Tuned mass absorbers on damped structures under random load

Probabilistic Engineering Mechanics 23 (2008) 408–415 www.elsevier.com/locate/probengmech

Tuned mass absorbers on damped structures under random load Steen Krenk ∗ , Jan Høgsberg Department of Mechanical Engineering, Technical University of Denmark, Nils Koppels Alle, Building 403, DK-2800 Kgs. Lyngby, Denmark Received 27 April 2007; accepted 6 June 2007 Available online 21 February 2008

Abstract A substantial literature exists on the optimal choice of parameters of a tuned mass absorber on a structure excited by a force or by ground acceleration with random characteristics in the form of white noise. In the absence of structural damping the optimal frequency tuning is determined from the mass ratio alone, and the damping can be determined subsequently. Only approximate results are available for the influence of damping in the original structure, typically in the form of series expansions. In the present paper it is demonstrated that for typical mass ratios in the order of a few percent the classic stochastic frequency tuning gives the same standard deviation of the response amplitude within a margin of 0.001 as when using the classic frequency tuning for harmonic load variation, and then optimizing the damping separately. Simple approximate, but very accurate, expressions are obtained for the response variance of a structure with initial damping in terms of the mass ratio and both damping ratios. Within this format the optimal tuning of the absorber turns out to be independent of the structural damping, and a simple explicit expression is obtained for the equivalent total damping. c 2008 Elsevier Ltd. All rights reserved.

Keywords: Tuned mass absorber; Random vibration; Structural dynamics

1. Introduction Tuned mass absorbers constitute an efficient means of introducing damping into structures prone to vibrations, e.g. bridges and high-rise buildings. The original idea is due to Frahm [1] who introduced a spring supported mass, tuned to the natural frequency of the oscillation to be reduced. It was demonstrated by Ormondroyd and Den Hartog [2] that the introduction of a damper in parallel with the spring support of the tuned mass leads to improved behavior, e.g. in the form of amplitude reduction over a wider range of frequencies. The standard reference to the classic tuned mass absorber is the textbook of Den Hartog [3] describing optimal frequency tuning for harmonic load, and the procedure of Brock [4] for the optimal damping. Many of the vibration problems involving tuned mass absorbers involve random loads, e.g. due to wind or earthquakes. The two standard problems are a structural mass excited by a random force, and the combined system of ∗ Corresponding author.

E-mail addresses: [email protected] (S. Krenk), [email protected] (J. Høgsberg). c 2008 Elsevier Ltd. All rights reserved. 0266-8920/$ - see front matter doi:10.1016/j.probengmech.2007.04.004

structural and absorber mass excited by motion of the support of the structure, Fig. 1. Crandall and Mark [5] obtained explicit results for the variance of the response in the case of support acceleration represented by a stationary white noise process, while Jacquot and Hoppe [6] treated the corresponding force excitation problem. In the design of tuned mass absorbers the mass, the stiffness and the applied damping must be selected, and the standard procedure is to select suitable stiffness and applied damping, once the mass ratio has been selected. This problem was studied by Warburton and Ayorinde [7–9] under the assumption that the initial structure is undamped. The approach was to derive optimal values of frequency tuning and applied damping for a given mass ratio that minimize the response variance. It turns out that the optimal frequency tuning and level of applied damping depend on the type of random loading and are also different from the values obtained by Den Hartog [3] for harmonic force loading. Real structures posses structural damping, and it is of considerable interest to find the optimal tuning of tuned mass absorbers on damped structures, and to assess the magnitude of the combined damping. While analytic expressions can be obtained for the response variance also for systems with structural damping [5,6], it has turned out to be quite difficult

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409

satisfying the equation of motion M¨x + C˙x + Kx = F(t)

Fig. 1. Tuned mass absorber. (a) Force excitation, (b) Support excitation.

