Dynamics 8

Although there are many significant applications of free vibrations, the most important class of vibration problems is that where the motion is continuously excited by a disturbing force. The force may be externally applied or may be generated within the system by such means as unbalanced rotating parts. Forced vibrations may also be excited by the motion of the system foundation

Although there are many significant applications of free vibrations, the most important class of vibration problems is that where the motion is continuously excited by a disturbing force. The force may be externally applied or may be generated within the system by such means as unbalanced rotating parts. Forced vibrations may also be excited by the motion of the system foundation

An important analogy exists between electric circuits and mechanical spring-mass systems. Figure 8/15 shows a series circuit consisting of a voltage E which is a function of time, an inductance L, a capacitance C, and a resistance R. If we denote the charge by the symbol q, the equation which governs the charge is (8/24) This equation has the same form as the equation for the mechanical system. Thus, by a simple interchange of symbols, the behavior of the electrical circuit may be used to predict the behavior of the mechanical system, or vice versa. The mechanical and electrical equivalents in the following table are worth noting:

An important analogy exists between electric circuits and mechanical spring-mass systems. Figure 8/15 shows a series circuit consisting of a voltage E which is a function of time, an inductance L, a capacitance C, and a resistance R. If we denote the charge by the symbol q, the equation which governs the charge is (8/24) This equation has the same form as the equation for the mechanical system. Thus, by a simple interchange of symbols, the behavior of the electrical circuit may be used to predict the behavior of the mechanical system, or vice versa. The mechanical and electrical equivalents in the following table are worth noting:

First, we treat the case where damping is negligible (c 0). Our basic equation of motion, Eq. 8/13, becomes (8/15) The complete solution to Eq. 8/15 is the sum of the complementary solution xc, which is the general solution of Eq. 8/15 with the right side set to zero, and the particular solution xp, which is any solution to the complete equation. Thus, x xc xp. We developed the complementary solution in Art. 8/2. A particular solution is investigated by assuming ¨x n 2x F0 m sin t ¨x 2n˙x n 2x kb sin t m k(x xB) cx˙ mx¨ ¨x 2n˙x n 2x F0 sin t m kx cx˙ F0 sin t mx¨ Article 8/3 Forced Vibration of Particles 601 x c k m F = F0 sin ωt F0 sin ωt (a) (b) cx m kx · x k B Neutral position xB = b sin ωt c m cx m k(x - xB) · Figure 8/9 c08.qxd 2/10/12 2:16 PM Page 601 that the form of the response to the force should resemble that of the force term. To that end, we assume (8/16) where X is the amplitude (in units of length) of the particular solution. Substituting this expression into Eq. 8/15 and solving for X yield (8/17) Thus, the particular solution becomes (8/18) The complementary solution, known as the transient solution, is of no special interest here since, with time, it dies out with the small amount of damping which is always unavoidably present. The particular solution xp describes the continuing motion and is called the steady-state solution. Its period is 2/, the same as that of the forcing function. Of primary interest is the amplitude X of the motion. If we let st stand for the magnitude of the static deflection of the mass under a static load F0, then st F0 /k, and we may form the ratio (8/

First, we treat the case where damping is negligible (c 0). Our basic equation of motion, Eq. 8/13, becomes (8/15) The complete solution to Eq. 8/15 is the sum of the complementary solution xc, which is the general solution of Eq. 8/15 with the right side set to zero, and the particular solution xp, which is any solution to the complete equation. Thus, x xc xp. We developed the complementary solution in Art. 8/2. A particular solution is investigated by assuming ¨x n 2x F0 m sin t ¨x 2n˙x n 2x kb sin t m k(x xB) cx˙ mx¨ ¨x 2n˙x n 2x F0 sin t m kx cx˙ F0 sin t mx¨ Article 8/3 Forced Vibration of Particles 601 x c k m F = F0 sin ωt F0 sin ωt (a) (b) cx m kx · x k B Neutral position xB = b sin ωt c m cx m k(x - xB) · Figure 8/9 c08.qxd 2/10/12 2:16 PM Page 601 that the form of the response to the force should resemble that of the force term. To that end, we assume (8/16) where X is the amplitude (in units of length) of the particular solution. Substituting this expression into Eq. 8/15 and solving for X yield (8/17) Thus, the particular solution becomes (8/18) The complementary solution, known as the transient solution, is of no special interest here since, with time, it dies out with the small amount of damping which is always unavoidably present. The particular solution xp describes the continuing motion and is called the steady-state solution. Its period is 2/, the same as that of the forcing function. Of primary interest is the amplitude X of the motion. If we let st stand for the magnitude of the static deflection of the mass under a static load F0, then st F0 /k, and we may form the ratio (8/

In order to solve the equation of motion, Eq. 8/9, we assume solutions of the form Substitution into Eq. 8/9 yields which is called the characteristic equation. Its roots are Linear systems have the property of superposition, which means that the general solution is the sum of the individual solutions each of which corresponds to one root of the characteristic equation. Thus, the general solution is

In order to solve the equation of motion, Eq. 8/9, we assume solutions of the form Substitution into Eq. 8/9 yields which is called the characteristic equation. Its roots are Linear systems have the property of superposition, which means that the general solution is the sum of the individual solutions each of which corresponds to one root of the characteristic equation. Thus, the general solution is

The equation of motion for the body of Fig. 8/1a is obtained by first drawing its free-body diagram. Applying Newton's second law in the form ΣFx gives (8/1) The oscillation of a mass subjected to a linear restoring force as described by this equation is called simple harmonic motion and is characterized by acceleration which is proportional to the displacement but of opposite sign. Equation 8/1 is normally written as (8/2) where (8/3) is a convenient substitution whose physical significance will be clarified shortly.

The equation of motion for the body of Fig. 8/1a is obtained by first drawing its free-body diagram. Applying Newton's second law in the form ΣFx gives (8/1) The oscillation of a mass subjected to a linear restoring force as described by this equation is called simple harmonic motion and is characterized by acceleration which is proportional to the displacement but of opposite sign. Equation 8/1 is normally written as (8/2) where (8/3) is a convenient substitution whose physical significance will be clarified shortly.

The ratio M is called the amplitude ratio or magnification factor and is a measure of the severity of the vibration. We especially note that M approaches infinity as approaches n. Consequently, if the system possesses no damping and is excited by a harmonic force whose frequency approaches the natural frequency n of the system, then M, and thus X, increase without limit. Physically, this means that the motion amplitude would reach the limits of the attached spring, which is a condition to be avoided.

The ratio M is called the amplitude ratio or magnification factor and is a measure of the severity of the vibration. We especially note that M approaches infinity as approaches n. Consequently, if the system possesses no damping and is excited by a harmonic force whose frequency approaches the natural frequency n of the system, then M, and thus X, increase without limit. Physically, this means that the motion amplitude would reach the limits of the attached spring, which is a condition to be avoided.

