Lecture notes, lectures 1-21 PDF

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MEC201: Dynamics of Structures and Machines Summary MEC201 comprises the following learning activities. 

21 Lectures



22 Tutorial sessions (5 tutorial sheets)



Vibration experiment and report



Numerical study, technical note and peer marking exercise



5 online quizzes

Assessment involves coursework and a 2 hour exam at the end of the module. The coursework element is worth 20% and involves activities related to the vibration exercise and the numerical study. Quizzes are not part of the assessment but there is a mark penalty if they are not completed on time.

MOLE The most up to date information, class times, deadlines and latest versions of documents are available on MOLE. You should therefore contact the module leader if you are unable to see ‘MEC201’ in your course list. You will also need to access MOLE to do the following: 

View lecture slides



Complete online quizzes



Submit the report and the technical note

 Participate in the Peer Marking exercise As material is released at different times, you are advised to visit the site at least once per week.

Hardcopy This pack contains: 

Lecture summaries with key equations



Tutorial sheets



Information on coursework



Questions from the Spring 2015 exam paper

Help and Feedback It is expected that you will manage your learning appropriately. 

The best times to get help and feedback are during the timetabled tutorial sessions.



You are encouraged to ask relevant questions during lectures, NOT after they have finished.



The online quizzes and the peer marking exercise will also give you valuable feedback on your understanding of the subject.

If you need to move your lab session, give information regarding extenuating circumstances or spot a mistake, it is best to email the module leader.

Module Leader Dr Jem Rongong Email: [email protected] Room: Mappin Building, Rc10c (but if you turn up unannounced, I may not be able to see you at that time)

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Spring Semester 2016

MEC201: Dynamics of Structures and Machines

Lecture and tutorial information Arrangements may change. Please check MOLE for the most up to date information.

Lectures Lectures are in Diamond LT-01, Tuesday 15.00-15.50 and Wednesday 12.00-12.50. Week 1 2 3 4 5 6

7 8 9 10 11

Lecture 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Date 09-Feb-16 10-Feb-16 16-Feb-16 17-Feb-16 23-Feb-16 24-Feb-16 01-Mar-16 02-Mar-16 08-Mar-16 09-Mar-16 15-Mar-16 16-Mar-16 12-Apr-16 13-Apr-16 19-Apr-16 20-Apr-16 26-Apr-16 27-Apr-16 03-May-16 04-May-16 10-May-16 11-May-16

Title Introduction to dynamics Mechanical vibrations SDOF system models Free vibration 1 Free vibration 2 Steady-state forced vibration 1 Steady-state forced vibration 2 Steady-state forced vibration 3 General forced vibration Time domain vibration studies System models 1 System models 2 EASTER BREAK – 3 WEEKS Damping Vibration case study Rigid body dynamics Kinematics with vector algebra Translating and rotating reference Kinetics Balancing of rotors Precession of gyroscopes NO LECTURE Review of 2015 Exam Paper

Supporting activities Quiz 1 Tutorial Sheet 1, Quiz 2 (indirect), Vibration experiment Tutorial Sheet 2, Quiz 2 (indirect), Vibration experiment Quiz 3, Numerical study Quiz 4, Tutorial Sheet 3 Quiz 4, Tutorial Sheet 3 Quiz 5, Tutorial Sheet 4 Tutorial Sheet 5

Tutorial sessions Group tutorial sessions run from Week 2 to Week 12 on Thursday from 14.00-14.50 and Friday from 14.00-14.50. These sessions are your chance to get one-to-one assistance and advice regarding ANY aspects of the course. Participation in tutorial sessions will be monitored. The class will be split between Diamond Workroom 1, 2 and 3. To balance numbers, you will be pre-assigned to one of these rooms. Please go to the room that you are assigned to.

