exponentially-Decaying Oscillation Ae tτ Exponentially Decaying Oscillation A=2 τ=2 w=10 φ=π/4 0.5 y(t) y(t)=ae t/τ sin(wt+φ) Time, sec. Logarithmic decrement: δ = ln y(t) y(t+t), T = 2π/w is the period. For non-consecutive peaks: δ = 1 B1 n ln B n+1 (see Palm, p.150)

17 Some Properties of y e (t) = Ae t/tau τ is called the time constant. y e (0) = A. At t = τ, y e has decreased to 37% of the initial value. At t = 4τ, y e 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.