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for a free vibration with linear damping, and Fig. 2.7b represents a model for a free vibration with torsional damping. The equation of motion for the linear displacement, x,is m x kx c xY 2X32 or m x c x kx 0X 2X33 The equation of motion for the angular displacement, y,is m y ky c yY 2X34 or m y c y k y 0X 2X35 Consider the single degree of freedom model with viscous damping shown in Fig. 2.8. The differential equation of motion is m x c x kx 0Y 2X36 or x c m x k m 0Y 2X37 or x 2a x o 2 x 0Y 2X38 where o k am is the undamped natural (angular) frequency, and a ca2m is the damping ratio. The characteristic equation in r for Eq. (2.38) is r 2 2ar o 2 0Y 2X39 with the solutions r 1Y2 a a 2 o 2 X 2X40 Figure 2.8 Mechanical model of free damped vibration with viscous damping. 348 Theory of Vibration Vibration Case 1. a 2 o 2 ` 0 the roots r 1 and r 2 are complex conjugate (r 1 Y r 2 , where is the set of complex numbers). Case 2. a 2 o 2 b 0 the roots r 1 and r 2 are real and distinct (r 1 Y r 2 Y r 1 r 2 , where is the set of real numbers). Case 3. a 2 o 2 0 the roots r 1 and r 2 are real and identical, r 1 r 2 . In this case, the damping coef®cient is called the critical damping coef®cient, c c cr , and a o c cr 2m k m r X 2X41 The expression of the critical damping coef®cient is c cr 2 km 2moX 2X42 One can classify the vibrations with respect to critical damping coef®cient as follows: Case 1. c ` c cr complex conjugate roots (low damping and oscilla- tory motion). Case 2. c b c cr real and distinct roots (great damping and aperiodic motion). Case 3. c c cr real and identical roots (great damping and aperiodic motion). Case 1: Complex Conjugate Roots a 2 o 2 ` 0 The term b o 2 a 2 2X43 is the quasicircular frequency. The roots of Eq. (2.39) are r 1Y2 a ibX 2X44 The solution of the differential equation is x e at C 1 sin bt C 2 cos btY 2X45 or x Ae at sinbt jY 2X46 where C 1 , C 2 (A and j) are constants. If we use the initial condition t 0 x x 0 x v 0 & 2X47 and the derivative of Eq. (2.45) with respect to time, x ae at C 1 sin bt C 2 cos btbe at C 1 cos bt C 2 sin btY 2X48 2. Linear Systems with One Degree of Freedom 349 Vibration the constants are C 2 x 0 Y C 1 v 0 ax 0 b Y 2X49 or A C 2 1 C 2 2 q x 2 0 v 0 ax 0 b 2 s 2X50 tan j C 2 C 2 bx 0 v 0 ax 0 X 2X51 The solution is x e at v 0 ax 0 b sin bt x 0 cos bt Y 2X52 or x x 2 0 v 0 ax 0 b 2 s e at sin bt arctan bx 0 v 0 ax 0 X 2X53 The exponential decay, Ae at , decreases in time, so one can obtain a motion that is modulated in amplitude. In Eq. (2.43) the quasicircular (or quasian- gular) frequency b is b o 2 a 2 k m c 2m 2 r o 1 c c cr 2 s X 2X54 The quasiperiod of motion is T b 2p b 2p o 2 a 2 X 2X55 The diagram of motion for x is shown in Fig. 2.9, for x 0 0m,v 0 0X2mas, a 0X1s 1 , and o 0X8 radas. The rate of decay of oscillation is xt xt T b Ae at sinbt j Ae at T b sinbt T b j 1 e 2paab e 2paab constX 2X56 Figure 2.9 Oscillation decay for free damped vibration with viscous damping: x 0 0m, v 0 0X2m, a 0X1s 1 , o 0X8 radas. 350 Theory of Vibration Vibration Equation (2.56) shows that the displacement measured at equal time intervals of one quasiperiod decreases in geometric progression. To characterize this decay, the logarithmic decrement d is introduced d ln xt xt T b ln e 2paab 2pa o 2 a 2 pc mb X 2X57 The manner of oscillation decay can be represented using x and o as axes, (Fig. 2.10). The oscillation continues until the amplitude of motion is so small that the maximum spring force is unable to overcome the friction force. Case 2: Real and Distinct Roots a 2 o 2 b 0 The roots of the characteristic equation are negative r 1 l 1 Y r 2 l 2 Y l 1 and l 2 b 0X 2X58 The solution of Eq. (2.38) in this case is x C 1 e l 1 t C 2 e l 2 t X 2X59 The motion is aperiodic and tends asymptotically to the rest position (x 0 when t ). Figure 2.11 shows the diagrams of motion for different initial conditions. Case 3: Real Identical Roots a 2 o 2 0 The roots of the characteristic equation are r 1 r 2 aX 2X60 Figure 2.10 Oscillation decay represented using x and o axes. Figure 2.11 Diagram of damped motion for different initial conditions: (a) x 0 b 0,v 0 b 0; (b) x 0 b 0, v 0 ` 0 (low); (c) x 0 b 0, v 0 ` 0 (great). 