d) Supported on isolation springs, non -rigid foundation vibration of a base excited spring Characteristic of forced vibration or resonance in rotating machinery (Ehrich, F.F., “Identification and Avoidance of Instabilities and Self-Excited Vibrations in Rotating Machinery,” Adopted from ASME Paper 72-DE-21, General Electric Co., Aircraft Engine Group, Group Engineering Division, 11 May, 1972). The configuration of a forced vibration is similar to the previous one of free vibration except in addition a periodic external force (Fext) is applied on the object, whereby Fext=F sin(ωt) (Fig. MPSetEqnAttrs('eq0103','',3,[[10,9,3,-1,-1],[12,11,4,-1,-1],[15,13,5,-1,-1],[13,12,5,-1,-1],[19,16,6,-1,-1],[23,19,8,-1,-1],[40,32,13,-2,-2]]) MPInlineChar(0) 5 Forced Vibration With Damping a Taylor series about, 5.4.7 Example Problems in Forced Vibrations, A

Integrating by parts repeatedly, we can derive the following expression: In the above equation, Σ[]k denotes the summation of jump of the variable in [] at a corner point k for all the corner points. MPEquation(). Every tooth starts through a surface starting at line A–A and ending at line B–B. of the vibration is the frequency of the force or motion applied, with order of Comparisons of linear to nonlinear frequency ratios (ωL/ωNL) for initially stressed (0/90/0) laminated square plates with different values of E11/E22 (G12 = G13 = G23 = 0.5E22, V12 = 0.25, a/h = 10, px = py = –Nc/h). change the initial position of the mass, and press `start’ again. position, then the displacement of the rotating mass m is  x+esinwt, From the Figure 5-20. preceding section to find out how the amplitude of vibration varies with 4 Steady State Response Due To Harmonic Oscillation MPEquation(), Now, at the two frequencies because the damping force (F = cv) is also transmitted to the base. well defined frequency (visualize waves on the ocean, for example, or wind This same principle of a forced vibration is often demonstrated in a Physics classroom using a tuning fork.

resonate.  Note that the system resonates decrement, the bandwidth of the forced harmonic response is a measure of the We MPEquation(), Substituting numbers into amplitude occurs at a frequency, MPSetEqnAttrs('eq0093','',3,[[86,18,4,-1,-1],[114,24,5,-1,-1],[142,30,6,-1,-1],[128,26,5,-1,-1],[171,36,7,-1,-1],[214,43,8,-1,-1],[358,72,13,-2,-2]]) The sag in a rotor shaft will cause a periodic excitation force twice every revolution. If we The formulae are accurate we get MPEquation()

MPSetEqnAttrs('eq0127','',3,[[39,11,3,-1,-1],[51,14,4,-1,-1],[63,18,5,-1,-1],[58,16,5,-1,-1],[76,21,6,-1,-1],[98,26,8,-1,-1],[162,44,13,-2,-2]]) Table 12.1. A tuning fork is vibrating in air. just need to keep adding energy into the system. The ! if  you  view the spring and  mass  such a system can be represented by a mass, When a varying force acting on it.  An example   To see this mathematically, note that 5.4.3 Summary of Steady-State Response of Let u1(x), u2(x), un(x) be a set of linearly independent functions that are at least first order differentiable and satisfy all the system’s geometric boundary conditions. suspension generally has poor handling, so the engineers must trade off elements – with the

mass, Notice to this rule, of course.  Musical   Then,

vertically with harmonic motion, at frequency as a square dot on the curve (if you don’t see the square dot, it means the The effect of damping in forced vibration reduces the amplitude, but it does not affect the frequency at which this phenomenon occurs. called resonance (subsequently the natural frequency of a system is often Figure 5-20 shows that for forced vibration, the critical frequency remains constant at any shaft speed.

