How can we analyse undamped vibration problems.

In our previous article on harmonic motion, we looked at how we can analyse problems involving simple harmonic motion, now we’re going to look at solving vibration problems with a single degree of freedom.

Vibration Analysis

In vibration analysis, there are two approaches we are interested in.  One is to control undesirable vibration, for example, the loosening of bolts.  The other is to enhance desirable vibration, such as mechanical agitators.

To look at either, we need to think about what the cause of vibration is.  Any disturbance in a mechanical system in a state of equilibrium results in a vibration response.  Disturbance can be from wind gusts on a bridge or aircraft, or periodic impulses from firing events in an internal combustion engine.

Basic concepts

Let’s think about a typical motor vehicle with a wheel and suspension system at each corner.  It will look something like this:

The system is shown as an idealised lumped parameter system.  What this means is that all the mass, stiffness, and damping of the real structure have been ‘lumped’ together into idealised lumped parameters.

A lumped parameter system is made up of:

• Mass (kg): the inertia of the real structure is represented and modelled.  The idealised mass has a rigid body where work done is stored as kinetic energy – this is where all the mass of the real suspension system is modelled.
• Spring (N/m): this is where the stiffness of the real structure is represented and modelled. The idealised spring has no mass, models an elastic component of the structure, the linear spring obeys Hooke’s Law and spring constant = stiffness = k = force / unit deflection.
• Damper (Ns/m): this is where the damping of the real structure is represented and modelled. The idealised damper has no mass, no elasticity, viscous damping force proportional to velocity, viscous damping coefficient, c = force / unit velocity, work done and energy dissipated as heat (non-conservative system)

Effect of Damping on Response

The response of a system that is undamped typically looks like this:

where the amplitude of the response, y, is A and the time to complete one revolution is the so-called time period, T, and the equation is as we have seen previously, as follows:

y=Asinωt

Now let us look at the response with damping included:

It is clear from the above that damping takes energy out of the system, allowing the mass to decay and rest. The energy is lost as heat in the damper.

Degrees of Freedom (DOF)

The degrees of freedom of a dynamic model are the number of independent coordinates necessary to describe its motion. The mass on a spring example discussed above is known as a single degree of freedom (SDOF) model because there is only one independent coordinate.

Below, we can see a simple pendulum model. It can be seen that the system is constrained to move in the x-y plane. The configuration can be defined by [x(t) , y(t)] or by θ(t)  alone. The  (x,y) coordinates are NOT independent as they are related by the following equation of constraint:

x² + y² = L²

Therefore, the system is SDOF.

If you find this interesting, make sure to check out our next article on damped vibrations.

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