Deadlines All deadlines for this module are 12.00 (midday) on a Wednesday. Type

Description

Quiz 1 Quiz 2 Quiz 3 Quiz 4 Quiz 5 Report Technical note Peer Marking

Basic dynamics concepts Matlab and vibrations Arbitrary forcing of SDOF systems Models and damping Kinematics with vector notation Report on vibration experiment Report on numerical study Comment on lab reports written by others

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Week

Date

2 4 6 8 10 7 9 11

17-Feb 2-Mar 16-Mar 20-Apr 4-May 13-Apr 27-Apr 11-May

Spring Semester 2016

MEC201: Dynamics of Structures and Machines

Introduction to Dynamics (Lecture 1) In engineering, dynamics is the study of motion and associated forces in a system. This course concentrates on the dynamics of structures and machines. The aims of this module are to: 

Understand the main concepts of vibration



Extend capability in rigid body dynamics



Learn how to create and use simple, valid models

The ability to predict the effects of force and motion is vital for engineering design. Catastrophic failures of machines and structures have occurred when understanding of the system’s dynamics has been inadequate. The equation underpinning the dynamics topics studied in this module is Newton’s Second Law which relates the resultant force applied to a body to its acceleration. For systems that involve rotation rather than translation, the equivalent equation links resultant moment to angular acceleration. Newton’s Second Law:

 F  ma

Equivalent for rotation:

M  I 

The first step in the dynamic analysis of a system is the development of an appropriate mathematical model – this usually involves simplifications . Consideration must be given to the following points: 

Intended purpose, scale of model and external influences



Representation of elements as rigid or flexible and hence the degrees of freedom (possible motions) to be considered Kinematic constraints

 

Linearity: a system is said to be linear when it satisfies the Principle of Superposition. This is expressed in the following way: if f  a  A and f  b  B , then f a  b  A  B

Figure 1.1

Note that homogeneity, which requires f Ca   CA if C is a constant, is a necessary but not sufficient condition for linearity. A dynamic system may often be assumed linear over a limited range of operation.

A free body diagram that summarises the motions and forces involved is a useful tool for developing the equations of motion.

Mechanical vibrations (Lecture 2) The study of vibration is concerned with the oscillatory motions of bodies and the forces associated with them. Vibrations are present throughout the universe, ranging in scale from inter-atomic motion to astronomic events. This module focuses on the types of vibrations that are commonly associated with machines and engineering structures – these are generally referred to as mechanical vibrations. All bodies possessing mass and elasticity can vibrate. Vibration is useful in some applications and very dangerous in others. In either case, because most engineering machines and structures experience mechanical vibration, their design requires consideration of this. Hazards associated with high vibration levels in machines include performance degradation, noise pollution and fatigue failures. Vibratory systems include means for storing potential energy (stiffness of spring elements or structural elasticity) and kinetic energy (mass elements or inertia). Vibration is a process which involves the transfer of energy between these two forms, potential and kinetic. The mechanism by which energy is removed from a system is known as damping. In analysing a vibrating structure, it is usually helpful to identify the following features   

Source: force or motion generation point Path: the way in which the vibration arrives at the structure in question Response: the way in which the structure responds depending on its characteristics (mass, stiffness and damping).

Where the system is linear, the vibration source and the structure response can be analysed independently.

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Forcing / excitation

Response path

structure mass

source

stiffness damping

Figure 2.1

Spring Semester 2016

MEC201: Dynamics of Structures and Machines

Harmonic motion The simplest type of vibration is simple harmonic motion. Figure 2.2 shows sine and cosine functions. We can define general sinusoidal motion as,

a 0

x  a sin t    or,

x  a sin  t  

2

t

3

x  a cos  t 

a

x  A cos t   B sin t  where A  a sin  and B  a cos

0



2

The term  is the frequency (in radians per second) of the vibration, a is the amplitude and  is the phase shift.



 a sin  t  2

t

3



Figure 2.2

Expressions for velocity and acceleration can be obtained by differentiating the displacement with respect to time. This gives: displacement:

x  A cos t  B sin t

velocity:

x   A sin t  B cos t

acceleration:

 Acos  t 

2 B sin  t     x

This shows that differentiation with respect to time results in a 90 degree (i.e.

