Lecture notes Mechanical vibrations Part I PDF

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Helwan University Faculty of Engineering – Mattaria Mechanical Design Department Lecture notes Mechanical vibrations Part I Prepared by Dr. Heba Hamed El-Mongy Prospective students: Second year, Automotive department Semester: Jan 2015 Chapter I Basic concepts of mechanical vibrations This chapter i...

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Helwan University Faculty of Engineering – Mattaria Mechanical Design Department

Lecture notes

Mechanical vibrations Part I Prepared by

Dr. Heba Hamed El-Mongy

Prospective students: Second year, Automotive department Semester: Jan 2015

Chapter I Basic concepts of mechanical vibrations

This chapter introduces the subject of mechanical vibrations including the following topics:         

Importance of the study of vibration; Vibration and simple harmonic motion; Characteristics of vibration; Main elements of a vibrating system; Classification of vibration; Harmonic analysis; Machine health monitoring; Degrees of freedom; Modelling of vibrating systems.

1.1 Importance of the study of vibrations: We experience many examples of vibration in our daily lives. A pendulum set in motion vibrates. A plucked guitar string vibrates. Vehicles driven on rough roads vibrate, and geological activity can cause massive vibrations in the form of earthquakes. All machines and structures are subject to vibrations which arise due to internal and external forces applied to the machine. There are good and bad effects of mechanical vibrations. The following are examples of the bad effects of vibration: 1- Increased wear of machine components (Ex. Bearings, couplings, etc.) 2- Looseness of fasteners (may cause catastrophic accidents in vehicles and aircrafts). 3- Failure of machine components due to fatigue. 4- Poor surface finish due to tool chatter in metal cutting processes. 5- Excessive noise. 6- Earthquakes. 7- Resonance. 8- Instability. 9- Discomfort in operating machines or vehicles. Figure 1.1 shows some examples of bad machine vibration. There are some applications where vibrations are intentionally generated to obtain useful functions. Some applications of good vibrations include (Fig. 1.2):  Vibratory conveyors;  Shaking table for earthquake testing of buildings;  Vibratory sieves; 1

 Grinding machines;  Energy harvesting using piezoelectric elements. Therefore, it is important to study vibration in order to reduce its bad effects through proper design of machines and their mountings (vibration isolation and vibration absorption) and also to design new systems that use the vibration phenomenon to perform useful functions.

Fig. 1.1 Examples of bad machine vibrations (a)

(b)

(c)

Fig. 1.2 Examples of good vibration applications, (a) Vibratory conveyor, (b) Shaking table, (c) Piezoelectric energy harvesting device

1.2 Vibration and simple harmonic motion: 1.2.1 What is vibration? Vibration can be defined as any motion that repeats itself in a certain interval of time around a certain equilibrium position. The study of vibration is concerned with the oscillatory motion of bodies and the forces associated with them. Most engineering machines and structures experience vibration to some degree and their design generally requires consideration of their oscillatory behavior.

1.2.2 Simple harmonic motion: If the motion is repeated after equal intervals of time, it is called periodic motion. The simplest type of periodic motion is harmonic motion. Harmonic motion can be represented as shown in Fig. 1.3 by means of vector OP of magnitude A rotating at a constant angular velocity ω. The projection of the end of the vector OP is given by: 2

x  A sin  t

(1.1)

This is considered the harmonic displacement. Velocity and acceleration can be obtained by differentiating Eq. (1.1) with respect to time once and twice respectively as follows:

x  A sin  t

(1.2)

x   A 2 sin  t

(1.3)

Figure 1.4 shows the three measures of vibration; displacement, velocity and acceleration waveforms and vectors.

Fig. 1.3 Harmonic motion as the projection of the end of a rotating vector

Fig. 1.4 Displacement, velocity and acceleration (a) waveforms, (b) vectors

1.3 Characteristics of vibration: 1- Cycle of vibration: This is the motion of a vibrating body as shown in Fig. 1.5: from its neutral position (Position O) to the maximum position in one direction (Position A), then passing through the neutral position to the maximum position in the other direction (Position B) and finally to the neutral position again.

3

Fig. 1.5 One cycle of vibration 2- Period (T): It is the time taken to complete one cycle of motion and is denoted by T and measured in seconds.

