GENERAL SPONSORED RESEARCH COLLEGE OF ENGINEERING THE UNIVERSITY OF MICHIGAN ANN ARBOR Final Report A NETWORK APPROACH TO VIBRATION ANALYSIS T..E o. Krie'wall J. C..-.Johnson Supervisor; Acoursti, s Laboratory OFFICE OF NAVAL RESEARCH UNDERSEA WARFARE BRANCH (CODE 466) CONTRACT NO. N6onr-23219 August 1957

ACKNOWLEDGMENTS This investigation, which was supported by the Undersea Warfare Branch, Code 466, of the Office of Naval Research under Contract No. N6onr-23219, was carried out within the Acoustics and Optics Group of The University of Michigan. The administrative cognizance for this contract rested initially with the Engineering Research Institute and subsequently with the College of Engineering of the University. The author wishes to express his appreciation to H. F. Reiher for his many helpful suggestions and technical editing of the paper, to R. N. Hamme who acted as technical consultant to the project, to G. E. Eberle for his valuable aid in the computational work, to C. C. Marcucci for preparation of the figures, and to the staff of the Engineering Research Institute Reports Office for their preparation of the manuscript. The author also wishes to acknowledge his indebtedness to S. W. Doroff, Mechanical Engineer, Code 466, Undersea Warefare Branch, for his motivation of this study and his continued interest and encouragement throughout the investigation. ii

TABLE OF CONTENTS Page LIST OF FIGURES iv ABSTRACT vi SECTION I. NETWORK THEORY1 Introduction 1 Objectives of the "Network" Approach 1 Four-Terminal Networks 3 Properties of the Mechanical Elements 6 The Machine 6 The Foundation 8 The Mount 9 Translation from Mechanical to Electrical Quantities 14 Expressions for the Isolation Parameters 19 SECTION II. APPLICATIONS TO SIMPLE ISOLATION PROBLEMS 24 Rigid-Machine, Simple-Elastic-Mount, Rigid-Foundation Model 24 The Mechanical Filter 32 SECTION III. NUMERICAL ANALYSIS OF A TYPICAL SYSTEM 40 Background and Definition of the Problem 41 Analysis of the System 41 Numerical Results 45 SECTION IV. SUMMARY AND RECOMMENDATIONS 55 Evaluation of the Network Approach 55 SECTION Vo REFERENCES 57 DISTRIBUTION LIST iii

LIST OF FIGURES No. Page 1 General four-terminal network. 3 2 Transmission line. 5 3 Simple machine idealization 7 4 Idealization of the mounto 10 5 Forces acting on a cross-sectional element of the mount. 11 6 Electrical circuit for machine directly attached to foundationo 21 7 Electrical circuit for pedestal mounted machine. 22 8 Machine mounted on a rigid foundation, 25 9 Illustration of circuit construction corresponding to a given mechanical system. 25 1.0 Electrical circuit after removal of transformers ' 26 11 Lumped parameter model of machine mounted against a rigid foundation 28 12 Graphical solution for resonance and minimum transmissibility points. 31 13 Transmissibility of simple mount and compound lines, 33 1-4 Machine-compound mount-foundation model, 54 15 Electrical circuit corresponding to Figure 14. 35 16 Pedestal mounted machine on a resilient foundation. 42 17 Machine with two point suspension. 43 18 Machine impedance measured at point of support. 46 19 Foundation impedance measured at point of support, 46 20 Force transmissibility of mounto 47 iv

LIST OF FIGURES (Concluded) No, Page 21 Real and imaginary parts of mount characteristic impedance. 49 22 Mount attenuation function, _. 50 23 Mount phase function,,. 50 24 Young's modulus of mount material, E1. 51 25 Velocity of sound in mount material, co 52 26 Visco-elastic coefficient of mount material, i. 53 27 Foundation velocity response to unit force generated by machine. 54

ABSTRACT This report discusses the application of impedance-network theory to the analysis of mechanical vibrating systems. A description of the relevant mechanical impedance parameters of a machine-mount-foundation system, a translation from mechanical to electrical network concepts, and a review of pertinent four-terminal-network theory led to a method of computing the one-dimensional vibration-response characteristics of distributed as well as lumped parameter systems. The usefulness of the network approach is demonstrated by the analysis of mechanical filters, and by the computation of the response of a sample machine-mount-foundation system. The capabilities and limitations of the method are discussed. vi

SECTION I. NETWORK THEORY INTRODUCTION In the continuing search for better engineering methods directed toward achieving a higher degree of vibration isolation in mechanical systems, the "mechanical filter" concept has been proposed1 as a possibly fruitful source of further investigation. This paper is the result of carrying out the suggested investigation. In the course of the research, it became apparent that the results obtained with respect to the filter evaluation were equivalent to conclusions reached by authors of several earlier articles.6,7 However, by arriving at these conclusions independently and in a somewhat more comprehensive context, a method of solution was evolved which is likely to be of general interest inasmuch as: (1) the procedure involves a conceptually simple method of analyzing, in one dimension, the behavior of complex vibrating systems, and (2) the criteria for evaluating the degree of isolation achieved in the system emphasize those design parameters which are of current interest. Thus it is primarily the purpose of this paper to report this method in a sufficiently elementary and complete form to permit its use without previous familiarity -on the part of the user, and secondarily to summarize the body of well-known results which constitute a reply to the memorandum of Ref. 1. OBJECTIVES OF THE "NETWORK" APPROACH The problem with which we shall concern ourselves arises from the basic question: how can the isolation of a vibration source from its surrounding environment be best achieved? This question leads naturally to the need for parameters which best describe the degree of isolation. The selection of these parameters, in turn, stimulates the search into the mechanics of the system to determine design criteria which permit optimization of the isolation parameters. A present, the art of vibration isolation of heavy machinery supported by complex structures has not been developed sufficiently to provide a satisfactory solution to the fundamental problem. Many questions remain unsettled with regard to which parameters best measure the degree of isolation of the system, and consequently which physical variables ought to be controlled to optimize the degree of isolation. In the past, a clarification of these difficulties was attempted by resorting to the theoretical analysis of one- or two-degree-of-freedom, lumped parameter idealizations of the real physical system. This "harmonic oscillator" idealization provided a fairly suitable means of mount analysis for low-frequency 1

vibration sources, i.e., frequencies corresponding to wavelengths which are large compared to the dimensions of the mount. However, it has not provided any insight into the higher frequency phenomena termed wave effects nor has it proved adequate in the analysis of typical systems in which the machine and foundation do not behave as a simple combination of masses, springs, and dashpots. Thus it represents an oversimplification of the problem since in many cases information of primary interest is suppressed. An extension of this method is suggested by the fact that sound possesses, in addition to an oscillatory property, a wave property or propagation characteristic, as it is called. Thus one may be led to an attempt to formulate the problem in terms of the wave equation subject to appropriate boundary conditions. This also has been successfully attempted with regard to the analysis of mount behavior, but it has not been applied, insofar as the author is aware, to the analysis of an entire vibrating complex. This may be due in part to the algebraic difficulty encountered in determining and applying the correct boundary conditions to the solution of the wave equation for this type of problem. To avoid this difficulty, a modified approach is suggested when one recalls the electrical engineer's preference for sOlving steady-state circuit problems. Rather than solve simultaneous differential equations subject to boundary conditions, he prefers to introduce the concept of impedance and reduce the differential equations to algebraic equations. This "network" approach has its analog in treating mechanical systems also, and is particularly promising in its application to vibration-isolation problems in view of the current work being done to improve techniques for measuring mechanical impedance. Hence the method is intended to perform the following functions: 1. To provide a better approximation than lumped parameter theory allows in the theoretical prediction of an entire vibrating system's behavior. 2. To emphasize the specific types of impedance measurements needed to compute, on the basis of this theory, the system response to oscillatory forces. 35. To provide a clearer understanding of the relation of isolation parameters currently used in evaluating the performance of system components to impedance quantities. 4. To demonstrate a conceptually simple approach to the analysis of entire systems. Since this method is similar to the procedures employed in the analysis of certain types of electrical circuits, and relies heavily on the concepts and terminology of electrical engineering, it seems appropriate to begin with a review of those electrical concepts which will be used most frequently. Specifically, the electrical results needed will be those of four-terminal-network theory and the related transmission-line theory. It might be well at the out2

set to keep in mind that the force-voltage, current-velocity analog will be employed later in the development. FOUR-TERMINAL NETWORKS J. --- -,-I --..1, GENERAL FOUR-TERMINAL NETWORK FIGURE I Figure 1 represents the general four-terminal network. The network within the "blackbox" is arbitrary, subject to the restrictions that the elements be linear and bilateral and that no sources are contained therein. Reference conditions for voltage and current are indicated on the diagram. General loopanalysis methods yield the equations (see, e.g., Ref. 3) E1 = Z11I1 - Z12I E2 = z2lI1 - z22I2 Interpreting these equations, we see that if the output end is open, I2 = 0, and El = z1lIl, E2 = z2li1 Consequently zll may be interpreted as the input impedance of end 1 with end 2 open-circuited. Similarly, z21 represents the transfer impedance under the same conditions. If the input end is open, I, = O, and E2 = -z22I2 El1 = - zl2I2 Thus z22 and z12 are, respectively, the input and transfer impedances of end 2 with end 1 open. The relation between input voltage and current and output voltage and current may be put in a convenient form by defining the following parameters: 5

