Carnegie Mellon University

Department of Mechanical Engineering

24-352 Dynamic Systems and Control

Spring 2001

 

Lab 3 - Sensors and Vibration Measurement

Overview

 

The objective of this laboratory is for you to become familiar with accelerometers and impact hammers, which are commonly used to measure vibration response of a structure. In this lab, you will make several simple measurements of transient vibration that will illustrate the concepts of natural frequency, damping ratio and elastic wave speed in solids. You will also learn how to measure a frequency response function and identify the resonant frequency.


Part I: Theory

 

1. Natural Frequency and Damping Ratio

 

Damped vibration is described in terms of the natural frequency wn and the damping ratio z . A typical free response can be modeled by the expression

 

 

 

where A and f are the peak amplitude and phase angle, as determined by the initial conditions. We can note that if we sample the response at the peak times tk of each oscillation cycle, then the amplitude of those points simply decays exponentially. Since the cosine function will be one at those points, we have

 

 

Thus, the envelope of the vibration decays in a manner that looks exactly like a first-order system. We know that the time between successive oscillation cycles is given by

 

 

We can therefore write

 

 

which provides the fractional amount of amplitude reduction that we see after one cycle of motion has elapsed. After N cycles,

 

 

Thus, by measuring the reduction in amplitude over some number of cycles, we can determine the value of the damping ratio. In practice, we make measurements for several values of N, and average the results. Note that in the above equations, z is usually small when compared with unity for underdamped motions.

2. Elastic Wave Speed

 

The speed for the propagation of tension/compression waves is given by

 

 

where E is the materialís elastic modulus, and r is the density.

 

3. Frequency Response Function

 

Frequency response function is the response amplitude of the system as a function of forcing frequency. It turns out that, in the sense of a Fourier series, an impulse (or hammer hit) has components at many different frequencies. Thus, striking a structure with a hammer is analogous to exciting it with sinusoidal excitation at many frequencies all at once. The spectrum of such a response is called the frequency response function.

 


Assignment:

1. Plot the free-vibration response of a structure using the equation given in these instructions (the very first equation). Assume that the structure's natural frequency is 120 Hz. Assume three cases of damping; undamped (z=0), slightly damped (z=0.03), and highly damped (z=0.10). Plot all cases for a period of 50ms with at least 10-5 time steps. For ease of calculation, you can set A=1, f=0). After plotting, compare there cases of damping and comment on the effect of damping on vibration response.

2. Using the equation given for elastic wave speed in solids, calculate the wave speeds in steel, cast iron, aluminum, concrete and wood. Compare the wave speed in these solids, in which medium the waves propagate fastest and slowest? You can pick up the properties of these materials from any Strength / Mechanics of Materials textbook.


Part II: Measurement

 

Using the Accelerometers and Force Hammers

The accelerometers and impact hammers at each laboratory station contain built-in micro-electronic amplifiers, and piezoelectric crystals. These are rigid quartz crystals that, when subjected to strain, produce an electric charge. This charge can be measured to indicate the force that is acting on the crystal (as in the impulse hammer), or the acceleration to which the transducer is subjected (as in the accelerometer). These transducers are small, fragile, and expensive. In particular, it is relatively easy to damage the wires, and so be sure to use with caution when handling them. In addition, power supply units are battery powered, please turn them off when you finish your measurements.

1. Natural Frequency and Damping Ratio Measurement

 

2. Elastic Wave Speed Measurement

 

3. Frequency Response Function Measurement