weeks 8-9 dynamics of machinerykisi.deu.edu.tr/hasan.ozturk/makina dinamigi/makina...weeks 8-9...

WEEKS 8-9 Dynamics of Machinery

• References Theory of Machines and Mechanisms, J.J.Uicker,

G.R.Pennock ve J.E. Shigley, 2011 Mechanical Vibrations, Singiresu S. Rao, 2010 Mechanical Vibrations: Theory and Applications, S. Graham Kelly, 2012

Prof.Dr.Hasan ÖZTÜRK 1

Prof.Dr.Hasan ÖZTÜRK

-Vibrations are oscillations of a mechanical or structural system about an equilibrium position.

Vibration Analysis

-Any motion that exactly repeats itself after a certain interval of time is a periodic motion and is called a vibration.

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we encounter a variety of vibrations in our daily life. For the most part, vibrations have been considered unnecessary.

washing machine 3

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Condition monitoring

Linear vibration sieve is also called linear sieve, which is one of the most widely used vibrating creening equipment. Linear vibration sieve can easily finish all kinds of material removing impurity, grading, screening.

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Free Vibration of Single Degree of Freedom Systems

Figure A. The space needle (structure)

The structure shown in Figure A can be considered a cantilever beam that is fixed at the ground. For the study of transverse vibration, the top mass can be considered a point mass and the supporting structure (beam) can be approximated as a spring to obtain the single-degree-of-freedom model shown in Figure B.

Figure B. Modeling of tall structure as spring-mass system

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The figure shows an idealized vibrating system having a mass m guided to move only in the x direction. The mass is connected to a fixed frame through the spring k and the dashpot c. The assumptions used are as follows:

1. The spring and the dashpot are massless. 2. The mass is absolutely rigid. 3. All damping is concentrated in the dashpot.

Consider next the idealized torsional vibrating system of the below figure. Here a disk having a mass moment of inertia I is mounted upon the end of a weightless shaft having a torsional spring constant k, defined by


where T is the torque necessary to produce an angular deflection θ of the shaft. In a similar manner, the torsional viscous damping coefficient is defined by

Next, designating an external torque forcing function by T = f (t), we find that the differential equation for the torsional system is

EXAMPLE:

The below figure illustrates a vibrating system in which a time-dependent displacement y= y(t) excites a spring-mass system through a viscous dashpot. Write the differential equation of this system.


VERTICAL MODEL: the external forces zero


FREE VIBRATION WITHOUT VISCOUS DAMPING Free vibrations are oscillations about a system’s equilibrium position that occur in the absence of an external excitation.

Consider the configuration of the spring-mass system shown in the Figure

D Alembert s Principle.

The constants A and B can be determined from the initial conditions of the system.

the system s natural frequency of vibration

9

00

n

vA , B x= =ω


The ordinate of the graph of the above Figure is the displacement x, and the abscissa can be considered as the time axis or as the angular displacement ωnt of the phasors for a given time after the motion has commenced. The phasors x0 and ν0 /ωn are shown in their initial positions, and as time passes, these rotate counterclockwise with an angular velocity of on and generate the displacement curves shown. The figure illustrates that the phasor ν0 /ωn starts from a maximum positive displacement and the phasor x0 starts from a zero displacement. These, therefore, are very special, and the most general form is that given by , in which motion begins at some intermediate point.

the system s natural frequency of vibration

the period of a free vibration is


The above equation is harmonic function of time. The motion is symmetric about the equilibrium position of the mass m.

the solution can be written as

Equation can also be expressed as

Harmonic Motion

11

where X0 and φ are the constants of integration whose values depend upon the initial conditions.

1 0

0

nxtanv

− ωψ =


velocity

acceleration

displacement

12

There is a difference of 90 degrees between the equations

Phase relationship of displacement, velocity, and acceleration


Example:

(a) System of the example . A mass is dropped onto a fixed-free beam.