to reduce these expressions to explicit design oriented formulae including the structural damping. A geometric construction has been proposed by Thompson [10] and series representations proposed e.g. by Tsai and Lin [11,12]. Surveys of various series expansions have been given by Soong and Dargush [13] and Asami et al. [14]. However, in spite of the large number of terms, it is easily seen that they capture the effect of the structural damping in a fairly indirect way. For the typical problem in practice the mass ratio will be of the order of a few percent, and the structural damping will similarly not exceed a few percent, and often be lower. Under these conditions most of the terms in the series representations are negligible. It is therefore of interest to find a different format for the combined influence of structural and applied damping. It was recently demonstrated by Krenk [15] that for a system without structural damping the classic frequency tuning for harmonic load splits the original single-degree-of-freedom oscillation of the structural mass into two modes with exactly equal damping, a damping which is approximately half of the applied damping. This is a system property and suggests that it may be of interest to investigate this particular frequency tuning also for systems with structural damping. The present paper presents such an investigation. It is demonstrated that, while the exact optimal frequency tuning under random load is different from the classic tuning for harmonic load, the influence of this difference on the resulting response variance will in most cases be negligible, if the applied damping is optimized. It is furthermore shown that the applied damping can be optimized in terms of the mass ratio alone, without influence from the structural damping and a compact expression can be found for the combined effect of structural and optimal applied damping. 2. Problem formulation The problem under consideration is illustrated in Fig. 1. The system consists of a structural mass m 1 , supported by a spring with stiffness k1 and a viscous damper with parameter c1 . The motion of the structural mass relative to the ground is denoted x1 (t). The structural mass supports an absorber mass m 2 via a spring with stiffness k2 and a viscous damper with parameter c2 . The motion of the absorber relative to the ground is denoted x2 (t). Two situations will be investigated: (a) motion due to a force F(t) acting on the structural mass, and (b) motion due to acceleration x¨0 (t) of the supporting ground. The motion of the two-degree-of-freedom system is described by the displacement vector x(t) = [x1 (t), x2 (t)]T ,

(1)

where the mass, damping and stiffness matrices are given by     c + c2 −c2 m1 0 , C= 1 M= −c2 c2 0 m2   k + k2 −k2 K= 1 . (2) −k2 k2 For harmonic excitation with angular frequency ω the equation of motion can be written as   K + iωC − ω2 M x = F (3) where F is the force amplitude vector, and x is the complex response amplitude vector. In the analysis it is convenient to represent the system properties in terms of parameters representing time scales and relative damping. First the stiffness parameters are expressed in terms of representative frequencies, p p ω1 = k1 /m 1 , ω2 = k2 /m 2 (4) and the damping parameters are expressed in terms of the damping ratios, c1 , ζ1 = √ 2 k1 m 1

c2 ζ2 = √ . 2 k2 m 2

(5)

The relative mass and time scale of the secondary mass are then described by the mass ratio µ and the frequency tuning parameter α, µ=

m2 , m1

α=

ω2 . ω1

(6)

In the typical tuned mass design problem the final damping is controlled by µ, and optimal properties are obtained by proper selection of the tuning parameter α. The present paper deals with input from forcing or support acceleration in the form of stationary wide-band stochastic processes. Analytic results will be obtained for the idealized case of white noise, representing wide-band excitation. The quality of the absorber system is defined via its ability to limit the variance of the response of the primary mass, σ12 = Var[x1 ]. The analysis is therefore based on the frequency transfer function for the component x1 alone. It is convenient to introduce the frequency ratio r = ω/ω1 and to introduce the normalized force f = F/k1 , whereby (1) takes the form   1 + µα 2 − r 2 + 2i(ζ1 + µαζ2 )r −µα 2 − 2iµαζ2r −µα 2 − 2iµαζ2r µα 2 − µr 2 + 2iµαζ2r     x f × 1 = 1 . (7) x2 f2 Only special load processes will be treated here, and it is therefore convenient to discuss the two cases separately.

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2.1. Forced motion In the case of forced motion there is only one load vector component F(t), acting on the structural mass. The corresponding normalized load vector is,     f F/k1 f= = (8) 0 0 where the component f = F1 /k1 corresponds to the quasistatic displacement of the structural mass. The response x1 of the structural mass follows from (7) as x1 = H f (ω) f.