2. Linear Systems with One Degree of Freedom 351 Vibration The solution of the differential equation, namely the law of motion, in this case becomes x e at C 1 t C 2 C 1 t C 2 e at 2X61 When t , one obtains the undeterminate x a. The l'Hospital rule is applied in this case: x lim t C 1 t C 2 e at lim t C 1 ae at C 1 0Y 2X62 namely, the motion stops aperiodically. The diagrams of motion are similar to the previous case, with the same boundary conditions. Critical damping represents the limit of periodic motion; hence, the displaced body is restored to equilibrium in the shortest possible time, and without oscillation. Many devices, particularly electrical instruments, are critically damped to take advantage of this property. 2.4 Forced Undamped Vibrations 2.4.1 RESPONSE OF AN UNDAMPED SYSTEM TO A SIMPLE HARMONIC EXCITING FORCE WITH CONSTANT AMPLITUDE Common sources of harmonic excitation imbalance in rotating machines, the motion of the machine itself, or forces produced by reciprocating machines. These excitations may be undesirable for equipment whose operation may be disturbed or for the safety of the structure if large vibration amplitudes develop. Resonance is to be avoided in most cases, and to prevent large amplitudes from developing, dampers and absorbers are often used. General Case An elastic system is excited by a harmonic force of the form F p F 0 sin ptY 2X63 where F 0 is the amplitude of the forced vibration and p is the forced angular frequencies. The differential equation of motion for the mechanical model in Fig. 2.12 is m x kx F 0 sin ptY 2X64 or x k m x F 0 m sin ptX 2X65 Figure 2.12 Mechanical model of forced undamped vibrations. 352 Theory of Vibration Vibration The following notation is used: k m o 2 Y F 0 m qX 2X66 Equation (2.65) can be written as x o 2 x q sin ptY 2X67 with the solution x C 1 sin ot C 2 cos ot x p X 2X68 The particular solution x p of the nonhomogeneous differential equation is x p C sin ptY 2X69 and the second derivative with respect to time is x p Cp 2 sin ptX 2X70 The constant C is determined using Eqs. (2.67), (2.69), and (2.70) for o p: Cp 2 sin pt o 2 C sin pt q sin ptY 2X71 or C o 2 p 2 ÀÁ q C q o 2 p 2 X 2X72 Introducing in Eq. (2.68) the obtained value of C, one can get x C 1 sin ot C 2 cos ot q o 2 p 2 sin ptY 2X73 which may be written as x A sinot j q o 2 p 2 sin ptX 2X74 The constants C 1 and C 2 (A and j) are determined from the initial conditions t 0 x 0 x 0 & 2X75 The derivative with respect to time of Eq. (2.73) is x C 1 o cos ot C 2 o sin ot qp o 2 p 2 cos ptX 2X76 With the help of Eqs. (2.75), (2.73), and (2.76) the constants are C 1 p o q o 2 p 2 C 2 0X The vibration equation is x q o 2 p 2 sin pt p o sin ot hi X 2X77 2. Linear Systems with One Degree of Freedom 353 Vibration Equation (2.77) is a combined motion of two vibrations: one with the natural frequency o and one with the forced angular frequency p. The resultant is a nonharmonic vibration (Fig. 2.13), here for o 1 radas, q 1Nakg, p 0X1 radas. The amplitude is A q o 2 p 2 F 0 m o 2 p 2 F 0 m 1 o 2 1 p 2 o 2 F 0 m m k 1 p o 2 F 0 k 1 1 p o 2 x st 1 p o 2 x st A 0 Y 2X78 Figure 2.13 Combined motion of two vibrations: o 1 radas, q 1Nakg, p 0X1 radas. 