The remainder of Chapter 8 is divided into four sections: Article 8/2 treats the free vibration of particles and Art. 8/3 introduces the forced vibration of particles. Each of these two articles is subdivided into undamped- and damped-motion categories. In Art. 8/4 we discuss the vibration of rigid bodies. Finally, an energy approach to the solution of vibration problems is presented in Art. 8/5.

The remainder of Chapter 8 is divided into four sections: Article 8/2 treats the free vibration of particles and Art. 8/3 introduces the forced vibration of particles. Each of these two articles is subdivided into undamped- and damped-motion categories. In Art. 8/4 we discuss the vibration of rigid bodies. Finally, an energy approach to the solution of vibration problems is presented in Art. 8/5.

When a spring-mounted body is disturbed from its equilibrium position, its ensuing motion in the absence of any imposed external forces is termed free vibration. In every actual case of free vibration, there exists some retarding or damping force which tends to diminish the motion. Common damping forces are those due to mechanical and fluid friction. In this article we first consider the ideal case where the damping forces are small enough to be neglected. Then we treat the case where the damping is appreciable and must be accounted for.

When a spring-mounted body is disturbed from its equilibrium position, its ensuing motion in the absence of any imposed external forces is termed free vibration. In every actual case of free vibration, there exists some retarding or damping force which tends to diminish the motion. Common damping forces are those due to mechanical and fluid friction. In this article we first consider the ideal case where the damping forces are small enough to be neglected. Then we treat the case where the damping is appreciable and must be accounted for.

On the other hand, if the frequency ratio /n is small, then M approaches unity (see Fig. 8/11) and X/b (/n) 2 or X b(/n) 2 . But b2 is the maximum acceleration of the frame. Thus, X is proportional to the maximum acceleration of the frame, and the instrument may be used as an accelerometer. The damping ratio is generally selected so that M approximates unity over the widest possible range of /n. From Fig. 8/11, we see that a damping factor somewhere between 0.5 and 1 would meet this criterion.

On the other hand, if the frequency ratio /n is small, then M approaches unity (see Fig. 8/11) and X/b (/n) 2 or X b(/n) 2 . But b2 is the maximum acceleration of the frame. Thus, X is proportional to the maximum acceleration of the frame, and the instrument may be used as an accelerometer. The damping ratio is generally selected so that M approximates unity over the widest possible range of /n. From Fig. 8/11, we see that a damping factor somewhere between 0.5 and 1 would meet this criterion.

An important and special class of problems in dynamics concerns the linear and angular motions of bodies which oscillate or otherwise respond to applied disturbances in the presence of restoring forces. A few examples of this class of dynamics problems are the response of an engineering structure to earthquakes, the vibration of an unbalanced rotating machine, the time response of the plucked string of a musical instrument, the wind-induced vibration of power lines, and the flutter of aircraft wings. In many cases, excessive vibration levels must be reduced to accommodate material limitations or human factors.

An important and special class of problems in dynamics concerns the linear and angular motions of bodies which oscillate or otherwise respond to applied disturbances in the presence of restoring forces. A few examples of this class of dynamics problems are the response of an engineering structure to earthquakes, the vibration of an unbalanced rotating machine, the time response of the plucked string of a musical instrument, the wind-induced vibration of power lines, and the flutter of aircraft wings. In many cases, excessive vibration levels must be reduced to accommodate material limitations or human factors.

As a further note on the free undamped vibration of particles, we see that, if the system of Fig. 8/1a is rotated 90 clockwise to obtain the system of Fig. 8/3 where the motion is vertical rather than horizontal, A2 B2 ψ ωnt ωnt τ C A B x t +x −x −C C x 0 0 x0 2 —- n ω = Figure 8/2 c08.qxd 2/10/12 2:16 PM Page 586 the equation of motion (and therefore all system properties) is unchanged if we continue to define x as the displacement from the equilibrium position. The equilibrium position now involves a nonzero spring deflection st. From the free-body diagram of Fig. 8/3, Newton's second law gives At the equilibrium position x 0, the force sum must be zero, so that Thus, we see that the pair of forces kst and mg on the left side of the motion equation cancel, giving which is identical to Eq. 8/1. The lesson here is that by defining the displacement variable to be zero at equilibrium rather than at the position of zero spring deflection, we may ignore the equal and opposite forces associated with equilibrium.*

As a further note on the free undamped vibration of particles, we see that, if the system of Fig. 8/1a is rotated 90 clockwise to obtain the system of Fig. 8/3 where the motion is vertical rather than horizontal, A2 B2 ψ ωnt ωnt τ C A B x t +x −x −C C x 0 0 x0 2 —- n ω = Figure 8/2 c08.qxd 2/10/12 2:16 PM Page 586 the equation of motion (and therefore all system properties) is unchanged if we continue to define x as the displacement from the equilibrium position. The equilibrium position now involves a nonzero spring deflection st. From the free-body diagram of Fig. 8/3, Newton's second law gives At the equilibrium position x 0, the force sum must be zero, so that Thus, we see that the pair of forces kst and mg on the left side of the motion equation cancel, giving which is identical to Eq. 8/1. The lesson here is that by defining the displacement variable to be zero at equilibrium rather than at the position of zero spring deflection, we may ignore the equal and opposite forces associated with equilibrium.*

As an illustrative example, consider the rotational vibration of the uniform slender bar of Fig. 8/16a. Figure 8/16b depicts the free-body diagram associated with the horizontal position of static equilibrium. Equating to zero the moment sum about O yields where P is the magnitude of the static spring force. Figure 8/16c depicts the free-body diagram associated with an arbitrary positive angular displacement . Using the equation of rotational motion ΣMO as developed in Chapter 6, we write where IO md2 ml2 /12 m(l/6)2 ml2 /9 is obtained from the parallel-axis theorem for mass moments of inertia. For small angular deflections, the approximations sin and cos 1 may be used. With P mg/4, the equation of motion, upon rearrangement and simplification, becomes ¨ (8/25) c m ˙ 4 k m (F0 l/3) cos t ml2/9 I (F0 cos t) l 3 cos 1 9 ml2 ¨ (mg) l 6 cos cl 3 ˙ cos l 3 cos P k 2l 3 sin 2l 3 cos IO ¨ P l 2 l 6 mg l 6 0 P mg 4 614 Chapter 8 Vibration and Time Response O′ y O′ x = 0 Oy O mg P mg (c) (b) (a) Ox F0 cos ωt θ P + k sin — 2l 3 θ c ) = ( cos —d dt —l 3 —cl 3 sin θ θ · θ F0 cos ωt —l 3 —l 3 —l 3 —l 2 l - 6 k c m O Figure 8/16 c08.qxd 2/10/12 2:17 PM Page 614 The right side has been left unsimplified in the form M0(cos t)/IO, where M0 F0l/3 is the magnitude of the moment about point O of the externally applied force. Note that the two equal and opposite moments associated with static equilibrium forces canceled on the left side of the equation of motion. Thus, it is not necessary to include the static-equilibrium forces and moments in the analysis.

