VIBRATION ISOLATION: MOMENTS AND ROTATIONS INCLUDED

VIBRATION ISOLATION: MOMENTS AND ROTATIONS INCLUDED

Journal of Sound and Vibration (1996) 198(2), 171–191 VIBRATION ISOLATION: MOMENTS AND ROTATIONS INCLUDED M. A. S Department of Applied Acous...

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Journal of Sound and Vibration (1996) 198(2), 171–191

VIBRATION ISOLATION: MOMENTS AND ROTATIONS INCLUDED M. A. S Department of Applied Acoustics, Chalmers University of Technology, S-412 96 Gothenburg, Sweden (Received 5 July 1994, and in final form 23 April 1996) By using simplified expressions, measures of vibration isolation efficiency, for any number of isolators situated between two structures, source and receiver, including all six degrees of freedom (DOFs) of motion can be predicted. These expressions can be used to calculate, for example, transmissibilities and power input and transmission to the receiving structure. Here, two DOFs of motion are considered to simplify the study of translations contra rotations. Generally, the structures are described by their receptance (mobility or accelerance) matrices and the isolators are described as massless springs with losses in order further to simplify the analysis. Two theoretical cases are examined. The first is a parameter study for a single isolator connecting different structures. The second example consists of two beams connected via two isolators to simulate the behaviour of a resonant machine mounted on a resonant foundation. Level errors, in the vibrational power transmitted to the receiving structure, are predicted for when rotational isolator stiffness values are not considered. The effects of disregarding the power, associated with rotational DOFs, from one structure to another via isolators can lead to over- and under-estimations of the total power transmission. Finally, some results from verification measurements, for two beams connected via two isolators, with a force excitation and a combined force–moment excitation, are presented. In order to obtain good agreement with the measurements, it was found necessary to include the mass of the isolator in the theoretical calculations. 7 1996 Academic Press Limited

1. INTRODUCTION

This study deals with measures of vibration isolation in which rotations and moments are included in the analysis, either theoretical or experimental. Methods and instruments for experimentally determining rotations and moments exist and the added information that can be gained by including these quantities need not be ignored. The behaviour of two structures connected by a single isolator is considered in order to examine the effects of rotations in association with translations in a manner comparable to that of previous studies of vibration isolation problems in which only translations are considered (or sometimes a torsional motion). Two DOFs of motion are considered; a translational and a rotational associated with bending waves in one plane. A simple and easily programmable set of equations for general multi-point connections between two structures for all six DOFs of motion is given in Appendix 1. The behaviour of two isolators in combination with two beams is examined numerically and experimentally. This case is used to represent the behaviour of a more complex system such as a resonant machine mounted on a resonant foundation, where a correlation between the isolators and non-rigid motion exists. Often, when choosing isolators only the mass and operational frequency range of the machine to be isolated (and sometimes the 171 0022–460X/96/470171 + 21 $25.00/0

7 1996 Academic Press Limited

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center of gravity) are considered: i.e., chosen only for a force acting on the isolators in one direction. This works well if the machine and the foundation can be considered to be rigid in the frequency range of interest and if the excitation truly acts in only one direction. One also has to assume that an individual isolator can be used to represent the behaviour of a group of isolators. However, a coupling between force and rotation, as well as moment and displacement, is pronounced in certain structures, such as finite beams and finite thin plates. Classical (here meaning including only one DOF of motion or only the translational DOFs of motion) vibration isolation prediction will not suffice, especially if the excitation covers a broad frequency range and also includes moments. Two types of ideal sources exist: excitation sources (generalized forces) and motion sources (generalized displacements). For measurement purposes it is simple to measure accelerations (easily converted to displacements or velocities) but in a theoretical analysis both types of source are readily used in calculations. In reality, a combination of both an excitation and a motion source can be expected to exist in a typical installation. Theories for vibration isolation or the attenuation of vibration date back to the early years of the twentieth century. It was found that isolators made of rubber or rubber-like materials, when designed and used in a certain way, could somewhat insulate the vibratory motion of a machine from a foundation. Comprehensive reviews of the literature concerning many aspects of vibration isolation can be found in references [1, 2]. The work described in, for example, references [3–5] demonstrates that vibration isolation predictions can be improved and can be made more reliable at higher frequencies by including the dynamic characteristics of the source, receiver and isolators which can be represented by their mobilities or blocked impedances. However, it is often the case that only translatory motion is considered and that moment excitations are dealt with only cursorily for the case of rigid body motion. This development can mainly be attributed to four reasons: only single-DOF systems are studied; symmetry is used in such a manner that DOFs are uncoupled; moments and rotations are not considered as important as forces and translations especially at lower frequencies; and translational motion with no cross-coupling between DOFs is assumed. It is shown in, for example, references [6, 7] that moments and rotations play an important role in power transmission and should generally be considered in vibration isolation analyses. If one considers only a single DOF, the power can be calculated as one half of the real part of the generalized force times the complex conjugate of the velocity in the same DOF. Therefore, for the purpose of comparison, the frequency dependence of the absolute velocities (translational and rotational) of bending waves, for unit forces and moments, at the free end of a semi-infinite beam and in an infinite thin plate [8] are shown in Figure 1. In calculating the rotational velocity of the plate due to a moment excitation, simple theory requires a force couple separated by a distance, which was taken arbitrarily as 0·01 m. It can be seen that the translational velocity decreases with frequency for beams and is constant for plates, and that the rotational velocity increases with frequency for both beams and plates. This demonstrates that rotations become increasingly important (and thus also their role in power transmission) with increasing frequency for common engineering structures. Previous workers have studied vibration isolation including more than one DOF of motion, often considering them as uncoupled or including only the translational stiffness values (see, for example, reference [9]). Usually, however, moments and rotations are not included, or they are considered only for rigid body motion. In this study they are treated as cognate to forces and translations and, from the outset, no biased assumptions are made.