A = Z Z21 B = M Z21 Zl w1 Z12 where |z | = Z |= ZlZ22 - Z21Z12 C = Z21 Z22 Z21 D = 2 Z21 In terms of these parameters, the following identities are readily verified: E1 = AE2 + BI2; (1) I1 = CE2 + DI2. (2) Solving for E2 and I2 gives E2 = DE1 - BIi; (3) -I2 = CE1 - AIl. (4) The characteristic impedance, ZO, of the network is defined to be that terminating impedance which causes the input impedance of the network to have the value of the terminating impedance. Z0 may be found from the expression for input impedance in the following way. By definition, the input impedance looking into the left of the network is = E1 = AE2 + BI2 I CEz + DI2 Since the receiving (output) end impedance is Zr = E2 r I2 12 then = AZr + B Z1 =.. C Zr + D For a symmetrical network, zll = zas2 and consequently A = D. Then, by definition of characteristic impedance, Z1 = Zr = ZO, and it is found immediately that 4

O = (5) The propagation constant, y, may be defined by E2 -2 = A +4 T 1 = e7, (6) E2 I2 where the ratios are taken for a network terminated in its characteristic impedance. It is then easily;-shown (see Ref. 3) that the following relations hold for a symmetrical network: A = D = cosh; (7a) B = Z0 sinh 7; (7b) C = sinh y (7c) zo The equations describing the properties of four-terminal networks are closely related to those describing the properties of "transmission lines." For if the section of transmission line depicted lnFigure 2 were treated as a four-terminal network, precisely the same conclusions would be reached as above except that each 7 that appears would be replaced by yL. Thus the four-terminal network treated above represents a unit length of transmission line. Therefore all the relations, especially Eqs. (1) through (7), may be applied directly to a transmission line simply by substituting yL for 7. ~TA MS1 LIN2 L _ _ _ _ _ _ _ _ L 1 TRANSMISSION LINE FIGURE 2 5

This brief review of the electrical theory will be sufficient for the present, pending a somewhat detailed account of the analysis of the components of a typical mechanical system. This will consist essentially of two parts. In the first, the machine and foundation will be treated immediately in terms of their electrical analogs. In the second part, a conventional mechanical analysis of the mount will be presented to compare results with the summary of the four-terminal-network theory given above. PROPERTIES OF THE MECHANICAL ELEMENTS To fix our ideas in the following considerations, we shall address our attention to the problem of analyzing, as completely as possible in one dimension, the characteristics of a vibrating mechanical complex composed essentially of three basic elements: 1. sources of vibratory energy; 2. elastic elements which serve to transmit the vibratory energy throughout the system (transmission paths); and 5. terminations of the transmission paths. Since such a model represents a highly idealized mechanical system which might apply equally well to an endless variety of physical realizations, we may choose without loss of generality the labels machine, mount, and foundation, to refer respectively to the elements of this system with the understanding that these terms are used in a symbolic rather than real sense. The Machine.-A machine may be defined, in the sense mentioned above, as any active component of a vibrating system. The term "active" has a meaning analogous to the electrical concept, implying here that a machine contains sources in its make-up as contrasted with the entirely passive nature of all other elements in the system. The idealizations of a machine may assume two forms: (a) a simple vibratory source, e.g., a constant amplitude force or velocity generator; or (b) a mechanical network of simple sources connected together by mechanical elements displaying some combination of the properties of inertia, compliance, and dissipation. The first of these machine models may be symbolized simply by a vector labeled to indicate the amplitude and frequency of the causal mechanism creating the vibrations, as in Figure 3a. The model representing the second possibility, Figure 5b, may be deduced from the use of the mechanical equivalent of Thevenin's orem To apply this theorem, we must first digress briefly to consider the concept of mechanical impedance. Mechanical impedance is the complex quotient of the alternating force applied to the system by the resulting linear velocity in the direction of the force at its point of application. With symbols, one writes Zm = F/v (mechanical ohms), (8) 6

Feq. Zeq, Fo sinwt 4 > 0 MECHANICAL MECHANICAL SYSTEM ST a. b. SIMPLE MACHINE IDEALIZATION FIGURE 3 where Zm is the mechanical impedance, F is the root-mean-square value of the applied sinusoidal force, and v is the root-mean-square value of the velocity. Because Zm is complex the phase angle between F and v must be known. The similarity between mechanical impedance and the corresponding electrical quantity Is immediately apparent by comparing Eq. (8) with the electrical definition z7 E "e I= Since the statement of the definition will be sufficient for the present, further comment on this concept will be reserved until it is again needed in connection with mount analysis. The form of Thevenin's theorem applicable to the mechanical network idealization of a machine states that any linear mechanical network containing any number of force and velocity sources of the same frequency can be viewed as an equivalent constant force source, Feq, acting through an equivalent (series) impedance, Zeq, as depicted inFigure 3bo The impedance of the machine must in general be experimentally measured and expressed in the complex form Zm = Rm + iXn ' where Rm is the mechanical resistance and Xm is the mechanical reactance. 7

It may be noted that the measurement of this particular type of impedance entails certain nontrivial difficulties which arise from the physical dissimilarity between mechanical and electrical networks. To satisfy the requirements of Thevenin's theorem, one can, in some types of electrical networks, replace each voltage generator by its internal impedance and then measure the total impedance as seen looking into the two output terminals of the network. The interpretation of this procedure in mechanical terms becomes immediately complicated since the force generators in a machine cannot be easily isolated to measure their respective internal impedances. Indeed, even if this could be done, there yet would remain the serious difficulty of physically replacing these sources by their impedances to make the total impedance measurement of the machine. Thus it appears that any direct measurement of machine internal impedance analogous to the type made in some electrical circuits is at best an unlikely possibility. Certainly a procedure involving measuring the velocity occurring at the feet of a machine in response to a force applied to the feet does not provide the impedance value to be used in connection with an application of Thevinints theorem. One possible method of circumventing these difficulties is suggested by the electrical technique employed when the previously mentioned method is not applicable. This is done by terminating the electrical network with a variable known impedance and determining the load impedance for which maximum power transfer is obtained. Because the condition of maximum power transfer requires that the load impedance be the complex conjugate of the source impedance, obtaining the condition of maximum power transfer establishes the source impedance in terms of a readily measured load impedance. The problem of the feasibility of this procedure for mechanical systems or of the determination of other possible procedures will be left to the enterprise of the experimentalists for solution. The purpose of the discussion at this point is simply to emphasize that a very specific type of machine internal impedance measurement is needed in connection with the general method of system analysis to be presented. With this brief look at the properties of a machine, we next consider the foundation, leaving the more extended analysis of the mount to the last. The Foundation.-It might be well to stress at the outset that the elements of the system which are termed "foundations" may appear physically in a form quite different from that intuitively suggested by the term. For example, suppose the problem consisted of isolating a vibration-sensitive, floor-mounted piece of equipment from a strongly vibrating machine attached to the same floor. In this case, the floor, whose transmission characteristics are to be modified, plays the role of a transmission path or "mount." Hence the vibration-sensitive item must be considered the termination to the transmission path, and therefore is termed the "foundation." Analyzing such a system further, it might develop that mounting the vibrating source machine on an isolation device is the only change permissible in the system. The problem then reassumes the conventional 8

configuration of a machine-mount-foundation by treating the floor as the foundation. This apparent difficulty in deciding what is to be called the termination of a transmission path may be resolved in a given situation by applying the following two rules: 1. Starting from the machine, one selects the number of connected transmission paths whose characteristics are expected to be modified or controlled according to certain specifications. Whatever then exists in physical connection with the terminal end of the last connected transmission path will be termed the foundation. 2. The impedance of all the (passive) elements thus composing the foundation may be expressed in terms of an equivalent foundation impedance Zf = Rf + iXf by means of Thevenin's theorem for passive, linear elements. In general it will be necessary to measure experimentally the mechanical impedance of a foundation in the force direction at the mount's point of connection. Utilization of this concept permits the possibility of considering to some degree the interacting effects of a complex foundation on a machine-mount combination. The Mount.-Since the problem is being considered in one dimension only, the mount may be treated as a series of connected, individually homogeneous, elastic cylinders of constant cross section, i.e., we consider only the propagation of plane waves in the mount. The model chosen to represent a homogeneous section of the mount may be described, due to its homogeneity, by the following quantities. p = density c = velocity of sound in the mount material E1 = Young:s modulus of elasticity S = cross-sectional area L = length of the resilient element m = pSL = mass of the element Consider the element shown in Figure 4. Let a(x) be the positive (tensile) stress acting on a cross-sectional element of thickness dx, and g(x) denote the displacement of this cross-sectional element from its equilibrium position in the positive x direction. Hence, |(x) will symbolize particle velocity at a point, x, along the rod. At the end x = 0, the element is being acted upon by a sinusoidal force creating a stress and velocity, denoted by cr and A1, respectively. This activity sets up a stress and velocity distribution along the rod. The respective quantities thus transmitted to the end x = L are labeled 02 and i2. These longitudinal vibrations are described by a second-order partial differential equation which may easily be derived., by considering the forces acting 9

