Chemical Engineering - Apllied Mechanics

Transmission Factor

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5. Force transmitted to the system constraint Let us consider the SDOF system illustrated in Figure 5.1. The mass m is subjected to a harmonic force of frequency 2 f π = Ω . The maximum amplitude of the force is F o=while β is the phase with respect to initial time. Therefore, the force can be expressed as: () o F( ) F it te β Ω+ = =(5.1)= = = = Figure 5.1 Forced SDOF system = Figure 5.2 Force transmitted to the system constraint = = The steady state response is given by:= ( ) ( ) ()() () o o 22 1F ( ) =XUX 12 i ititit nn xt e eee k ih ϕβββ ωω Ω+Ω+Ω+ == −Ω + Ω  = (5.2)= That is:= () () () 2 2 22 o X1 X12 nn h ωω = −Ω + Ω =(5.3)= Now we are interested in evaluating the dynamic force,= θF() t , transmitted by the system to the constraint.= This force is obviously the sum of the elastic and damping forces.=That is:= θF()=F()+F() ed t t t k x cx= + = (5.4)= In the steady state condition this force becomesΩ= ( ) () () TTF()=F()+F()X F i itit ed t t t k ic e e e ϕβ β Ω+Ω+ = +Ω =  = (5.5)= Then, the maximum magnitude of the force= θF()t =is given by:= () () () () 22 o 22 θ2 2 22 F () F =X ( ) 12 nn kk c kc h ωω +Ω + Ω= −Ω + Ω  = (5.6)= Eq.(5.6) can be rewritten as: () () () () () 2 θ2 2 22 o 12 F Tr= F12 n nnn h h ω ωωω +Ω  Ω =    −Ω + Ω  = (5.7)= The dimensionless function Tr, that is called Transmission Factor, gives the magnitude of the force transmitted to the constraint, as a function of the dimensionless frequency.= Eq.(5.5) can=also be written asΩ= ( ) ( ) ( ) ( ) ( ) o ()() To 22 F1 2 F ( ) =Tr F 12 n itit n nn ih t ee ih ββ ω ω ωω Ω+Ω+ +Ω = Ω −Ω + Ω =(5.8)= Figure=5.3 shows the magnitude and phase of the Transmission Factor as a function of the dimensionless frequency nω Ω .= = 0 0 .5 1 1 .5 2 -18 0 -13 5 -90 -45 0 D im e ns ionle ss fre q ue ncy P ha s e [ de g r ee ] T ransm iss io n F ac tor ( F T /F 0 ) h: 0 .04 h: 0 .05 h: 0 .07 5 h: 0 .1 h: 0 .2 h: 0 .4 0 0 .5 1 1 .5 2 0 2 4 6 8 10 12 14 D im e ns ionle ss fre q ue ncy Amplitude F T /F 0 X: 1.41 Y: 1.005 h: 0 .04 h: 0 .05 h: 0 .07 5 h: 0 .1 h: 0 .2 h: 0 .4 Figure 5.3 Transmission Factor vs. dimensionless frequency Note that when the dimensionless frequency nω Ω =is= 2=numerator and denominator of eq.(5.7) are equal, independently of the damping ratio,=h. Therefore, the Transmission Factor is unitary. This means that the magnitude of the force transmitted to the constraint is lower than that of the exciting force only for dimensionless frequency values higher than 2.= Conversely, for significantly low values of the dimensionless frequency ( 1 nω Ω  ) the magnitude of the= force transmitted to the constraint is equal or higher than that of the exciting force. In the end, when the= dimensionless frequency is close to unity (resonance region) the magnitude of the force transmitted to the constraint is considerably amplified by the system vibration. In general, high levels of the force transmitted to the constraint of a vibrating system are harmful, as they may cause high mechanical stresses and they may generate considerable pressure waves in the supporting structure which may be transmitted to other adjacent mechanical systems (e.g. machine tools) or parts of the building where the vibrating system is installed (e.g. pillars, beams and walls). 6. Vibrating constraint Often, the frame which the vibrating system are connected to, are affected by time-varying displacements. Consider the SDOF system illustrated in Figure 6.1. The absolute displacement of the mass, evaluated around the static equilibrium position, is denoted as ()xt , while ()yt =is the constraint displacement. Consider a harmonic displacement= ()yt =expressed asΩ= o () Y it yt e Ω = = (6.1)= = = Figure 6.1 SDOF system supported on a vibrating= constraint = = The equation of motion of the system is:= ( ) ( ) 0 mx cxy kxy+ −+ − =   = (6.2)= Note that the elastic force is proportional to the relative displacement between the spring ends, while the damping force is proportional to the relative velocity=between the ends of the viscous damper. Eq.(6.2) can= be rewritten as:= () oY it mxcxkxcyky ick e Ω + + = + = Ω+    = (6.3)= The steady=solution of eq.