3- Frequency (f): It is the number of cycles per unit time and is denoted by f and measured in Hz or cps.

f

1 T

(1.4)

4- Angular (Circular) frequency (ω): This is the angular velocity of the cyclic motion measured in rad/s.

  2 f 

2 T

(1.5)

5- Amplitude of vibration: It is the maximum displacement of a vibrating body from its equilibrium position.

6- Phase angle (φ): It is the angular difference between the occurrence of similar points of two harmonic motions. If we consider the two waves shown in Fig. 1.6 denoted by:

x1  A1 sin  t

and

x2  A2 sin ( t   )

(1.6) (1.7)

Fig. 1.6 Phase difference between two waves It can be noticed that there is a lag (difference) between the peak of x1 and the peak of x2. The difference is called phase lag and is measured by the phase angle φ. The two waves are described to be out of phase by φ degrees. 4

1.4 Main elements of a vibrating system: Figure 1.7 shows the mass-spring-damper system (MSD) which demonstrates the main elements of a vibrating system. All machines and structures have these fundamental properties (mass, stiffness, damping) that combine to determine how the machine will react to the forces that cause vibrations just like the MSD system. Table 1.1 shows the function, forces and energies associated with each element. It may be noted that there are other forms of damping such as fluid resistance, friction, etc. however, in this course focus will be given to viscous damping shown in Fig. 1.7 which resembles shock absorbers used in vehicles.

Fig. 1.7 Mass-spring-damper system Table 1.1 Main elements of a vibrating system details Notation

Mass m (mass in kg)

Function

Moving element

Force

Inertia force 

Energy

1.4.1

F = m a = mx Kinetic energy 1 1 2 2 K.E. = m v  m x 2 2 P.E. = m g h (If the cg level of the body is changed by h)

Spring k (Spring constant or stiffness in N/m) Restoring element Spring force F=kx Potential energy 1 2 P.E. = k x 2

Damper C (Damping coefficient in N.s/m) Energy dissipating element Damping force  F=Cv= Cx Dissipated energy D.E. =

2 1 Cx 2

Elastic elements as springs:

Stiffness has other forms in vibrating systems than the simple linear spring. Elastic elements like beams also behave as springs. Consider a cantilever beam with an end mass m, as shown in Fig. 1.8. We assume for simplicity that the mass of the beam is negligible in comparison with the end mass m.

5

Fig. 1.8 Equivalent system of a cantilever beam with end mass From strength of materials, we know that the static deflection of the beam at the free end is given by:  st 

W L3 3E I

(1.8)

where W = mg is the weight of the mass, E is Young’s modulus and I is the moment of inertia of beam cross-section. Hence, the spring constant is: k

W

 st



3E I L3

(1.9)

Similar results can be obtained for beams with different end and different loading conditions as shown in Table 1.2. Table 1.2 Equivalent stiffness formulae for different cases *: Configuration Longitudinal vibration in rod/cable

Equivalent stiffness k

EA L

k

3E I L3

Simply-supported beam

k

48 E I L3

Fixed-fixed beam:

k

192 E I L3

Cantilever beam (Fixed – free beam)

*(For more information on the derivation of these formulae refer to Appendix B in Ref. [1])

6

1.4.2 Combinations of springs: In many practical applications, several linear springs are used in combination. The springs can be combined into a single equivalent spring as indicated below: Case (1): Springs in parallel (Common displacement): For the springs shown in Fig. 1.9, in equilibrium position:

W = k1δst + k2δst

(1.10)

where W is the weight of the mass and δst is the static deflection in the springs due to the weight of the mass and it is shown to be equal in both springs, i.e. displacement is common for parallel springs. For the equivalent spring:

W = keq δst

(1.11)

keq = k1 + k2

(1.12)

Substituting Eq. (11) in Eq. (10) yields: Case (2): Springs in series (Common force): For the springs shown in Fig. 1.10, the deflections in the springs are not equal. However, both springs are subjected to the same force (W). In equilibrium position:

δst = δ1 + δ2

(1.13)

where δ1 is the deflection in spring k1 and δ2 is the deflection in spring k2. For k1: δ1 = W/k1, for k2: δ2 = W/k2 and for the equivalent spring keq: δst = W/keq Substituting in Eq. (13) we get:

1 1 1   keq k1 k2 keq 

k1 k2 k1  k2

7

(1.14)

Fig. 1.9 springs in parallel

Fig. 1.10 Springs in series

1.5 Vibratory motion and transfer of energy: The vibration of a system involves the transfer of its potential energy to kinetic energy and vice versa. If a damper is present in the system, some energy will be lost. Example: Consider the simple pendulum shown in Fig. 1.11 and the corresponding oscillation in Fig. 1.12: 1- The mass is given an initial displacement and then released, the pendulum starts oscillation.

8

2- When the mass m reaches its maximum position, the velocity will be zero and therefore all the kinetic energy is converted to potential energy due to the elevation of the mass from its neutral position. 3- This will cause a moment that will restore the mass to return to the neutral position. 4- The mass will not stop but will continue to the maximum position in the other direction. 5- The same as step (2) is repeated. The sequence of events will continue but the magnitude of oscillation will decrease gradually and the pendulum will stop due to the resistance (damping) offered by the surrounding medium (air).

Fig. 1.11 Vibration as an interchange between kinetic ad potential energies

Fig. 1.12 Oscillation of the simple pendulum expressed in velocity

1.6 Classification of vibration: 9

Vibration can be classified in several ways. The following are the most important classifications:

1.6.1 Undamped and damped vibration (Fig. 1.13):  Undamped vibration occurs when no energy is lost or dissipated in friction or other resistance during oscillation (No damping is present).

 Damped vibration occurs when some energy is lost due to damping.

(a)

(b)

Fig. 1.13 Classification of vibration according to damping (a) Undamped, (b) Damped vibration

1.6.2 Free and forced vibration:  Free vibration is the condition when the system is given an initial disturbance (displacement, velocity or force) and then released to vibrate on its own.

 Forced vibration is the condition when the system is subjected to an external force. 1.7 Harmonic analysis: A vibration or system response can be represented in both time and frequency domains as shown in Fig. 1.14 and Fig. 1.15. It can be inferred from Fig. 1.16 that using time domain view for complex signals is less useful while the frequency spectrum is clear and more informative. Time waveform: Amplitude varies with time. Frequency spectrum: Amplitude varies with frequency. Fast Fourier transform (FFT) is used to convert vibration from time domain to frequency domain. It is based on the Fourier series principle that indicates that any periodic function in time can be represented as an infinite series of sine and cosine terms.

10

Fig. 1.14 Example of a single harmonic signal in time and frequency domains

Fig. 1.15 Example of a two-harmonic vibration signal in time and frequency domains

Fig. 1.16 Frequency spectrum versus time waveform for complex signals

1.8 Machine health monitoring: Predictive maintenance is basically a condition-based preventive maintenance. Various techniques are used to assess the equipment condition such as: 1- Vibration analysis; 2- Oil analysis; 11

3- Ultrasonics; 4- Temperature measurement, etc. However, vibration is considered the best indicator of machine condition due to the following reasons: 1- Its effectiveness in detecting various faults that may occur in a machine such as unbalance, misalignment, bearing defects, gear defects, mechanical looseness, shaft cracks and flow related problems. 2- It is a non-destructive method that does not disturb machine operation. 3- It can be used during normal operation and during starts and stops of machines. Therefore, vibration analysis is primarily used on rotating equipment such as steam turbines, pumps, motors, compressors, rolling mills, machine tools and gearboxes. The vibrations caused by the defects occur at specific frequencies which are characteristic of the components. Hence, the vibration amplitudes at particular frequencies are indicative of the presence and severity of the faults as illustrated in Fig. 1.17. (Ex: for faulty gears, vibration is high at gear mesh frequency which equals = rotational frequency of gear * No. of teeth).