(9)

The frequency response function H f (ω) of the forced motion is a rational function of the form p f (ω) H f (ω) = q(ω)

(10)

with numerator p f (ω) = α 2 − r 2 + 2iαζ2r

(11)

2.2. Support acceleration When the system is loaded via support acceleration the total motion is x + x0 , where x0 represents the support motion. The support motion leads to translations that do not directly activate elastic and damping forces. Thus, the support motion only contributes to the inertial term, and the equation of motion can be expressed in the form M¨x + C˙x + Kx = −M¨x0 .

and denominator q(ω) = [1 + µα 2 − r 2 + 2i(ζ1 + µαζ2 )r ] × [α − r + 2iαζ2r ] − µ[α + 2iαζ2r ] . 2

the result will be analyzed in two steps. In Section 3 the solution is analyzed for systems without structural damping. This is the classic case, which has typically been used to derive optimal frequency tuning α and applied damping ratio ζ2 , [5–9]. It will be demonstrated that considerably simpler results are obtained, if the frequency is tuned according to the classic result for harmonic excitation [3], and the applied damping ratio ζ2 is optimized subsequently. The difference in the resulting response variance from the exact optimum is negligible, and the new results have the great advantage that they permit development of a set of accurate explicit formulae covering the case including structural damping ζ1 as shown in Section 4.

2

2

2

(12)

Let the force be represented by a white noise process, and let the normalized force process f (t) = F(t)/k1 have the spectral density S f . The variance of the response σ12 can then be evaluated from the integral [16], Z ∞ Z ∞ 2 2 σ1 = S f |H f (ω)| dω = S f H f (ω) H f (−ω)dω. −∞

−∞

(13) The poles of the rational frequency transfer function H f (ω) all lie in the upper complex half-plane. This type of integral can be evaluated directly from the coefficients of the polynomials in the numerator and denominator [17,18], P f (µ, α, ζ1 , ζ2 ) π σ12 = S f ω1 2 Q(µ, α, ζ1 , ζ2 )

(14)

where P f and Q are polynomials in the coefficients of p f (r ) and q(r ), respectively, and thereby in the indicated arguments. After some algebra the result can be written as P f = µα 2 (αζ1 + ζ2 ) + [1 − (1 + µ)α 2 ]2 ζ2 + 4αζ2 [ζ1 + (1 + µ)αζ2 ](αζ1 + ζ2 )

The support acceleration leads to an equivalent load vector   m F(t) = −M¨x(t) = − 1 x¨0 (t). (18) m2 The corresponding normalized force is obtained by division with the stiffness k1 of the primary structure, whereby   1 x¨0 f=− . (19) µ ω2 1 It is convenient to consider the normalized acceleration x¨0 /ω12 as input in the following to obtain the most direct analogy with the case of force excitation. The response x1 of the structural mass follows from (7) as x1 = Ha (ω)x¨0 /ω12 .

(20)

The frequency response function Ha (ω) of motion due to support acceleration then follows in the form Ha (ω) =

pa (ω) . q(ω)

(21)

The numerator is found as (15)

pa (ω) = (1 + µ)α 2 − r 2 + 2i(1 + µ)αζ2r

(16)

while the denominator is the same as in the case of force excitation, already given in (12). Let the support acceleration be represented by a white noise process, and let the normalized support acceleration x¨0 /ω12 have the spectral density Sa . The variance of the response σ12 then follows as in the case of force excitation in the form of a rational function,

and Q = µα(αζ1 + ζ2 )2 + [1 − (1 + µ)α 2 ]2 ζ1 ζ2 + 4αζ1 ζ2 [ζ1 + (1 + µ)αζ2 ](αζ1 + ζ2 ).

(17)

It is seen that all terms in P f are linear or cubic in the damping ratios ζ1 and ζ2 , while all terms in Q are quadratic or quartic. In spite of this property, and the fact that several of the factors occur repeatedly, the general form of the result seems to be intractable analytically in exact form. In the following

σ12 =

π Pa (µ, α, ζ1 , ζ2 ) Sa ω1 2 Q(µ, α, ζ1 , ζ2 )

(22)

(23)

S. Krenk, J. Høgsberg / Probabilistic Engineering Mechanics 23 (2008) 408–415

where the polynomial Pa is obtained from the coefficients of the frequency response function Ha (ω) as described in [17,18]. Pa = µ(1 + µ)2 α 2 (αζ1 + ζ2 ) + µ2 αζ1 + [1 − (1 + µ)2 α 2 ]ζ2 + 4α(1 + µ)2 ζ2 × [ζ1 + (1 + µ)αζ2 ](αζ1 + ζ2 ).

(24)

411

The optimal frequency follows by differentiation of this expression with respect to α 2 , when keeping the other independent variable αζ2 constant, ∂σ12 = 0. ∂(α 2 )

(29)

The polynomial Q has already been given in (16). These polynomials correspond to a rearrangement of the expressions given in [5].