,(o) (2(L) zq tO a~tI (. J)4 dx X:O XzL I DEALIZATION OF THE MOUNT FIGURE 4 upon a cross-sectional element of thickness dx. Since the forces are due to the elasticity and viscosity of the material, we consider them individually, Let tj be the coefficient of viscosity, defined so that it represents the factor of proportionality between the viscous force and the product of area times the time rate of change of strain in the element. Thus the component of the force due to viscosity is defined by Fv = =Sa ) = at x A similar role is assumed by Youngts modulus of elasticity which relates the elastic force and the product of area times strain in the element. Hence, the force component due to elasticity is defined by Fe = Ei i S Fe = The total force acting on the element dx is depicted in Figure 5o Writing Newtonts second law for the element shown, we have 62 e dx + dx = 6x ox m * a Then by performing the indicated differentiation on Fv And Fez the equation of motion becomes El a +1~ at 8x2) (9) 10 X = O ' X=L IDEALIZAIONOTHMUT

F --- > F + V dx 8x Fe 0<- ------ Fe + -^dx ' -— X --- dx FORCES ACTING ON A CROSS-SECTIONAL ELEMENT OF THE MOUNT FIGURE 5 This may be put into the form of the familiar wave equation by assuming the steady-state solution to Eq. (9) is of the form -= (x) e, X = 2if, f = excitation frequency, (10) and defining a complex modulus E so that E = E+ i E2, (11) where E2 = Uo. Then differentiating a2S/ox2 with respect to time, Eq. (9) becomes (EI + ipw) x2 = P t2, and substituting the modulus E defined in Eq. (11), we finally obtain pSt2 X = ~ 0 x L, t >0 (12) Equation (12) completely describes the propagation of plane longitudinal waves in a linear homogeneous, elastic, viscous element of constant cross section. The solution to this equation, which may be verified by substitution, is a linear combination of expressions describing waves traveling in negative and positive x directions, respectively: g(x,t) = (CleYmx + C2e-Ymx) ei t (13) 11

where y, the mechanical propagation function is defined by Ym = n + ~im (14a) oX = attenuation constant (14b) Pm = phase constant (14c) (Note: The subscript m is maintained to differentiate between the mechanical and electrical propagation functions only until equivalence is established. It will then be dropped in the interest of notational simplicity.) We now proceed to formulate the steady-state solution to Eq. (12). Specifically, we seek an expression relating the stress and velocity at any point x along the mount section to the generating stress and velocity, ca and |o The total stress at a point x is defined in terms of the complex modulus E as follows: a(x) = E () The expression for velocity at any point x may be computed from Eq. (13) by performing the differentiation with respect to time: f(x) = ic (C1 e7mx + C2 e-7mx) eit (15) It now remains only to evaluate C1 and C2 in terms of al and 01 (the boundary values at x = 0). '_ 7mx -Ymx icct = 7m (Ci em - C2 e Ymx) ie (16) Thus 7mx 7mx icwt a(x) = E = E o m (C1 emx - Cz e ). (17) Applying the boundary conditions at x = 0, C(O) '= o, |(0) = ti i to Eqs. (15) and (17) yields e (0) = (1 = ic7 (C1 + Ca) e 12

and ict (0O) = l = E 7m (Cl - C2) e Hence, -icut C1 + C2 = and -iot C1 e-t C1 - C2 = E ym Adding these expression, we obtain. C, - t +- -i)t 2 iC E 7w) Similarly, subtraction yields C = - -L _ - e -ict 2 (j E ym Substituting these values of Ci and C2 into Eq. (17), we have a(x) = E 7m + ) _ ) e E y 2 iW E 2 2 Then using the definitions eYmX + e-7mx cosh ymx = - and eYmX - e-YmX sinh yx = we may combine terms and write: E 7m a(x) E Ym l sinh y x + cosh ymx. (18) An in the same manner, Eq. (15) becomes 13

t(x) = lj cosh ymx + W aIs sinh ymx. (19) E 1m The relation between a2, s2, and 1, l, may be obtained from Eqs. (18) and (19) immediately by substituting the second set of boundary conditions: for x = L, c(L) = a2, and |(L) = I2o Thus S a2 = S al cosh ymL + i ZO sinh YmL (20) 1S c(21) a = SJZ sinh ymL + tj cosh mL (21) ZO0 where ZO = EyS/icu. It will be demonstrated later that ZO represents the "characteristic impedance" of the resilient element. Inspection of Eqs. (20) and (21) shows that the transmission characteristics of the mount are determined by the parameters ZO and y. These equations may now be compared directly with those listed in the summary of four-terminal-network theory. From this comparison, a table can then be drawn up to facilitate the translation from mechanical to electrical terms which will be needed in sections on application of the method. TRANSLATION FROM MECHANICAL TO ELECTRICAL QUANTITIES The essential point to note at the outset is that the homogeneous mount section may be regarded as a mechanical transmission line which has as its electric analog the electrical transmission line. To support this assertion, the equations of motion of the parent mechanical transmission line will be shown to have the same form as the analogous electrical transmission line under certain specified conditions. It may be shown by elemental analysis similar to the above mechanical analysis that the equations describing the voltage and current distributions along the electrical transmission line are given by 2aE 2 ax -e E = (22a) a2I =, (22b) where ye = ae + i{e is the electrical propagation function. We now determine the conditions under which Eqo (12) describing velocity distribution along a mechanical transmission line has the same form as Eq. (22b). Substituting Eq. (10) into Eqo (12), and performing the differentiation with respect to time, we may write Eq. (12) in the form 14

ax2 (25) E This form then becomes identical with Eq. (22b) by equating 2Pt 2 C = Y 2 (24) E m Assuming the validity of Eq. (24) for the moment, we may use this relation as a method of evaluating the attenuation and phase functions in terms of mechanical parameters as follows. Recalling that E = E1 + ipzo, the left side of Eq. (24) may be rationalized and rearranged to read: _PW2 + ipPw3 -pa2 -p a2 (E1 - i p) E, E2___ E1 + ifp El2 + +2 X ( E( + ) For most cases of interest pD/E1 < lo Using this approximation, Eq. (24) reduces to 2 2 3 -P p + i p12 = y 2 (22 Am) ~E E E = m m Equating the corresponding real and imaginary parts yields the simultaneous equations - 2 o _,. m m - _ El 2 Cmnm = ~ El From the first of these two equations, om2 may be neglected for small damping coefficients in comparison with m2; thus ~ 2 p2 ~2 W2 12\2 ~m E E E = c = \hy) (25) where X = wavelength, and where c, the velocity of sound, is conventionally computed from the relation c = JI~. (26) Substitution of gm into the second equation results in 15

Z = i~^- c (27) m 2c El It is sufficient to note that these values of am and Pm are also obtainable in a similar fashion by substitution of Eqo (13) into Eq. (12)o This fact justifies the assumption of the equality expressed in Eq. (24). Since the electrical transmission line is the analog of the mechanical mount, one would expect to find some relation between the mount characteristics and a four-terminal network also. Indeed, this is shown to be the case upon comparison of the sets of Eqs. (20) and (3), and (21) and (4): - S a2 = - S oa ~ cosh yL - ES l sinh yL (20) E2 = DE1 - BIl (3) - = - E. S al sinh yL - 01 cosh yL (21) EyS -2 = C o El - AI (4) As mentioned earlier, we shall use the force-voltage analog, which provides then the following correspondence: F > E, | e - I o Due to the commonly accepted convention which regards tensile stresses as positive. the relation between a positive force and a positive stress is given by F = - S o Therefore, in terms of stress, we may identify the following two analogous quantities - S-a - E Now comparing coefficients of corresponding quantities in the above two sets of equations, we find that the mechanical coefficients corresponding to the ABCD parameters are Am = Dm = cosh yL, (7'a) B = ES sinh L, (7b) Cm = ENS sinh yL (7'c) EyS 16

