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free Damped and Un-damped Vibration lab report

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Vibration Experiments

• First Online: 23 February 2023

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• C. Sujatha 2  

This chapter describes seventeen fundamental experiments in vibration which can easily be conducted in the laboratory to understand the basic vibration theory which has been described in Chapter 2. The presentation is such that an academician can mix and match these experiments to suit the laboratory courses being taught at undergraduate or postgraduate levels. The material will also be useful for researchers who would like to set up basic experiments in the laboratory. Simple rigs that can easily be fabricated in the workshop for specific experiments are suggested. The description of every experiment starts with the aim, theory behind the experiment, test setup and procedure, as well as how to report the results. Suggestions are given for the choice of transducers (described in Chapter 3) and for alternate experimental setups.

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Scheffer, C., & Girdhar, P. (2004). Practical machinery vibration analysis and predictive maintenance (practical professional) . Newnes.

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Venkateshan, S. P. (2008). Mechanical measurements . ANE Books India, and CRC Press.

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Department of Mechanical Engineering, IIT Madras, Chennai, India

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Sujatha, C. (2023). Vibration Experiments. In: Vibration, Acoustics and Strain Measurement. Springer, Cham. https://doi.org/10.1007/978-3-031-03968-3_10

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5.2 Free vibration of conservative, single degree of freedom, linear systems.

First, we will explain what is meant by the title of this section.

It turns out that all 1DOF, linear conservative systems behave in exactly the same way.  By analyzing the motion of one representative system, we can learn about all others.

We will follow standard procedure, and use a spring-mass system as our representative example.

There is a standard approach to solving problems like this

(i) Get a differential equation for s using F=ma (or other methods to be discussed)

(ii) Solve the differential equation.

The picture shows a free body diagram for the mass.

 Newton’s law of motion states that

This is our equation of motion for s .

5.2.1 How to solve equations of motion for vibration problems

Note that all vibrations problems have similar equations of motion.  Consequently, we can just solve the equation once, record the solution, and use it to solve any vibration problem we might be interested in.  The procedure to solve any vibration problem is:

1.       Derive the equation of motion, using Newton’s laws (or sometimes you can use energy methods, as discussed in Section 5.3)

2.       Do some algebra to arrange the equation of motion into a standard form

3.       Look up the solution to this standard form in a table of solutions to vibration problems.

We have provided a table of standard solutions as a separate document that you can download and print for future reference.

We will illustrate the procedure using many examples.

5.2.2 Solution to the equation of motion for an undamped spring-mass system

We would like to solve

We therefore consult our list of solutions to differential equations, and observe that it gives the solution to the following equation

Finally, we see that if we define

then our equation is equivalent to the standard one.

The solution for x is

When we present the solution, we have a choice of writing down the solution for x, and giving formulas for the various terms in the solution (this is what is usually done):

Alternatively, we can express all the variables in the standard solution in terms of   s

But this solution looks very messy (more like the Mathematica solution).

Observe that:

5.2.3 Natural Frequencies and Mode Shapes.

We saw that the spring mass system described in the preceding section likes to vibrate at a characteristic frequency, known as its natural frequency. This turns out to be a property of all stable mechanical systems. 

All stable, unforced, mechanical systems vibrate harmonically at certain discrete frequencies, known as natural frequencies of the system.

A system with three masses would have three natural frequencies, and so on.

In general, a system with more than one natural frequency will not vibrate harmonically.

For example, suppose we start the two mass system vibrating, with initial conditions

In general, the vibration response will look complicated, and is not harmonic. The animation above shows a typical example (if you are using the pdf version of these notes the animation will not work - you can download the matlab code that creates this animation here and run it for yourself)

then the response is simply

i.e., both masses vibrate harmonically, at the first natural frequency, as shown in the animation to the right.   (To repeat this in the MATLAB code, edit the file to set A1=0.3 and A2=0)

Similarly, if we choose

i.e., the system vibrates harmonically, at the second natural frequency. (To repeat this in the MATLAB code, edit the file to set A2=0.3 and A1 = 0)

The special initial displacements of a system that cause it to vibrate harmonically are called  `mode shapes’ for the system.

If a system has several natural frequencies, there is a corresponding mode of vibration for each natural frequency.

The natural frequencies are arguably the single most important property of any mechanical system .  This is because, as we shall see, the natural frequencies coincide (almost) with the system’s resonant frequencies .  That is to say, if you apply a time varying force to the system, and choose the frequency of the force to be equal to one of the natural frequencies, you will observe very large amplitude vibrations.

When designing a structure or component, you generally want to control its natural vibration frequencies very carefully.  For example, if you wish to stop a system from vibrating, you need to make sure that all its natural frequencies are much greater than the expected frequency of any forces that are likely to act on the structure.  If you are designing a vibration isolation platform, you generally want to make its natural frequency much lower than the vibration frequency of the floor that it will stand on.  Design codes usually specify allowable ranges for natural frequencies of structures and components.