This leads to the optimal frequency tuning ratio determined by

3. Undamped primary structure

The optimal damping ratio follows from (27),

The response variance in both the two cases is given by a rational function in the parameters µ, α, ζ1 and ζ2 . While an exact analysis of the general expressions is difficult, the case without structural damping, ζ1 = 0, is considerably simpler. In practice the structural damping will often be considerably smaller than the imposed damping, ζ1  ζ2 , and it therefore seems reasonable to obtain information of the relative importance of detailed frequency tuning from the simplified case of ζ1 = 0 before proceeding to the case of combined damping. For both the cases of force excitation and support acceleration it is demonstrated that in the absence of structural damping the classical frequency tuning for harmonic load leads to a response variance which only deviates negligibly from the exact optimum. The consequences of this are then developed in Section 4.

ζ22 =

3.1. Force excitation

α2 =

1 + 21 µ . (1 + µ)2

1 + 34 µ µ . 4(1 + µ) 1 + 21 µ

These are the classic expressions for optimal frequency and damping, and resulting response variance [5,7,9]. It has recently been demonstrated in [15] that the classic frequency tuning for harmonic load leads to the special property that the two modes generated by mounting the secondary mass have identical damping ratio, very closely represented by 21 ζ2 . This suggests special properties associated with this particular frequency tuning 1 . 1+µ

α=

The minimum value of the variance is conveniently found by considering the combinations αζ2 and α 2 as independent variables. Optimal damping then leads to the condition

The corresponding value of the response variance is

(26)

from which  2  1 µ 1 2 2 (αζ2 ) = α + 1 − (1 + µ)α . 4 1+µ µ 2

(27)

This relation determines the optimal damping ratio ζ2 for any value of the frequency tuning parameter α. Substitution of this optimal damping parameter into the expression (25) for the response variance gives  2  π S f ω1 2 1  2 2 α + σ1 = 1 − (1 + µ)α . (28) αζ2 µ

(31)

The minimum response variance is found by substituting αζ2 from (27) and α 2 from (30) into the response variance expression (25), s 1 + 43 µ 2 σ1,min = 2π S f ω1 . (32) µ(1 + µ)

In the case of force excitation the expression (14) for the variance simplifies considerably for ζ1 = 0,  2 π S f ω1 2 1  1 − (1 + µ)α 2 α + σ12 = 2 αζ2 µ  4 2 + (1 + µ)(αζ2 ) . (25) µ

∂σ12 =0 ∂(αζ2 )

(30)

(33)

When using this frequency tuning, the optimal damping ratio follows from (27) as √ µ . (34) ζ2 = 2 π 2π σ12 = √ S f ω1 = S f ω1 . µ ζ2

(35)

These expressions are remarkably simple. Furthermore the ratio of the standard deviation σ1,min to that of the frequency tuning (33) from harmonic load is s 3 σ1,min 1 4 1 + 4µ = ' 1 − µ + ···. (36) σ1 1+µ 16 In most practical cases the mass ratio is of the order of a few percent, leading to a relative difference in the standard deviation of the response of the order 0.001. In view of this the simple results (33)–(35) are used as basis for the development of an approximate but accurate set of formulae for the general case including structural damping in Section 4.

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3.2. Support acceleration In the absence of structural damping the expression (23) takes the following simpler form,  2 1 π Sa ω1 2 (1 + µ)2 α 2 + 1 − (1 + µ)2 α 2 σ1 = 2 αζ2 µ  4 + (1 + µ)3 (αζ2 )2 . (37) µ The procedure is the same as for force excitation treated above. After selecting αζ2 and α 2 as independent variables, partial differentiation with respect to αζ2 gives the optimal damping ratio h 1 µ (αζ2 )2 = (1 + µ)2 α 2 4 (1 + µ)3 2  1 + 1 − (1 + µ)2 α 2 . (38) µ The optimal frequency tuning is then found from partial differentiation with respect to α 2 , whereby α2 =

1 − 21 µ . (1 + µ)2

(39)

The corresponding optimal damping follows from (38) as ζ22 =

1 − 14 µ µ . 4(1 + µ) 1 − 12 µ

(40)