Comparing these with the ABCD parameters derived from considering an electrical transmission line as a four-terminal network, as obtained from (7a), (7b), and (7c) by replacing y with 7L, namely, At = Dt = cosh yL, (7"a) Bt = ZO sinh 7L, (7"b) Ct = sinh yL,(7"c) Z0 we note the following two factso lo The characteristic mechanical impedance of a homogeneous elastic element treated either as a mechanical four-terminal network or as a mechanical transmission line is defined by Zo = E as indicated earlier. 2o Since Am =.Dm and At = Dt, the homogeneous resilient element and its electrical analog display the property of symmetry, i.eo, zll = z22 in both cases. With the correspondence between an electrical four-terminal network thus specified, the minimum information sufficient to characterize the mechanical element may be deduced. Six parameters associated with the element are involved, namely A,BC,D,Zo, and y7 However, since A = D, this list is immediately reduced to five. There are three conditions imposed on the remaining variables as stated by Eqs. (5), (6), and the following identity: A * D -B C = 1 which is readily verifiable by substitution. We are thus left with the choice of any two of the five as independent variables. Usually the most convenient pair will be ZO and y. Relations (5) through (7) provide a method of computing these functions once the necessary input and transfer impedances of the mount are measured. Since these quantities are generally complicated functions of frequency, it will be necessary to perform a point-wise analysis of the system. Once again it must be emphasized that very specific conditions must be met in the measurement of these impedance values. With reference to the voltagecurrent relationships in a four-terminal network, the input and transfer impedances on each side of the network must be obtained with the opposite side opencircuited. Since the current on this side will then be zero, we may interpret the "open circuit" condition mechanically by saying that the input and transfer impedances on each side of the mount must be obtained with the opposite side rigidly constrained so that the velocity on this side will be zero. 17

It might also be well to clarify at this point the units of mechanical impedance used in this report inasmuch as some variation is found among different authors regarding this matter, To begin with, we note from the expression for ZO: ZO = e i = i that if damping is neglected, then as p goes to zero, y goes to ip. Hence ZO reduces to Zo = iS ( i ) = E_ = pcS which is the well-known acoustic resistance for a plane wave of wave front area S. The medium in which the wave is being propagated may be solid or fluid. Now it is conceivable that the entire transmission path to be considered in the problem may have sections which are solid and others which are fluid, and thus it will be necessary to consider the impedances of each case, which are customarily separated into two categories termed mechanical and acoustic impedance, respectively. It is imperative that, if one is to be able to work with these quantities mathematically, they must have the same physical dimensions. Now it is generally agreed that the fundamental definition is r7 F in accordance with its electrical analog. The divergence of views arises when the impedance is defined in terms of other quantities, such as stress. This seems to be caused by the two facts: (1) the convention regarding positive (tensile) stress is violated on occasion, and (2) the requirement that mechanical impedance have the dimensions FD F 1 MLT2 M LvJ LT' T is not met but is compensated for in a variety of ways. A consistent approach with regard to these two considerations dictates that the definition of impedance in terms of stress be written Z - = S_ (lb (force) - sec) v in. in English units. So far we have discussed the similarity between electrical and mechanical systems only on the basis of the comparable network equationso However, in regard to the method of constructing the electrical circuit corresponding to a 18

mechanical system, some difficulty may seem to be pending due to our departure from the customary analogy which usually treats the equivalent of mechanical elements as two-terminal networkso However, an excellent system of circuit construction which is highly suited to the consideration of mechanical elements as four-terminal networks has been developed by Bo B. Bauer and will be adopted throughout the remaining analysis (see Refo 4). Table 1 shows the equivalent four-terminal-network form of the various mechanical components. The method of constructing the general circuit corresponding to a mechanical complex consists, briefly, of (1) placing the appropriate four-terminal networks (as obtained from Table 1) in the same relative geometrical position as displayed by their corresponding mechanical elements; (2) providing the coupling between elements by means of ideal transformers of 1:1 turns ratio, The transformers can often be removed afterward by considering the voltage-current relations in such a circuit and placing jumpers across their terminals when possible to do so without upsetting these relationships. Examples of the use of this method will be included later. For further information on the method of this analog and the currentforce analog, see Ref. 4. This essentially completes the groundwork necessary for understanding the application of network methods to specific vibration-isolation problems. We shall at this point include, however, a summary of electrical equations which appear to be applicable to the analysis of a mechanical system. EXPRESSIONS FOR THE ISOLATION PARAMETERS General information which shall be of interest later may be derived by considering the circuit in Figure 60 This circuit may be interpreted mechanically as a vibrating machine directly attached to a resilient foundation. The generator and the source impedance, Zs = Zm, represent the resolution of a general mechanical or electrical network into an equivalent constant voltage generator and series impedance by means of Thevenin's theorem. The impedance of the foundation, measured at the point of connection with the machine, is represented by the complex quantity Zr = Zfo Now, suppose the general four-terminal network were inserted in this electrical circuit as shown in Figure 7. Inspection of the circuit indicates the following relations hold: El = Eg - IlZs and E2 = I2Zr o Using these results and Eqs. (7") in Eqs. (1) and (2) yields the important ratio E_ (Zs + ZO) (Zr + Zo)eYL - (Zs - ZO) (Zr - ZO)e yL -------------------------------------. (28) Ea 2 Zo Zr This may be rearranged to the form 19

0 w 2 i4 w zr w I LJ~~~~~~~~~~~~I w w w 4 bJ C,> z 2 4 - 4 4 U ~ U I- 4 w IIJw w w w o o u CD 0 IL cn CD 2 W~~~~ zz S w~~~~ w 49 0~~~~ Co 2 0 (D -j 3 L N 2~~~~ w z o 0 r~~~~~~U z r r I-L) Qt:CD 2LI 4gr 2: aor

r!;~4 0 U z I z 0 0 ~ 4~~ LL~~~~~I t 0 LL ZI 8~~~~ U. IL >5.~~~~~~~~~L V C.)w C0 5 z 0 mmmt 5w C.) C.) 21 d NI 2 I-. LJ 111~~~~~~~~~(. 21

z FL-r' I -, 0 0 O ~ 0 w A 0 0I 4.. z T T! (,.)n,' 0 - Ln I L ~H |oH 'Lt ----- - C l W ~~~LJ 0 Z2 i O2 A -J 22

g aX S (n 3 L(}i_ zs Zow 1 (r { -AyL 22 2 lrsZo re r (29) where the reflection coefficients for the sending and receiving end (or input and output end) are given by rIs = S, (30a) Zs + ZO rr = Zr - Z (30b) Zr + ZO Equation (29) places in evidence the various correction factors created by insertion of the four-terminal network. The first term, 2 ~Zs/Zr, on the right accounts for the modification of the voltage (or force) ratio due to the difference between the source and terminal impedance. The factor e7L is seen to represent, by definition, the voltage ratio that would prevail if ZO = Zs = Zro Since each of the three remaining factors reduces to unity when ZO = Zs = Zr, they represent corrections due to impedance mismatch between the four-terminal network and the terminations. The third and fourth factors partially compensate for mismatch at the input and output ends, respectively, while the last term includes the remainder of mismatch effects. Since the fifth term involves the effects at one end due to mismatch at the other. it is called the interaction term. This term is near unity for small mismatch and high attenuation, but becomes large for the opposite conditions of large mismatch and small attenuation. Now by writing the term 2 4Zs/Zr in the form (Zs + Zr)/Zr 2i = 1 ( S r Zr = Equation (29) becomes E J(Zs + Zr)/Zr, ey L + Irl — -II- ) (1 -.s * r e-21L) Then noting from Figure 6 that Eg Zs + Zr E2% Zr 23

the insertion properties of the network can be described by 1 Z( 0 1 Zr e7 11!7^ 0^ lt A7 (n rr e-27L E2 o Neil E2 f/2 + Tz (32a) This equation gives the ratio of what the load (or foundation) potential would be, if the source and load were directly connected, to what it is when the network is inserted. The magnitude of the ratio, N, is termed the insertion ratio, and r is called the insertion angle. For computational purposes, the following form is often more desirable: E2 - (Zs + ZO) (Zr + ZO) eYL - (Zs - ZO) (Zr - Z0) e-yL --- ___ ___ __ ___ ___ __ ___ ____....(32b) E2 2 Z0 (Zs + Zr) Either Eq. (28) or (32) constitutes a criterion for the evaluation of a given mechanical system. Equation (28) will lead to the conventional notion of force transmissibility, while Eq. (32) is the analog of a recently introduced mechanical concept termed "mount effectivenesso" The former provides the overall attenuation achieved in a given mechanical configuration, while the latter determines precisely what is gained (or lost) in the way of isolation by inserting a mount between the machine and its foundation, An extensive treatment of the insertion properties of a mount at low frequencies may be found in Ref. 5~ The formulas collected in this section are sufficient to permit a relatively complete analysis of a mechanical vibration problem in one dimension. The mechanical ingredients needed to feed into this machinery are the particular foundation and machine impedances, discussed earlier, and the two independent parameters (ZO, y) necessary to characterize the mount, With these remarks, the development of the fundamentals will be concluded. SECTION IIo APPLICATIONS TO SIMPLE ISOLATION PROBLEMS RIGID-MACHINE, SIMPLE-ELASTIC-MOUNT, RIGID-FOUNDATION MODEL The simplest example which exhibits the characteristic features of the theory is that of a rigid vibrating mass separated from an infinite foundation by a conservative, homogeneous, resilient element of small, constant cross section. The model is shown in Figure 8, Using a general four-terminal network to represent the resilient element, the corresponding electrical circuits is constructed as follows: the machine is reduced by Thevenines theorem to a constant force 24

