EN4: Dynamics and Vibrations

Division of Engineering
Brown University


6.4 Free vibration of a conservative, single degree of freedom, linear spring mass system.

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.

Problem: The spring mass system is released with velocity from position at time . Find .

There is a standard approach to solving problems like this

The picture shows a free body diagram for the mass. Newton II states that

This is our equation of motion for s.

Now, we need to solve this equation. We will cheat. Since we do not have time to cover the necessary mathematics, we will look up the solution. If you are curious and/or highly advanced in math, you can download a handout  (pdf format) that describes the solution procedure in detail. If not, you can wait until this material is covered in AM33 and AM34.

We therefore consult a handy list of solutions to differential equations , (This list of solutions is available in pdf format if you wish to print it out.  The list of solutions may be used in examinations)  and observe that it gives the solution to the following equation

This is very similar to our equation, but not identical. We need to massage our equation a bit to make it look right. To this end, let

Note that d is constant, so when these are substituted into our equation of motion, it reduces to the standard form. The solution for x is

Here, and are the initial value of x and its time derivative, which must be computed from the initial values of s and its time derivative

Finally, we can compute s

Observe that:

6.5 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.

For the spring—mass system, we found only one natural frequency. More complex systems have several natural frequencies. For example, the system of two masses shown below has two natural frequencies, given by

A system with three masses would have three vibration 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

The response may be shown to be


In general, the vibration response will look complicated, and is not harmonic. However, if we choose the special initial conditions:

then the response is simply

i.e., both masses vibrate harmonically, at the first natural frequency. Similarly, if we choose


i.e., the system vibrates harmonically, at the second natural frequency.

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 and mode shapes 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 mechanical system, you want to make sure that the natural frequencies of vibration are much larger than any excitation frequency that the system is likely to see.

A beautiful example of this kind of design exercise can be found at the homepage of NASA’s Goddard Space Flight Center, where natural frequencies and vibration modes are calculated for the Mars Orbiter Laser Altimeter. Tha animations below show some of the mode shapes for the system, predicted using a computer simulation technique known as Finite Element Analysis. The amplitude of vibration has been greatly exaggerated for clarity - the real system could never withstand displacements as large as those shown below!

Modes 1 and 2:



Modes 5 and 6


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, but vibration consultants like to call them `impulse hammers’ so they can charge more for their services). 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.

Impulse hammer tests can even be used on big structures like bridges or buildings – but you need a big hammer. In a recent test on a new cable stayed bridge in France, the bridge was excited by first attaching a barge to the center span with a high strength cable; then the cable was tightened to raise the barge part way out of the water; then, finally, the cable was released rapidly to set the bridge vibrating.



6.6 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.


Example 1: A linear system.

Calculate the natural frequency of vibration for the system shown below

Assume that the cylinder rolls without slip on the wedge.

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

The last equation is almost in one of the standard forms given on the list of solutions, except that the right hand side is not zero. There is a trick to dealing with this problem – simply subtract the constant right hand side from s, and call the result x. (This only works if the right hand side is a constant, of course). Thus let

and substitute into the equation of motion:

This is now in the form

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


Summary of procedure for calculating natural frequencies:

Example 2: A nonlinear system.

We will illustrate the procedure with a second example, which will demonstrate another useful trick.

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

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

We will follow the steps outlined earlier:

(1) We describe the motion using the angle

(2) We write down T and V:

(3) Differentiate with respect to time:

(4) Arrange the EOM into standard form. Houston, we have a problem. There is no way this equation can be arranged into standard form. This is because the equation is nonlinear ( is a nonlinear function of ). There is, however, a way to deal with this problem. We will show what needs to be done, summarizing the general steps as we go along.



We follow the same procedure as before.

The potential and kinetic energies of the system are


Once again, we have found a nonlinear equation of motion. This time we know what to do. We are told to find natural frequency of oscillation about , so we don’t need to solve for the equilibrium configurations this time. We set , with and substitute back into the equation of motion:

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)

We now have an equation in standard form, and can read off the natural frequency

Question: what happens for ?