Finally, substitution of the optimal frequency tuning and damping ratio back into (37) gives the minimum response variance as s   1+µ 1 2 σ1,min = 2π Sa ω1 (1 + µ) 1− µ . (41) µ 4 These expressions are also classic for optimal frequency and damping and resulting response variance [7,9]. It is of interest to investigate the effect of using the frequency tuning (33) for harmonic load also in this case. The corresponding optimal damping follows from (38) as r 1 µ ζ2 = . (42) 2 1+µ The response variance then follows from (37), s (1 + µ)3 π(1 + µ) σ12 = 2π Sa ω1 = Sa ω1 . µ ζ2

(43)

Also in this case the damping and response variance comes out with simpler formulae than the exact optimum. The ratio between the exact minimum standard deviation and that obtained by using harmonic tuning of the frequency ratio is given by r σ1,min 1 1 4 = 1 − µ ' 1 − µ + ··· (44) σ1 4 16

i.e. asymptotically the same as for the case of force excitation and for most practical cases giving a negligible difference between exact and approximate minimum response. 4. Damped primary structure It was demonstrated in the last section that use of the classic frequency tuning (33) for harmonic load followed by optimization of the applied damping ζ2 leads to a response variance that is essentially equal to the exact minimum. In this section it is demonstrated that for this particular frequency tuning the general expression including both structural and applied damping can be reduced to a simple factored form by a slight approximation involving a small term in the structural damping ζ1 . 4.1. Force excitation In the case of the classic frequency tuning for harmonic load (33) the rational expression (14) for the response variance simplifies. The polynomial in the numerator now takes the form P f = µα(α 2 ζ1 + ζ2 ) + 4αζ2 (ζ1 + ζ2 )(αζ1 + ζ2 ).

(45)

When the mass ratio is small, the frequency tuning parameter is close to 1, and for optimal damping the total damping will √ be in the order of µ. In typical applications the structural damping ζ1 will furthermore be small relative to the applied damping ζ2 . Under these conditions exchange of the factor α 2 in the first parenthesis with α will have only modest effect on the numerical value. However, this approximation leads to the following factored form P f ' α [µ + 4αζ2 (ζ1 + ζ2 )] (αζ1 + ζ2 ).

(46)

The polynomial in the denominator of (14) can be factored by a similar approximation. The classic harmonic frequency tuning gives h i Q = µα α 2 ζ12 + ζ22 + (1 + α)ζ1 ζ2 + 4αζ1 ζ2 (ζ1 + ζ2 )(αζ1 + ζ2 ).

(47)

Again, replacement of the factor α 2 with α in the first term leads to a factored form, Q ' α(µ + 4ζ1 ζ2 )(ζ1 + ζ2 )(αζ1 + ζ2 ).

(48)

When these factored forms are used in the expression (14) for the response variance, the following simple expression is obtained σ12 '

π S f ω1 µ + 4ζ2 (ζ1 + ζ2 ) . 2 ζ1 + ζ2 µ + 4ζ1 ζ2

(49)

This approximation contains the exact result in both the limit of vanishing structural damping ζ1 = 0 and in the absence of imposed damping,  π S f ω1   , ζ2 = 0  2 ζ1   σ12 = (50) 4 2 π S f ω1    1 + ζ2 , ζ1 = 0. 2 ζ2 µ

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413

It is seen that the case of no applied damping gives a denominator of 2ζ1 , just like the case of force excitation of a single-degree-of-freedom system [16]. In contrast, for vanishing structural damping the optimal applied damping √ ζ2 = µ/2 gives a denominator of ζ2 . Thus, when applied damping dominates, its effect is only half of the initial structural damping. For a general combination of damping its effect can be expressed in terms of an effective damping ratio ζeff , defined by analogy with the formula for structural damping alone, as σ12 =

π S f ω1 . 2 ζeff

(51)

It follows from (49) that the effective damping is given by ζeff = ζ1 +

µ ζ2 . µ + 4ζ2 (ζ1 + ζ2 )

(52)

Minimum response variance is obtained for maximum effective damping, and thus the last term in (52) should be maximized. When minimizing its reciprocal, the result can be read off directly as √ µ , ζ1 6= 0. (53) ζ2,opt = 2 This leads to the interesting conclusion that the magnitude of the optimal applied damping depends only on the mass ratio, but is independent of the structural damping. The corresponding effective damping ratio is ζeff ,opt = ζ1 +

ζ22 (ζ1 + ζ2 )2 = ζ1 + 2ζ2 ζ1 + 2ζ2

(54)

with the optimized response variance 2 σ1,opt =

π ζ1 + 2ζ2 S f ω1 . 2 (ζ1 + ζ2 )2

(55)