(b) The system is modeled as a mass hanging from a spring of equivalent stiffness. Since x is measured from the equilibrium position of the system, the initial displacement is the negative of the static deflection of the beam.

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φ0

φ0

14


Example:

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16 Prof.Dr.Hasan ÖZTÜRK


Simple Pendulum (Approximate Solution)

19 - 18

• Results obtained for the spring-mass system can be applied whenever the resultant force on a particle is proportional to the displacement and directed towards the equilibrium position.

for small angles,

( )

gl

tlg

nn

nm

πωπτ

φωθθ

θθ

22

sin

0

==

+=

=+

:tt maF =∑

• Consider tangential components of acceleration and force for a simple pendulum,

0sin

sin

=+

=−

θθ

θθ

lg

mlW

Example:

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The equivalent spring constant of a parallel spring arrangement (common displacement) is the sum of the individual constants.

The equivalent spring constant of a series spring arrangement (common force) is the inverse of the sum of the reciprocals of the individual constants.

Combination of springs

19


a compacting machine 20

STEP INPUT FORCING


21

Let us assume that this force is constant and acting in the positive x direction. we consider the damping to be zero.

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PHASE-PLANE REPRESENTATION

We have already observed that a free undamped vibrating system has an equation of motion, which can be expressed in the form


PHASE-PLANE ANALYSIS


TRANSlENT DISTURBANCES

Any action that destroys the static equilibrium of a vibrating system may be called a disturbance to that system. A transient disturbance is any action that endures for only a relatively short period of time.


Construction of the phase-plane and displacement diagrams for a four-step forcing function.

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EXAMPLE

(1) Prof.Dr.Hasan ÖZTÜRK

Phase-Plane Graphical Method.



Free Vibration with Viscous Damping

we assume a solution in the form

where A and s are undetermined constants. The first and second time derivatives of are

34

Inserting this function into Equation leads to the characteristic equation


Thus the general solution

Critical Damping Constant and the Damping Ratio.

Thus the general solution

35

1 2C A, C B= =


and hence the solution becomes

The constants of integration are determined by applying the initial conditions

is called the frequency of damped vibration

36

0 0x v=

And the solution can be written


Thus, the general solution is

Free vibrations with ζ= 1 are called critically damped because the damping force is just sufficient to dissipate the energy within one cycle of motion. The system never executes a full cycle; it approaches equilibrium with exponentially decaying displacement. A system with critical damping returns to equilibrium the fastest without oscillation. A system that is overdamped has a larger damping coefficient and offers more resistance to the motion.

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Thus, the general solution is

The response of a system that is overdamped is similar to a critically damped system. An overdamped system has more resistance to the motion than critically damped systems. Therefore, it takes longer to reach a maximum than a critically damped system, but the maximum is smaller. An overdamped system also takes longer than a critically damped system to return to equilibrium.

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Prof.Dr.Hasan ÖZTÜRK Logarithmic Decrement:

The logarithmic decrement represents the rate at which the amplitude of a free-damped vibration decreases. It is defined as the natural logarithm of the ratio of any two successive amplitudes. Let t1 and t2 denote the times corresponding to two consecutive amplitudes (displacements), measured one cycle apart for an underdamped system

we can form the ratio

40

0

0

0

0

0


The logarithmic decrement δ can be obtained as:

If we take any response curve, such as that of the below figure, and measure the amplitude of the nth and also of the (n+N)th cycle, the logarithmic decrement δ is defined as the natural logarithm of the ratio of these two amplitudes and is

Example:

12

3

2

1 22 1

N

xlnx

=

πζδ = =

− ζ

2

1 21

n

n N

xlnN x +

πζδ = =

− ζ

N: is the number of cycles of motion between the amplitude measurements.