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Figure 1. Normalized translational (——) and rotational (- - -) velocity magnitudes for a semi-infinite beam and an infinite plate. The beam has a cross-sectional area of t × t and the plate has thickness t.

2. THEORY

2.1.   In Figure 2 is shown an isolator (considered as a massless spring) inserted between two structures, the positive directions of the translational and rotational motions due to force and moment excitations are defined. Harmonic time dependence ejvt with circular frequency v is assumed. Note that structure 1 is shown as a beam- or plate-like object. If structure 1 is a machine or an object with a general configuration, then the forces, moments, displacements and rotations are considered at the locations at which the isolators are connected. In terms of the notations and sign conventions in Figure 2, the equations of motion for the system can be conveniently written by using receptance formulations: i.e., matrix [A] for the isolator connection point to structure 1, matrix [B] for the isolator connection point to structure 2 and a stiffness formulation, matrix [K], for the isolator (see the discussion in reference [10]). One then has, in compact notation, {x1 } = [A]{F1 − F2 },

[K]{x1 − x2 } = {F2 },

{x2 } = [B]{F2 },

(1–3)

Figure 2. A model of an isolator inserted between two structures. Structure 1 is considered as the source and gives rise to force, moment, translational and rotational components in all DOFs (x, y, z, a, b and g) of motion. Structure 2 is the receiving structure and also has response components in all DOFs of motion.

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where x = {x, y, z, a, b, g} is a vector of displacements and rotations, F = {Fx , Fy , Fz , Ma , Mb , Mg }T is a vector of forces and moments, and the subscripts 1 and 2 refer to the structure number. Matrices [A], [B] and [K] are of the order 6 × 6 in the general case for a single isolator, corresponding to six DOFs of motion. The equations of motion are readily expanded for N isolators involving transfer receptance matrices between isolator connection points, giving the total matrix size of the order 6N × 6N (see Appendix 1). The number of DOFs of motion can be reduced for some common engineering structures, such as beams and plates, or if the vibratory machine is considered as a rigid body, by using its mass and moments of inertia. In experimental work the elements of the matrices [A], [B] and [K] can be determined. For many common engineering structural elements, expressions for the receptances can be found in the literature. Also, a number of different isolator models exist that can be used for calculating the elements of the dynamic stiffness matrix. A list of symbols is found in Appendix 3. The structures can be modelled as rigid bodies with mass and rotational inertia, beams or plates (finite, semi-infinite and infinite) with various boundary conditions, by using the finite element method or by using measured data where receptance (mobility or accelerance) matrices are the required quantity. Structure 1 is usually considered as a vibratory machine or a construction detail, such as a frame or footing, on which a machine is attached or fixed and structure 2 as a resilient foundation. A measure of stiffness K, for each possible response to excitation, is required. For certain types of isolators a number of elements in the stiffness matrix are non-existent or are not coupled to one another. This can be conceptually imagined by considering a cylindrical isolator in which a rotation would not arise from a central excitation force. For a general isolator, however, a force or a moment excitation gives responses in each DOF. For an excitation source only forces or moments or a combination of forces and moments are applied to structure 1 and are assumed as known quantities. By using expressions (1)–(3), the unknown forces, moments, displacements and rotations can be calculated as T

{F2 } = [C]{F1 },

{x2 } = [B][C]{F1 },

{x1 } = [A][[I] − [C]]{F1 },

(4–6)

where [C] = [[A] + [K]−1 + [B]]−1[A]. Damping can be introduced through the use of a complex Young’s modulus. The elements of the dynamic stiffness matrix, describing the isolator, can also be calculated by using a number of different isolator models, measured under the conditions in which they are to be used or as massless springs with constant stiffness values. 2.2.     A variety of measures of isolator performance are found in the literature; two are considered here. The first is the transmissibility, because its measurement can readily be used to validate theoretical models. The second is transmitted power, because it is a design quantity that it is appropriate to minimize, which takes into account both generalized forces and generalized velocities. 2.1.1. Transmissibility Transmissibility is defined as the magnitude of the ratio of quantities from each extremity of the isolator—for example, the displacement and rotation transmissibility levels in the x- and b-directions: Tdx = 20 log =x2 /x1 =

and

Trb = 20 log =b2 /b1 =

dB.

(7, 8)

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Figure 3. (a) A simple model of a force excited machine modelled as a rigid body with mass m, mounted via a lossy spring with stiffness K, on a rigid foundation. (b) The associated force transmissibility level.

