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2 Coordinates, displacement, velocity and acceleration

2

Structural dynamics and vibration in practice

θ1

y1

r1

m1

x1

θ2

y2

r2

m2

x2

Fig. 1.1 Double pendulum illustrating generalized coordinates.

Cartesian coordinates of this kind are not always suitable for defining the vibration

behavior of a system. The powerful Lagrange method requires coordinates known as

generalized coordinates that not only fully describe the possible motion of the system,

but are also independent of each other. An often-used example illustrating the

difference between Cartesian and generalized coordinates is the double pendulum

shown in Fig. 1.1. The angles 1 and 2 are sufficient to define the positions of

m1 and m2 completely, and are therefore suitable as generalized coordinates. All

four Cartesian coordinates x1 , y1 , x2 and y2 , taken together, are not suitable for use

as generalized coordinates, since they are not independent of each other, but are

related by the two constraint equations:

x21 ỵ y21 ẳ r21

and

x2 x1 ị2 ỵ y2 y1 ị2 ẳ r22

This illustrates the general rule that the number of degrees of freedom, and the

number of generalized coordinates required, is the total number of coordinates minus

the number of constraint equations. In this case there can only be two generalized

coordinates, but they do not necessarily have to be 1 and 2 ; for example, x1 and x2

also define the positions of the masses completely, and could be used instead.

Generalized coordinates are fundamentally displacements, but can also be differentiated, i.e. expressed in terms of velocity and acceleration. This means that if a

_

certain displacement coordinate, z, is defined as positive upwards, then its velocity, z,

and its acceleration, z€, are also positive in that direction. The use of dots above

symbols, as here, to indicate differentiation with respect to time is a common convention in structural dynamics.

1.3 Simple harmonic motion

Simple harmonic motion, more usually called ‘sinusoidal vibration’, is often

encountered in structural dynamics work.

Chapter 1. Basic concepts

3

1.3.1 Time History Representation

Let the motion of a given point be described by the equation:

x ẳ X sin !t

1:1ị

where x is the displacement from the equilibrium position, X the displacement

magnitude of the oscillation, ! the frequency in rad/s and t the time. The quantity

X is the single-peak amplitude, and x travels between the limits ỈX, so the peak-topeak amplitude (also known as double amplitude) is 2X.

It appears to be an accepted convention to express displacements as double amplitudes, but velocities and accelerations as single-peak amplitudes, so some care is

needed, especially when interpreting vibration test specifications.

Since sin !t repeats every 2 radians, the period of the oscillation, T, say, is 2=!

_ of

seconds, and the frequency in hertz (Hz) is 1=T ¼ !=2. The velocity, dx=dt, or x,

the point concerned, is obtained by differentiating Eq. (1.1):

x_ ẳ !X cos !t

1:2ị

The corresponding acceleration, d x=dt , or x€, is obtained by differentiating

Eq. (1.2):

2

2

x€ ¼ À!2 X sin !t

ð1:3Þ

_ and the acceleration x€, plotted

Figure 1.2 shows the displacement, x, the velocity, x,

against time, t.

Since Eq. (1.2),

x_ ¼ !X cos !t

can be written as

x_ ¼ !X sin !t ỵ

2

X

Displacement

t

x

/2

X

Velocity

t

x

2X

Acceleration

/2

t

x

Fig. 1.2 Displacement, velocity and acceleration time histories for simple harmonic motion.

4

Structural dynamics and vibration in practice

or

h

i

x_ ẳ !X sin ! t ỵ

2!

1:4ị

_ for example the maximum value, occurs at a

any given feature of the time history of x,

value of t which is =2! less (i.e. earlier) than the same feature in the wave representing x. The velocity is therefore said to ‘lead’ the displacement by this amount of time.

This lead can also be expressed as a quarter-period, T/4, a phase angle of =2 radians,

or 90°.

Similarly, the acceleration time history, Eq. (1.3),

x€ ¼ À!2 X sin !t

can be written as

x ẳ !2 X sin!t ỵ Þ

ð1:5Þ

so the acceleration ‘leads’ the displacement by a time =!, a half-period, T/2, or a

phase angle of radians or 180°. In Fig. 1.2 this shifts the velocity and acceleration

plots to the left by these amounts relative to the displacement: the lead being in time,

not distance along the time axis.