 2

) increase in phase.

Harmonic motion can also be described using rotating vectors known as phasors on an Argand diagram. In the single phasor representation, a vector of magnitude a rotating at a constant angular velocity  is used. Figure 2.3 shows that the projection along the horizontal axis for a rotating vector gives a sine function. Projection along the vertical axis would give a cosine function.

a

t

y = a sin ( t) 0

a 

2

t

3

Euler’s formula states that

e j  cos  j sin Figure 2.3

where j   1 .

The rotating vector can therefore be represented as a complex number with real and imaginary parts. Using Euler’s formula, the vector x is written in terms of the angle t between the vector and the real axis. Thus,

x  a e j t

Imaginary

x  j  a e j t  j  x

x   2a e j t   2 x

a cos t 

As,





Re a e j t  a cos t  a sin (t  2 ) it can be seen that harmonic motion expressed as an exponential is the same as the general sinusoidal motion considered earlier

a sin  t 

a

t

Real

Figure 2.4

a a j t and x   e j t e 2 2 which cancel out the imaginary terms from Euler’s equation. The relationships between displacement, velocity and acceleration however, remain the same. Another definition for harmonic motion exists: this uses contra-rotating phasors x 

Frequency domain Complicated signals can be represented by combinations of sinusoidal functions using the Fourier series. In dynamics, the related Fourier Transform is widely used to obtain the frequency spectrum of a signal. While the mathematical treatment of Fourier analysis is beyond the scope of this module, the ability it gives to move

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Spring Semester 2016

MEC201: Dynamics of Structures and Machines

between the time and frequency domains should be noted. The Fourier transform of the time signal x(t) is mathematically expressed as,

X    





xt e j t dt

where  is the radian frequency. The inverse Fourier transform is given by, xt  

1 2



 X e

j t

d

0.5 0 -0.5 -1 0

10 20 time, milliseconds

FREQUENCY 1 0.8 0.6 0.4 0.2 0 0

500 1000 1500 frequency, Hz

Figure 2.5(a) single frequency at 440 Hz

TIME 1 0.5 0 -0.5 -1 0

normalised amplitude

1

normalised amplitude

TIME

normalised amplitude

normalised amplitude

In the frequency domain, the signal a sin t  or a cos t  is represented by a single point at frequency  and amplitude a . See Figure 2.5a for a typical example. Figure 2.5b shows a more complex waveform. FREQUENCY 0.4 0.3 0.2 0.1 0

10 20 time, milliseconds

0

500 1000 1500 frequency, Hz

Figure 2.5(b) frequencies, 440, 880 and 1320 Hz

Note that in general, a frequency domain definition should contain both magnitude and phase information. For this reason, the Fourier transform of a signal is usually a complex number.

SDOF system models (Lecture 3) The minimum number of independent coordinates needed to determine the position of all parts of a system at any instant in time defines the degree of freedom (DOF) of the system. Each mass can have up to six possible motions unless kinematic constraints limit the movement directions. Systems with a finite number of degrees of freedom are called discrete, or lumped parameter systems. Systems such as beams and plates which have an infinite number of degrees of freedom are called continuous, or distributed systems. Some numerical analysis techniques (e.g. finite element modelling) involve representing continuous systems as a large number of discrete units. The response of a structure is often characterised by its frequency response function (FRF). This shows the amplitude of response at different frequencies for unit amplitude forcing. Figure 3.1 shows the mobility FRF (amplitude of velocity divided by amplitude of input force) for a typical structure that summarises its steady-state characteristics. A peak in the FRF is called a resonance. The complexity of the FRF depends on the number of degrees of freedom the structure has – one resonance exists per DOF. A single degree-of-freedom (SDOF) system is the simplest system for studying vibrations. It is often possible to reduce a complex system to SDOF form either by making assumptions about the structure or by focusing the study around a distinct resonance. Figure 3.1 The simplest SDOF system comprises a mass and a spring (Figure 3.2a). When the mass is displaced by x(t), the resulting free body diagram (FBD) is shown in Figure 3.2b. Applying Newton’s Second Law gives the equation of motion,

m x  k x  0

x(t) m

Free body diagram m

x , x , x

k st

The solution can be found by assuming x(t )  Ce where C and s are constants. This gives,