(6.3)=is given by:= () () X X i tit s xt e e ϕ Ω+Ω = = = (6.4)= Substituting this solution into eq.(6.3) we obtainΩ= () () 2() o XY i tit k m ic e i c k e ϕ Ω+Ω  −Ω + Ω= Ω +  = ( ) ( ) () o 2 Y X i tit ic k ee k m ic ϕ Ω+Ω Ω+ = −Ω + Ω  =(6.5) = Then, the maximum amplitude of the steady state response is: ( ) ( ) ( ) ( ) 2 2 o 22 2 Y Xkc km c+Ω = −Ω +Ω =(6.6) = Therefore, the dimensionless ratio= o XY =is given by: = ( ) ( ) ( ) ( ) ( ) 2 2 2 22 o 12 X = Y 12 n nn h h ω ωω +Ω −Ω + Ω  = (6.7)= Figure=6.2 shows the magnitude and phase of the ratio= o XY =as a function of the dimensionless frequency= nω Ω .= = 0 0 .5 1 1 .5 2 -18 0 -13 5 -90 -45 0 D im e ns ionle ss fre q ue ncy P ha s e [ de g r ee ] D im e ns ionle ss Re s po ns e (X /Y 0) h: 0 .04 h: 0 .05 h: 0 .07 5 h: 0 .1 h: 0 .2 h: 0 .4 0 0 .5 1 1 .5 2 0 2 4 6 8 10 12 14 D im e ns ionle ss fre q ue ncy Amplitude X/Y 0 h: 0 .04 h: 0 .05 h: 0 .07 5 h: 0 .1 h: 0 .2 h: 0 .4 Figure 6.2 Dimensionless response vs. dimensionless frequency of a vibrating system, caused by time-varying displacements of the constraint Note that when the dimensionless frequency nω Ω =is= 2=numerator and denominator of eq.(6.7) are equal, independently of the damping ratio,=h. Therefore, for this condition, the amplitude of the mass vibration are equal to the constraint vibration. This means that the amplitude of the mass vibration is lower than the constraint vibration only for dimensionless frequency values higher than 2.= Conversely, for significantly low values of the dimensionless frequency ( 1 nω Ω ) the amplitude of the= mass vibration=is equal or higher than the constraint vibration. In the end, when the dimensionless frequency= is close to unity (resonance region)=the amplitude of the mass vibration=is considerably magnified= Note that eq.(6.7) coincides with that of the Transmission Factor given by eq.(5.7), however, the physical parameters=at the left hand side of eqs.(5.7) and (6.7) are quite different.= In general, it is very=important to mitigate the vibration transmitted by the constraint to the=mechanical= system. Therefore, it is necessary=that the=dimensionless frequency nω Ω =is=higher than 2.=In the more general=case of a nonϑharmonic displacement of the constraint, it is necessary that the lowest frequency of the not null harmonic components contained in the frequency spectrum satisfies the condition: 2 nω Ω> . This can be obtained reducing the system natural frequency= nω .=θhis needs to insert elastic elements of sufficiently low stiffness=(vibration isolators, resilients) between the mechanical system and the constraint= (supporting structure).= = Consider the relative displacement, z, given by: z xy = − = (6.8)= Then, eq.(6.2) can be rewritten in the following form:= () 0 m z y cz kz++ + =  = (6.9)= Being 2 oY it ye Ω =−Ω  , eq.(6.9) becomesΩ = 2 oY it m z cz kz m e Ω ++ =Ω   =(6.10)= The steady state=solution of eq.(6.10) isΩ= () () ω ω i tit szt e e θ Ω+Ω = =  = (6.11)= Substituting this solution into eq.(6.11) we obtain:= ( ) 2() 2 o ZY i tit k m ic e m e ϑ Ω+Ω  −Ω + Ω = Ω  = ( ) 2 () o 2 Y ω i tit m ee k m ic θ Ω+Ω Ω =  −Ω + Ω  =(6.12) = Then, the maximum amplitude of the steady state response, expressed by means=of the relative vibration, is:= () () 2 o 22 2 Y ωm km cΩ = −Ω +Ω =(6.13)= Therefore, the dimensionless ratio= o ZY =is given by:= () () () 22 2 2 22 o ω = Y12 n nn h ω ωω Ω −Ω + Ω  = (6.14)= Figure 6.3 shows the magnitude and phase of the ratio o ZY as a function of the dimensionless frequency nω Ω . 0 0 .5 1 1 .5 2 2 .5 3 -18 0 -13 5 -90 -45 0 D im e ns ionle ss fre q ue ncy P ha s e [ de g r ee ] Frequency Response (Z/Y 0) h: 0 .04 h: 0 .05 h: 0 .07 5 h: 0 .1 h: 0 .2 h: 0 .4 0 0 .5 1 1 .5 2 2 .5 3 0 2 4 6 8 10 12 14 D im e ns ionle ss fre q ue ncy Amplitude Z/Y 0 h: 0 .04 h: 0 .05 h: 0 .07 5 h: 0 .1 h: 0 .2 h: 0 .4 Figure 6.3 Dimensionless response (relative displacement) vs. dimensionless frequency of a vibrating system, caused by time-varying displacements of the constraint 7. RIGID AND ELASTIC FOUNDATIONS Many mechanisms, machines and, more in general, mechanical systems are subjected to time-varying unbalanced forces. These systems can be connected to the floor of an industrial building or to a supporting structure (e.g. metallic frames, concrete foundations, etc.). A considerable amount of the above mentioned unbalanced forces can be transmitted from the mechanical system to the supporting structure, causing significant problems like excessive mechanical stresses, high vibration levels, transmission of high dynamic forces to other adjacent machines (or structures). These problems can be mitigated designing a suitable apparatus, called machine foundation in the following, which connect the mechanical system to its supporting structure. Depending on the system characteristics and the corresponding technical requirements, two main types of foundations can be made: i) rigid foundations; ii) elastic foundations. 