It is most convenient to illustrate the combined effect of structural and applied damping in terms of the effective damping ratio. Fig. 2 shows the development of the effective damping ratio as a function of the applied damping ζ2 for different values of the structural damping ζ1 . Note, that for ζ2 = 0 the effective damping is equal to the structural damping, and thus the structural damping for each curve can be read off its intersection with the vertical axis. The fully drawn curves give the results from the approximate formula (54) where ζ2 is defined from the mass ratio by (53). The dots indicate what the effective damping would be, if this mass ratio was given, and frequency tuning α as well as damping ratio ζ2 were then optimized to find the precise minimum of the response variance σ12 . The optimal values were found by a simple numerical search. It is seen that the approximate procedure consisting in use of classic frequency tuning, followed by optimized damping ζ2 by (53) and the approximate formula (54) gives a response variance that is just about indistinguishable from the exact minimum value. The curve for zero structural damping is the straight line ζeff = 12 ζ2 , also known from the case of harmonic loading [15].

Fig. 2. Effective damping ratio ζeff in % for force excitation. Full lines for explicit solution (54) and dots for optimal numerical solution for given µ. Structural damping: ζ1 = 0%, 2%, 5% and 10%.

The curves for cases including finite structural damping appear to exhibit asymptotic behavior for increasing applied damping ζ2 parallel to this line but at a slightly lower level than suggested by the initial value of structural damping alone. An explicit asymptotic formula can be found by writing the formula (54) for the optimal effective damping in the alternative form 1 1 ζ1 ζ2 ζeff ,opt = ζ1 + ζ2 − . 2 2 ζ1 + 2ζ2

(56)

It follows from this formula that for the typical case of ζ2  ζ1 the last term contributes − 14 ζ1 , leaving the effective damping as 1 3 ζ1 + ζ2 , ζ1  ζ2 . (57) 4 2 Thus, in the typical case of relatively small structural damping it contributes with 34 ζ1 , while the applied damping contributes 1 2 ζ2 to the combined effective damping. ζeff ,opt '

4.2. Support acceleration In the case of support acceleration use of the classic frequency tuning for harmonic load (33) leads to the following form of the numerator of the expression (23) for the response variance, Pa = µ(αζ1 + ζ2 ) + µ2 αζ1 + 4(1 + µ)ζ2 (ζ1 + ζ2 )(αζ1 + ζ2 ).

(58)

Typically the mass ratio is small and the structural damping considerably smaller than the applied damping. Therefore the second term is small relative to the first. Omission of the second term leads to the factored form, Pa ' [µ + 4(1 + µ)ζ2 (ζ1 + ζ2 )] (αζ1 + ζ2 ).

(59)

The polynomial Q in the denominator of (23) has already been factored in (48). When using the approximate factored forms the following expression is obtained for the response variance σ12 '

π Sa ω1 (1 + µ) µ + 4(1 + µ)ζ2 (ζ1 + ζ2 ) . 2 ζ1 + ζ2 µ + 4ζ1 ζ2

(60)

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Fig. 3. Effective damping ratio ζeff in % for support acceleration. Full lines for explicit solution (64) and dots for optimal numerical solution for given µ. Structural damping: ζ1 = 0%, 2%, 5% and 10%.

This expression contains the exact limit for ζ1 = 0. A substantial simplification follows, if the factor (1 + µ) is introduced in the last term of the denominator. For small mass ratio this term is of higher order, and the exact limit for ζ1 = 0 is still recovered. The resulting expression σ12 '

π Sa ω1 (1 + µ) 2 ζ1 + ζ2

µ 1+µ

+ 4ζ2 (ζ1 + ζ2 )

µ 1+µ

+ 4ζ1 ζ2

(61)

is very similar to (49) for the case of force excitation. In fact the only two differences are the factor (1 + µ) and replacement of the mass ratio µ with µ/(1 + µ) in the last fraction. This implies that the optimal value of the applied damping follows from (53) by a simple parameter replacement, r µ 1 , ζ1 6= 0. (62) ζ2,opt = 2 1+µ It is seen that the introduction of higher-order corrections for ζ1  1 has led to an optimal applied damping that is independent of the structural damping. In the case of support acceleration it is convenient to define the effective damping by the relation σ12 =