Fig. 1.17 Fault detection using vibration

1.9 Degrees of freedom: Degrees of freedom can be defined as the minimum number of independent coordinates that describe the motion of a system completely. To specify the DOF for any system, you need to observe the number of independent displacements for each mass. DOF = ∑ (Independent displacements of masses) The systems shown in Fig. 1.18, represent single-degree-of-freedom systems. For example, the motion of the simple pendulum can be stated either in terms of the angle or in terms of the linear coordinates. In this example, we find that the choice of θ as the independent coordinate will be more convenient than the choice of x or y. For the torsional system (long bar with a heavy disk at theend) shown in Fig. 1.18, the angular coordinate can be also used to describe the motion. The 12

mass-pulley system motion can be represented by either the linear displacement of the mass or the angular displacement of the pulley because the two coordinates are dependent on each other through this equation (x = r θ) assuming no slip condition where r is the radius of the pulley. Fig. 1.19 and Fig. 1.20 show examples of two-DOF and three-DOF systems respectively. Some systems, especially those involving continuous elastic members, have an infinite number of degrees of freedom. As a simple example, consider the cantilever beam shown in Fig. 1.21. Since the beam has an infinite number of mass points, we need an infinite number of coordinates to specify its deflected configuration. Thus the cantilever beam has an infinite number of degrees of freedom. Most structural and machine systems have deformable (elastic) members and therefore have an infinite number of degrees of freedom.

Fig. 1.18 Examples of 1-DOF systems

Fig. 1.19 Examples of 2-DOF systems

Fig. 1.20 Examples of 3-DOF systems

13

Fig. 1.21 A cantilever beam having infinite number of DOF

1.10 Modelling of a vibrating system: Vibration analysis of an engineering system usually involves the following steps: 1-

234-

Mathematical modelling: First specify the purpose of the analysis. Second identify the elements of the system based on the purpose of the analysis. Derivation of the governing equations. Solution of the governing equation. Interpretation of the results.

Steps 1 and 4 depend on the experience and judgment of the analyzer while steps 2 and 3 depend on mathematical treatment. Example: Modelling of an automobile: Model 1 (Fig. 1.22.a): The simplest quarter-model (1 DOF)   

Masses of the engine, gearbox, body, frame, passengers can be lumped as a single mass. Tires and suspension system can be modeled as one spring and one damper. One coordinate: the vertical displacement of the mass.

Model 2 (Fig. 1.22.b): More sophisticated quarter model (2 DOF)   

The masses of the wheel axles are considered. The suspension and tires are considered separately as springs and dampers. Two coordinates: the vertical displacements of the two masses.

Model 3 (Fig. 1.22.c): Simple half-vehicle model (2 DOF)    

The rotational motion of the car body is taken into account (center of gravity of car body is in the middle). The suspension stiffness values are not equal. The masses of the wheel axles are not considered. Two coordinates: Vertical and rotational displacements of the mass. 14

Model 4 (Fig. 1.22.d): Half-vehicle model (4 DOF)    

The rotational motion of the car body is taken into account (center of gravity of car body is not in the middle). The masses of the wheel axles are considered. The suspension and tires are considered separately as springs and dampers. Four coordinates: Vertical and rotational displacements of the vehicle mass and the two displacements of the two masses.

Model 5 (Fig. 1.22.e): Half-vehicle model (5 DOF)

 The car seat is considered as a mass, spring and damper in addition to the assumptions of model 4 also.  Five coordinates: Vertical and rotational displacements of the vehicle mass, the two displacements of the two masses and the displacement of the car seat. More sophisticated models are shown in Fig. 1.23 as extra information. (b)

(a)

(c)

15

(e)

(d)

Fig.1.22 Models of an automobile using different assumptions

(b)

(a)

(c)

Fig. 1.23 Sophisticated models of automobile, (a) Two passengers are taken into account (6 DOF), (b) One passenger body is taken as several masses, springs and dampers (9 DOF) and (c) Full car model (10 DOF)

16

Chapter II Single degree of freedom systems

2.1 Free vibration of undamped SDOF system: Figure 2.1 shows a spring-mass system that represents the simplest possible vibratory system. The motion is called free vibration because there is no external force applied to the mass. Motion is started by initial displacement and/or velocity at t = 0. The system is undamped because there is no element that causes dissipation of energy during the motion of the mass. Therefore, the amplitude of motion remains constant with time. The governing equation of a vibrating system is called the equation of motion (EOM) and it is a second order ordinary differential equation whose solution gives the displacement of the system at any instant ...