— S<rg % C2 MACHINE MOUNTED ON A RIGID FOUNDATION FIGURE 8 generator, -Sag, acting through an impedance presented by the rigid masso The elements of the system are arranged geometrically in series; therefore we so arrange the corresponding elements chosen from Table 1. The coupling is provided by ideal transformers of 1:1 turns ratio, indicated by vertical straight lines in Figure 9. Inspection of the circuit shows that attaching four jumpers at ab, cd, ef, and gh will not disturb the voltage-current relationshipso Hence the transformers are easily removed in this case. The final circuit is given in Figure 10o * X z c d g h Eg - l —o: r1 Egf^J )I I ABCD I 1r Zr =C a b e f XFMR XFMR ILLUSTRATION OF CIRCUIT CONSTRUCTION CORRESPONDING TO A GIVEN MECHANICAL SYSTEM FIGURE 9 25

Egg. 0+ Egt^Vjt 9 ABCD 2 Zr o ELECTRICAL CIRCUIT AFTER REMOVAL OF TRANFORMERS FIGURE 10 This circuit is identical to that of Figure 70 Therefore Eq. (28) may be taken over directly. To obtain the transmissibility, the values of Z0, Zs, Zr, and y are needed to substitute into Eq. (28). Since the foundation has been assumed to be infinite, Eq. (28) may be written in the form Eg = Zs + e ZL sZs \ e "7L E2 \7.o ) 2 \LQ j 2 Also since the line has no damping,. = 0, and ZO reduces to pcS. The impedance of the machine is obtained simply from its equation of motion. Thus S (ai - g) = ME = iu ME inM = s a - al) = Zs (34) Substituting these values into the above form of Eq. (28), we obtain after some rewriting E, ag _Fg IM E = ' F2 =.cosh yL + S sinh 7L. (5) Since lp = O, = a + ip reduces to y = ipo Using the relations between the hyperbolic functions and the circular trigonometric quantities given by cosh ipL = cos PL, and sinh ipL = i sin ~L, 26

the conventional form of force transmissibility in decibels then becomes 1 T = 20 loglo CcM decibels (56) cos PL - sin L ( pcS This may be put in computational form by introducing the dimensionless ratios PL = f, (37a) ( V M (37b) LDM fX, n (57b) pcS 0 m where EjS c. %D0 = V, K = = c = = M L 'V PThe transmissibility then takes the form T1 = 20 log cos (o (m~ I (38) C c~ oMV c8o Vm VCD^ V sin_ We shall defer computation of other isolation parameters until we consider the effects of resilient foundations. We note in particular that the computation of mount "effectiveness" provides no new information for this case since it has been assumed that the foundation impedance, Zr, is infinite. From Figure 6, it is seen that this assumption equates E2 with Eg. Therefore E2 = Eg E2 E2 and the "effectiveness" thus becomes the reciprocal of transmissibilityo The analysis of Eq. (36) has been performed by various authors (see, eggo, Ref. 2). It will be included here for the sake of completeness. The salient features of the system will be shown to be: lo That at low frequencies, the system response is identical to that predicted by lumped parameter theoryo 2. That the transmission characteristics exhibit attenuation bands alternating with pass bands which occur at frequencies for which the length of the mount is an integral number of half wavelengths (wave effects). 27

3. That a low value of c and a high mass ratio tends to produce attenuation beginning at low frequencies, To demonstrate the relation between the transmissibility as deduced from this theory and that of lumped parameter theory, it is sufficient to note, from Eq. (36), that since 6 == ' = 2< c X where \ = wavelength, then 2i PL = * L And for >> L, sin PL + PL cos PL -+ 1 Hence the transmissibility at low frequencies becomes, using Eqs. (36), (37a), and (37b), T = 20 loglo l _Q 2 \ This is the familiar result obtained from analysis of the lumped parameter model of Figure 11. Fo co wt _ ~^ M K LUMPED PARAMETER MODEL OF MACHINE MOUNTED AGAINST A RIGID FOUNDATION FIGURE II 28

The isolation properties of the model as described by Eqo (36) are placed in evidence by solving for those values of X for which T = 0, T = oo, and T = minimum. The points at which T = 0 indicate the limits of attenuation and pass bands, Thus WM T = 0 for cos PL - - sin L = + 1 pcS This equation is satisfied under any of the following three conditions: PL = nst (n = 0,1,2,...), (40a) cos PL- M sin PL = + 1 for PL ~ ni, (40b) pcS cos PL - M sin PL = - 1 for - L ~ ni. (40c) pcS Equations (40b) and (40c) above may be simplified as follows. From Eq. (40b), uM cos PL - 1 for L n in pcS sin BL Recalling that m = pSL is the mass of the resilient element, we may rewrite the above equation with the help of the standard trigonometric identities as -M iM) PL = tan L, L i nx (40'b) pcS \m/ 2 Similarly Eq. (40c) becomes WM /M) +cL = +cot (40c) pcS \m/ 2 Maximum transmission occurs for T = oo, i.e., when cos PL - - sin PL = 0 (41) pcS or (M) PL = cot PL, pt n_ (resonance points) W'M~~~~ ~ 2 In addition, minimum T is observed when 29

o rcos PL - -- - sin L = 0. D {^ pcS J Hence, substituting w = w/c, and differentiating after c, we have L nDL M. wL wL C L = - sin - -- sn - cos = O c c pcS c pe2S c which, after combining terms, becomes /P SLc + Mc) ML (p- Isin L = - -- cos PL. \ pcS pc2S Finally, this can be rearranged to read / (D 2MLc) = (M - fmPL = - tan PL (minimum transmissibility) o (42) \Mc + me M + my The values of ~L (and thus w) which denote band limits, resonances, and minimum transmissibility are most readily obtained by graphical solution as in Figure 12. In addition to providing the over-all picture of the system's transmission characteristics, consideration of the graphical method of solution is also instructive in regard to the relative influence of the various material parameters M, m, p, c, S, and L. From the upper graph, it is apparent that all attenuation bands terminate, and pass bands begin, at the points PL = nit (n = 0,1,2,...). Thus the intersection of the lines y ~ (M) ~L M. \ with the cot ~L/2 and tan PL/2 curves corresponding to Eqs. (40'b) and (40'c) determines the inception frequencies of the attenuation bands onlyo It is interesting to examine methods of reducing the frequencies at which the attenuation bands begin. From Figure 12 we see that by increasing the slope of the straight lines the intersection points are moved toward the left, the y axis. This corresponds to the lowering of frequency for which attenuation is obtained. Since the slope of the line is proportional to the ratio of machine mass to mount mass, increasing this mass ratio will achieve the desired effect. In addition, it must be recognized that the scale of the abscissa, along which frequency is measured, can be varied. Examining this change of scale more closely, the form (DL/ce where nL/c = ~L, shows that fundamentally the abscissascale unit is controlled by the length of the mount, L, and the velocity of sound in the mount material, c. Hence, a given point on the abscissa may be made to correspond to a lower frequency by expanding the scale of the abscissa. For example, suppose wl. L/c = constant. Then for an increase in the L/c ratio, which essentially expands the scale, the value of cxl corresponding to the point decreases in inverse proportion. 30

GRAPHICAL SOLUTION FOR BAND PASS LIMITS 3 tan --- L1 -3 I - / ________ | PASS I AeU6 P.ATTENUATION -.-L-=, c p-cot iIp-tanPL I I| ____! _l L\ I / _I / _ _ I -I 0 R --- —-- -— ]L -2 -3 GRAPHICAL SOLUTION FOR RESONANCE AND MINIMUM TRANSMISSIBILITY POINTS FIGURE 12 51 -m ~ MNIU T [ ---S"',,~SIB.o~,;,. ----. F~~~IGURE 12~~~~~~~~ 31~~~~4

Thus we have determined that the inception points of the attenuation bands may be lowered by increasing the mass ratio or the mount length along with selecting a mount material that propagates sound at a lower velocity. In a particular application, however, it may be more important to control the transmission characteristics in a more elaborate fashion, e.g., to avoid the occurrence of standing waves at certain specified frequencies, than to achieve attenuation at a very low frequency. Therefore one cannot specify on the basis of a general analysis what constitutes the "best" mass ratio, mount length, etc. The theory may be used to indicate the general features of a vibration-isolation problem, from which design information is available to obtain optimum performance. It is apparent that for this simple case the most important contribution mechanical transmission line theory has offered is its prediction of the pass bands which occur at higher frequencies. The loss of attenuation due to this phenomena, termed "wave effects," had been first observed in practical vibrationisolation devices. Efforts to explain this marked deviation from the predictions of lumped parameter theory encouraged the use of a distributed parameter treatment of the resilient element which had been applied to similar mechanical systems as early as 1930 by R. Bo Lindsay.7 The consideration of distributed parameters associated with a resilient element permits the examination of a resonant condition which does not occur in lumped systems, i.e., the standing waves which are found at frequencies for which the resilient element's length is an integral number of half wavelengths. The elementary treatment of our first example indicated this occurrence of standing wave resonances at regular frequency intervals to indefinitely high frequencies. A plot of the transmissibility, curve (a) in Figure 13, in db attenuation thus reveals, in addition to the "normal mode" peak at low frequencies, the fundamental and higher harmonics of the "wave effect" peaks. THE MECHANICAL FILTER In the general formulation of the problem, the possibility is provided for examining connected transmission paths which are individually homogeneous but different from each other in composition. Our next case will involve the simplest example of such a "compound" line. For the time being, the machine and foundation will be relegated, as in the first example, to rather unimportant roles in the system. We may anticipate the results somewhat by considering the fact that constructing a compound line is analogous to the connection of electrical transmission lines having different characteristic impedances. Electrically, this "impedance mismatch" causes a loss in the power transmitted. We therefore expect that some increase in attenuation is to be achieved. The various sections of a compound line are usually treated as elastic elements. However, one may ignore the elasticity of certain sections as a first approximation, and consider simply the impedance presented by the mass of these sections. We shall now consider an example of this type. In keeping with the desire to maintain conceptual and algebraic simplicity, the machine and founda32

