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. 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
Prof.Dr.Hasan ÖZTÜRK

3. Thus the general solution
Critical Damping Constant and the Damping Ratio.
Thus the general solution
Prof.Dr.Hasan ÖZTÜRK

4. 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
Prof.Dr.Hasan ÖZTÜRK

5. Prof.Dr.Hasan ÖZTÜRK

6. Thus, the general solution is
Free vibrations with = 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.
Prof.Dr.Hasan ÖZTÜRK

7. 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.
Prof.Dr.Hasan ÖZTÜRK

8. Logarithmic Decrement:
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

9. 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:
Prof.Dr.Hasan ÖZTÜRK

10. 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
Prof.Dr.Hasan ÖZTÜRK

11. EXAMPLE
Prof.Dr.Hasan ÖZTÜRK

12. Prof.Dr.Hasan ÖZTÜRK

13. Prof.Dr.Hasan ÖZTÜRK

14. RESPONSE TO PERIODIC FORCING
Prof.Dr.Hasan ÖZTÜRK

15. 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.
Prof.Dr.Hasan ÖZTÜRK

16. h p
Prof.Dr.Hasan ÖZTÜRK

17. Steady-State Solution
Because the forcing is harmonic, the particular part is obtained by assuming a solution in the form
Steady-State Solution
Prof.Dr.Hasan ÖZTÜRK

18. lagging the direction of the positive cosine by a phase angle of
Prof.Dr.Hasan ÖZTÜRK

19. Prof.Dr.Hasan ÖZTÜRK

20. Prof.Dr.Hasan ÖZTÜRK

21. only in the steady-state term
and find the successive derivatives to be
Prof.Dr.Hasan ÖZTÜRK

22. These equations can be simplified by introducing the expressions

23. Relationship of the phase angle to the damping and ffequency ratios
Relative displacement of a damped forced system as a function of the damping and frequency ratios.
Prof.Dr.Hasan ÖZTÜRK

24. FORCING CAUSED BY UNBALANCE
If the angular position of the masses is measured from a horizontal position, the total vertical component of the excitation is always given by
Prof.Dr.Hasan ÖZTÜRK 24
Doç.Dr.Hasan ÖZTÜRK

25. A plot of magnification factor versus frequency ratio
Prof.Dr.Hasan ÖZTÜRK

26. Z
Prof.Dr.Hasan ÖZTÜRK

27. Force Transmission 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.
Prof.Dr.Hasan ÖZTÜRK

28. 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.
Prof.Dr.Hasan ÖZTÜRK

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

30. Prof.Dr.Hasan ÖZTÜRK

31. EXAMPLE
Prof.Dr.Hasan ÖZTÜRK

32. Prof.Dr.Hasan ÖZTÜRK

33. Prof.Dr.Hasan ÖZTÜRK

34. TORSIONAL SYSTEMS kt I1 I2 I1 I2
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 1 and 2 as the angular displacements of mass 1 and mass 2, respectively, with respect to the rotating axes. kt I1 I2 I1 I2
Prof.Dr.Hasan ÖZTÜRK

35. Prof.Dr.Hasan ÖZTÜRK

36. Therefore, the masses rotate together without any relative displacement and there is no vibration.
Prof.Dr.Hasan ÖZTÜRK