354 Theory of Vibration Vibration where x st F 0 ak is the static deformation of the elastic system, under the maximum value F 0 , and A 0 1 1 p o 2 2X79 is a magni®cation factor. In Fig. 2.14 is shown the magni®cation factor function of pao. From Fig. 2.14 one can notice: j Point a ( p 0, A 0 1) and x x st . The system vibrates in phase with force. j Point b ( p , A 0 0), which corresponds to great values of angular frequency p. The in¯uence of forced force is practically null. j Point c ( p o, A 0 ). This phenomenon called resonance and is very important in engineering applications. The curve in Fig. 2.14 is called a curve of resonance. Resonance When the frequency of perturbation p is equal to the natural angular frequency o, the resonance phenomenon appears. The resonance is char- acteristic through increasing amplitude to in®nity. In Eq. (2.77) for o p, the limit case is obtained, lim po x 0. Using l'Hospital's rule one can calculate the limit: lim po x q lim po sin pt p o sin ot o 2 p 2 q lim po t cos pt 2p q 2o cos otX 2X80 A diagram of the motion is shown in Fig. 2.15 for q 1Nakg and o 0X2 radas. The amplitude increases linearly according to Eq. (2.80). The Beat Phenomenon The beat phenomenon appears in the case of two combined parallel vibrations with similar angular frequency (o p). One can introduce the Figure 2.14 Curve of resonance. 2. Linear Systems with One Degree of Freedom 355 Vibration factor e o p, and in this case pao 1 and p o 2o. The vibration becomes x q o 2 p 2 sin pt sin ot q o po p 2 sin p ot 2 cos p ot 2 2q 2oe sin et 2 cos 2ot 2 q oe sin e 2 t cos otX 2X81 The amplitude is in this case is ft q oe sin e 2 t X 2X82 The vibration diagram is shown in Fig. 2.16 for q 2Nakg, e 0X12 radas and o 0X8 radas. 2.4.2 RESPONSE OF AN UNDAMPED SYSTEM TO A CENTRIFUGAL EXCITING FORCE Unbalance in rotating machines is a common source of vibration excitation. Frequently, the excited (perturbation) harmonic force came from an Figure 2.15 Diagram of resonance phenomenon: q 1Nakg, o 0X2 radas. Figure 2.16 Diagram of beat phenomenon: q 2Nakg, E 0X12 : radas, o 0X8 radas. 356 Theory of Vibration Vibration unbalanced mass that is in a rotating motion that generates a centrifugal force. For this case the model is depicted in Fig. 2.17. The unbalanced mass m 0 is connected to the mass m 1 with a massless crank of lengths r. The crank and the mass m 0 rotate with a constant angular frequency p. The centrifugal force is F 0 m 0 rp 2 Y 2X83 and represents the amplitude of the forced vibration (F p F 0 sin pt). The amplitude of the combined vibration is A q o 2 p 2 F 0 k 1 1 P o 2 m 0 rp 2 k 1 1 p o 2 m 0 m rp 2 k m 1 1 p o 2 m 0 m r p o 2 1 p o 2 m 0 r m A 0 Y 2X84 where A 0 is a magni®cation factor, and m m 1 m 0 . The magni®cation factor A 0 is A 0 p o 2 1 p o 2 p o 2 A 0 X 2X85 The variation of the magni®cation factor is shown in Fig. 2.18 where the representative points are a, b, and c. Figure 2.17 Mechanical model of undamped system with centrifugal exciting force. Figure 2.18 Variation of the magni®cation factor. 2. Linear Systems with One Degree of Freedom 357 Vibration [...]... the elementary mechanical work of the perturbation force is denoted by 2X206 Because the mechanical work of the perturbation force is an active mechanical work, it results in dEmec b 0, that is, the mechanical energy of the system increases because of the perturbation force The equation of motion is q p 2X2 07 sin pt À sin ot X x 2 o À p2 o 2.8.5 DAMPED FORCED VIBRATION The mechanical model... 2X 173 2X 174 or v v 2 2 u u u u a2 p 2 c p u u 14 2 2 14 u u o o ccr o2 u u 2 2 2 2 X u u t p2 a2 p 2 t p2 c p 1À 2 4 2 2 1À 2 4 ccr o2 o o o o 2X 176 From Eq (2. 