_ _ _ _ ___ ___ u A -m -.__-___ _ _ __.... L....- - -. --- = =I_ z A,==._. _. _ -_ 3W '.

tion will again be considered rigid and the effects of viscosity neglected. The model to be considered is similar to that used in the first example and is given in Figure 14o The machine mass is M1 and the mass of the interposed mount section is labeled M20 We shall refer to M2 as the loading mass. The over-all length of the resilient transmission line will be considered the same as in the first example with the loading mass attached at x = L/2. The electrical circuit, shown in Figure 15, is obtained in the same manner as before, using the corresponding elements from Table 1. In this case, the four-terminal networks representing the two halves of the transmission line will naturally have ABCD parameters involving L/2 instead of L as before. The analysis of this circuit can be conveniently carried out using matrix algebra. Recalling the usual rules of matrix addition and multiplication, consider the E,I relations across the last four-terminal network on the right in Figure 15. We have by Eqs. (1) and (2) Em2 = A E2 + B I2 Im2 = C E2 + D I2 In matrix form these equations may be written Em2 A B E2 -S~g C ID M, L --- — ' // -- ' C m2 Ml, — r2, M2 MACHINE- COMPOUND MOUNT- FOUNDATION MODEL FIGURE 14 34

8 N X 0 ( _______ Im-I l f < g H i 9 I ' _,___| U s t~z _) o m4

Proceeding to the left in Figure 15, similar relations are written for each successive four-terminal networko Hence, for the impedance Z2, Eml = Em2 + ZzIm Iml = Im2 which, written in matrix form, becomes Emr 1 1 Z2 Em2 Iml 0 1 Iml Substituting for the Em., Im2 matrix, we have Emi 1 Z2 A B E2 I Iml.0 1 C D I2 Again E1I 1 A B Eml A B 1 Z2 A E Ii I C D Imi C D 0 1 C D I2 and finally, since |Eg 1 Z1 EI Ig 01. I10 then Eg 1 Z1 A B A 1 Z2 A,B E2 I g1CD 0 1 C D Il (43) Carrying out the indicated matrix multiplication yields Eg| [A+ZiC] [A+Z2C] + C [B+Z1D] [A+Z1C] [B+Z2D] + D [B+Z1D] E2 Ig| c [A + Z2C] + CD C [B + ZzD] + D2 12 36

Since I2 = 0, for this case, the above expression reduces to: Eg [A2 + AC (Z1 + Z2) + Z1ZZCZ + CB + Z1CD] E2 Ig [AC + Z2C2 + DC] E2 The voltage and current may immediately be read off as follows: Eg = [A + AC (Zi + Z2) + ZIZaC2 + CB + ZiCD] E2 Ig= [(CA + Z2C2 + DC) Ez] From the voltage equation: Eg/E2 = A2 + AC (Z1 + Z2) + Z1Z2C2 + BC + Z1CD Utilizing the assumptions that the masses M1 and M2 are rigidd, the impedances Zl and Z2 are Z1 = i M1, Z2, = ic M The ABCD parameters are given by Eqs. (7"a), (7"b), and (7"c), noting that L/2 must be substituted as the length of each resilient element. D = A = cosh Z = cos L. (since t = 0) 2 2 B = ZO sinh L = Z i sin 2 0 2 C = sinh yL/2 = sin _L ZO ZO 2 Substitution into the voltage ratio results in E = cos2 + i(Zl+Z) sin -L cos L -P sin Lsn2 + isin cos L Ea 2 ZO 2 2 Zo 2 2 ZO 2 2 = cos L + i sin L + i sin L Z Z in2 s L Zo 2 Z ZO 2 Finally 37

- = cos PL - - - sin PL FinL (44) E2 pcS pcS L pcS 2 j The transmissibility in decibels attenuation becomes: T = 20 logo E2 = 20 logo cos L - o- sin ~L - M in sL - sin pcS pcS L pcS 2 (45) The dimensionless form of Eq. (45), using the following relations 0o = i — C02 = m = pLS Y Ml M2 PL = (t-, K = ES ES 0ol yM1 L (U 1 wM2 - u lf pcS m pcS O V m becomes T= 20 log3( o IW 1 c 1 4 m T == 2 o - sin -[sin -. 1 X M01 o1 cW m 0ol VMJ ol m ' ol M OZV m 2cobl IiJ (46) Although the algebraic complexity has already become impressive for this relatively simple problem, the essential characteristics of the system are easily obtained from Eqs. (44) to (46). For instance, it is encouraging to note that Eqs. (45) and (46) reduce to Eqs. (56) and (38), respectively, if M2 (the loading mass) is chosen negligibly small. The effects of adding M2 is most easily seen from Eq. (45) by noting that the first of the band-limit equations of the previous example, Eq. (40a), is now changed to T = 0 for PL = 2nx (n = 0,1,2,....). (47) The significance of the factor 2 appearing here is that pass bands occur only half as many times as in the previous system~ Physically, this means that the standing waves which previously entertained an antinode at the point of attachment of the loading mass (odd harmonics) have now been suppressed by this mass. Hence, only those standing waves which have the point x = L/2 as a node are possible with the mass attached. Thus it is apparent that a single loading mass attached at the center of the rod tends to suppress every other wave resonance, providing in this way relatively wide attenuation bands between the "wave-effect" pass bands as shown by curve (b) in 38

Figure 13. In fact, attaching the mass at the center may be shown to provide the widest attenuation bands possible for a simple mass-loaded lineo Shifting the loading mass off center causes each of the remaining wave-effect peaks to split into two peaks, their separation being controlled by the distance of the mass from the center As the loading mass moves away from the center, the pairs of transmissibility peaks increase their separation until they finally arrive at the positions of the simple line peaks as the mass is removed from the line, For certain points of attachment, for example, as with the loading mass at L/3 from the bottom as shown by curve (c), Figure 13, a peak moving up in the frequency scale becomes superimposed on the adjacent descending peak, resulting in a transmission characteristic similar to that of center loading. The difference is that the fundamental frequency has been reduced due to lower frequency standing waves occurring in the longer portion of the resilient element. Since the higher peaks occur at regular harmonic intervals, the attenuation bandwidths are reduced for off-center loading. In addition we note that by introducing impedance mismatch at the center of the transmission line, the amount of attenuation obtained in the attenuation bands has been approximately doubled over that observed in the first example as anticipated. Evidence of this fact is provided in Eqo (45) wherein the coefficient of the predominant term at high frequency, sin2 PL/2, is proportional to the square of the frequency rather than simply proportional as in the first case o Although more detailed analysis of these equations is necessary when the elasticity of the interposed section is considered, the essential behavior has been sufficiently indicated in the discussion thus far, For further analysis of the suppression of odd harmonic "wave effects" in compound lines and the supporting experimental evidence, see Refs. 6 and 7. Comparing the foregoing examples a bit further, the occurrence of wave effects as a natural consequence of the elastic element's continuity is again emphasized. It is possible to show, from Eq. (45), that the transmissibility as obtained from this theory reduces at low frequency to that obtained by conventional lumped parameter methods. The equation obtained from Eq. (45) by assuming the length of the resilient element to be small compared to the wavelength is quadratic in n, yielding the two normal modes of vibration of this system. The two peaks attributable to these modes are clearly evident in curves (b) and (c) in Figure 13. The entire resilient section, pictured in Figure 14, consisting of a length of elastic element with a loading mass rigidly attached to the center has long been known in acoustics as a "section" of a mechanical filtero It is possible to construct a mechanical filter consisting of an arbitrary number of such sections. The general behavior of such an n-section filter is very much like that of the analogous electrical filter, As additional sections are added to the filter, some attenuation is lost at low frequencies due to the additional normal mode resonances created by the additional "degrees of freedom." However, this 39