The ‘single-peak’ and ‘peak-to-peak’ values of a sinusoidal vibration were introduced above. Another common way of expressing the amplitude of a vibration level is

the root mean square, or RMS value. This is derived, in the case of the displacement,

x, as follows:

Squaring both sides of Eq. (1.1):

x2 ẳ X2 sin2 !t

1:6ị

The mean square value of the whole waveform is the same as that of the first halfcycle of X sin !t, so the mean value of x2 , written hx2 i, is

hx2 i ẳ X2

2

T

Z

T=2

sin2 !t dt

1:7ị

0

Substituting

tẳ

1

!tị;

!

!

hx i ẳ X

dt ẳ

Z

=!

1

d!tị;

!

Tẳ

2

;

!

X2

1:8ị

2

0

p

Therefore the RMS value of x is X 2, or about 0.707X. It can be seen that this

pﬃﬃﬃ

ratio holds for any sinusoidal waveform: the RMS value is always 1 2 times the

single-peak value.

The waveforms considered here are assumed to have zero mean value, and it should

be remembered that a steady component, if present, contributes to the RMS value.

2

2

sin2 !t Á dð!tÞ ¼

Chapter 1. Basic concepts

5

Example 1.1

The sinusoidal vibration displacement amplitude at a particular point on an engine

has a single-peak value of 1.00 mm at a frequency of 20 Hz. Express this in terms of

single-peak velocity in m/s, and single-peak acceleration in both m/s2 and g units. Also

quote RMS values for displacement, velocity and acceleration.

Solution

Remembering Eq. (1.1),

x ẳ X sin !t

Aị

we simply differentiate twice, so,

x_ ẳ !X cos !t

Bị

x ẳ !2 X sin !t

ðCÞ

and

The single-peak displacement, X, is, in this case, 1.00 mm or 0.001 m. The value of

! ¼ 2f, where f is the frequency in Hz. Thus, ! ẳ 220ị ¼ 40 rad/s.

From Eq. (B), the single-peak value of x_ is !X, or 40 0:001ị ẳ 0:126 m/s or

126 mm/s.

From Eq. (C), the single-peak value of x€ is !2 X or ẵ40ị2 0:001 =15.8 m/s2 or

(15.8/9.81) = 1.61 g.

p

Root mean square values are 1 2 or 0.707 times single-peak values in all cases, as

shown in the Table 1.1.

Table 1.1

Peak and RMS Values, Example 1.1

Displacement

Velocity

Acceleration

Single peak value

RMS Value

1.00 mm

126 m/s

15.8 m/s2

1.61g

0.707 mm

89.1 mm/s

11.2 m/s2

1.14g rms

1.3.2 Complex Exponential Representation

Expressing simple harmonic motion in complex exponential form considerably

simplifies many operations, particularly the solution of differential equations. It is

based on Euler’s equation, which is usually written as:

ei ẳ cos ỵ i sin

pﬃﬃﬃﬃﬃﬃﬃ

where e is the well-known constant, an angle in radians and i is À1.

ð1:9Þ

6

Structural dynamics and vibration in practice

ω

Im

AXIS

ωX

X

π/2

π/2

ωt

Re AXIS

t

ω 2X

x = Im(Xeiωt ) = X sin ωt

x˙ = Im(ωXeiωt ) = ωX cos ωt

x˙˙= Im(ω 2Xeiωt ) = – ω 2X sin ωt

˙˙

x = Re(ω 2Xeiωt ) = – ω 2X cos ωt

x˙ = Re(ωXeiωt ) = – ωX sin ωt

x = Re(ωXeiω t ) = X cos ωt

t

Fig. 1.3 Rotating vectors on an Argand diagram.

Multiplying Eq. (1.9) through by X and substituting !t for :

Xei!t ¼ X cos !t ỵ iX sin !t

1:10ị

When plotted on an Argand diagram (where real values are plotted horizontally,

and imaginary values vertically) as shown in Fig. 1.3, this can be regarded as a vector,

of length X, rotating counter-clockwise at a rate of ! rad/s. The projection on the

real, or x axis, is X cos !t and the projection on the imaginary axis, iy, is iX sin !t. This

gives an alternate way of writing X cos !t and X sin !t, since

À

Á

ð1:11Þ

X sin !t ¼ Im Xei!t

where Im ( ) is understood to mean ‘the imaginary part of ( )’, and

À

Á

X cos !t ¼ Re Xei!t

ð1:12Þ

where Re ( ) is understood to mean ‘the real part of ( )’.