Note that expressions equivalent to equation (8) are defined for rotating structures such as shafts and axles by previous workers; however, here the rotation is not limited to a certain DOF of motion: i.e., bending moments and their associated rotations are considered. Also, similar expressions for forces and moments can be defined. A simple and also classic vibration isolation example is of a rigid machine, with mass m, mounted on a rigid foundation, and having a single DOF, as is depicted in Figure 3(a). Expression (4) can be used with the following substitutions: [A] = −1/v 2m, [B] = 0 and v02 = K/m, with f0 = v0 /2p to give the force transmissibility; see Figure 3(b). Damping can be included in the isolator stiffness value if required. A number of tendencies that are found in many transmissibility plots can also be observed in Figure 3(b). For example, much below the fundamental resonance frequency f0 , the machine and foundation act as if rigidly connected; as the frequency increases there is an amplification of the excitation force at the foundation until f/f0 = z2; and with a further increase in frequency an isolation of the excitation force from the foundation occurs which increases at a rate of 12 dB per octave. Expressions (7) and (8) relate quantities for one DOF and for an individual isolator. It is possible, for example, to consider the transmissibilities for a number of isolators considered as a group, a number of DOFs of motion lumped together, between different isolators, or a combination of these options, to calculate the transmissibility; however, this gives rise to the possibility that significant responses may be smeared and lost, or that insignificant responses occurring at individual isolators or in a single DOF of motion are exaggerated.

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Expressions for calculating the transmissibilities for two structures connected with a single general isolator for two DOFs of motion are found in Appendix 2. 2.2.2. Power The input power, superscript i, and transmitted power, superscript t, are important measures in isolation design [11] and can be calculated by using Pi = (v/2) Re ({F1 }T{x1 }*)

and

P t = (v/2) Re ({F2 }T{x2 }*).

(9, 10)

To observe better the effects of including rotational stiffness in calculations, the error in the power transmission can be determined by setting all the rotational stiffness values in the stiffness matrix, [K], essentially to zero (exactly zero is not possible, due to the fact that [K] must be inverted). The following expression is therefore used for determining the error level in the power transmission when rotational stiffness values are not included in predictions: Le = 10 log (P t) − 10 log (P t=Ka ,b ,g = 0 ).

(12)

The real parts of the receiving structure’s force and moment mobilities are used to define a non-dimensional stiffness ratio, k, relating the isolator’s translational and rotational stiffness values, here in two DOFs, as k = Kxx Re (jvBxx )/Kbb Re (jvBbb ).

(13)

In this manner, k incorporates the receiving structure’s dynamic characteristics that are related to the active power, and its value can attest to the degree of the receiving structure’s sensitivity to the rotational stiffness of an isolator. Higher values generally indicate less sensitivity. Another measure of isolation performance that is inversely proportional to transmissibility is effectiveness (insertion loss) [4], but is not dealt with in this study. 2.3.   Two basic limits, when using simple bending wave theory, can be stated according to reference [8]. The first is when the contact area (described by a diameter D) between an isolator and a structures can be considered a point, kD E 2p/10,

(14)

and the second is when simple bending wave theory for beams or plates is applicable, kD E 2pD/6t,

i.e. lB e 6t,

(15)

where k is the bending wavenumber, t is the thickness of a beam or plate, and lB is the bending wavelength. These limits were not exceeded in the experimental part of the study. A further limitation is involved by considering the isolators as simple massless springs. This places a limit on the maximum frequency range, depending on the relationship between the structure mass and isolator mass. This fact is less important for the theoretical considerations of this study, but not for the experimental part, since the main focus is towards the effects of including moments and rotations. In a more detailed study, the isolators should be treated as black boxes, where the blocked impedance characteristics of the input and output terminals and the transfer impedance between them must be determined. At low frequencies the dynamic stiffness of an elastomer can often be approximated well by using 1–2 times the value of its static stiffness and allowing its damping factor to vary only slightly with frequency [1, 12].

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3. NUMERICAL EXAMPLES

The question of how important it is to include rotational stiffness values and moment excitations depends on a number of factors, such as source characteristics, receiver characteristics and the frequency range of interest. Simple examples are studied which illuminate some situations in which it is important to include moments and rotations in vibration isolation prediction. Since the massless springs are allowed to have rotational stiffness values, the cross-receptances, for example, Axa , must also be considered even for pure force or moment excitations, since in general, a force or moment excitation will cause both translational and rotational motions. However, the isolator’s cross-stiffness value is set to zero as for an ideal cylindrical isolator. The first example (section 3.1) is an examination of the power error level, as given in expression (12) for different combinations of source and receiver structures and for a pure force and moment excitation in respect to k, expression (13). The second example (section 3.2) is of a more realistic nature, in that the model can be modified to represent situations and complexities that can be found in typical applications. Here a machine is considered which provides symmetrical force excitations at two points. Three excitation cases are compared: equal forces in phase, equal forces 180° out of phase and non-equal forces acting in unison. For both examples a translational and a rotational DOF, sufficient to describe bending wave motion in the structures, are considered. The results are specific for the cases studied and should be viewed as guiding. It is complex to examine all possible structure–isolator–structure combinations with different source characteristics and all six DOFs of motion in a meaningful way. This is due to the variety of combinations and the different dimensions of the quantities involved. The following structure and isolator data were used: thickness, t = 0·02 m; width, isolator diameter and length, w = D = l, 0·1 m; beam length, L = 1 m; density, r = 7850 kg/m3; Young’s modulus E = 205 × 109(1 + jh) Pa, with loss factor h = 0·05. The static isolator stiffness values are based on Young’s modulus E = 7 × 106(1 + jh) Pa, with loss factor h = 0·08 for a cylindrical elastomer, where Kxx = pED 2/4l and Kbb = pED 4/64l. 3.1.   Excitations at the free end of a semi-infinite beam are used as the source. This is similar to a non-rigid structure in which both translations and rotations are imparted to the isolator. Three cases of configurations of the receiving structure are examined, each comparing the finite behaviour with a corresponding infinite behaviour; see Figure 4. The first is when the receiving structures, a semi-infinite beam and a finite beam with length L, have the same orientation; see Figures 4(a) and (b). The second case—see Figures 4(c) and (d)—is similar to the previous one except that the beams have been rotated 180°. The only difference is the sign of the cross-receptances of the receiving beams. The third case—see Figures 4(e) and (f)—is when the isolator is connected at the middle of an infinite beam and a finite beam: i.e., the cross-receptances of the receiver beams are zero. There exists an infinite group of excitation possibilities involving combinations of forces and moments as well as their placement. As examples, a pure force and a pure moment, respectively, are applied at the isolator to examine their effects, combined with the exclusion of the rotational stiffness values, on the power transmission error levels. An ideal force excitation is often considered in studies concerning vibration isolation; therefore it