is compensated for by the increased attenuation received above the highest normal mode resonance, the attenuation predicted being approximately n times the amount provided by a single filter sectiono Again when sufficiently high frequencies are reached so that the mass spacing becomes of the order of a half wavelength, a loss of attenuation is observed due to standing waves in the resilient element. Although the pass bands caused by standing waves become increasingly narrow and less detrimental at higher frequencies, their effect is primarily to slow the rate of increase of attenuation with increasing frequencyo This type of mechanical filter corresponds roughly to the electrical lowpass filters. Attempts have been made to try to design a mechanical high-pass filter with very limited success (see Refo 6). The difficulty encountered in attempting to obtain attenuation starting at zero frequency is essentially that the presence of an infinite mass is needed at some point in the system. Thus the mechanical analog of the electrical high-pass filter is most unlikely to provide any fruitful source of investigation for those interested in obtaining practical results in noise isolation. It appears that the best one can do is to design his isolation system to obtain attenuation of a few frequency bands which are most troublesome. In general, the lower the frequency to be attenuated, the larger the masses involved will have to be, This, of course, is not a novel result, but is essentially a restatement of the well-known results of lumped parameter theory that the larger the mass and the softer the spring, the lower the resonant frequency will be, and hence the lower the frequency at which attenuation is obtained, Thus it is clear that the "mechanical filter" provides no new mechanism of isolating vibration sourceso SECTION III. NUMERICAL ANALYSIS OF A TYPICAL SYSTEM Thus far, the examples considered have been elementary to display conveniently the salient features of the theory without unnecessary complications. However, the methods discussed are just as easily applied to entire systems consisting of several vibration sources, transmission paths which have damping, and terminations possessing arbitrary impedanceso The difficulties encountered are largely computational rather than conceptual. As an extension of the theoretical work contained in the previous two sections, this laboratory has attempted an application of network theory to the analysis of a typical heavy-machine noise-isolation problem. The intention was to use representative values of machine, mount, and foundation impedance as the basis for a numerical example illustrating: (1) the applicability of network analysis to a typical system; and (2) the magnitude of the computational problem involved in the analysis of a complex systemo However, upon embarking on this investigation it became apparent that the scarcity of experimental impedance data applicable to this method of analysis precluded the possibility of analyzing 40

an actual system. Thus it became necessary to construct an artificial system employing the type of data that are available, to furnish in an indirect manner the basis for the computationo Before reporting the details of this computation, we shall discuss briefly some of the considerations in -selecting the characteristics of the artificial system. BACKGROUND AND DEFINITION OF THE PROBLEM The best data that were found to be available for computational purposes were the following: (1) impedance curves which were considered to be typical of the internal mechanical impedance characteristics of a heavy machine as measured at four points of support; (2) transmissibility data on several commercial mounts under rated loading; and (3) force-to-velocity ratios which were characteristic of four points on a machine foundation structure. It originally had been hoped that the system analysis could be carried out for the case of a machine having several points of support. However, because values for the transfer impedances appearing between the normal machine-support points were not available, the analysis had to be restricted to a single-pedestal machine mounted through an isolator to a single point on a resilient foundation as shown in Figure 16, Had the transfer-impedance data been available, the analysis would have been carried out using a circuit configuration similar to that of Figure 17 which shows schematically a general form of the mechanical network for a machine having two points of support, The effects of the multiple mounting and of the transfer impedances can be considered through the use of one circuit of the type shown in Figure 16 for each point of support, with the points of support on both the machine and the foundation side, respectively, interconnected through networks having suitable transfer-impedance characteristicso One could then apply excitation to each of the circuits in succession and compute the response at each foundation terminal. Then by assuming a linear response, the individual responses could be superimposed to obtain a first approximation of the total foundation response due to the vibratory effects of the machine, ANALYSIS OF THE SYSTEM Employing the theory developed in the first two sections, we may draw the circuit analog for a pedestal-mounted machine as in Figure 16. The force-velocity relationships at each end of the mount may immediately be written in matrix form as follows: Eg |1 Z A B Er -- =o: i i (48) I0 1 C D I 41

L o z 0 -.'-fr\~~~F::) *ji~~ w N I Z C u LLL 0CD 0 ~ ~ _ w OIA w< N Z xLLJ 0N 0. 42LlLJ

OL 5 I I I i 0X ToFC z N 1 N ( ' (.l;:: w- 0:cn u a: 5r4 a a u~r uJ w I CQQ~~~~~~: =4 >I I —

where Er = Ir Zr. (49) Performing the matrix multiplication indicated yields gI CEr + DIr The ABCD parameters are given by Eqs. (7a), (7b), and (7c), or, more fundamentally, by their definitions in terms of the mount-impedance values, zll, Z12, Z21, and Z22. Force and velocity transmissibility may be obtained from Eq. (50) by taking 20 log io of the voltage and current ratios. The complex power ratio is obtained by taking the product of these ratios. The real part of this complex number represents the real power ratio, as it is conventionally usedo These ratios are given below. Force ratio: fg - Eg = ( + z) cosh 7L + r + sinh yL (51) Fr Er Z \Zr ZJ Velocity ratio: g I Zr -g = g = cosh yL + -- sinh yL (52) Ir Z0 Complex power ratio: P Egg 1 F Z\ s Z(r Z 0 g E 2 [KZ+ r)+ Z 0 + cosh 2 yL Pr =ErIr 2 = r/ ZO 1 Zr Z Zs zo 1 Zs Zrzs 2 Zo \ ZI. Zo Zr/J 2 (53) The mount "effectiveness parameter may be obtained from (2b) Effectiveness = (Zs + ZO) (Zr + ZO) eL - (Zs - ZO) (Zr - ZO) e 7L 2 Zo (Zs + Zr) 44

Using the hyperbolic function identities, this reduces to Effectiveness = sZr +Z0 sinh yL + cosh yL. (54) Zo (Zs+Zr) Still another parameter of possible value is the system transfer impedance, ZT, i.e,, the ratio of force generated by the machine to the foundation velocity: System F ( cZh + Zcr Transfer = = (Zr + Zs) cosh L +sinh L Impedance J I2 () The preceding expressions provide the framework for numerically computing the values of the isolation parameters as a function of frequency once the impedance values which properly characterize the machine, mount and foundation are determined as a function of frequency. We shall now consider the application of these results to the fictitious system which is intended to characterize the degree of complexity found in a typical physical system~ NUMERICAL RESULTS The magnitude of the machine and foundation impedances were taken from the curves shown in Figures 18 and 19 which for the present purpose were considered to be typical. Although only the impedance modulus is plotted in each case, the real and imaginary parts of the machine impedance were obtained by estimating the resistance at resonant points and computing the reactance from this, assuming the resistance to be relatively constant in the vicinity of the peaks. In the case of the foundation, only one resonance existed, at about 1200 cps. The resistance was estimated at this point and the reactance computed for the 10-to800-cps range, using this value of resistance. Considerably more effort was necessary to obtain comparable data for the mount. In the past, mount measurements have consisted almost entirely of obtaining "transmissibility" data under specified (usually rated) loading. These data are virtually useless for inclusion in a system analysis using a networkimpedance approach. What of course is needed in the way of mount data is the input and transfer impedance of each end with the opposite end rigidly constrained. The procedure followed in this case was essentially to select by trial and error the physical parameters that a simple cylinder of elastic material would have to possess to duplicate the transmissibility curve shown in Figure 20. This curve is the result of experimental measurements made on a conventional commercial mount. The equation for transmissibility was obtained by application of the theory presented in this paper to the physical system on which the measurements were made. Using this theory, one finds the equation for transmissibility to be: 45

i'| | 1111111 1 1 1111111 i i, ~,I.. ---. a L::::_S,===:::::-====::::==== ^ 8 - Z * 8 U _i 0 a: [:::======= n::2!Iy =::I I3 i HiI g!

., I:.1 ' 1 ' I -1z z- _Z-"b 0, o ___ _ In: - )l / 414 ~. Cm o ILLW ~ ~ ~ ~ ~ ~ ~ I II~~ w 4 ____ 5 2 ~ w CM - -- - - -o Or + + (40qa) AlI119 ISSIWSNVaiL 47

