Mechanical Vibration
Web Book
Chapter 1
Introduction
Learning objectives
- Recognize phenomena and applications of vibration in practice.
- Model a vibrating system with mass, spring and damper elements.
- Understand the concept of amplitude, frequency and phase.
Sections
1.1 Why study vibration?
In June 29, 1995 people in Seoul, South Korea were mourned with the deadliest disaster of a building collapse which killed 502 people and 937 were injured. The structural failure was caused by vibration from the air-conditioned chiller installed on the roof, which provided energy load by four times the design limit of the roof structure (Figure 1.1).
On April 24, 2013 similar tragedy took place in Bangladesh, where an eight-storey building collapsed leaving 1,129 people died and 2,515 were injured. The building which was designed for shops and offices, was later used as a garment factory. The structure unfortunately could not sustain the vibration energy coming from heavy machineries. Figure 1.2 shows the scene when the disaster happened.
Figure 1.1 Building structural damage due to machine vibration and load in South Korea.
The two examples reveal the capacity of vibration to the damage and destruction; which gives similar result of damage due to ’natural’ vibration energy from an earthquake (scary, isn’t it?).
The receiving structure should have an allowable level of vibration energy injected by vibrating machines. For this purpose vibration isolators are used at the contact points between the machine and the structure to block some of the vibration input power from the machine.
Figure 1.2 Building collapse in Dhaka, Bangladesh due to vibration impact excitation from machineries.
Not only to the receiving structures, the machinery system itself can also suffer mechanical failure due to the internal vibration. Rotating machineries which are commonly found in oil and gas industry; electric motors, compressors, centrifugal pump, gas turbine, must be maintained properly to ensure the system runs smoothly for production.
Any failure due to vibration from mass unbalance and misalignment, for example, can damage bearings and shafts and results in shutting down the operation unexpectedly for machine replacement and unscheduled maintenance
(see Figure 1.3).
The company can lose millions of dollar a day, besides bad image of company to customers due to late delivery. If you are the engineer in charge, you could straight away get fired!
Figure 1.3 Possible faults in rotating machines which can be indicated by high level of vibration.
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Other negative effects from vibration are discomfort and disturbance. Apart for structural safety, cars, trains and airplanes are also designed to have minimum transmitted vibration which can create discomfort to the passengers. The car engine, for example, is mounted on vibration-isolated mountings to minimise transfer of vibration waves across the car structure. The car floor is usually treated with damping layer to reduce vibration and noise radiation into the cabin.
Vibration can damage bearings and shaft in a rotating machine.
Railway track should be properly treated in order to reduce the ground-borne vibration transmission to nearby dwelling which can vibrate the walls, ceilings and floors of houses and finally radiates an annoying low frequency noise causing sleep disturbance (Figure 1.4).
Figure1.4 Transfer of vibration energy not only creates noise in the train cabin, but also to the environment.
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Direct exposure to vibration has also been known to affect human health. In vehicles, frequent experience to low frequency vibration is found to cause back pain to the driver. This is found especially for a tractor operator working in an agriculture sector. A regulation is thus enforced by a government for the tractor manufacturer to comply with minimum level of vibration.
A worker with a prolonged use of vibrating hand-held tools such as pneumatic jack hammers, drills or grinders could suffer had-arm vibration syndrome and vibration induced white finger. The latter causes a person to end up with the fingers becoming numb due to poor blood circulation.
Figure 1.5 Tractor’s operator can be susceptible to vibration exposure to the back of the body.
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In this book, the content is aimed at providing the fundamental concepts of vibration. In practice, a vibration engineer tasks are:
- to design a mechanical system to avoid high vibration amplitude in an engineering structure, and
- to perform analysis and troubleshoot to eliminate vibration in a rotating machine or in a structure.
Here, both math and physics are used as the tools for the deep analysis to understand the concept. You will find in certain chapters that you have to deal with a rather ‘abstract system’ which is sometimes difficult to be related in a real situation. This is not merely the mathematical exercise, but to help improve the cognitive skill, strengthening the concept of mass, spring and damper elements, and also the amplitude and frequency.