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is often assumed that the rotational DOFs of motion can be neglected. Moment excitations are in theory equally possible and are also considered here; see, for example, references [7, 13]. In Figures 5(a)–(c) the power transmission error levels with respect to the stiffness ratio, k, are plotted for a force excitation, and in Figures 5(e) and (f) for a moment excitation. Figures 5(a) and (d) correspond to the configurations, Figures 5(b) and (e) to the configurations and Figures 5(c) and (f) to the configurations. It can be seen in Figures 5(a)–(c) that rotational stiffness values, for finite structures, should be considered for values of k less than approximately 10–100 in order to obtain a negligible power error level. This value is typical for the cases examined in this paper; however, it would be venturesome to state that this range is universal without further study. What is certain, though, is that for individual values of k the error level can be positive or negative. By comparing Figures 5(a) and (b) or Figures 5(d) and (e), the effect of the sign of the cross-mobility on the error level can be seen. When the cross-mobilities have the same sign (Figures 5(a) and (d)), the error level for the infinite system is monotonically increasing with a decreasing k. When the cross-mobilities have opposite signs (Figures 5(b) and (e)), there is a dip in the error level at approximately k = 0 before monotonically increasing. In Figure 5(b) the error level is positive for k between values of 1 and 2, while in the same range in Figure 5(e) the error level varies between positive and negative values, suggesting that a type connection is more sensitive to moment excitation than force excitation. When the cross-mobilities of the receiver structures are zero—see Figures 5(c) and (f)—the character of the error level changes completely. For the case of a pure force excitation (Figure 5(c)), the error level depends on the resulting moment excitation on the receiver beam due to the neglect of the isolator’s rotational stiffness. There is no coupling between rotational motion and force excitation at the isolator on the receiver beam because the cross-mobility of the beam and the cross-stiffness of the isolator are zero. By comparing Figures 5(a) and (d) or Figures 5(c) and (f), the effect of the type of excitation can be seen. The peak error levels are shifted by approximately 5 dB when changing from a force excitation to a moment excitation. This is not the case for the configuration. The behaviour of this beam configuration becomes more and more similar to that of a centrally excited beam as the connection between the receiver and source

Figure 4. Structure–isolator–structure configurations with a semi-infinite beam source. configurations: (a) semi-infinite beam receiver and (b) finite beam receiver; configurations; (c) semi-infinite beam receiver and (d) finite beam receiver; configurations: (e) infinite beam receiver and (f) finite beam receiver.

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Figure 5. Theoretical power transmission error levels as a function of k for the configurations. Force excitation: (a) ; (b) ; (c) . Moment excitation: (d) ; (e) ; (f) . ——, Finite receiver beams; - - -, semi-infinite or infinite receiver beams.

structure becomes increasingly more rigid: i.e., the cross-mobility of the configuration tends to zero. The error levels are shown as a function of kL for the respective configurations for a pure force excitation in Figures 6(a)–(c) and for a pure moment excitation in Figures 6(d)–(f). k was kept constant at unity.

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In general, the error levels are increasing with increasing kL until kL 1 30, with the exception as seen in Figures 6(c) and (f) where, because of the chosen central connection point, the error levels are less than for the other configurations. In Figure 6(b) the error level decreases quickly above kL 1 30 and in Figures 6(e) the error level increases in the negative direction only at high values of kL. Also, at kL 1 0·6 there is a peak in the error level for the configuration with the finite beam. This is thought to be due to a rigid body-like motion behaviour, where the lower beam is rocking

Figure 6. Theoretical power transmission error levels as a function of kL for the configurations with k = 1. Force excitation: (a) ; (b) ; (c) . Moment excitation: (d) ; (e) ; (f) . ——, Finite receiver beams; - - -, semi-infinite or infinite receiver beams.