7.1 Rigid foundations Many machines need to be affected by very low vibration levels in order to fulfil specific practical and technical requirements. Assume that the machine must be mounted on the ground floor of an industrial building (Figure 7.1). The average stiffness of the soil is denoted as k=while eqc =is the corresponding= equivalent viscous damping coefficient. In this case the foundation is represented by the soil itself.= Figure=7.2=shows a very simple approximated model of the system. The mechanical system modelled with a= rigid body of mass=m is assumed to be subjected only to vertical displacements. Figure 7. 1 Mechanical system subjected to a time -varying force Figure 7. 2 Approximated model of the mechanical system = The soil characteristics can be various, as=the soil=can be rocky, sandy, gravelly, clayey, etc. Therefore, the= values of the stiffness coefficient kcan be=rather=different. However they are contained in a range of very= high=stiffness values. Conversely, the damping properties of the soil=are=commonly poor.= The average pressure exerted on the contact area= 1A =between machine and floor, caused by the machine= weight, may exceed a maximum allowable limit (e.g.=in the case of sandy and clayey soils). Therefore, it can= be necessary to enlarge the contact area on the soil. This can be obtained inserting a concrete block, between= machine and soil, whose area= 2A =is=significantly larger than that of the machine contour=(Figure=7.3). Thus, the pressure on the soil becomesΩ= 21 21 AA mg mg pp= .= Besides, the lower the damping ratio=h, the higher the decrease of the Transmission Factor caused by increasing values of the dimensionless frequency. Therefore, it is necessary that the lowest frequency 1 Ω =of= the harmonic component of interest of the frequency spectrum of the system excitation F( ) t =fulfils the condition:= 1 2 nω Ω> . As the frequency= 1 Ω =can be rather low, it is important to reduce the value of the natural frequency nω =of the system.=This=can be obtained installing a set of vibration isolators=(resilients), of sufficiently low stiffness, between the machine and the supporting structure. In this case, the stiffness and viscous damping coefficients of Figure 7.1 are those of the vibration isolator. Figures from 7.5 to 7.14 show some examples of vibration isolators. Some of them are properly designed to reduce the dynamic forces transmitted from the machinery to the supporting structure. Other vibration isolators are designed to limit the vibrations transmitted from the supporting structure to machineries or buildings (e.g. to limit the harmful effects caused by earthquakes or underground railway lines). Some vibration isolators are designed to be deformed along their longitudinal direction while other ones are designed to be subjected to shear forces. Figure 7.5 Vibration isolators Figure 7.6 Vibration isolators Figure 7.7 Vibration isolators Figure 7.8 Vibration isolators Figure 7.9 Resilients Figure 7.10 Vibration isolation devices for railways Figure 7.11 Vibration isolation devices for buildings Figure 7.12 Spring and dampers used to mitigate the forces transmitted from the spin basket of a washing machine to the machine frame. Figure 7.13 Additional mass connected to the housing of a washing machine drum Figure 7.14 Seismic isolation bearings (anti -seismic devices) Figure 7.15 Comparison between the dynamic behaviour of two buildings that are not equipped (left) and equipped (right) with anti-seismic devices (isolation bearings) Figure 7.16 Anti -seismic device installed in the supporting structure of a statue (spherical bearing) Figure 7.17 behaviour of a spherical bearing