T = 20 log10 cosh yL + iM sinh L (56) z 0 Thus by considering various trial values of the physical parameters of the material (Youngls modulus, velocity of sound, visco-elastic coefficient, crosssectional area of the cylinder, and length of the cylinder), the proper values were selected so that the number computed from the above equation matched the experimental curve at about twenty points between 10 and 800 cps. These values were then used to compute Z0, a, and P from the formulas given in Section I. The graphs of these three variables as a function of frequency are given in Figures 21, 22, and 23, respectively. The data used to characterize the mount are drawn entirely from these three figures without further consideration of their possible significance. The graphs of the physical properties of the fictitious mount material are included in Figures 24, 25, and 26 for the sake of completeness. The radius of the cylindrical mount was selected as 2.8 ino and its length as 5 ino It might be mentioned that the length is determined by the frequencies at which the first wave effects occur and by the velocity of sound in the material at these frequencies. Once these machine-mount-foundation data were compiled, a decision had to be made as to which one of the several isolation parameters listed earlier was to be computed since lack of time prohibited more than one sample computation. Somewhat arbitrarily, the system transfer impedance was selected. Equation (55) was put in the proper form for the computation simply by expanding all complex quantities, rationalizing individual terms, and collecting all real terms and all imaginary terms, The modulus of this complex number was then determined by computation involving some 34 distinct, elementary operations on a standard desk calculatoro The time required for the computation of about 25 characteristic points by a moderately experienced laboratory technician was approximately 20 hours. Further practice with the computational procedures might reasonably be expected to reduce the time for a problem of this degree of complexity to about 12 hours. It should be pointed out, however, that the inclusion of effects due to transfer impedance would greatly increase the time of computation. Although the operations still would remain within the scope of a desk calculatorts function, the time element would probably encourage the use of automatic computers if there were a considerable number of runs to be made on the same general program of computation. The results of the computation are shown in Figure 27 where the reciprocal of ZT is shown as a function of frequency. This quantity, 1/ZT, is equal to the foundation velocity response to a unit force generated by the machineo It should be noted that the general behavior of this function bears little similarity to the mount transmissibility or to the reciprocals of the machine and foundation impedanceso Rather it represents, as one would expect, a composite 48

M _7 _ 13 \. 0 100 20 300 400 500 600 700 SW FREQUENCY (C.P.S.) REAL PART OF MOUNT CHARACTERISTIC IMPEDANCE 1.4 31e~ i i it111 i iL11~ II0.6 0.9 _ ' 1.,1 - TE 2 L| ~ 1 T - _ POSITIVE VALUES 0 100 200 300 400 500 600 700 800 FREQUENCY (C. P.S.) IMAGINARE PART OF MOUNT CHARACTERISTIC IMPEDANCE 49 4-9

0.7 -0,.5 0.4 --- — - 0. - 0.2 0.1 -O 100 200 300 400 500 600 700 800 FREQUENCY (C.P.S.) MOUNT ATTENUATION FUNCTION, a FIOURE 22 35O - - -I -' -I -- II- - -- — I — 300 - 20 to lS.. SO- 15-~o L ZlZ I I I I! i I! 0 100 200 300 400 500 600 700 800 FREQUENCY (C. P. S.) MOUNT PHASE FUNCTION, f FIGURE 23 50

r 1 \ I — 3 11 i1 1!!!!X LL I |,M 0 CY_ cw Ir:_ ~ IOr /I Oi _ r ___ 0 8:!/8 tO 8 8 8 o X 8 8 g o (smnnow SNnoOA) Zul/ql '3 51

8 -: ------ __ --- —--- 0 C I — o z ---- ---- --- z -- _ 1 =E o _ _~/ I l I.OIL! O UJ So O d 1: '- I I1O (aNnOS iO A11O1303^ ) *S/UI 3 52

->~~~~ - ( - - - - - - I. i i i I I I I I. II! l i l l l El I U.. * * - * _ - - 4 IJ i v30: O11SV13-03SIA~, IqF...W Io w w f cu q No w t 0 N ' - - - d d d 0 5 ~ ru

U -1 --- - -- - - - - - — __ — --- i- --- - - -i — ___ — -- —. -" 0 10. 10 100 1000,w FREQUENCY (C.-P.-S.) FOUNDATION VELJDCITY RESPONSE TO UNIT FORCE GENERATED BY MACHINE FIGURE 27 ]r~~~~~~~~~~5 wr Ii. I-.~ ~ ~ ~ ~ ~~FRQEC -C.. I'm.FU DTO EI) VRESOS OUNTFR 0 ~ ~ ~ ~~GNRTDB AHN 0 ~ ~ ~ ~ ~~~~IU~2

of the properties of all three elements in the system. A brief look at the behavior of the transfer-impedance function shows that the foundation velocity response in the vicinity of 10 cps is high due to the normal mode.resonance of the system. In the frequency range immediately above this, the foundation velocity decreases until a minimum is reached at 40 cpSo The curve then shows an increase in foundation velocity which may be attributed largely to a strong machine resonance just above 40 cpSo After a relative maximum is reached at 70 cps, the response decreases until a broad minimum is reached in the 250-to-400 -cps range. The fact that little increase in foundation velocity appears at the first wave resonance of the mount is apparently due to the high machine and foundation impedances obtained at 360 cps. The irregular behavior found in the 400-to-700-cps range seems to follow the machine impedance characteristic most closely, while at 800 cps an extreme increase in foundation velocity is noted due to a combination of resonance characteristics in all three elements of the system. The fact that the properties of no one element of the system correspond to the properties of the system over a wide frequency range indicates the need for considering the system as a whole when attempting to evaluate analytically the importance of component properties such as mount or foundation damping, wave effects, and the like. SECTION IVo SUMMARY AND RECOMMENDATIONS EVALUATION OF THE NETWORK APPROACH The analysis of complex vibratory systems has no formulation that is at once simple and complete. In the approach described by this paper, we have only a better approximation to the explanation of vibratory phenomena than the conventional and simpler lumped parameter theory provideso Many obvious limitations remain imposed on the distributed parameter analysis, and these must be kept in mind when evaluating the results of the theory. Specifically, it should be constantly remembered that this is still a one-dimensional theory which considers only the propagation of plane longitudinal waves in structures. In general, there is little reason to assume that this mode of vibration is the most prominent causative factor of detrimental effects in the system. It is simply the easiest mode to treat analytically. However, the fact that this one-dimensional theory cannot provide all the answers should not give rise to pessimism. For we have here the framework on which to build a better understanding of difficult problems that demand solution. Extensions of the capability of the theory are always possible as, for example, by the addition of empirical corrections which experience suggests or by implicitly taking into account other effects known to be important through better methods of applying the techniques which the present theory affords. 55

Such better methods in turn help to shorten or improve the experimental labors required to achieve useful practical resultso The functions that this network approach might be expected to serve are two-fold. First of all, the fact that it employs the various impedance quantities outlined in Section I serves to clarify the objectives sought in apply.ing the impedance concept to the analysis of vibrating mechanical systems. For one must have some knowledge of the relation such a concept bears toward the parameters which measure the quality of a systemfgs design, and this is most readily obtained via the mathematical analysis of the system, Since the relation between the isolation parameters and the various impedance quantities is given by network theory, the way is opened to new design approaches in which certain preferred impedance characteristics may be sought in the construction of the system components. Secondly, the network approach imposes specific conditions which must be met in the experimental measurement of the appropriate impedance quantities. Hence it serves to define the type and form of impedance measurement that must be obtained for fruitful analysis of a vibrating system. The theory also provides the mechanical interpretation of the particular impedances employed which forms the basis of the experimental technique used to measure their values. Thus it gives direction to the experimentalist in making impedance measurements by making clear the specific use that will be made of his datao Before any real benefits can be derived from the prediction of system behavior provided by this network approach, better impedance measurements must be devised and be made available, especially with regard to the Thevenin equivalent series impedance of the machine and the input and transfer impedances of each end of the mount with the opposite end rigidly constrained. Once these measurements are made available, a better evaluation of the significance of the network approach can be made by comparing its predictions with experimental results. In regard to the recommendations made on behalf of the "mechanical filter" concept in BuShips Memorandum, Ser 371-M91, it may be stated that the one-dimen-.sional theory of mechanical filters has been worked out in considerable detail by several investigators and is duplicated in a large part in Section II of this report. To summarize the result of this work, mechanical filters do not provide any new mechanism for achieving vibration isolation. Nevertheless the principles involved may find useful application in certain problems since they provide essentially one technique of altering the impedance characteristics of the mount structure. The value of this technique can be determined only after such time as impedance considerations become a significant factor in the design of isolation systems. 56

SECTION V. REFERENCES 1. Thorpe, H. A., Application of Acoustic Filtration in Solids to Ship Noise Problems, Bureau of Ships, Washington, Do Co, BuShips Code 371 Memorandum for File Ser. 371-M91, 19 May 1954o 2. Harrison, M., and Sykes, Ao 0., Wave Effects in Isolation Mounts, David W. Taylor Model Basin, Washington, D. Co, DTMB Report No. 766, BuShips Project No. NS 713-219, 44 p. inclo illus., tables, graphs, May, 1951. 35. LePage, W. R,, and Seely, S., General Network Analysis, New York, McGrawHill Book Company, Inc., 1952. 4. Bauer, B. B., "Equivalent Circuit Analysis of Mechano-Acoustic Structures, Transactions of the I.R.E. Professional Group on Audio, Vol. AU-2, No. 4, July-August, 1954. 5. Sykes, Ao 0O, "The Evaluation of Mounts Isolating Nonrigid Machines from Nonrigid Foundations," Shock and Vibration Instrumentation, New York, The American Society of Mechanical Engineers, 1956. 6. Epstein, D., Filtration of Sound, National Bureau of Standards, Washington, D. C., NBS Report No. 1494, Part III, February, 1952. 7. Lindsay, R. B., and White, Fo Eo, "Theory of Acoustic Filtration in Solid Rods," Journal of the Acoustical Society of America, Vol. 4, No, 2, October, 1932. 57

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