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Figure 7. Two isolators, a and b, connecting two finite beams with lengths L and 2L; case 1.

back and forth. This effect can also be observed in Figure 6(f) for the case of moment excitation. The error level peaks, for the resonant structures, occur at the resonance and anti-resonance frequencies. For the infinite structures, the average behaviour is observed except as noted for Figures 6(b) and (e). The errors at the resonance frequencies must be considered dimensionally, since this concerns the active power transmission. 3.2.       In Figure 7 are depicted two free–free beams with lengths L and 2L connected via two identical isolators a and b. This example is used to simulate some of the complexities of a vibratory machine (upper beam) mounted on a foundation (lower beam). Three excitation cases are compared: equal forces 180° out of phase, i.e., a moment equal to FL/3 at the center (case 1); equal forces in phase acting in unison, i.e., a force equal to 2F at the center (case 2); and different forces with a three to one ratio, i.e., a moment equal to FL/6 and a force equal to 2F at the center (case 3). These three excitation cases can approximately represent a rotating machine mounted on four isolators. The first excitation case models two isolators on opposite sides of the rotating axle, the second excitation case models two isolators on the same side of the axle, and the third is a more general excitation somewhere between cases 1 and 2. The excitations at the isolators will thus be comprised of generalized motions. The non-dimensional stiffness ratio is used to examine the effects of including rotational stiffness values. In Figure 8(a)–(c) are shown the power transmission error levels as a function of k for excitation cases 1–3 respectively. Some observations can be made by studying Figure 8. As k increases the error level generally becomes smaller. Case 1 results in error levels that are mostly positive and case 2 results in error levels that are mostly negative, while case 3 results in error levels that are approximately equally positive and negative. Under the right conditions the additive effect of cases 1 and 2 may be thought to result in negligible total power transmission error levels; however, case 3 suggests instead an equipartition of the error level between positive and negative values. The error level associated with a particular value of k is multi-valued, depending on the behaviour of the structure at a certain frequency. This is examined next. In Figure 9 are shown the power transmission error levels as a function of a non-dimensional wavenumber for the three excitation cases with k = 1. For comparison the force-, moment- and cross-mobilities at isolator a on the lower beam, normalized by the mobilities at the end of a semi-infinite beam with the same properties, are shown in Figure 10. In Figure 9, in the range of kL and for k = 1, it can be seen that the error level is small for low values of kL, implying that very little power is transmitted to the foundation via the rotational DOFs, and thus the rotational stiffness values do not need to be considered. For example, at the first structural resonance at kL = 1 there is virtually no error seen for cases 2 and 3, and an error of −1 dB for case 1. In the middle range a maximum and

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Figure 8. The theoretical power transmission error level as a function of k: (a) case 1; (b) case 2; (c) case 3.

minimum error level are evident. With increasing kL the positive error level is generally decreasing and the negative error level is generally constant to slightly decreasing. For increased values of k the results are approximately the same in form except that the error levels are decreased. Case 3 is here better indicated as giving average results that never exceed the peak error levels of cases 1 and 2.

Figure 9. Theoretical power transmission error levels as a function of kL: k = 1. ——, Case 1; – – –, case 2; – · – , case 3.

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Figure 10. Normalized mobilities of the receiver beam calculated at isolator a. ——— , force mobility; ——, moment mobility; – – –, cross mobility.

As suggested previously, a correlation between receiver mobility level and the transmission power error level can be found by comparing Figures 9 and 10. The maximum and minimum error levels occur at resonance and anti-resonance frequencies. For example, the cross-mobility anti-resonance frequencies correspond well with the minimum error levels for case 2 for values of kL greater than 10. At these frequencies, anti-resonances are also found for the force and moment mobilities, or their magnitudes are equal. For case 1, a moment-like excitation, the lowest error level peak values occur when the normalized moment mobility has the largest peak values. The maximum error levels occur at the structure’s resonance frequencies, where peak values in the active power transmission normally occur.

4. EXPERIMENTS

The experimental set-up is depicted in Figure 11. The measured and predicted velocities at c on the lower beam caused by a force excitation at c on the top beam are shown in Figure 12. It was found to be necessary to include the mass of the isolators (twice 0·312 kg) as loads; however, their rotary inertia was not found to be essential in the frequency range studied. The data used in the calculations are as follows: top beam, length L = 1·4 m, thickness t = 0·012 m, width w = 0·05 m; lower beam, length L = 1 m, thickness t = 0·01 m, width w = 0·04 m; both beams, density r = 7850 kg/m3 and Young’s modulus E = 203 ×

Figure 11. Two isolators, a and b, connecting two finite beams. Transmissibilities, for force and force–moment excitations at isolator a, are measured and calculated. The velocity at point c on the lower beam is measured and calculated for a force excitation on the upper beam. Dimensions are given in millimeters.

. . 

184 –10 –15

Lv (dB re 1 m/s)

–20 –25 –30 –35 –40 –45 –50 –55 –60

50

100

150

200 250 300 350 400 450 Frequency (Hz) Figure 12. The velocity level, Lv , calculated at point c on the lower beam for a force excitation at point c on the top beam. ——, Measured; – – –, predicted.

