Mechanical Vibration

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Chapter 1


Learning objectives

  1. Recognize phenomena and applications of vibration in practice.
  2. Model a vibrating system with mass, spring and damper elements.
  3. Understand the concept of amplitude, frequency and phase.


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1.2  Terminologies and definitions

Vibration can be defined simply as a repetitive motion of an object around a reference point. This reference point where the object is rested (when it stops moving) is called equilibrium position.

1.2.1  Free and forced vibrations

A system needs energy to set it into motion which is provided by an acting force to the system. Motion of a system which is due to an external force is called forced vibration, for example vibration of a reciprocating engine. The vibration you feel on the seat of your car once you turn on the engine (your car is not moving yet) is because there is a vibration force transmitted from the engine through the mountings and propagates throughout the car structures.

Now let us see a simple example of a traditional baby hammock (usually found in Asia). The baby is placed inside a sarong and then hung to the ceiling through a helical spring. The baby feels comfortable and quickly falls asleep as the hammock is bouncing. The mother can either pull the hammock down and let the hammock bounce itself, or keep her hand on the hammock to bounce it with the wanted speed.

The first one is an analogy of free vibration, where the hammock keep bouncing although the mother’s hand is off from it. Of course the bounce will slow down and eventually stop (due to the presence of damping in the spring) and the mother will pull the hammock down again to start over. In this case, the frequency of the bounce (the speed) is the natural frequency of the hammock itself, which depends on the weight of the baby and the stiffness of the spring. Play with Animation 1.1.

Animation 1.1 The traditional baby hammock. Illustration of free and forced vibrations.


The second one is the analogy for forced vibration, where the speed of the bounce follows the speed of the mother’s hand pulling up and down the hammock. We will see later why it will be easy to control the frequency of the bounce if the the mother pulls it down and up slowly (below the natural frequency), and why it will be difficult to control the speed of the hammock at higher frequency (at or above the natural frequency). Can you feel it?

1.2.2  Damped and undamped vibrations

In a free vibration, the system will eventually stop vibrating due to mechanism of energy dissipation through damping. For such case the motion is called damped vibration. If the damping is absent, then it is called undamped vibration (although damping equals zero is only used for assumption).

Can you imagine there is a scooter that keeps bouncing after it goes through a bump? The illustration is in Animation 1.2.

Animation 1.2 Body motion of a scooter after passing through a bump: with and without damping.


1.2.3  Deterministic and random vibrations

If the behaviour of the motion can be predicted at any given time, this is called deterministic vibration. For example a swinging pendulum or a rotating machine due to purely mass unbalance, the vibration signal will be like sinusoidal function with the frequency is the speed of the machine.

However in several cases, we cannot predict mathematically the motion of a system. For example vibration of a car due to a rough road. For this case, the vibration is called non-deterministic or random vibration. For random vibration, usually we are not interest in the exact value at a specific time, but rather a “statistical” value at a time interval, e.g. the mean or mean square values. Play Animation 1.3. 

Animation 1.3 Illustration of deterministic and random vibration.

1.2.4  Degree of freedom

The vibration of the rotating machine in Animation 1.3 is assumed to be only in vertical direction (all other directions are restrained), and thus we deal with the motion of the system in one direction only. This kind of system is said to have single or one degree of freedom.

Singledegree-of-freedom system

Now let us see the motion of a pendulum with length, l in Animation 1.4. If we want to state the position of the ball, how many coordinates can we use? As we can see, there are three coordinates we may use: the horizontal position x , the height of the ball relative to the horizontal rest position y , and the angle of the string relative the vertical rest position, \theta .

Animation 1.4 The swinging pendulum.

Are these three coordinates independent to each other? The horizontal position can be expressed as x=l\sin\theta  and the vertical position is y=l(1-\sin\theta) . From here, we can see that x and y depend on \theta and vice versa, which means they are not independent coordinates.

We can express the position of the ball just with one coordinate, that is \theta . We thus can simply state the position of the ball at, for example \theta=30^{\circ} or \theta=-45^{\circ} . We do not need to use coordinate x and y , do we!

Because the motion of the pendulum can be represented with only one independent coordinate, therefore the pendulum is a ‘one-degree-of-freedom’ (1-DOF) system.

Twodegree-of-freedom system

Two-degree-of-freedom (2-DOF) system has therefore two independent coordinates to represent the motion of the system. As shown in Animation 1.5, the bottom mass B and the upper mass A has their “freedom”to move together or against each other, also with different amplitude (We call it different vibration modes. Each mode vibrates at different frequency). Because the coordinates are independent, we need two sets of equation to explain the motion of the system (We will discuss this in Chapter: Multi-Degree-of-Freedom System). 

For the other 1-DOF system, the two masses, A and B are connected with a rigid link, and thus they move together. The masses are not independent. We need only one coordinate to represent the motion of both masses. 

How many degrees of freedom can a sailing ship in the ocean have? The answer is six, but we leave it this one for you to think.

(Hint: We have also rotational motions, instead of translational motions)

Animation 1.5 Examples of single and two-degree-of-freedom systems.

Figure 1.6 A boat in the ocean can have up to six degrees of freedom.

1.2.5  Mass, stiffness, and damping elements

When we observe a vibrating system, we can observe at least two things regarding its motion. First, there is a rigid body moving up and down and second, there is a flexible element that can stretch and being suppressed while the rigid body is moving.  

The first property is called mass, which is a rigid body that can accelerate under an action of force. The second property is called stiffness, which corresponds to a flexible element that deforms under an action of force, and returns to its original shape when the force is absent. For example a helical spring. See Animation 1.6.

The third property is called damping, which is responsible in absorbing the vibration energy and converts it into heat or sound energy. A bouncing mass on a spring (without continuous external driving force) will eventually stop vibrating because of the presence of damping in the spring. This inherent damping is not visible, however the dashpot or shock absorber in a motorbike is also one of the damping elements (in the form of viscous damping).

1.2.6  Discrete and continuous systems

A vibrating system of an electric motor mounted on springs as in Animation 1.6 is called discrete system or lumped parameter system. This is because we can easily breach the system into mass (the motor) and stiffness components (the spring mounts), and the system is limited in terms of degree of freedom. 

Animation 1.6 A vertically vibrating motor on springs, an example of discrete or lumped parameter system.

The opposite of discrete system is called continuous system or distributed parameter system. This refers to a structure which has vibration waves propagating inside the structure. If a discrete system can have one or two independent coordinates, a continuous system thus has an infinite number of independent coordinates to represent the dynamics of the system. 

A simple example is a vibrating beam or plate. Animation 1.7 shows the first two modes of vibration of a fixed-free beam out of its infinite number of modes.

Points No. 1 and No. 2 are chosen here as the points of observation. At these points, there are an elemental mass of the beam that moves up and down, and also there is an elemental stiffness determining the degree of bending at each corresponding point. The closer the observation point to the fixed edge, the greater the value of the stiffness at that particular location.

These elemental mass and stiffness are distributed across the beam (distributed parameter system).

Animation 1.7 A vibrating beam with fixed-free edges, an example of a continuous or distributed parameter system.