MEMS 431 (FL11) Lab 5

Natural Frequencies and Modes of a 5-DOF Spring-Mass System

Objective

To determine the natural frequencies and mode shapes of a system composed of discrete mass and stiffness elements.

Background

The five degree-of-freedom (5-DOF) system is connected by a tensioned elastic cable to an electromagnetic shaker. The shaker is driven by a function generator and power amplifier, producing an approximately sinusoidal force. Measurements of the accelerations and displacements of the masses will be obtained. First the natural frequencies of the system will be found from the frequencies present in the transient response. Then the mode shapes will be approximated from the operational deflection shapes exhibited by the system in response to harmonic forcing at each natural frequency.

Reading

Review lecture notes and text chapters on the free and forced response of multi-DOF systems.

Procedure

Step 1: Record system parameters

Record the mass of each cart. Do not disassemble the system; the mass of each cart has been measured ahead of time. Measure the force required to extend the representative spring suspended from the air track. Make several measurements of force (weight) and displacement and perform a least squares fit to estimate the relationship. Note that tension springs often have a pre-load. Turn on the air supply. Record the initial location of each mass relative to one end of the air track. AFTER THE LAB: Estimate spring stiffness from the force-extension data.

Step 2: Find the natural frequencies

Record the transient response of the system to an initial displacement of one mass. Use the data-acquisition system to sample the accelerometer signal and find the power spectrum (amplitude vs frequency) of the response. Repeat this step twice, applying the initial displacement to a different mass each time. From the spectral amplitudes, estimate the five natural frequencies of the system.

Step 3: Find the displacement amplitudes

Attach a slip of paper (or "Post-It" note) to each mass. Turn on the function generator and power amplifier, in that order. Set the power amplifier amplitude to its minimum value. Tune the function generator to a natural frequency while monitoring the frequency input with the digital multimeter. Adjust the amplitude if necessary (you can use the amplitude knobs on both the function generator and the amplifier) to produce moderate size oscillations. Don't allow motion to get too large.

While the system is moving, position a pencil over a mass and lower it until it leaves a light mark as the mass moves below it. Try not to affect the motion of the mass. Leave the pencil in light contact for several cycles. Do not move the pencil horizontally. The length of the pencil line is the peak-to-peak amplitude of oscillation. Repeat this procedure for each mass.

Locate all the nodes of the system. Nodes are points where there is no motion; they should be sought between two masses moving in opposite directions. (NOTE: The first mode shape has no nodes.)

Repeat this procedure for each of the five natural frequencies.

Step 4: Find the mode shapes

AFTER THE LAB: Mass-normalize your operational deflection shapes (approximated mode shapes). Plot the shapes on separate plots. On the horizontal axis mark the equilibrium location of each mass. On the vertical axis plot the amplitude of the mass. Because the direction of motion changes at every node, the sign of the amplitude on your plot should change at every node (i.e., some amplitudes should be plotted as negative values).

Theory

Derive the linear equations of motion for the system, neglecting any friction or damping. Determine mass and stiffness matrices. Find the natural frequencies and mode shapes predicted by the mathematical model (use MATLAB). Mass-normalize and plot your mode shapes.

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. Write clearly and concisely. The report should contain: (1) A schematic diagram of the experiment, including measurement apparatus; (2) Parameters (particularly masses and spring stiffnesses) of the experimental system; (3) Tables of the natural frequencies obtained from experiment and theory; and (4) Plots of mass-normalized mode shapes obtained from experiment and theory. Can you explain discrepancies between observed data and theoretical predictions? What assumptions did you make? Were these assumptions justified?