As an illustrative example, consider the rotational vibration of the uniform slender bar of Fig. 8/16a. Figure 8/16b depicts the free-body diagram associated with the horizontal position of static equilibrium. Equating to zero the moment sum about O yields where P is the magnitude of the static spring force. Figure 8/16c depicts the free-body diagram associated with an arbitrary positive angular displacement . Using the equation of rotational motion ΣMO as developed in Chapter 6, we write where IO md2 ml2 /12 m(l/6)2 ml2 /9 is obtained from the parallel-axis theorem for mass moments of inertia. For small angular deflections, the approximations sin and cos 1 may be used. With P mg/4, the equation of motion, upon rearrangement and simplification, becomes ¨ (8/25) c m ˙ 4 k m (F0 l/3) cos t ml2/9 I (F0 cos t) l 3 cos 1 9 ml2 ¨ (mg) l 6 cos cl 3 ˙ cos l 3 cos P k 2l 3 sin 2l 3 cos IO ¨ P l 2 l 6 mg l 6 0 P mg 4 614 Chapter 8 Vibration and Time Response O′ y O′ x = 0 Oy O mg P mg (c) (b) (a) Ox F0 cos ωt θ P + k sin — 2l 3 θ c ) = ( cos —d dt —l 3 —cl 3 sin θ θ · θ F0 cos ωt —l 3 —l 3 —l 3 —l 2 l - 6 k c m O Figure 8/16 c08.qxd 2/10/12 2:17 PM Page 614 The right side has been left unsimplified in the form M0(cos t)/IO, where M0 F0l/3 is the magnitude of the moment about point O of the externally applied force. Note that the two equal and opposite moments associated with static equilibrium forces canceled on the left side of the equation of motion. Thus, it is not necessary to include the static-equilibrium forces and moments in the analysis.

Because 0 , the radicand (2 1) may be positive, negative, or even zero, giving rise to the following three categories of damped motion: I. 1 (overdamped). The roots 1 and 2 are distinct, real, and negative numbers. The motion as given by Eq. 8/10 decays so that x approaches zero for large values of time t. There is no oscillation and therefore no period associated with the motion. II. 1 (critically damped). The roots 1 and 2 are equal, real, and negative numbers (1 2 n). The solution to the differential equation for the special case of equal roots is given by x (A1 A2t)ent A1e(21)nt A2e(21)nt x A1e1t A2e2t 1 n( 2 1) 2 n( 2 1) 2 2n n 2 0 x Aet ¨x 2n˙x n 2x 0 c/(2mn) k/m, 588 Chapter 8 Vibration and Time Response c08.qxd 2/10/12 2:16 PM Page 588 Again, the motion decays with x approaching zero for large time, and the motion is nonperiodic. A critically damped system, when excited with an initial velocity or displacement (or both), will approach equilibrium faster than will an overdamped system. Figure 8/5 depicts actual responses for both an overdamped and a critically damped system to an initial displacement x0 and no initial velocity ( 0). III. 1 (underdamped). Noting that the radicand (2 1) is negative and recalling that ea eb , we may rewrite Eq. 8/10 as where i It is convenient to let a new variable d represent the combination Thus, Use of the Euler formula eix cos x i sin x allows the previous equation to be written as (8/11) where A3 (A1 A2) and A4 i(A1 A2). We have shown with Eqs. 8/4 and 8/5 that the sum of two equal-frequency harmonics, such as those in the braces of Eq. 8/11, can be replaced by a single trigonometric function which involves a phase angle. Thus, Eq. 8/11 can be written as or x Ce (8/12) nt sin (dt ) x {C sin (dt )}ent {A3 cos dt A4 sin dt}ent {(A1 A2) cos dt i(A1 A2) sin dt}ent x {A1(cos dt i sin dt) A2(cos dt i sin dt)}ent x {A1eidt A2eidt }ent n1 2. 1. x {A1ei12nt A2ei12nt }ent e(ab) ˙x0 Article 8/2 Free Vibration of Particles 589 c = 15 N·s/m ( = 2.5), overdamped c = 6 N·s/m ( = 1), critically damped Conditions: m = 1 kg, k = 9 N/m x0 = 30 mm, x0 = 0 x, mm t, s 30 20 10 0 0 234 1 · ζ ζ Figure 8/5 c08.qxd 2/10/12 2:16 PM Page 589 Equation 8/12 represents an exponentially decreasing harmonic function, as shown in Fig. 8/6 for specific numerical values. The frequency is called the damped natural frequency. The damped period is given by d 2/d It is important to note that the expressions developed for the constants C and in terms of initial conditions for the case of no damping are not valid for the case of damping. To find C and if damping is present, you must begin anew, setting the general displacement expression of Eq. 8/12 and its first time derivative, both evaluated at time t 0, equal to the initial displacement x0 and initial velocity respectively.

Because 0 , the radicand (2 1) may be positive, negative, or even zero, giving rise to the following three categories of damped motion: I. 1 (overdamped). The roots 1 and 2 are distinct, real, and negative numbers. The motion as given by Eq. 8/10 decays so that x approaches zero for large values of time t. There is no oscillation and therefore no period associated with the motion. II. 1 (critically damped). The roots 1 and 2 are equal, real, and negative numbers (1 2 n). The solution to the differential equation for the special case of equal roots is given by x (A1 A2t)ent A1e(21)nt A2e(21)nt x A1e1t A2e2t 1 n( 2 1) 2 n( 2 1) 2 2n n 2 0 x Aet ¨x 2n˙x n 2x 0 c/(2mn) k/m, 588 Chapter 8 Vibration and Time Response c08.qxd 2/10/12 2:16 PM Page 588 Again, the motion decays with x approaching zero for large time, and the motion is nonperiodic. A critically damped system, when excited with an initial velocity or displacement (or both), will approach equilibrium faster than will an overdamped system. Figure 8/5 depicts actual responses for both an overdamped and a critically damped system to an initial displacement x0 and no initial velocity ( 0). III. 1 (underdamped). Noting that the radicand (2 1) is negative and recalling that ea eb , we may rewrite Eq. 8/10 as where i It is convenient to let a new variable d represent the combination Thus, Use of the Euler formula eix cos x i sin x allows the previous equation to be written as (8/11) where A3 (A1 A2) and A4 i(A1 A2). We have shown with Eqs. 8/4 and 8/5 that the sum of two equal-frequency harmonics, such as those in the braces of Eq. 8/11, can be replaced by a single trigonometric function which involves a phase angle. Thus, Eq. 8/11 can be written as or x Ce (8/12) nt sin (dt ) x {C sin (dt )}ent {A3 cos dt A4 sin dt}ent {(A1 A2) cos dt i(A1 A2) sin dt}ent x {A1(cos dt i sin dt) A2(cos dt i sin dt)}ent x {A1eidt A2eidt }ent n1 2. 1. x {A1ei12nt A2ei12nt }ent e(ab) ˙x0 Article 8/2 Free Vibration of Particles 589 c = 15 N·s/m ( = 2.5), overdamped c = 6 N·s/m ( = 1), critically damped Conditions: m = 1 kg, k = 9 N/m x0 = 30 mm, x0 = 0 x, mm t, s 30 20 10 0 0 234 1 · ζ ζ Figure 8/5 c08.qxd 2/10/12 2:16 PM Page 589 Equation 8/12 represents an exponentially decreasing harmonic function, as shown in Fig. 8/6 for specific numerical values. The frequency is called the damped natural frequency. The damped period is given by d 2/d It is important to note that the expressions developed for the constants C and in terms of initial conditions for the case of no damping are not valid for the case of damping. To find C and if damping is present, you must begin anew, setting the general displacement expression of Eq. 8/12 and its first time derivative, both evaluated at time t 0, equal to the initial displacement x0 and initial velocity respectively.