175 ) one can observe that the transmissibility coef®cient t does not depend on the amplitude of perturbation force Equation (2. 175 )... force is not in the same phase with Figure 2.33 Mechanical model of transmissibility in the case of a machine on a foundation with an elastic element and a damper 373 2 Linear Systems with One Degree of Freedom the perturbation force In this case, the transmitted force is Ftrmax kx c x max X 2X 170 x A sin pt À fY 2X 171 x Ap cos pt À fX 2X 172 The exciting vibration is and Therefore,... elastic force of a linear spring is is Fe Àkx Y 2X 179 where x is the linear displacement The elementary mechanical work of an elastic force is dL Fe dx Àkxdx Y and the mechanical work for a displacement from 0 to x is x x x2 L dL Àk xdx Àk X 2 0 0 2X180 2X181 The elastic force is a conservative force and 1 U À kx 2 C X 2 2X182 375 2 Linear Systems with One Degree of Freedom One... is the same as Eq (2.112) In the case of n mechanical impedances Z1 , Z2 Y F F F , Zn in parallel, one can write Z Z1 Z2 Á Á Á Zn Y 2X159 or Z n Zi X i1 2X160 For mechanical impedances in series the total impedance is Z 1 Y 1 1 1 ÁÁÁ Z1 Z2 Zn 2X161 or Z 1 X n 1 i1 Zi 2X162 Vibration 2 .7 Vibration Isolation: Transmissibility 2 .7. 1 FUNDAMENTALS A machine with mass m excited... Rayleigh method is an approximative method used to compute the circular frequency of conservative mechanical systems with one or more DOF One may consider a conservative mechanical system with one DOF The kinetic energy T and the potential energy V were shown in Fig 2.35 Hence, Emax Vmax X 2X2 27 With Eq (2.2 27) one can compute the approximative circular frequency Vibration From Eqs (2.222) and (2.224)... plane of the wheel at O, (Fig 2.37a), and the shaft center is de¯ected with r OA The lateral view of a general position of the rotating wheel of mass m is shown in Fig 2.37b A particular case is shown in Fig 2.37b: OA and AG are in extension The elastic force and the centrifugal force are in relative equilibrium, kr m r ep 2 mrp 2 mep 2 Y 2X31 Figure 2. 37 (a) Shaft with a wheel in rotational... is m x c x kx 0Y 2X190 376 Theory of Vibration or m x kx Àc x X Multiplying by x gives d dt 1 1 2 2 m x kx Àc x 2 2 2 2X191 2X192 or d dx E V Àc x X dt dt 2X193 dEmec Àc x dx dLdamp Y 2X194 Therefore, where Àc x dx dLdamp is the elementary mechanical work of the viscous damped force, which is the resistant mechanical work The solution of the... Àa sin bt j cos bt jX 2X196 Hence, The quasiperiod is given by Tb 2pab In considering instances when the displacement x 0, the mechanical energy of system is equal to the maximum kinetic energy, Emech Emax 1 m x 2 X 2 2X1 97 With Eq (2.1 97) , this gives the following results: Vibration For t 0 A E0 1 mA 2 Àa sin j b cos j2 Y 2 2X198 For t Tb 2pab A ETb !2 1... cos b j 2 b b 1 mA 2 e À8apab Àa sin j b cos j2 X 2X200 2 377 2 Linear Systems with One Degree of Freedom For comparison one can use the ratio ETb aE0 e À4apab and E2Tb aETb e À4apab and the same values are obtained With the logarithmic decrement one can get Et Tb Et e À2d X 2X201 2.8.4 UNDAMPED FORCED VIBRATION The mechanical model was shown in Fig 2.12 The differential equation of . ptX 2X70 The constant C is determined using Eqs. (2. 67) , (2.69), and (2 .70 ) for o p: Cp 2 sin pt o 2 C sin pt q sin ptY 2X71 or C o 2 p 2 ÀÁ q C q o 2 p 2 X 2X72 Introducing. derivative with respect to time of Eq. (2 .73 ) is x C 1 o cos ot C 2 o sin ot qp o 2 p 2 cos ptX 2X76 With the help of Eqs. (2 .75 ), (2 .73 ), and (2 .76 ) the constants are C 1 p o q o 2 . vibration equation is x q o 2 p 2 sin pt p o sin ot hi X 2X 77 2. Linear Systems with One Degree of Freedom 353 Vibration Equation (2 .77 ) is a combined motion of two vibrations: one with the natural frequency