Example: 32

3 2

2

1 22 1( )

N

xlnx +

=

πζδ = = − ζ

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Measurements of many damping ratios indicate that a value of under 20% can be expected for most machine systems, with a value of 10% or less being the most probable. For this range of values the radical in the below equation can be taken as approximately unity, giving

EXAMPLE



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EXAMPLE

2

1 2ln1

nn d

n N

x TN x +

πξδ = = ξω =

− ξ

2 22 , 1 , , 1 2 ,2d n R n n

c kTmkm

π= = ω = ω −ξ ξ = ω = ω − ξ ω =ω

,n

r ω=λ =

ω2

2 2 2

1 (2 )

(1 ) (2 )

rXY r r

+ ξ=

− + ξ

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RESPONSE TO PERIODIC FORCING


The first term on the right-hand side of the above equation is called the starting transient. Note that this is a vibration at the natural frequency ωn, not at the forcing frequency ω. The usual physical system will contain a certain amount of friction, which, as we shall see in the sections to follow, will cause this term to die out after a certain period of time. The second and third terms on the right represent the steady-stale solution and these contain another component of the vibration at the forcing frequency ω.


Computer solution of the above equation for ωn = 3 ω; the amplitude scales for x and y are equal.


Examination of the Equation indicates that, for ω/ωn =0, the solution becomes

Harmonically Excited Vibration


h p



Because the forcing is harmonic, the particular part is obtained by assuming a solution in the form

Steady-State Solution

Steady-State Solution


lagging the direction of the positive cosine by a phase angle of


only in the steady-state term

and find the successive derivatives to be

60

These equations can be simplified by introducing the expressions


Relative displacement of a damped forced system as a function of the damping and frequency ratios.

Relationship of the phase angle to the damping and ffequency ratios

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FORCING CAUSED BY UNBALANCE

Doç.Dr.Hasan ÖZTÜRK

If the angular position of the masses is measured from a horizontal position, the total vertical component of the excitation is always given by

62 Prof.Dr.Hasan ÖZTÜRK


A plot of magnification factor versus frequency ratio

Relative Motion

Automobiles have the input vibratory motion from the ground and hence it comes under Support motion

Response of a Damped System Under the Harmonic Motion of the Base

Sometimes the base or support of a spring-mass-damper system undergoes harmonic motion, as shown in Fig. 1(a). Let y(t) denote the displacement of the base and x(t) the displacement of the mass from its static equilibrium position at time t. Then the net elongation of the spring is x-y and the relative velocity between the two ends of the damper is From the free-body diagram shown in Fig. 1(b), we obtain the equation of motion:

Fig.1


Using trigonometric identities, the above equation can be rewritten in a more convenient form as



Z

Force Transmitted and ISOLATION


The steady-state solution

The transmissibility T is a nondimensional ratio that defines the percentage of the exciting force transmitted to the frame.

70

71

This is a plot of force transmissibility versus frequency ratio for a system in which a steady-state periodic forcing function is applied directly to the mass. The transmissibility is the percentage of the exciting force that is transmitted to the frame.

rotating unbalance



A plot of acceleration transmissibility versus frequency ratio. In a system in which the exciting force is produced by a rotating unbalanced mass, this plot gives the percentage of this force transmitted to the frame of the machine. mu

mu


We shall choose the complex-operator method for the solution of the system of the figure. We begin by defining the forcing function as

EXAMPLE



EXAMPLE

78


EXAMPLE

79


EXAMPLE

80


EXAMPLE

81


TORSIONAL SYSTEMS We wish to study the possibility of free vibration of the system when it rotates at constant angular velocity. To investigate the motion of each mass, it is necessary to picture a reference system fixed to the shaft and rotating with the shaft at the same angular velocity. Then we can measure the angular displacement of either mass by finding the instantaneous angular location of a mark on the mass relative to one of the rotating axes. Thus, we define θ1 and θ2 as the angular displacements of mass 1 and mass 2, respectively, with respect to the rotating axes.

I1 I2

kt

I1 I2


Therefore, the masses rotate together without any relative displacement and there is no vibration.

weeks 8-9 dynamics of machinerykisi.deu.edu.tr/hasan.ozturk/makina dinamigi/makina...weeks 8-9...

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