π Sa ω1 (1 + µ) . 2 ζeff

(63)

When using the optimal applied damping (62) the corresponding effective damping ratio is ζeff ,opt =

(ζ1 + ζ2 )2 ζ1 + 2ζ2

(64)

leading to the optimized response variance 2 σ1,opt =

π ζ1 + 2ζ2 (1 + µ) Sa ω1 . 2 (ζ1 + ζ2 )2

(65)

In the case of support acceleration the expressions for optimal imposed damping and the resulting optimized response variance are exact for ζ1 = 0, but do not recover the exact results in the opposite limit ζ2 = 0. However, the expressions are quite accurate for cases of most interest for design of

tuned mass absorbers, where the mass ratio is in the order of a few percent, and the structural damping is small. The accuracy is illustrated in Fig. 3. The full lines are the results of the approximate formula (64), and the points are the results of full optimization with given mass ratio. The approximate formulation is seen to be very accurate for the full range of imposed damping, when the structural damping is less than about 5%. When the structural damping is larger, some difference occurs for small imposed damping. There are two approximations involved in the explicit expression for the effective damping ratio ζeff , the classic frequency tuning for harmonic load is used, and some approximations are introduced in the rational expression (23) for the variance. The dashed curves show√the results from using the classic frequency tuning and ζ2 = 12 µ/(1 + µ) in the exact formula for the response variance. It is seen that the deviation between true optimum and the explicit approximation is due to the different tuning. In fact, the approximations introduced to obtain the factored form leads to a slight improvement of the approximation. 5. Conclusions An investigation has been made on a system consisting of a structural mass with damping and stiffness, and a tuned mass absorber. It has been demonstrated that for systems without structural damping the traditional frequency tuning based on harmonic load [3,15] leads to response variances that in most cases of practical interest are so close to the exact minimum, that simplifications obtained by this frequency tuning seem justified. Based on this frequency tuning a simple set of formulae is developed for the optimal applied damping and the resulting combined damping. The optimal value of the imposed damping is independent of the structural damping and given by √ the mass ratio as ζ2,opt ' 12 µ, i.e. a value somewhat lower than for harmonic force loading [3,15]. The approximate, yet quite accurate, formula for the effective damping provides an interpolation from the case of structural damping alone, acting with its full value ζeff ' ζ1 to the case where the imposed damping dominates and the effective damping of the structure is well represented by ζeff ' 43 ζ1 + 12 ζ2 . Thus, the absorber can be calibrated without considering the structural damping, that simply shows up in the form of a reduced resulting response variance due to the contribution of the structural damping to the effective damping of the combined system. References [1] Frahm H. Device for damped vibrations of bodies. U.S. Patent No. 989958. October 30, 1909. [2] Ormondroyd J, Den Hartog JP. The theory of the dynamic vibration absorber. Transactions of ASME 1928;50(APM-50-7):9–22. [3] Den Hartog JP. Mechanical vibrations. 4th edn. New York: McGraw-Hill; 1956. (Reprinted by Dover, New York, 1985). [4] Brock JE. A note on the damped vibration absorber. Journal of Applied Mechanics 1946;13:A284. [5] Crandall SH, Mark WD. Random vibration. New York: Academic Press; 1967.

S. Krenk, J. Høgsberg / Probabilistic Engineering Mechanics 23 (2008) 408–415 [6] Jacquot RG, Hoppe DL. Optimal random vibration absorbers. Journal of the Engineering Mechanics Division ASCE 1973;99(E3):936–42. [7] Ayorinde EO, Warburton GB. Minimizing structural vibrations with absorbers. Earthquake Engineering and Structural Dynamics 1980;8: 219–36. [8] Warburton GB. Optimum absorber parameters for minimizing vibration response. Earthquake Engineering and Structural Dynamics 1981;9: 251–62. [9] Warburton GB. Optimum absorber parameters for various combinations of response and excitation parameters. Earthquake Engineering and Structural Dynamics 1982;10:381–401. [10] Thompson AG. Optimum tuning and damping of a dynamic vibration absorber applied to a force excited and damped primary system. Journal of Sound and Vibration 1981;77:403–15. [11] Tsai H-C, Lin G-C. Optimum tuned-mass dampers for minimizing steady-state response of support-excited and damped systems. Earthquake Engineering and Structural Dynamics 1993;22:957–73.

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