Later in industry, you will use a sophisticated numerical software to design or to solve vibration problems. However if the fundamental concept is not strong, the software becomes just a black-box, and accurate engineering solution is difficult to obtain.
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.
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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.
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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.
Single–degree-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.
Two–degree-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.
1.3 Modeling a vibrating system
To solve a vibration problem (simple word: reducing the vibration amplitude), there are four main steps involved:
- Development of mathematical model
- Generation of governing equations
- Prediction of response of vibration
- 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.
Solution
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.
(a)
(b)
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.
1.4 Amplitude, frequency and phase
When we discuss about vibration, we are talking about a motion which fluctuates around an equilibrium point. Observe the vibration of a vertical motor pump in Figure 1.15 below.
Figure 1.15 A vibrating, vertical motor pump.
Someone may ask, ‘What is the level of vibration?’
You reply, ’8 mm/s’.
‘At what frequency?’
’At 13 Hz’
‘How does the machine vibrate?’
You reply again, ‘Mostly in horizontal direction (x-axis), rocking motion’.
The conversation reveals three important variables in vibration:
Amplitude: represents the level of vibration
(How severe?)
Frequency: represents the speed of vibration
(How fast?)
Phase: represents the behaviour of vibration
(How does it vibrate?)
1.4.1 Amplitude
Let us see an example of motion of a mass on a spring, where its centre moves with a distance from the equilibrium line (position when the mass is not in motion). Suppose there is a pen fixed to the mass and is placed on top of a paper which runs from left to right (with the same speed as that of the mass), we can see in Animation 1.19 that the line drawn on that paper looks like a sinusoidal function.
So from our observation, the instantaneous displacement y(t) of the moving mass can be expressed as
where T is the period (i.e. the time taken from a peak to the next peak, in second), t is the instantaneous time and \omega=2\pi f is the angular frequency (in rad/s), with f the frequency (in Hertz).
Animation 1.9 Motion of a mass-spring system showing a sinusoidal pattern.
Since maximum of \cos(\omega t) across the time is 1, thus the maximum of the displacement of the mass from Eq. (1.1) is y_{max}= A . This is called the peak amplitude of vibration.
In Eq. (1.1), the position of the mass is changing with time. And therefore instead of stating the amplitude of vibration with displacement, we can also state it with velocity.
From Eq. (1.1). the velocity is given by
The maximum amplitude of the velocity is |v_{max}|=\omega A
We do not state the vibration amplitude in ‘negative’ value. This is a dynamic problem where mass is oscillating around the equilibrium position. We are only interested in its magnitude, i.e. how much the mass goes beyond the equilibrium line, either in positive or negative directions.
In terms of acceleration, the amplitude is given by
where the peak acceleration is |a_{max}|=\omega^2 A .
Example Problem 1.2
A system vibrating harmonically at 2 Hz has a maximum displacement of 2 mm.
What is the maximum velocity?
Solution
For a harmonic motion, the displacement can be expressed as y(t)=A\cos\left(\omega t\right) as shown in Eq. (1.1).
The peak displacement is thus A = 2 mm.
The velocity is
From this, the peak velocity is therefore
v_{max}=|A\omega|=A(2\pi f)=2(2\pi 2)=8\pi mm/s.
Example Problem 1.3
What is the peak displacement of a system known to have peak velocity of 10 mm/s at frequency 4 Hz? The system is known to vibrate harmonically.
Solution
From Example Problem 1.2, we know that the peak velocity is
The peak displacement is thus
A=y_{max}=\displaystyle\frac{v_{max}}{\omega}=\frac{10}{2\pi(4)}=0.4 mm
Example Problem 1.4
What is the peak displacement of a system with harmonic vibration with amplitude of acceleration of 50 mm/s^2 at 10 Hz?
Solution
From Eq. (1.3), we know that the peak acceleration is
The peak displacement is therefore
A=y_{max}=\displaystyle\frac{a_{max}}{\omega^2}=\frac{50}{(2\pi(10))^2}=0.013 mm