Because we anticipate an oscillatory motion, we look for a solution which gives x as a periodic function of time. Thus, a logical choice is (8/4) or, alternatively, (8/5) Direct substitution of these expressions into Eq. 8/2 verifies that each expression is a valid solution to the equation of motion. We determine the constants A and B, or C and , from knowledge of the initial displacement x0 and initial velocity of the mass. For example, if we work with the solution form of Eq. 8/4 and evaluate x and at time t 0, we obtain Substitution of these values of A and B into Eq. 8/4 yields (8/6) The constants C and of Eq. 8/5 can be determined in terms of given initial conditions in a similar manner. Evaluation of Eq. 8/5 and its first time derivative at t 0 gives x0 C sin and ˙x0 Cn cos x x0 cos nt ˙x0 n sin nt x0 A and ˙x0 Bn ˙x ˙x0 x C sin (nt ) x A cos nt B sin nt n k/m ¨x n 2x 0 kx mx¨ or mx¨ kx 0 mx¨ Article 8/2 Free Vibration of Particles 585 c08.qxd 2/10/12 2:16 PM Page 585 Solving for C and yields Substitution of these values into Eq. 8/5 gives (8/7) Equations 8/6 and 8/7 represent two different mathematical expressions for the same time-dependent motion. We observe that C and tan1 (A/B).

Because we anticipate an oscillatory motion, we look for a solution which gives x as a periodic function of time. Thus, a logical choice is (8/4) or, alternatively, (8/5) Direct substitution of these expressions into Eq. 8/2 verifies that each expression is a valid solution to the equation of motion. We determine the constants A and B, or C and , from knowledge of the initial displacement x0 and initial velocity of the mass. For example, if we work with the solution form of Eq. 8/4 and evaluate x and at time t 0, we obtain Substitution of these values of A and B into Eq. 8/4 yields (8/6) The constants C and of Eq. 8/5 can be determined in terms of given initial conditions in a similar manner. Evaluation of Eq. 8/5 and its first time derivative at t 0 gives x0 C sin and ˙x0 Cn cos x x0 cos nt ˙x0 n sin nt x0 A and ˙x0 Bn ˙x ˙x0 x C sin (nt ) x A cos nt B sin nt n k/m ¨x n 2x 0 kx mx¨ or mx¨ kx 0 mx¨ Article 8/2 Free Vibration of Particles 585 c08.qxd 2/10/12 2:16 PM Page 585 Solving for C and yields Substitution of these values into Eq. 8/5 gives (8/7) Equations 8/6 and 8/7 represent two different mathematical expressions for the same time-dependent motion. We observe that C and tan1 (A/B).

Complex dashpots with internal flow-rate-dependent one-way valves can produce different damping coefficients in extension and in compression; nonlinear characteristics are also possible. We will restrict our attention to the simple linear dashpot. The equation of motion for the body with damping is determined from the free-body diagram as shown in Fig. 8/4a. Newton's second law gives kx cx˙ mx¨ or mx¨ cx˙ kx 0 (8/8) cx˙. ˙x. N s/m mx¨ kx 0 kst mg 0 k(st x) mg mx¨ Article 8/2 Free Vibration of Particles 587 *For nonlinear systems, all forces, including the static forces associated with equilibrium, should be included in the analysis. m m k x mg k( st δ + x) δst Equilibrium position Figure 8/3 k x c m Equilibrium position (a) (b) Fd kx cx mg N · x · Figure 8/4 c08.qxd 2/10/12 2:16 PM Page 587 In addition to the substitution n it is convenient, for reasons which will shortly become evident, to introduce the combination of constants The quantity (zeta) is called the viscous damping factor or damping ratio and is a measure of the severity of the damping. You should verify that is nondimensional. Equation 8/8 may now be written as

Complex dashpots with internal flow-rate-dependent one-way valves can produce different damping coefficients in extension and in compression; nonlinear characteristics are also possible. We will restrict our attention to the simple linear dashpot. The equation of motion for the body with damping is determined from the free-body diagram as shown in Fig. 8/4a. Newton's second law gives kx cx˙ mx¨ or mx¨ cx˙ kx 0 (8/8) cx˙. ˙x. N s/m mx¨ kx 0 kst mg 0 k(st x) mg mx¨ Article 8/2 Free Vibration of Particles 587 *For nonlinear systems, all forces, including the static forces associated with equilibrium, should be included in the analysis. m m k x mg k( st δ + x) δst Equilibrium position Figure 8/3 k x c m Equilibrium position (a) (b) Fd kx cx mg N · x · Figure 8/4 c08.qxd 2/10/12 2:16 PM Page 587 In addition to the substitution n it is convenient, for reasons which will shortly become evident, to introduce the combination of constants The quantity (zeta) is called the viscous damping factor or damping ratio and is a measure of the severity of the damping. You should verify that is nondimensional. Equation 8/8 may now be written as

Conservation of energy may also be used to determine the period or frequency of vibration for a linear conservative system, without having to derive and solve the equation of motion. For a system which oscillates with simple harmonic motion about the equilibrium position, from which the mx¨ kx 0 ˙x d dt (T V) mx˙¨x kxx˙ 0 T V 1 2mx˙2 1 2kx2 V 1 2kx2 V Ve Vg 1 2k(x st)2 1 2kst 2 mgx 624 Chapter 8 Vibration and Time Response m m k x Equilibrium position st δ Figure 8/17 c08.qxd 2/10/12 2:17 PM Page 624 displacement x is measured, the energy changes from maximum kinetic and zero potential at the equilibrium position x 0 to zero kinetic and maximum potential at the position of maximum displacement x xmax. Thus, we may write The maximum kinetic energy is and the maximum potential energy is For the harmonic oscillator of Fig. 8/17, we know that the displacement may be written as x xmax sin (nt ), so that the maximum velocity is nxmax. Thus, we may write where xmax is the maximum displacement, at which the potential energy is a maximum. From this energy balance, we easily obtain This method of directly determining the frequency may be used for any linear undamped vibration. The main advantage of the energy approach for the free vibration of conservative systems is that it becomes unnecessary to dismember the system and account for all of the forces which act on each member. In Art. 3/7 of Chapter 3 and in Arts. 6/6 and 6/7 of Chapter 6, we learned for a system of interconnected bodies that an active-force diagram of the complete system enabled us to evaluate the work U of the external active forces and to equate it to the change in the total mechanical energy T V of the system

