Mechanical Vibration

Web Book

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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1.3  Modeling a vibrating system

To solve a vibration problem (simple word: reducing the vibration amplitude), there are four main steps involved:

  1. Development of mathematical model
  2. Generation of governing equations
  3. Prediction of response of vibration
  4. Analysis

The amplitude of vibration in Part 3) can also be directly obtained through measurement. And in engineering practice, Part 1) and 2) usually involves computer modeling and simulation, especially when we deal with a complex structure.

For example, in an automobile manufacturer, the team in the Computer Aided Engineering department will first design the structure of the car body in a computer, where the software can simulate the flow of vibration and noise inside the car interior. Once the desired vibro-acoustics design has been fulfilled, the prototype of the body can be manufactured. The experimental testing is then conducted to validate the model. Figure 1.7 shows the illustration.

Figure 1.7 Modeling and validating a vibrating system.

Modeling a vibrating system is basically how to represent the system into three lumped parameters namely mass, spring and damper (MSD) elements. 

Mass: A ’rigid body’ which moves or accelerates with the action of force (mass property).

Spring: A flexible element where it deforms with the action of force and returns to its original shape when the force is absent (stiffness property). 

Damper: An element responsible to ’absorb’ the vibration energy (damping property)

In this book the discussion is emphasized on modeling a discrete system, i.e. a system having one, two or three degrees of freedom which can be easily breached into MSD components. For a complex structure, the modeling method is also built on the same foundation as the discrete system. 

The mathematical symbols for the three lumped elements used in this book are shown in Figure 1.8.

Figure 1.8 Symbols used for mass, spring, and damper elements.

The MSD model for a single-degree-of-freedom (SDOF) system can be represented by various configurations as in Figure 1.9. It is important to note that Figure 1.9 represents the mathematical model of a system, not really to depict the actual situation. 

Thus for a SDOF system that vibrates in vertical direction in the real application (such as the motor resting on spring mountings in Animation 1.6), the represented MSD model is correct from either one in Figure 1.9. 

Figure 1.9 Various MSD models for single-degree-of-freedom system.

1.3.1  Modeling a discrete system

How can we model a vibrating system consisting of a motor rigidly mounted on a flexible beam as seen in Animation 1.8?

The challenge is usually to identify each element of mass, spring and damper in the system, what engineering assumption we use to approach the problem and whether the proposed model can provide an acceptable result.

For the motor-beam system in Animation 1.8, the motor is the rigid body that moves up and down in vertical direction, so in this system, the motor is the element of mass. The supported beam structure bends during the vibration and thus this serves as stiffness/spring element.  Some portion of the beam’s mass (we call it effective mass) also moves with the mass of the motor, and so it is also included in the total mass of the system.

Animation 1.8 A motor bolted on a flexible beam and vibrates vertically.

Which element is responsible as the damper? Imagine if the machine is shut down. The vibration will eventually stop as there is no source exciting the vibration energy to the supported structure. The deflection of the beam decays and the beam returns to its original shape. 

The damping is therefore provided by the beam through its inherent damping, which is due to the friction of the molecule/atom inside the structure or due to the developed strain and stress that convert the motion energy into heat energy.

The MSD model for the system is as shown in Figure 1.10.

Figure 1.10 MSD model of the motor-beam system in Animation 1.8.

Example Problem 1.1

Propose the MSD model of the car as shown in Figure 1.11 which focuses on the rigid body motion of the car. Assume only vertical vibration when the car is running on a rough road and where the front and rear wheels to have the same and in-phase displacement.  Consider two cases of MSD models:

a. One-degree-of-freedom system.
b. Two-degrees-of-freedom system.

Figure 1.11 A car to be modeled in terms of mass-spring-damper element.


a.  For a SDOF model, the MSD model is shown in Figure 1.12. Here the total stiffness is the combination of elasticity of the tyre (rubber+air) and the spring of the suspension. The  mass of the wheels are also included in the total mass of the system. The vibration amplitude of the car body as the function of time is denoted as y(t) .

This is a very crude model with imposed assumption, by neglecting the independent motion of the wheels with respect to the motion of the car body.

Figure 1.12 The MSD model of a car for a single degree of freedom. 

If the effect of vibration to the driver is of interest, a 3-DOF system can be modeled by considering the stiffness of the seat.

b.  For 2-DOF system, the wheels are now treated to have independent vertical motion due to the stiffness of the tyre, so they have independent coordinates y_2(t) and have relative phase with the body of the car. The MSD model is shown in Figure 1.13. 

This model provides a more accurate prediction, but again the model assumes the front and rear wheels have in-phase motion, in other words this model neglects the roll and pitch motions of the car body (this will be discussed in details in Chapter 4: Multi-degree-of-freedom System).

Figure 1.13 The MSD model of a car for two degrees of freedom. 

Watch Video 1.1 with other examples to further strengthen your concept on mass-spring-damper model of a vibrating system.

Video 1.1 Modeling a vibrating system with mass-spring-damper elements.

1.3.2  Modeling a continuous system

A continuous system, such as a vibrating plate, also has mass, stiffness and damping elements, but these are distributed across the structure. Therefore in principle, the panel can also be divided into ‘finite’ number of mass, spring and damper elements. 

By subdividing the structures into elements, each of the element represents the sub-equation to the global equation. The greater the number of the elements, the more accurate the model to determine the vibration behaviour of the plate. 

From here, a set of mass, stiffness and damping components are produced in terms of matrices. These are then solved numerically to obtain the natural frequencies and mode shapes of the plate (We will discuss the basic concept in Chapter: Multi-Degree-of-Freedom System).

For more complex structures, modelling the vibration of engineering structures can be conveniently performed using software tools, usually called Finite Element Analysis (FEA) software. Figure 1.14 shows the example of an I-beam modeled using FEA. Here, it animates the vibration behavior (vibration mode) at the natural frequency where the beam is bending across x-axis.



Figure 1.14 An I-beam structure modeled in FEA: (a) meshing the structure into ‘finite’ elements and determination of boundary conditions before calculation, and (b) simulating the vibration mode of the structure after calculation.