Once a prototype has been built, it is usual to measure the natural frequencies and mode shapes for a system.  This is done by attaching a number of accelerometers to the system, and then hitting it with a hammer (this is usually a regular rubber tipped hammer, which might be instrumented to measure the impulse exerted by the hammer during the impact).  By trial and error, one can find a spot to hit the device so as to excite each mode of vibration independent of any other.  You can tell when you have found such a spot, because the whole system vibrates harmonically.  The natural frequency and mode shape of each vibration mode is then determined from the accelerometer readings.

5.2.4 Calculating the number of degrees of freedom (and natural frequencies) of a system

When you analyze the behavior a system, it is helpful to know ahead of time how many vibration frequencies you will need to calculate.  There are various ways to do this.   Here are some rules that you can apply:

The number of degrees of freedom is equal to the number of independent coordinates required to describe the motion .  This is only helpful if you can see by inspection how to describe your system.  For the spring-mass system in the preceding section, we know that the mass can only move in one direction, and so specifying the length of the spring s will completely determine the motion of the system.  The system therefore has one degree of freedom, and one vibration frequency.   Section 5.6 provides several more examples where it is fairly obvious that the system has one degree of freedom.

For a 2D system, the number of degrees of freedom can be calculated from the equation

p is the number of particles in the system

To be able to apply this formula you need to know how many constraints appear in the problem.  Constraints are imposed by things like rigid links, or contacts with rigid walls, which force the system to move in a particular way.   The numbers of constraints associated with various types of 2D connections are listed in the table below.  Notice that the number of constraints is always equal to the number of reaction forces you need to draw on an FBD to represent the joint

For a 3D system, the number of degrees of freedom can be calculated from the equation

where the symbols have the same meaning as for a 2D system.   A table of various constraints for 3D problems is given below.

5.2.4 Calculating natural frequencies for 1DOF conservative systems

In light of the discussion in the preceding section, we clearly need some way to calculate natural frequencies for mechanical systems.  We do not have time in this course to discuss more than the very simplest mechanical systems.  We will therefore show you some tricks for calculating natural frequencies of 1DOF, conservative, systems. It is best to do this by means of examples.

 Our first objective is to get an equation of motion for s .  We could do this by drawing a FBD, writing down Newton’s law, and looking at its components.  However, for 1DOF systems it turns out that we can derive the EOM very quickly using the kinetic and potential energy of the system. 

The potential energy and kinetic energy can be written down as:

Differentiate our expressions for T and V (use the chain rule) to see that

Finally, we must turn this equation of motion into one of the standard solutions to vibration equations.

Our equation looks very similar to

and substitute into the equation of motion:

By comparing this with our equation we see that the natural frequency of vibration is

Summary of procedure for calculating natural frequencies:

(1)  Describe the motion of the system, using a single scalar variable (In the example, we chose to describe motion using the distance s );

(2) Write down the potential energy V and kinetic energy T of the system in terms of the scalar variable;

(4) Arrange the equation of motion in standard form;

(5) Read off the natural frequency by comparing your equation to the standard form.

Example 2: A nonlinear system.

Find the natural frequency of vibration for a pendulum, shown in the figure.

We will idealize the mass as a particle, to keep things simple.

We will follow the steps outlined earlier:

  (2)  We write down T and V :

 (3) Differentiate with respect to time :

(i) Find the static equilibrium configuration(s) for the system .

(ii) Assume that the system vibrates with small amplitude about a static equilibrium configuration of interest.

(iii) Linearize the equation of motion, by expanding all nonlinear terms as Taylor Maclaurin series about the equilibrium configuration.

Now, recall the Taylor-Maclaurin series expansion of a function f(x) has the form

Apply this to the nonlinear term in our equation of motion

Finally, we can substitute back into our equation of motion, to obtain

(iv) Compare the linear equation with the standard form to deduce the natural frequency.

We can do this for each equilibrium configuration.

Next, try the remaining static equilibrium configuration

If we look up this equation in our list of standard solutions, we find it does not have a harmonic solution.  Instead, the solution is

No wonder the equation is predicting an instability…

Here is a question to think about.  Our solution predicts that both x and dx/dt become infinitely large.  We know that a real pendulum would never rotate with infinite angular velocity.  What has gone wrong?

We follow the same procedure as before.

The potential and kinetic energies of the system are

Now, expand all the nonlinear terms (it is OK to do them one at a time and then multiply everything out.  You can always throw away all powers of x greater than one as you do so)

If you prefer, you can use Mathematica to do the Taylor series expansion:

(The number 4 in the ‘series’ command specifies the highest power of x that should appear in the expansion). We now have an equation in standard form, and can read off the natural frequency

Our first objective is to get an equation of motion for s .   We do this by writing down the potential and kinetic energies of the system in terms of s .

The potential energy is easy:

The first term represents the energy in the spring, while second term accounts for the gravitational potential energy.

The kinetic energy is slightly more tricky.   Note that the magnitude of the angular velocity of the disk is related to the magnitude of its translational velocity by

Thus, the combined rotational and translational kinetic energy follows as

Now, note that since our system is conservative

Differentiate our expressions for T and V to see that

This is now in the form

and by comparing this with our equation we see that the natural frequency of vibration is