Conservation of energy may also be used to determine the period or frequency of vibration for a linear conservative system, without having to derive and solve the equation of motion. For a system which oscillates with simple harmonic motion about the equilibrium position, from which the mx¨ kx 0 ˙x d dt (T V) mx˙¨x kxx˙ 0 T V 1 2mx˙2 1 2kx2 V 1 2kx2 V Ve Vg 1 2k(x st)2 1 2kst 2 mgx 624 Chapter 8 Vibration and Time Response m m k x Equilibrium position st δ Figure 8/17 c08.qxd 2/10/12 2:17 PM Page 624 displacement x is measured, the energy changes from maximum kinetic and zero potential at the equilibrium position x 0 to zero kinetic and maximum potential at the position of maximum displacement x xmax. Thus, we may write The maximum kinetic energy is and the maximum potential energy is For the harmonic oscillator of Fig. 8/17, we know that the displacement may be written as x xmax sin (nt ), so that the maximum velocity is nxmax. Thus, we may write where xmax is the maximum displacement, at which the potential energy is a maximum. From this energy balance, we easily obtain This method of directly determining the frequency may be used for any linear undamped vibration. The main advantage of the energy approach for the free vibration of conservative systems is that it becomes unnecessary to dismember the system and account for all of the forces which act on each member. In Art. 3/7 of Chapter 3 and in Arts. 6/6 and 6/7 of Chapter 6, we learned for a system of interconnected bodies that an active-force diagram of the complete system enabled us to evaluate the work U of the external active forces and to equate it to the change in the total mechanical energy T V of the system

Every mechanical system possesses some inherent degree of friction, which dissipates mechanical energy. Precise mathematical models of the dissipative friction forces are usually complex. The dashpot or viscous damper is a device intentionally added to systems for the purpose of limiting or retarding vibration. It consists of a cylinder filled with a viscous fluid and a piston with holes or other passages by which the fluid can flow from one side of the piston to the other. Simple dashpots arranged as shown schematically in Fig. 8/4a exert a force Fd whose magnitude is proportional to the velocity of the mass, as depicted in Fig. 8/4b. The constant of proportionality c is called the viscous damping coefficient and has units of or lb-sec/ft. The direction of the damping force as applied to the mass is opposite that of the velocity Thus, the force on the mass is

Every mechanical system possesses some inherent degree of friction, which dissipates mechanical energy. Precise mathematical models of the dissipative friction forces are usually complex. The dashpot or viscous damper is a device intentionally added to systems for the purpose of limiting or retarding vibration. It consists of a cylinder filled with a viscous fluid and a piston with holes or other passages by which the fluid can flow from one side of the piston to the other. Simple dashpots arranged as shown schematically in Fig. 8/4a exert a force Fd whose magnitude is proportional to the velocity of the mass, as depicted in Fig. 8/4b. The constant of proportionality c is called the viscous damping coefficient and has units of or lb-sec/ft. The direction of the damping force as applied to the mass is opposite that of the velocity Thus, the force on the mass is

Harmonic movement of the base is equivalent to the direct application of a harmonic force. To show this, consider the system of Fig. 8/9b where the spring is attached to the movable base. The free-body diagram shows the mass displaced a distance x from the neutral or equilibrium position it would have if the base were in its neutral position. The base, in turn, is assumed to have a harmonic movement xB b sin t. Note that the spring deflection is the difference between the inertial displacements of the mass and the base. From the free-body diagram, Newton's second law gives or (8/14) We see immediately that Eq. 8/14 is exactly the same as our basic equation of motion, Eq. 8/13, in that F0 is replaced by kb. Consequently, all the results about to be developed apply to either Eq. 8/13 or 8/14.

Harmonic movement of the base is equivalent to the direct application of a harmonic force. To show this, consider the system of Fig. 8/9b where the spring is attached to the movable base. The free-body diagram shows the mass displaced a distance x from the neutral or equilibrium position it would have if the base were in its neutral position. The base, in turn, is assumed to have a harmonic movement xB b sin t. Note that the spring deflection is the difference between the inertial displacements of the mass and the base. From the free-body diagram, Newton's second law gives or (8/14) We see immediately that Eq. 8/14 is exactly the same as our basic equation of motion, Eq. 8/13, in that F0 is replaced by kb. Consequently, all the results about to be developed apply to either Eq. 8/13 or 8/14.

If the frequency ratio /n is large, then X/b 1 for all values of the damping ratio . Under these conditions, the displacement of the mass relative to the frame is approximately the same as the absolute displacement of the frame, and the instrument acts as a displacement meter. To obtain a high value of /n, we need a small value of n which means a soft spring and a large mass. With such a combination, the mass will tend to stay inertially fixed. Displacement meters generally have very light damping.

If the frequency ratio /n is large, then X/b 1 for all values of the damping ratio . Under these conditions, the displacement of the mass relative to the frame is approximately the same as the absolute displacement of the frame, and the instrument acts as a displacement meter. To obtain a high value of /n, we need a small value of n which means a soft spring and a large mass. With such a combination, the mass will tend to stay inertially fixed. Displacement meters generally have very light damping.

In Arts. 8/2 through 8/4 we derived and solved the equations of motion for vibrating bodies by isolating the body with a free-body diagram and applying Newton's second law of motion. With this approach, we were able to account for the actions of all forces acting on the body, including frictional damping forces. There are many problems where the effect of damping is small and may be neglected, so that the total energy of the system is essentially conserved. For such systems, we find that the principle of conservation of energy may frequently be applied with considerable advantage in establishing the equation of motion and, when the motion is simple harmonic, in determining the frequency of vibration. Determining the Equation of Motion To illustrate this alternative approach, consider first the simple case of the body of mass m attached to the spring of stiffness k and vibrating in the vertical direction without damping, Fig. 8/17. As previously, we find it convenient to measure the motion variable x from the equilibrium position. With this datum, the total potential energy of the system, elastic plus gravitational, becomes

In Arts. 8/2 through 8/4 we derived and solved the equations of motion for vibrating bodies by isolating the body with a free-body diagram and applying Newton's second law of motion. With this approach, we were able to account for the actions of all forces acting on the body, including frictional damping forces. There are many problems where the effect of damping is small and may be neglected, so that the total energy of the system is essentially conserved. For such systems, we find that the principle of conservation of energy may frequently be applied with considerable advantage in establishing the equation of motion and, when the motion is simple harmonic, in determining the frequency of vibration. Determining the Equation of Motion To illustrate this alternative approach, consider first the simple case of the body of mass m attached to the spring of stiffness k and vibrating in the vertical direction without damping, Fig. 8/17. As previously, we find it convenient to measure the motion variable x from the equilibrium position. With this datum, the total potential energy of the system, elastic plus gravitational, becomes

In many cases, the excitation of the mass is due not to a directly applied force but to the movement of the base or foundation to which the mass is connected by springs or other compliant mountings. Examples of such applications are seismographs, vehicle suspensions, and structures shaken by earthquakes.