109(1 + jh) Pa with loss factor h = 5 × 10−4. The isolator data used in the calculations are as follows: diameter and length D = l = 0·05 m, translational stiffness Kxx = 5·5 × 105(1 + jh) N/m, cross-stiffness Kxb = 0, rotational stiffness Kbb = 86(1 + jh) Nm with h = 0·08. The static stiffness values are based on Young’s modulus E = 7 × 106 Pa for a cylindrical isolator. It was assumed that the dynamic stiffness value was twice the static stiffness value and that the isolator functions ideally, so that force excitations would not give rise to a rotational motion. The measured and the theoretical transmissibilities for both isolators are shown in Figures 13(a)–(h). The displacement transmissibilities were measured directly on the beams using two accelerometers located above and below the respective isolator. The rotational transmissibilities were, however, derived from two consequent measurements by a single Kistler TAPTM, which was used to measure directly the rotational acceleration. Since the rotational amplitudes are smaller than the translational amplitudes, especially with a force excitation, the signal to noise ratio was found to be poor in some frequency ranges, especially the lower ones. The excitations supplied to the structure were a force and a combined force–moment. The force and combined force–moment excitations were applied only at isolator a, this being due to the amount of available equipment. This, however, allowed for two situations for possible comparison: a direct excitation at isolator a and a remote excitation for isolator b. The force–moment excitation (see reference [14]) was accomplished by using a T-like configuration with different force amplitudes delivered by the two exciters. In Figures 13(a)–(h) it is demonstrated that both displacement and rotational transmissibilities can be predicted with reasonable accuracy when using simplified theory for the isolators. Two frequency ranges, however, gave noticeably poor results: f 1 300–500 Hz in Figure 13(d) and f 1 350–450 Hz in Figure 13(h). This could be due to a slight misplacement of the rotational transducer, which is very sensitive to positioning, or to difficulties in modelling the actual excitation. Also, a frequency shift that increases with frequency can be seen when using the assumption that the isolators are massless. It can also be seen in Figures 13(a)–(h) that the transmissibilities are quite different for both isolators, and that one could not consider them as lumped together as a single isolator. For example, by comparing Figures 13(a) and (b) it can be seen that the peaks at

 

185

Figure 13. Displacement transmissibilities for a force excitation: (a) isolator a; (b) isolator b. Rotational transmissibilities for a force excitation: (c) isolator a; (d) isolator b. Displacement transmissibilities for a combined force–moment excitation: (e) isolator a; (f) isolator b. Rotational transmissibilities for a combined force–moment excitation: (g) isolator a; (h) isolator b. ——, Measured; · · · , predicted; – – –, predicted with the isolator’s mass loading included.

approximately 240 Hz and 390 Hz for isolator a become anti-peaks for isolator b. Also, by comparing Figures 13(a) and (c) one can see that the same peaks in the displacement plots are troughs or flanks in the rotational plots.

186

. .  5. DISCUSSION

This study shows, for some examples of connected beam structures, that moments and rotations can be important factors in vibration isolation prediction. Also observed is the fact that rotational DOFs of motion can be important at any frequency, depending on the value of a non-dimensional stiffness ratio k. The extension of this work to general structures can be carried out by using the expressions given in Appendix 1. For predictions, the point and cross-mobilities of both structures at connection points (also excitation and response points) are used of necessity, since an isolator can couple translational to translational, translational to rotational, rotational to translational and rotational to rotational motion from one structure to the other. The transfer mobilities between excitation or response point to all connection points are also required. Thus changing the isolator position will give different results, depending on the mobilities at the new connection points. The mobilities of the source and receiver structure were emphasized by using three configurations of isolator connected beams. These three configurations demonstrate, for two DOFs, some of the behaviour of a single isolator connecting two structures where, in principle, the basic difference involves the relationship between the receiver and the source structures’ cross-mobilities. When the cross-mobility on the receiver structure is non-zero, the total error level is due to three effects for the case of a pure force excitation. The resulting moment excitation due to the neglect of the isolator’s rotational stiffness, the resulting rotational motion at the isolator on the receiver beam from the force excitation and the resulting translational motion at the isolator on the receiver beam that is caused by the supplementary moment excitation. The sum of these effects can be additive and can each contribute to the total error level, or they can somewhat or completely cancel each other out, resulting in low or zero error levels. That is why the great variations from positive to negative occur in the error levels for different values of k for resonant structures. This additive effect of the contributions to the total error level can also be expected to occur for those errors associated with the omission of the isolator’s cross-stiffness values. The parameter k can be used to examine the influence of rotational stiffness values with respect to the real part of the receiving structure’s point mobilities and to determine if rotational DOFs of motion need be considered in an analysis. For example, increasing the thickness of the receiving structure also increases the value of k. For the beam structures studied, rotational DOFs of motion should be considered for values of k less than approximately 100, in order to calculate the correct values of power transmission. This implies that rotational stiffness values become increasingly important with frequency. In the cases studied, a cylindrical isolator has been assumed. For a general isolator possessing cross-stiffness values, the situation becomes more complex. Neglecting cross-stiffness values is expected to give errors similar to when neglecting rotational stiffness values, and could be examined in a similar manner by using an appropriate definition of k. For a single cylindrical isolator mounted between infinite beams, k = (4/kb D)2, and infinite plates k = 32/(kp D)2; see Figure 14. With the aid of Figure 14, the frequency from which the rotational stiffness values need not be included in an analysis can be estimated for a given isolator diameter and chosen value of k. A value of k 1 100 would be sufficient for structures similar to those examined in this study that exhibit resonant behaviour, and a value of k 1 10 for structures that exhibit infinite behaviour. Studies of other structures and more complex isolators are needed to determine general analysis guidelines. A structure comprising two finite beams and two cylindrical isolators, simulating a machine–isolator–foundation combination, was studied to demonstrate some of the complexities involved and the effects of neglecting rotational stiffness values.