C m s2  k  0

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kx

Figure 3.2(a)

Figure 3.2(b)

Spring Semester 2016

MEC201: Dynamics of Structures and Machines

For a non-trivial solution, the terms inside the parentheses must equal zero. Thus, s   k

m

  j n .

The two values of s are known as the eigenvalues or characteristic values of the equation of motion. Since both satisfy the equation of motion, the general solution is of the form n t

x(t )  C1e j

 C2e  jnt

Euler’s formula can be applied to show that e j nt  cos( nt)  j sin( nt) therefore,

x( t)  C1cos( nt)  j sin( nt)  C2cos( nt)  j sin( nt)    C1  C2 cos( nt)  j C1  C2 sin( nt) Since C1 and C2 are complex constants, the general solution can be written as,

x(t)  A cos( nt) B sin( n t) For the general solution, the values of the constants A and B can be determined from initial conditions.

xt  0  A  x0 , x The solution can then be written as,

x (t )  x0 cos(n t )  The term n 

x 0

n

sin( nt )

k is known as the natural frequency of the system. m

In the absence of external forcing, a system given an initial excitation will oscillate at its natural frequency. The natural frequency is one of the most important properties of a vibrating system. It is a system property and affects both the free and forced vibration response. As there exists one natural frequency for each degree of freedom, in real structures, where there are many degrees of freedom, there are many natural frequencies. As resonance is caused by excitation near a natural frequency, correct identification is an important feature in the analysis and evaluation of dynamic systems.

Free vibration 1 (Lecture 4) Real systems do not continue to vibrate for ever – energy is dissipated through damping. The viscous damper is the simplest model used for studying SDOF free vibration. A viscously damped SDOF system and its FBD are presented in Figure 4.1. The resulting equation of motion is, m x  c x  k x  0

x(t) m

Free body diagram

The solution can again be found by assuming x (t )  Ce where C and s are constants. Substituting the assumed solution into the equation of motion gives,



k

c

kx



2

x , x , x

m

st

c x

C m s  cs  k  0 Figure 4.1(a)

Figure 4.1(b)

The characteristic equation is of second order so has two roots. Thus, 2

s1,2 

2

k  c  c  4mk  c  c       2m 2m  m  2m 

The roots of the characteristic equation can be real and distinct, real and equal or complex. The condition for the roots to be real and equal (terms inside the square root go to zero) is known as critical damping. Critical damping cc can then be defined as,

cc  2 km  2mn Other values of damping can be defined as a fraction of critical damping, namely



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c . cc Spring Semester 2016

MEC201: Dynamics of Structures and Machines

The roots of the characteristic equation can be rewritten to give, 2 s1,2       1  n  

The resulting general solution to the equation of motion depends on the fraction of critical damping . For the critically damped case, as  = 1, roots are real and equal: i.e. s1  s2   n The general solution of a second order differential equation with equal roots gives,

x (t )  C1  C2t  e n

t

Inserting the initial conditions to find the constants C1 and C2 gives,

x (t )  x0  x0  n x0  t  e n t The free response for critically damped motion with different starting velocities is given in Figure 4.2. Figure 4.2 When  > 1 the system is said to be overdamped and the system has two distinct real roots.

s1       2  1 n ,  

s 2       2  1 n  

The solution to the equation of motion then becomes   

x (t )  C1e

2 1  n t 

     2 1   t  n  

 C2 e 

The response is sluggish – see Figure 4.3.The constants C1 and C2 can be obtained from th...