In many cases, the excitation of the mass is due not to a directly applied force but to the movement of the base or foundation to which the mass is connected by springs or other compliant mountings. Examples of such applications are seismographs, vehicle suspensions, and structures shaken by earthquakes.

In the analysis of every engineering problem, we must represent the system under scrutiny by a physical model. We may often represent a continuous or distributed-parameter system (one in which the mass and spring elements are continuously spread over space) by a discrete or lumped-parameter model (one in which the mass and spring elements are separate and concentrated). The resulting simplified model is especially accurate when some portions of a continuous system are relatively massive in comparison with other portions. For example, the physical model of a ship propeller shaft is often assumed to be a massless but twistable rod with a disk rigidly attached to each end—one disk representing the turbine and the other representing the propeller. As a second example, we observe that the mass of springs may often be neglected in comparison with that of attached bodies.

In the analysis of every engineering problem, we must represent the system under scrutiny by a physical model. We may often represent a continuous or distributed-parameter system (one in which the mass and spring elements are continuously spread over space) by a discrete or lumped-parameter model (one in which the mass and spring elements are separate and concentrated). The resulting simplified model is especially accurate when some portions of a continuous system are relatively massive in comparison with other portions. For example, the physical model of a ship propeller shaft is often assumed to be a massless but twistable rod with a disk rigidly attached to each end—one disk representing the turbine and the other representing the propeller. As a second example, we observe that the mass of springs may often be neglected in comparison with that of attached bodies.

Near resonance the magnitude X of the steady-state solution is a strong function of the damping ratio and the nondimensional frequency ratio /n. It is again convenient to form the nondimensional ratio M X/(F0/k), which is called the amplitude ratio or magnification factor (8/23) An accurate plot of the magnification factor M versus the frequency ratio /n for various values of the damping ratio is shown in Fig. 8/11. This figure reveals the most essential information pertinent to the forced vibration of a single-degree-of-freedom system under harmonic excitation. It is clear from the graph that, if a motion amplitude is excessive, two possible remedies would be to (a) increase the damping (to obtain a larger value of ) or (b) alter the driving frequency so that is farther from the resonant frequency n. The addition of damping is most effective near resonance. Figure 8/11 also shows that, except for 0, the magnification-factor curves do not actually peak at /n 1. The peak for any given value of can be calculated by finding the maximum value of M from Eq. 8/23.

Near resonance the magnitude X of the steady-state solution is a strong function of the damping ratio and the nondimensional frequency ratio /n. It is again convenient to form the nondimensional ratio M X/(F0/k), which is called the amplitude ratio or magnification factor (8/23) An accurate plot of the magnification factor M versus the frequency ratio /n for various values of the damping ratio is shown in Fig. 8/11. This figure reveals the most essential information pertinent to the forced vibration of a single-degree-of-freedom system under harmonic excitation. It is clear from the graph that, if a motion amplitude is excessive, two possible remedies would be to (a) increase the damping (to obtain a larger value of ) or (b) alter the driving frequency so that is farther from the resonant frequency n. The addition of damping is most effective near resonance. Figure 8/11 also shows that, except for 0, the magnification-factor curves do not actually peak at /n 1. The peak for any given value of can be calculated by finding the maximum value of M from Eq. 8/23.

Not every system is reducible to a discrete model. For example, the transverse vibration of a diving board after the departure of the diver is 8/1 Introduction 8/2 Free Vibration of Particles 8/3 Forced Vibration of Particles 8/4 Vibration of Rigid Bodies 8/5 Energy Methods 8/6 Chapter Review CHAPTER OUTLINE 8 Vibration and Time Response c08.qxd 2/10/12 2:16 PM Page 583 a somewhat difficult problem of distributed-parameter vibration. In this chapter, we will begin the study of discrete systems, limiting our discussion to those whose configurations may be described with one displacement variable. Such systems are said to possess one degree of freedom. For a more detailed study which includes the treatment of two or more degrees of freedom and continuous systems, you should consult one of the many textbooks devoted solely to the subject of vibrations.

Not every system is reducible to a discrete model. For example, the transverse vibration of a diving board after the departure of the diver is 8/1 Introduction 8/2 Free Vibration of Particles 8/3 Forced Vibration of Particles 8/4 Vibration of Rigid Bodies 8/5 Energy Methods 8/6 Chapter Review CHAPTER OUTLINE 8 Vibration and Time Response c08.qxd 2/10/12 2:16 PM Page 583 a somewhat difficult problem of distributed-parameter vibration. In this chapter, we will begin the study of discrete systems, limiting our discussion to those whose configurations may be described with one displacement variable. Such systems are said to possess one degree of freedom. For a more detailed study which includes the treatment of two or more degrees of freedom and continuous systems, you should consult one of the many textbooks devoted solely to the subject of vibrations.

The motion may be represented graphically, Fig. 8/2, where x is seen to be the projection onto a vertical axis of the rotating vector of length C. The vector rotates at the constant angular velocity n which is called the natural circular frequency and has the units radians per second. The number of complete cycles per unit time is the natural frequency ƒn n/2 and is expressed in hertz (1 hertz (Hz) 1 cycle per second). The time required for one complete motion cycle (one rotation of the reference vector) is the period of the motion and is given by 1/ƒn 2/n. k/m, A2 B2 x x0 2 (˙x0/n)2 sin [nt tan1(x0n/˙x0)] C x0 2 (˙x0/n)2 tan1(x0n/˙x0) 586 Chapter 8 Vibration and Time Response We also see from the figure that x is the sum of the projections onto the vertical axis of two perpendicular vectors whose magnitudes are A and B and whose vector sum C is the amplitude. Vectors A, B, and C rotate together with the constant angular velocity n. Thus, as we have already seen, C and tan1 (A/B).

The motion may be represented graphically, Fig. 8/2, where x is seen to be the projection onto a vertical axis of the rotating vector of length C. The vector rotates at the constant angular velocity n which is called the natural circular frequency and has the units radians per second. The number of complete cycles per unit time is the natural frequency ƒn n/2 and is expressed in hertz (1 hertz (Hz) 1 cycle per second). The time required for one complete motion cycle (one rotation of the reference vector) is the period of the motion and is given by 1/ƒn 2/n. k/m, A2 B2 x x0 2 (˙x0/n)2 sin [nt tan1(x0n/˙x0)] C x0 2 (˙x0/n)2 tan1(x0n/˙x0) 586 Chapter 8 Vibration and Time Response We also see from the figure that x is the sum of the projections onto the vertical axis of two perpendicular vectors whose magnitudes are A and B and whose vector sum C is the amplitude. Vectors A, B, and C rotate together with the constant angular velocity n. Thus, as we have already seen, C and tan1 (A/B).

The phase angle , given by Eq. 8/21, can vary from 0 to and represents the part of a cycle (and thus the time) by which the response xp lags the forcing function F. Figure 8/12 shows how the phase angle varies with the frequency ratio for various values of the damping ratio . Note that the value of when /n 1 is 90 for all values of . To further illustrate the phase difference between the response and the forcing function, we show in Fig. 8/13 two examples of the variation of F and xp with t. In the first example, n and is taken to be /4. In the second example, n and is taken to be 3/4.

