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

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

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 Prof.Dr.Hasan ÖZTÜRK

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

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

VERTICAL MODEL: the external forces zero Prof.Dr.Hasan ÖZTÜRK

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

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

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 Prof.Dr.Hasan ÖZTÜRK

displacement velocity acceleration There is a difference of 90 degrees between the equations displacement velocity acceleration Phase relationship of displacement, velocity, and acceleration Prof.Dr.Hasan ÖZTÜRK

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 Prof.Dr.Hasan ÖZTÜRK

0 0 Prof.Dr.Hasan ÖZTÜRK

Prof.Dr.Hasan ÖZTÜRK

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. Prof.Dr.Hasan ÖZTÜRK

STEP INPUT FORCING Prof.Dr.Hasan ÖZTÜRK a compacting machine

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

Prof.Dr.Hasan ÖZTÜRK

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 Prof.Dr.Hasan ÖZTÜRK

Prof.Dr.Hasan ÖZTÜRK

PHASE-PLANE ANALYSIS Prof.Dr.Hasan ÖZTÜRK

Prof.Dr.Hasan ÖZTÜRK

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

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

Phase-Plane Graphical Method. Prof.Dr.Hasan ÖZTÜRK

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

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

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

Prof.Dr.Hasan ÖZTÜRK

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

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

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