 

187

——, Connecting two infinite beams; – – –, Figure 14. k as a function of kD for a single cylindrical isolator. —— connecting two infinite thin plates.

For a symmetrically applied excitation, consisting of a force pair acting 180° out of phase (moment-like excitation), the power transmission error levels tended towards positive values, while for a symmetrically applied excitation consisting of a force pair acting in unison (force-like excitation) the power transmission error levels tended towards negative values. For a non-symmetric and a combined force–moment excitation, the error level is equally positive and negative valued. For this case, it was also observed that the peak error levels generally decrease with increasing values of kL and are varied according to the source’s excitation. The peak values in the error levels occurred at structural resonance and anti-resonance frequencies. For prediction purposes the errors at the resonance frequencies are the most important, since these are associated with the active power. Agreement between measured and predicted velocity levels for the beam–isolator–beam structure was good. Also, agreement between theoretical and measured displacement and rotational transmissibilities was found to be adequate. For these measurements it was necessary to include the mass of the isolators as loads, which is easily incorporated in the theory. This demonstrates that the simplified theory is acceptable in the frequency range studied and can be used for prediction of quantities such as input and transmitted power, for structures and isolators displaying similar behaviour. The measurements show some basic qualities of translational and rotational transmissibilities. Even though the isolators are nominally identical, the transmissibilities have different appearances. This is dependent on the actual excitation position and the mobilities of the source and receiver beams at the isolator connection points. At some peaks in the translational transmissibility of one isolator, troughs are found for the other isolator. This is also true when comparing the rotational with the translational transmissibilities for the same isolator or between isolators. This may be attributed to the fact that, when translational motion is stopped, rotational motion might be favoured and vice versa. Also, when changing the excitation from a force to a combined force–moment excitation, similar changes in the transmissibilities were seen. Vibration isolation design and optimization are possible, based on a mechanical power concept, by using simple transmissibility measurements for model verification and the knowledge that a correct model is available for study.

188

. . 

It is also worthwhile to study the rotational transmissibilities in a comparable manner to the displacement transmissibilities. The possibility always exists that extra information, such as resonance frequency peaks, may be drawn from them that otherwise could not be deduced from the displacement transmissibilities alone. In the frequency range studied in this paper, the use of constant values of isolator stiffness and loss factors seems to be advantageous. However, measured dynamic stiffness values would be expected to give more accurate results, especially if they are varying with frequency.

ACKNOWLEDGMENT

The support of the Swedish Work Environment Fund has been greatly appreciated.

REFERENCES 1. J. C. S 1979 Journal of the Acoustical Society of America 66, 1245–1279. Vibration isolation: use and characteristics. 2. I. I. K 1979 Soviet Physics—Acoustics 25, 181–191. Vibration attenuation of resilient mounts and dampers underneath actively vibrating machines (review). 3. A. O. S 1956 Transactions of the American Society of Mechanical Engineers Shock and Vibration Instrumentation, 1–39. The evaluation of mounts isolating non-rigid machines from non-rigid foundations. 4. E. E. U and C. W. D 1966 Journal of Sound and Vibration 4, 224–241. High-frequency vibration isolation. 5. J. I. S and M. G. H 1968 Journal of Sound and Vibration 8, 329–351. Vibration isolation between non-rigid machines and non-rigid foundations. 6. H. G. D. G and R. G. W 1980 Journal of Sound and Vibration 68, 59–75. Vibration power flow from machines into built-up structures, part I: Introduction and approximate analysis of beam and plate-like foundations. 7. B. A. T. P 1993 Journal of Sound and Vibration 160, 43–66. Structural acoustic power transmission by point moment and force excitation, part I: beam- and frame-like structures. 8. L. C, M. H and E. U 1973 Structure-Borne Sound. Berlin: Springer-Verlag. 9. C. E. C and J. E. R 1988 in Shock and Vibration Handbook. New York: McGraw-Hill; third edition. Chapter 30: Theory of vibration isolation. 10. J. I. B-V 1987 Journal of the Acoustical Society of America 81, 1801–1804. A fundamental problem with mobility analysis of vibration isolation systems. 11. R. J. P and R. G. W 1981 Journal of Sound and Vibration 75, 179–197. Power flow through machine isolators to resonant and non-resonant beams. 12. C. E. C 1951 Vibration and Shock Isolation. New York: John Wiley. 13. H. G. D. G and R. G. W 1980 Journal of Sound and Vibration 68, 97–117. Vibrational power flow from machines into built-up structures, part III: power flow through isolation systems. 14. M. A. S 1995 Journal of Sound and Vibration 179, 685–696. Direct measurement of moment mobility, part II: an experimental study.

APPENDIX 1: MULTI-POINT CONNECTIONS

In Figure A1 are shown two structures connected by N isolators and the generalized forces.

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189

Figure A1. Two structures attached via N isolators. Structure 1 is excited by N arbitrary and independent forces and moments acting at the isolators.

In compact notation, a set of equations of motion corresponding to expressions (1)–(6) can be written as

F G j J G f

J K G G f=G F G G G xN1 j k xa1 xb1 . . .

K G G G G k

Aa Aba . . .

Aab Ab . . .