The phase angle , given by Eq. 8/21, can vary from 0 to and represents the part of a cycle (and thus the time) by which the response xp lags the forcing function F. Figure 8/12 shows how the phase angle varies with the frequency ratio for various values of the damping ratio . Note that the value of when /n 1 is 90 for all values of . To further illustrate the phase difference between the response and the forcing function, we show in Fig. 8/13 two examples of the variation of F and xp with t. In the first example, n and is taken to be /4. In the second example, n and is taken to be 3/4.

The subject of planar rigid-body vibrations is entirely analogous to that of particle vibrations. In particle vibrations, the variable of interest is one of translation (x), while in rigid-body vibrations, the variable of primary concern may be one of rotation ( ). Thus, the principles of rotational dynamics play a central role in the development of the equation of motion. We will see that the equation of motion for rotational vibration of rigid bodies has a mathematical form identical to that developed in Arts. 8/2 and 8/3 for translational vibration of particles. As was the case with particles, it is convenient to draw the free-body diagram for an arbitrary positive value of the displacement variable, because a negative displacement value easily leads to sign errors in the equation of motion. The practice of measuring the displacement from the position of static equilibrium rather than from the position of zero spring deflection continues to simplify the formulation for linear systems because the equal and opposite forces and moments associated with the static equilibrium position cancel from the analysis. Rather than individually treating the cases of (a) free vibration, undamped and damped, and (b) forced vibrations, undamped and damped, as was done with particles in Arts. 8/2 and 8/3, we will go directly to the damped, forced problem.

The subject of planar rigid-body vibrations is entirely analogous to that of particle vibrations. In particle vibrations, the variable of interest is one of translation (x), while in rigid-body vibrations, the variable of primary concern may be one of rotation ( ). Thus, the principles of rotational dynamics play a central role in the development of the equation of motion. We will see that the equation of motion for rotational vibration of rigid bodies has a mathematical form identical to that developed in Arts. 8/2 and 8/3 for translational vibration of particles. As was the case with particles, it is convenient to draw the free-body diagram for an arbitrary positive value of the displacement variable, because a negative displacement value easily leads to sign errors in the equation of motion. The practice of measuring the displacement from the position of static equilibrium rather than from the position of zero spring deflection continues to simplify the formulation for linear systems because the equal and opposite forces and moments associated with the static equilibrium position cancel from the analysis. Rather than individually treating the cases of (a) free vibration, undamped and damped, and (b) forced vibrations, undamped and damped, as was done with particles in Arts. 8/2 and 8/3, we will go directly to the damped, forced problem.

The topic of vibrations is a direct application of the principles of kinetics as developed in Chapters 3 and 6. In particular, construction of a complete free-body diagram drawn for an arbitrary positive value of the displacement variable, followed by application of the appropriate governing equations of dynamics, will yield the equation of motion. From this equation of motion, which is a second-order ordinary differential equation, you can obtain all information of interest, such as the motion frequency, period, or the motion itself as a function of time.

The topic of vibrations is a direct application of the principles of kinetics as developed in Chapters 3 and 6. In particular, construction of a complete free-body diagram drawn for an arbitrary positive value of the displacement variable, followed by application of the appropriate governing equations of dynamics, will yield the equation of motion. From this equation of motion, which is a second-order ordinary differential equation, you can obtain all information of interest, such as the motion frequency, period, or the motion itself as a function of time.

The value n is called the resonant or critical frequency of the system, and the condition of being close in value to n with the resulting large displacement amplitude X is called resonance. For n, the magnification factor M is positive, and the vibration is in phase with the force F. For n, the magnification factor is negative, and the vibration is 180 out of phase with F. Figure 8/10 shows a plot of the absolute value of M as a function of the driving-frequency ratio /n.

The value n is called the resonant or critical frequency of the system, and the condition of being close in value to n with the resulting large displacement amplitude X is called resonance. For n, the magnification factor M is positive, and the vibration is in phase with the force F. For n, the magnification factor is negative, and the vibration is 180 out of phase with F. Figure 8/10 shows a plot of the absolute value of M as a function of the driving-frequency ratio /n.

Thus, for a conservative mechanical system of interconnected parts with a single degree of freedom where U 0, we may obtain its equation of motion simply by setting the time derivative of its constant total mechanical energy to zero, giving Here V Ve Vg is the sum of the elastic and gravitational potential energies of the system. Also, for an interconnected mechanical system, as for a single body, the natural frequency of vibration is obtained by equating the expression for its maximum total kinetic energy to the expression for its maximum potential energy, where the potential energy is taken to be zero at the equilibrium position. This approach to the determination of natural frequency is valid only if it can be determined that the system vibrates with simple harmonic motion.

Thus, for a conservative mechanical system of interconnected parts with a single degree of freedom where U 0, we may obtain its equation of motion simply by setting the time derivative of its constant total mechanical energy to zero, giving Here V Ve Vg is the sum of the elastic and gravitational potential energies of the system. Also, for an interconnected mechanical system, as for a single body, the natural frequency of vibration is obtained by equating the expression for its maximum total kinetic energy to the expression for its maximum potential energy, where the potential energy is taken to be zero at the equilibrium position. This approach to the determination of natural frequency is valid only if it can be determined that the system vibrates with simple harmonic motion.

Various forms of forcing functions F F(t) and foundation displacements xB xB(t) are depicted in Fig. 8/8. The harmonic force shown in part a of the figure occurs frequently in engineering practice, and the understanding of the analysis associated with harmonic forces is a necessary first step in the study of more complex forms. For this reason, we will focus our attention on harmonic excitation.

Various forms of forcing functions F F(t) and foundation displacements xB xB(t) are depicted in Fig. 8/8. The harmonic force shown in part a of the figure occurs frequently in engineering practice, and the understanding of the analysis associated with harmonic forces is a necessary first step in the study of more complex forms. For this reason, we will focus our attention on harmonic excitation.

We begin by considering the horizontal vibration of the simple frictionless spring-mass system of Fig. 8/1a. Note that the variable x denotes the displacement of the mass from the equilibrium position, which, for this system, is also the position of zero spring deflection. Figure 8/1b shows a plot of the force Fs necessary to deflect the spring versus the corresponding spring deflection for three types of springs. Although nonlinear hard and soft springs are useful in some applications, we will restrict our attention to the linear spring. Such a spring exerts a restoring force kx on the mass—that is, when the mass is displaced to the right, the spring force is to the left, and vice versa. We must be careful to distinguish between the forces of magnitude Fs which must be applied to both ends of the massless spring to cause tension or compression and the force F kx of equal magnitude which the spring exerts on the mass. The constant of proportionality k is called the spring constant, modulus, or stiffness and has the units N/m or lb/ft.