Figure 1.3 also shows the velocity vector, of magnitude !X, and the acceleration

vector, of magnitude !2 X, and their horizontal and vertical projections

Equations (1.11) and (1.12) can be used to produce the same results as Eqs (1.1)

through (1.3), as follows:

If

1:13ị

x ẳ Im Xei!t ẳ ImX cos !t ỵ iX sin !tị ẳ X sin !t

then

x_ ẳ Im i!Xei!t ẳ Imẵi !X cos !t ỵ iX sin !tị ẳ !X cos !t

1:14ị

Chapter 1. Basic concepts

7

(since i2 ¼ À1) and

Â

Ã

À

Á

2

x€ ¼ Im À! 2 Xei!t ¼ Im !2 X cos !t ỵ iX sin !tị ẳ ! X sin !t

ð1:15Þ

If the displacement x had instead been defined as x ¼ X cos !t, then Eq. (1.12), i.e.

À

Á

X cos !t ¼ Re Xei!t , could have been used equally well.

The interpretation of Eq. (1.10) as a rotating complex vector is simply a mathematical device, and does not necessarily have physical significance. In reality, nothing is

rotating, and the functions of time used in dynamics work are real, not complex.

1.4 Mass, stiffness and damping

The accelerations, velocities and displacements in a system produce forces when

multiplied, respectively, by mass, damping and stiffness. These can be considered to be

the building blocks of mechanical systems, in much the same way that inductance,

capacitance and resistance (L, C and R) are the building blocks of electronic circuits.

1.4.1 Mass and Inertia

The relationship between mass, m, and acceleration, x€, is given by Newton’s second

law. This states that when a force acts on a mass, the rate of change of momentum (the

product of mass and velocity) is equal to the applied force:

d

dx

m

ẳF

1:16ị

dt

dt

where m is the mass, not necessarily constant, dx/dt the velocity and F the force. For

constant mass, this is usually expressed in the more familiar form:

F ẳ m

x

1:17ị

If we draw a free body diagram, such as Fig. 1.4, to represent Eq. (1.17), where F and

x (and therefore x_ and x€) are defined as positive to the right, the resulting inertia force,

m€

x, acts to the left. Therefore, if we decided to define all quantities as positive to the

right, it would appear as –m€

x.

Mass

m

Acceleration

˙x˙

Force

F

Inertia force

˙˙

mx

Fig. 1.4 D’Alembert’s principle.

8

Structural dynamics and vibration in practice

Fy

y

yG

θG

xi

mi

ri

yi

Mθ

G

xG

Fx

x

y

x

Fig. 1.5 Plane motion of a rigid body.

This is known as D’Alembert’s principle, much used in setting up equations of

motion. It is, of course, only a statement of the fact that the two forces, F and m€

x,

being in equilibrium, must act in opposite directions.

Newton’s second law deals, strictly, only with particles of mass. These can be

‘lumped’ into rigid bodies. Figure 1.5 shows such a rigid body, made up of a large

number, n, of mass particles, mi , of which only one is shown. For simplicity, the body

is considered free to move only in the plane of the paper. Two sets of coordinates are

used: the position in space of the mass center or ‘center of gravity’ of the body, G, is

determined by the three coordinates xG , yG and G . The other coordinate system, x, y,

is fixed in the body, moves with it and has its origin at G. This is used to specify the

locations of the n particles of mass that together make up the body. Incidentally, if

these axes did not move with the body, the moments of inertia would not be constant,

a considerable complication.

The mass center, G, is, of course, the point where the algebraic sum of the first

moments of inertia of all the n mass particles is zero, about both the x and the y axes, i.e.,

n

X

iẳ1

mi x i ẳ

n

X

mi yi ẳ 0,

1:18ị

iẳ1

where the n individual mass particles, mi are located at xi , yi (i = 1 to n) in the x, y

coordinate system.

External forces and moments are considered to be applied, and their resultants

through and about G are Fx , Fy and M . These must be balanced by the internal

inertia forces of the mass particles.

## Structural dynamics and vibration in practice an engineering handbook douglas thorby

## 4 Mass, stiffness and damping

## 1 Setting up the equations of motion for simple, undamped, multi-DOF systems

## 5 Complex (damped) eigenvalues and eigenvectors

## 1 Stationarity, ergodicity, expected and average values

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2 Coordinates, displacement, velocity and acceleration