1. 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
Dr.H.ÖZTÜRK-2010

2. Vibration Analysis
-Vibrations are oscillations of a mechanical or structural system about an equilibrium position.
-Any motion that exactly repeats itself after a certain interval of time is a periodic motion and is called a vibration.
Prof.Dr.Hasan ÖZTÜRK

3. we encounter a variety of vibrations in our daily life
we encounter a variety of vibrations in our daily life. For the most part, vibrations have been considered unnecessary.
washing machine
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4. 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
Prof.Dr.Hasan ÖZTÜRK

5. where T is the torque necessary to produce an angular deflection  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
Prof.Dr.Hasan ÖZTÜRK

6. VERTICAL MODEL:
the external forces zero
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7. FREE VIBRATION WITHOUT VISCOUS DAMPING
Free vibrations are oscillations about a system’s equilibrium position that occur in the absence of an external excitation.

8. 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 nt of the phasors for a given time after the motion has commenced. The phasors x0 and 0 /n 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 0 /n 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 period of a free vibration is
the system s natural
frequency of vibration
Prof.Dr.Hasan ÖZTÜRK

9. about the equilibrium position of the mass m.
Harmonic Motion
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
where X0 and  are the constants of integration whose values depend upon the initial conditions.
Equation can also
be expressed as
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10. displacement velocity acceleration
There is a difference of 90 degrees between the equations
displacement
velocity
acceleration
Phase relationship of displacement, velocity, and acceleration
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11. Example:
System of the example . A mass is dropped onto a fixed-free beam.
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.
Prof.Dr.Hasan ÖZTÜRK
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12. 0 0
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13. Prof.Dr.Hasan ÖZTÜRK

14. Combination of springs
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.
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15. STEP INPUT FORCING
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a compacting machine

16. 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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17. Prof.Dr.Hasan ÖZTÜRK

18. 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
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19. Prof.Dr.Hasan ÖZTÜRK

20. PHASE-PLANE ANALYSIS
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21. Prof.Dr.Hasan ÖZTÜRK

22. 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.
Prof.Dr.Hasan ÖZTÜRK

23. Construction of the phase-plane and displacement diagrams for a four-step forcing function.
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24. Phase-Plane Graphical Method.
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25. 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
Inserting this function into Equation leads to the characteristic equation
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26. Thus the general solution
Critical Damping Constant and the Damping Ratio.
Thus the general solution
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27. and hence the solution becomes
The constants of integration are determined by applying the initial conditions
is called the frequency of damped vibration
And the solution can be written
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28. Prof.Dr.Hasan ÖZTÜRK

29. Logarithmic Decrement:
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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

30. Example: Example: The logarithmic decrement  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  is defined as the natural logarithm of the ratio of these two amplitudes and is
N: is the number of cycles of motion between the amplitude measurements.
Example:
Example:
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31. can be taken as approximately unity, giving
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
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