ANa

··· ···

Ka

0

···

0

Kb

···

. . . 0

. . . . . . ··· ···

F G j J G f

xN2

F G j J G f

J K G G f=G F G G G FN2 j k

xa2 x . . .

b 2

Fa2 Fb2 . . .

J G f F G j

K G G G G k

0 . . . . . . KN

··· ··· . . .

LF F1 − F2 GG Fb1 − Fb2 Gj . GJ .. GG AN lf FN1 − FN2

AaN . . . . . .

LF xa1 − xa2 GG xb − xb 2 Gj 1 GJ .. GG . lf xN1 − xN2

a

J F G G f=j F J G G j f

a

Fa2 Fb2 . . . FN2

B . . .

B . . .

··· . . .

BNa

···

···

LF Fa2 GG Fb2 Gj . GJ .. GG BN lf FN2

Ca Cba . . .

Cab Cb . . .

··· ··· . . .

CaN . . . . . .

CNa

··· ···

Ba

Bab

···

ba

b

J G f, F G j J G f, F G j

BaN . . . . . .

J G f, F G j

LF Fa1 GG Fb1 Gj . GJ .. GG CN lf FN1

J G f, F G j

(A1.1, A1.2)

(A1.3, A1.4)

. . 

190

F G j J G f

xN2

J K G G f G F=G G G j k

F xa1 G xb1 j . J .. G f xN1

J K G G f= G F G G G j k

a 2

x xb2 . . .

BNa

··· ···

LK Ca Cab GG Cba Cb GG . GG ... . . GG N Na B lk C ···

Aa Aba . . .

Aab Ab . . .

··· ··· . . .

AaN . . . . . .

ANa

··· ···

AN

a

ab

B Bba

B Bb

. . .

. . .

··· ··· .

.

.

BaN . . . . . .

.

.

.

···

LKK I 0 GGG 0 I GGG . . GGG .. .. GGG lkk 0 · · ·

K G G − G G k

LF Fa1 GG Fb1 Gj GJ ... GG CN lf FN1

CaN . . . . . .

··· ···

Ca Cba . . .

Cab Cb . . .

CNa

··· ···

··· ··· . . .

J G f F, G j

(A1.5)

L G G G G ··· I l

··· 0 . ··· . . . . . . . .

LLF Fa1 J GGG Fb1 G GGj . f . (A1.6) GGJ .. F GGG G CN llf FN1 j

CaN . . . . . .

Here, as before, [C] = [[A] + [K]−1 + [B]]−1[A] and, for example, Aab is the transfer receptance matrix between isolators a and b.

APPENDIX 2: TRANSMISSIBILITY EXPRESSIONS, GIVEN A PURE FORCE EXCITATION, FOR TWO STRUCTURES CONNECTED VIA A SINGLE GENERAL TWO-DOF ISOLATOR 2 2 2 Tdx = 20 log =((Axb − Axx Abb − Axx Bbb )Bxx + Axx Bxb )(Kxb − Kxx Kbb )

+ iv(Axx Bxx Kxx + Axx Bxb Kxb + Axb Bxx Kxb + Axb Bxb Kbb )= 2 2 2 − 20 log =((Axb − Axx Abb − Axx Bbb )Bxx + Axx Bxb )(Kxb − Kxx Kbb ) 2 Kbb + Abb Axx Kbb ) − Axx v 2=, + iv(Axx Bxx Kxx + 2Axx Bxb Kxb + Axx Bbb Kbb − Axb

(A2.1) 2 2 Trb = 20 log =((Axb − Axx Abb + Axb Bxb )Bxb − Axb Bxx Bbb )(Kxb − Kxx Kbb )

+ iv(Axx Bxb Kxx + Axx Bbb Kxb + Axb Bxb Kxb + Axb Bbb Kbb )= 2 2 − 20 log =((Axb − Axx Abb + Axb Bxb )Bxb − Axb Bxx Bbb )(Kxb − Kxx Kbb ) 2 + iv(−Axx Abb Kxb + Axb Bxx Kxx + 2Axb Bxb Kxb + Axb Bbb Kbb + Axb Kxb ) − Axb v 2=.

(A2.2)

APPENDIX 3: LIST OF SYMBOLS Aij , Bij [A] [B] [C]

receptances (m/N, 1/N or 1/Nm) structure 1’s receptance matrix (m/N, 1/N or 1/Nm) structure 2’s receptance matrix (m/N, 1/N or 1/Nm) help matrix

  D f f0 E F j kb kp K [K] k l m L Le Lv M P pi pt t T, T w x, y, z a, b, g h lB r v v0

isolator diameter (m) frequency (Hz) natural frequency (Hz) Young’s modulus (Pa) force (N) = z−1 beam bending wavenumber (1/m) plate bending wavenumber (1/m) stiffness (N/m, N or Nm) stiffness matrix (N/m, N or Nm) stiffness ratio isolator length (m) mass (kg) beam length (m) error level (dB) velocity level (dB re 1/m/s) moment (Nm) power ratio input power (W) transmitted power (W) time (s), thickness (m) transmissibility (dB), transpose width (m) complex valued displacements (m), co-ordinate complex valued rotations (radians), co-ordinate loss factor bending wavelength (m) density (kg/m3) circular frequency (radians/s) natural circular frequency (radians/s)

Subscripts or 1, 2 a, b, . . . , N d, r

superscripts structure numbers isolator displacements or rotations

191