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
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 2
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 3
Condition monitoring Prof.Dr.Hasan ÖZTÜRK 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. 4
Free Vibration of Single Degree of Freedom Systems 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 A. The space needle (structure) Prof.Dr.Hasan ÖZTÜRK Figure B. Modeling of tall structure as springmass system 5
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 6
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. Prof.Dr.Hasan ÖZTÜRK 7
VERTICAL MODEL: the external forces zero Prof.Dr.Hasan ÖZTÜRK 8
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 system s natural frequency of vibration The constants A and B can be determined from the initial conditions of the system. Prof.Dr.Hasan ÖZTÜRK v 0 A =, B= x0 ωn 9
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 ω n t of the phasors for a given time after the motion has commenced. The phasors x 0 and ν 0 /ω 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 ν 0 /ω n starts from a maximum positive displacement and the phasor x 0 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 Prof.Dr.Hasan ÖZTÜRK 10
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 X 0 and φ are the constants of integration whose values depend upon the initial conditions. Equation be expressed as can also x ω 1 0 n ψ= tan v0 Prof.Dr.Hasan ÖZTÜRK 11
displacement There is a difference of 90 degrees between the equations velocity acceleration Phase relationship of displacement, velocity, and acceleration Prof.Dr.Hasan ÖZTÜRK 12
Example: (a) (b) System of the example. A mass is dropped onto a fixedfree 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 13
φ 0 φ 0 Prof.Dr.Hasan ÖZTÜRK 14
Example: Prof.Dr.Hasan ÖZTÜRK 15
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Example: Simple Pendulum (Approximate Solution) 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. Consider tangential components of acceleration and force for a simple pendulum, F t = ma t : W sinθ = ml θ g θ + sinθ = 0 l Prof.Dr.Hasan ÖZTÜRK for small angles, g + θ θ = 0 l θ = θ sin τ n m 2π = = ω n ( ω t + φ ) n 2π l g 18 19-18
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 19
STEP INPUT FORCING Prof.Dr.Hasan ÖZTÜRK a compacting machine 20
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 21
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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 Prof.Dr.Hasan ÖZTÜRK 23
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PHASE-PLANE ANALYSIS Prof.Dr.Hasan ÖZTÜRK 25
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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 27
Construction of the phase-plane and displacement diagrams for a four-step forcing function. Prof.Dr.Hasan ÖZTÜRK 28
EXAMPLE Prof.Dr.Hasan ÖZTÜRK (1) 29
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Phase-Plane Graphical Method. Prof.Dr.Hasan ÖZTÜRK 33
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 34
Thus the general solution C = A, C = B 1 2 Critical Damping Constant and the Damping Ratio. Thus the general solution Prof.Dr.Hasan ÖZTÜRK 35
and hence the solution becomes The constants of integration are determined by applying the initial conditions x = v 0 0 is called the frequency of damped vibration And the solution can be written Prof.Dr.Hasan ÖZTÜRK 36
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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. Prof.Dr.Hasan ÖZTÜRK 38
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 39
Logarithmic Decrement: Prof.Dr.Hasan ÖZTÜRK 0 0 0 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 t 1 and t 2 denote the times corresponding to two consecutive amplitudes (displacements), measured one cycle apart for an underdamped system we can form the ratio 0 0 40
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 1 x n 2πζ δ= ln = N x n + N 1 ζ 2 Example: N: is the number of cycles of motion between the amplitude measurements. N = 2 1 x 2πζ 1 δ= ln = 2 x 3 1 ζ 2 N = 2 Prof.Dr.Hasan ÖZTÜRK 1 x 2πζ δ= ln = x + Example: 3 2 ( 3 2) 2 1 ζ 41
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 42
EXAMPLE Prof.Dr.Hasan ÖZTÜRK 43
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EXAMPLE T 2π 2 c 2 = =, ω =ω 1 ξ, ξ=, ω =ω 1 2 ξ, ω = ω d n 2 km R n n k m 2 X 1 + (2 ξr) = Y (1 r ) + (2 ξr) 2 2 2 ω r =λ= ω n, 1 x n 2πξ δ = ln = ξω ntd = N x n + N 1 ξ 2 Prof.Dr.Hasan ÖZTÜRK 46
RESPONSE TO PERIODIC FORCING Prof.Dr.Hasan ÖZTÜRK 47
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 ω n, 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 48
Computer solution of the above equation for ω n = 3 ω; the amplitude scales for x and y are equal. Prof.Dr.Hasan ÖZTÜRK 49
Examination of the Equation indicates that, for ω/ω n =0, the solution becomes Prof.Dr.Hasan ÖZTÜRK 50
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Harmonically Excited Vibration Prof.Dr.Hasan ÖZTÜRK 52
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h p Prof.Dr.Hasan ÖZTÜRK 54
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 55
lagging the direction of the positive cosine by a phase angle of Prof.Dr.Hasan ÖZTÜRK 56
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only in the steady-state term and find the successive derivatives to be Prof.Dr.Hasan ÖZTÜRK 59
These equations can be simplified by introducing the expressions 60
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 61
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 Doç.Dr.Hasan ÖZTÜRK 62 62
A plot of magnification factor versus frequency ratio Prof.Dr.Hasan ÖZTÜRK 63
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 Prof.Dr.Hasan ÖZTÜRK 65
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Using trigonometric identities, the above equation can be rewritten in a more convenient form as Prof.Dr.Hasan ÖZTÜRK 67
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Z Prof.Dr.Hasan ÖZTÜRK 69
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. Prof.Dr.Hasan ÖZTÜRK 70
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 Prof.Dr.Hasan ÖZTÜRK 71
m u 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. m u Prof.Dr.Hasan ÖZTÜRK 72
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 73
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EXAMPLE Prof.Dr.Hasan ÖZTÜRK 75
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EXAMPLE Prof.Dr.Hasan ÖZTÜRK 78
EXAMPLE Prof.Dr.Hasan ÖZTÜRK 79
EXAMPLE Prof.Dr.Hasan ÖZTÜRK 80
EXAMPLE Prof.Dr.Hasan ÖZTÜRK 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 θ 1 and θ 2 as the angular displacements of mass 1 and mass 2, respectively, with respect to the rotating axes. k t I 1 I 2 I 1 I 2 Prof.Dr.Hasan ÖZTÜRK 82
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Therefore, the masses rotate together without any relative displacement and there is no vibration. Prof.Dr.Hasan ÖZTÜRK 84