MEMS 431 (FL11) Lab 2

Single DOF Response to Initial Conditions and Rotating Imbalance

Objective

To observe transient and forced vibrations in a simple system.

Background

The single degree-of-freedom (1-DOF) rotational system consists of a plate which pivots on a pair of ball bearings, carries a small motor and attaches to a spring. The motor turns a disk bearing a small eccentrically mounted mass. An Airpot device may be added to change the system dynamics. A metal strip can be used to provide Coulomb friction. Displacement of the plate is measured by a non-contact inductive gauging sensor (EX-110/202). Signals are acquired and stored using a PC-based data acquisition system.

Reading

Review textbook sections on 1-DOF system response to harmonic excitation, and the instructions below. READ THE INSTRUCTIONS BEFORE COMING TO LAB. BRING A FLASH DRIVE TO STORE EXPERIMENTAL DATA.

Procedure

Step 1: Create a schematic

Sketch the experimental apparatus. You should make a clean schematic diagram later. Disconnect the Airpot assembly from the system by unscrewing the right screw and pivoting the assembly away from the bar. Make sure the Coulomb damper is loosened so it does not touch the bar.

Step 2: Measure physical parameters

Measure and record the spring stiffness using calibrated weights and a scale. Load a weight on the end of the platform and record the deflection. Measure the mass and eccentricity of the unbalance mass. Also measure the calibration constant for the displacement sensor by recording sensor voltage and deflection. AFTER THE LAB: You will estimate the total equivalent mass of the system from the stiffness and the measured natural frequency.

Step 3: Free response to initial conditions

Open the data acquisition program for lab 2. Make sure the trigger is set to INPUT. Click Start and tap the plate lightly but firmly to initiate free vibration. View the displacement signal during free response and record the amplitudes of three successive peaks and the time intervals between them. Repeat the procedure and measurements. Record a typical transient response and save in a data file. You will need to transfer the saved data to a flash drive to take with you. AFTER THE LAB: Estimate the damped and undamped natural frequencies of the system and the damping ratio. Plot the transient response, showing angular displacement vs time.

Step 4: Forced response to unbalance excitation

Change the trigger to Free Run. Turn on the motor and increase speed to about 45% of the maximum. Click Start to measure the RMS amplitude and frequency of the displacement signal. Decrease speed by 1-2% of maximum, wait until steady state, and record amplitude and frequency again. Repeat measurement of frequency and amplitude until you have passed through resonance and are near 30% of maximum speed. You should have about 12-15 measurements of amplitude and frequency. AFTER THE LAB: Plot amplitude vs frequency.

Step 5: Effect of Airpot on free and forced vibration

Attach the Airpot assembly to the system using two #10-32 screws. Repeat the previous two steps (free response and unbalance response). Save one set of free response data. AFTER THE LAB: How has the free response changed? How has the forced response changed? Are the changes consistent?

Step 6: Estimate Coulomb Damping

Turn off the motor and remove the Airpot assembly. Tighten the screw on the Coulomb friction assembly until it touches the assembly, then tighten one or two more turns. Record two transient responses and observe the amplitudes of successive peaks. Save one transient response in a file to plot later. AFTER THE LAB: Compare the free response with Coulomb damping to the other free response plots. Estimate an equivalent viscous damping ratio and comment on its applicability. Can you determine the friction force (coefficient of friction times normal force) on the plate?

Theory

Formulate a mathematical model of the vibrating system. Consider a simple spring-mass-damper system. Base the parameters of your model in terms of physical properties that you measured (but leave your equation in simple symbolic form). Estimate the "effective mass" from the natural frequency and stiffness. Also estimate viscous damping from the observed damping ratio. Now, add the effects of the Airpot device to your model. How did it change? Obtain a formula for the amplitude of oscillation as a function of frequency in your model. For each of the two viscous damping cases, make a plot of amplitude vs. excitation frequency (solid lines). On the same plots, show the experimental data that you recorded (amplitude vs. frequency) for the corresponding damping ratios. Show experimental data with with markers and dashed lines.

Report

The lab report should be a full report. Please refer to the lab website for the sample report format. Organize your report into sections (e.g. Introduction, Procedures, Results, Discussion). Write concisely and clearly.

Include the following: (1) A schematic diagram and description of the apparatus. (2) Plots of the free response for each of the three damping cases [(i) no damping other than "natural" damping; (ii) with the Airpot device; and (iii) with Coulomb damping]. (3) Estimates of natural frequency and damping ratio for each case. (4) Free-body diagrams, and equations for your mathematical models. (5) Plots of the response to rotating unbalance from theory and experiment. (6) Assumptions and possible explanations for differences between theory and observation.