Introduction to Experiment 3: Vibrations

1.0 Overview:

From jackhammers to buildings, automotive suspension systems to aircraft gas turbine engines, it is obvious to even the most casual observer that vibrations are extremely important. In this experiment you will examine the vibrations of the spring, mass, damper system. In the experimental investigation you will look at the effect of sinusoidal excitation on a cantilever system with small damping. The thrust will be to gain an understanding of the natural frequency of a system. You will vary the dimensions (length) of the beam, hence affecting its frequency response characteristics. You will also use a delta impulse function for each case to determine the system’s damping coefficient. In the simulation portion of this investigation you will verify the theory behind the experimental investigation by comparing the system response using a simulated single degree of freedom system. You will expand your database by repeating the simulation using various materials, end masses and damping coefficients to quantify the system’s phase and amplitude response characteristics. This part of the lab provides insight into how the engineer and scientist can use damping to minimize the negative effects of vibrations. Before you begin this lab you should review second order differential equations with constant coefficients. Keep in mind what quantities in the governing second order differential equation for the spring mass damper system that you are changing throughout the experiment. This should be discussed in some detail in the lab write-up.

1.1 Elementary Theory:

1.1.1 A Simple Spring-Mass System

Many oscillating systems can be modeled as a spring-mass system using the differential equation of motion. The displacement, y(t), of such systems can be found using

                    (1)

where m is the mass of the object, c is the coefficient of viscous damping, k is the spring constant, and F(t) is some forcing function. Each term in this expression is actually a time-dependent force: my,, is the inertial force, cy, is the frictional or damping force, and ky is the spring force. Modeling the damping of a system in this way assumes that the damping force is proportional to the velocity of the mass; this is called viscous damping.

It is convenient to express damping non-dimensionally by calculating the viscous damping factor, , using

                (2)

Here,  is the natural frequency of the system as defined below. For  ,   can be estimated by

                (3)

where Y1 and Y2 are the values of any two consecutive maximum displacements from the time response that are one cycle apart.

The natural frequency of a system is the frequency at which an undamped system will freely vibrate, and can be calculated by using 

                  (3)

                        (4)

Elementary Beam Theory and The Solution to the Governing Differential Equation

1.2 Required Equipment:

1.3 Experiment Apparatus:

The apparatus consist of a steel cantilever beam mounted on a single degree of freedom shaker. Two accelerometers are mounted to the beam, one at the base (input) and one at the free end (response) of the beam. The charge amplifiers supply excitation to the transducers. In addition, the base of the beam is fixed using a removable clamp that can be used to adjust the length of the beam (which will change the natural frequency of the system).

The LabVIEW software will be used to acquire the experimental data. The format for some of the files is found following the "Topics for Discussion".

 

 

2.0 Experiment

Procedure:

Measure the dimensions of the beam for use in calculation of the theoretical natural frequency. We will be performing this test for three different bar lengths. Be sure to include the uncertainty associated with the measuring device. Let’s concentrate on the first bar length for now. Also, note the material of the beam (most likely carbon steel). The mass of the accelerometer is small and has little effect on the system’s response. Be sure to measure only the portion of the beam that will be vibrating. Do not include the portion of the beam located in the clamp.

 

2.1 The Shaker Test

In this experiment, the bars forced frequency is investigated. When the forcing frequency matches the beam's natural frequency, resonance is observed.

We begin by sweeping through several input functions to see where the different modes of the system’s natural frequency exist. .

 

2.2 The Accelerometer Test

In this test, the natural frequency and damping of the bar’s free response is investigated.

2.2.1 Recording the Natural Frequency of the System

We will be running this experiment for three different beams lengths – 16”, 20”, and 24”. Therefore, be sure to properly name the output file accordingly, as not to rewrite over previous data.

 

2.3 The Impulse Test

Remove the beam from the clamp and place it in the similar style clamp mounted to the solid bar located above the shaker. Try at best to preserve the same length used in the previous study, as you will be trying to compare natural frequencies between these two investigations. As an impulse, you will strike the end of the beam with an instrumented hammer.  In this part it is best that you practice the timing between the impulse and the data acquisition system before saving any data to file as you have only about 2 second window to get things right!

 

2.4 Exercises

 

2.5 Topics for Discussion

 

 

3.0 Simulation: Experiment validation, Effects of Changing Damping and End Mass

Note: Every time you save a run two .txt files will be created. The first will have the system’s response in voltage vs. time and the second in voltage vs. frequency.

3.1 Experimental Validation

In this simulation, the results obtained from the experimental part of the lab will be duplicated and compared. If one is performing this part of the lab first, it is advisable to first measure the dimensions and take note of the material types of the beam used in the experimental part of this lab.

 

3.2 The Effects of Damping

We will now quantify the sensitivity of the systems frequency response to varying damping coefficients.

 

3.3 The Effects of End Mass and Material Type

We will now quantify the sensitivity of the systems frequency response to varying weights applied to the end of the cantilever beam and to the beam’s material.

3.3.1 Effect of End-Mass

3.3.2 Effect of Material Type

 

3.4 Exercises