We begin by considering the horizontal vibration of the simple frictionless spring-mass system of Fig. 8/1a. Note that the variable x denotes the displacement of the mass from the equilibrium position, which, for this system, is also the position of zero spring deflection. Figure 8/1b shows a plot of the force Fs necessary to deflect the spring versus the corresponding spring deflection for three types of springs. Although nonlinear hard and soft springs are useful in some applications, we will restrict our attention to the linear spring. Such a spring exerts a restoring force kx on the mass—that is, when the mass is displaced to the right, the spring force is to the left, and vice versa. We must be careful to distinguish between the forces of magnitude Fs which must be applied to both ends of the massless spring to cause tension or compression and the force F kx of equal magnitude which the spring exerts on the mass. The constant of proportionality k is called the spring constant, modulus, or stiffness and has the units N/m or lb/ft.

We first consider the system of Fig. 8/9a, where the body is subjected to the external harmonic force F F0 sin t, in which F0 is the force amplitude and is the driving frequency (in radians per second). Be sure to distinguish between n which is a property of the system, and , which is a property of the force applied to the system. We also note that for a force F F0 cos t, one merely substitutes cos t for sin t in the results about to be developed.

We first consider the system of Fig. 8/9a, where the body is subjected to the external harmonic force F F0 sin t, in which F0 is the force amplitude and is the driving frequency (in radians per second). Be sure to distinguish between n which is a property of the system, and , which is a property of the force applied to the system. We also note that for a force F F0 cos t, one merely substitutes cos t for sin t in the results about to be developed.

We now reintroduce damping in our expressions for forced vibration. Our basic differential equation of motion is [8/13] Again, the complete solution is the sum of the complementary solution xc, which is the general solution of Eq. 8/13 with the right side equal to ¨x 2n˙x n 2x F0 sin t m M X st 1 1 (/n)2 xp F0 /k 1 (/n)2 sin t X F0 /k 1 (/n)2 xp X sin t 602 Chapter 8 Vibration and Time Response 0 6 5 4 3 2 1 0 1 |M| n ω /ω 2 3 Figure 8/10 c08.qxd 2/10/12 2:16 PM Page 602 zero, and the particular solution xp, which is any solution to the complete equation. We have already developed the complementary solution xc in Art. 8/2. When damping is present, we find that a single sine or cosine term, such as we were able to use for the undamped case, is not sufficiently general for the particular solution. So we try Substitute the latter expression into Eq. 8/13, match coefficients of sin t and cos t, and solve the resulting two equations to obtain (8/20) (8/21) The complete solution is now known, and for underdamped systems it can be written as (8/22) Because the first term on the right side diminishes with time, it is known as the transient solution. The particular solution xp is the steadystate solution and is the part of the solution in which we are primarily interested. All quantities on the right side of Eq. 8/22 are properties of the system and the applied force, except for C and (which are determinable from initial conditions) and the running time variable t.

We now reintroduce damping in our expressions for forced vibration. Our basic differential equation of motion is [8/13] Again, the complete solution is the sum of the complementary solution xc, which is the general solution of Eq. 8/13 with the right side equal to ¨x 2n˙x n 2x F0 sin t m M X st 1 1 (/n)2 xp F0 /k 1 (/n)2 sin t X F0 /k 1 (/n)2 xp X sin t 602 Chapter 8 Vibration and Time Response 0 6 5 4 3 2 1 0 1 |M| n ω /ω 2 3 Figure 8/10 c08.qxd 2/10/12 2:16 PM Page 602 zero, and the particular solution xp, which is any solution to the complete equation. We have already developed the complementary solution xc in Art. 8/2. When damping is present, we find that a single sine or cosine term, such as we were able to use for the undamped case, is not sufficiently general for the particular solution. So we try Substitute the latter expression into Eq. 8/13, match coefficients of sin t and cos t, and solve the resulting two equations to obtain (8/20) (8/21) The complete solution is now known, and for underdamped systems it can be written as (8/22) Because the first term on the right side diminishes with time, it is known as the transient solution. The particular solution xp is the steadystate solution and is the part of the solution in which we are primarily interested. All quantities on the right side of Eq. 8/22 are properties of the system and the applied force, except for C and (which are determinable from initial conditions) and the running time variable t.

We often need to experimentally determine the value of the damping ratio for an underdamped system. The usual reason is that the value of the viscous damping coefficient c is not otherwise well known. To determine the damping, we may excite the system by initial conditions and obtain a plot of the displacement x versus time t, such as that shown schematically in Fig. 8/7. We then measure two successive amplitudes x1 and x2 a full cycle apart and compute their ratio The logarithmic decrement is defined as From this equation, we may solve for and obtain For a small damping ratio, x1 x2 and 1, so that /2. If x1 and x2 are so close in value that experimental distinction between them is impractical, the above analysis may be modified by using two observed amplitudes which are n cycles apart

We often need to experimentally determine the value of the damping ratio for an underdamped system. The usual reason is that the value of the viscous damping coefficient c is not otherwise well known. To determine the damping, we may excite the system by initial conditions and obtain a plot of the displacement x versus time t, such as that shown schematically in Fig. 8/7. We then measure two successive amplitudes x1 and x2 a full cycle apart and compute their ratio The logarithmic decrement is defined as From this equation, we may solve for and obtain For a small damping ratio, x1 x2 and 1, so that /2. If x1 and x2 are so close in value that experimental distinction between them is impractical, the above analysis may be modified by using two observed amplitudes which are n cycles apart

lerometers are frequently encountered applications of harmonic excitation. The elements of this class of instruments are shown in Fig. 8/14a. We note that the entire system is subjected to the motion xB of the frame. Letting x denote the position of the mass relative to the frame, we may apply Newton's second law and obtain where (x xB) is the inertial displacement of the mass. If xB b sin t, then our equation of motion with the usual notation is which is the same as Eq. 8/13 if b2 is substituted for F0/m. Again, we are interested only in the steady-state solution xp. Thus, from Eq. 8/20, we have If X represents the amplitude of the relative response xp, then the nondimensional ratio X/b is where M is the magnification ratio of Eq. 8/23. A plot of X/b as a function of the driving-frequency ratio /n is shown in Fig. 8/14b. The similarities and differences between the magnification ratios of Figs. 8/14b and 8/11 should be noted.

lerometers are frequently encountered applications of harmonic excitation. The elements of this class of instruments are shown in Fig. 8/14a. We note that the entire system is subjected to the motion xB of the frame. Letting x denote the position of the mass relative to the frame, we may apply Newton's second law and obtain where (x xB) is the inertial displacement of the mass. If xB b sin t, then our equation of motion with the usual notation is which is the same as Eq. 8/13 if b2 is substituted for F0/m. Again, we are interested only in the steady-state solution xp. Thus, from Eq. 8/20, we have If X represents the amplitude of the relative response xp, then the nondimensional ratio X/b is where M is the magnification ratio of Eq. 8/23. A plot of X/b as a function of the driving-frequency ratio /n is shown in Fig. 8/14b. The similarities and differences between the magnification ratios of Figs. 8/14b and 8/11 should be noted.