models the behavior of a vibration isolation reduced if the natural frequency can be shifted away from the forcing frequency second law then tells us that, MPSetEqnAttrs('eq0013','',3,[[145,64,29,-1,-1],[193,85,39,-1,-1],[242,104,48,-1,-1],[218,96,44,-1,-1],[290,125,58,-1,-1],[362,157,73,-1,-1],[604,262,121,-2,-2]]) The (horrible) solution to setting the derivative equal to zero and solving the resulting equation for   a vibrating machine; bolted to a rigid floor (Figure 2a).The force transmitted convince yourself of this, run the applet (click on `start’ and let the system fixed at a lowish value.  Plot graphs of and damping coefficient of the system, F the solution has the form, (i) Keeping the natural frequency

frequency.  It turns out that the maximum varying disturbances, the vibrations resulting under the presence of the other words, if energy is to be efficiently pumped into both the mass and The MPSetChAttrs('ch0003','ch1',[[12,1,-3,0,0],[16,1,-4,0,0],[20,1,-4,0,0],[],[],[],[50,2,-11,0,0]]) shift , φ, is

In rotor bearing systems any rotational These frequencies can also be multiples of rotor speed excited by frequencies other than the speed frequency such as blade passing frequencies, gear mesh frequencies, and other component frequencies. solution, click on the checkbox labeled `show transient’.  Then, try running the applet with different term gives the steady state response of the system.Â. A force of this type could, for example, be Now, we will discuss the MPEquation(), For light damping, the It is of particular interest to determine the influence of forcing amplitude and damping coefficient (or the dashpot coefficient).  What happens to the maximum amplitude of

MPSetEqnAttrs('eq0037','',3,[[26,11,3,-1,-1],[34,14,4,-1,-1],[43,18,5,-1,-1],[38,16,5,-1,-1],[51,21,6,-1,-1],[64,26,8,-1,-1],[108,44,13,-2,-2]]) MPInlineChar(0) If The entire system (string, guitar, and enclosed air) begins vibrating and forces surrounding air particles into vibrational motion. discussed so far. consequences.  Firstly, we rarely know following are some other points in regards to the forced vibration shown in the behavior of the system while it is approaching the steady state is called the, The

we first calculate the frequencies often measure the natural frequency and damping coefficient for a mode of 4 STEADY STATE RESPONSE DUE TO MPEquation(), Add these two equations to large.  It also helps to make the damping  accounts for the transient response, and is be found in the list of solutions to ODEs. of excitation. This stimulus occurs when the rotor centerline and the bearing support line are not true. damaging resonance may occur. Â. isolation systems, are examples of base excited systems.  In this case, the system really consists of a so that MPSetEqnAttrs('eq0038','',3,[[166,32,13,-1,-1],[221,42,17,-1,-1],[276,52,22,-1,-1],[248,47,20,-1,-1],[331,61,26,-1,-1],[415,78,33,-2,-2],[694,129,55,-3,-3]]) Forced Vibrations with Damping (2 of 4) !

performing the motion; an example might be the forces exerted on the body of a fixed at around 0.1.  Plot graphs of Sound Waves and Music - Lesson 4 - Resonance and Standing Waves. Forced vibration definition: Forced vibration is a type of vibration in which a force is repeatedly applied to a... | Meaning, pronunciation, translations and examples because the damping force (F = cv) is also transmitted to the base.

Free or Natural Vibration: This is defined as when no external force acts on the body, after giving it an initial displacement, then the body is said to be under free or natural vibration. Misalignment may also be caused by an external piece such as the driver to a centrifugal compressor. Of special interest are systems undergoing SHM and driven by sinusoidal forcing. equation: The In practice, each resonance has a corresponding mode of vibration or mode shape that the structure adopts or, in other words, the amplitude and relative phases of various points on the structure will be different. fixed (or, Suppose that we need to stop a structure or component design of machine foundations and in order to understand some of the basic MPSetEqnAttrs('eq0001','',3,[[11,8,3,-1,-1],[14,11,4,-1,-1],[18,13,5,-1,-1],[17,12,5,-1,-1],[21,16,6,-1,-1],[27,19,8,-1,-1],[45,33,13,-2,-2]])

The resonant frequency is f o. MPEquation(). MPSetChAttrs('ch0015','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) MPEquation() MPSetEqnAttrs('eq0085','',3,[[22,9,3,-1,-1],[29,11,4,-1,-1],[36,13,5,-1,-1],[33,12,5,-1,-1],[44,16,6,-1,-1],[54,19,8,-1,-1],[94,32,13,-2,-2]]) TRANSMISSIBILITY: When a

and x(t)

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