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Frequency response functions for power and connectivity
Journal of Sound and Vibration (1995) 181(4), 709–725
FREQUENCY RESPONSE FUNCTIONS FOR POWER AND CONNECTIVITY L. L. K Department of Mechanical Engineering, Monash University, Clayton, Victoria 3168, Australia (Received 9 March 1993, and in final form 31 January 1994) Methods of calculating vibratory power transfer for complex structures by using frequency response functions for power are developed. Connectivity theory for mobility functions is employed to determine frequency response functions for power for connected structures and to develop connectivity relations for power transfer by employing frequency response functions for power. Mobility functions and frequency response functions are computed and compared for two complex structures with different types of damping. The comparisons show that both the mobility method and the frequency response function for power method yield identical results. An exact relationship for power sharing for all types of force input, equation (7), and a relationship which applies to cases of high frequencies and random force inputs, equation (9), are derived and employed for selected structural elements. General formulas relating power transfer and modal properties are developed by using the frequency response function for power concept. These formulas give further insight into the role of damping in power transfer relations; see, e.g., equation (36). Frequency response functions for power for many different structural elements are presented in Tables 1 and 3.
1. INTRODUCTION
In several previous papers [1–3], frequency response functions for power were defined and related to mobility functions. Frequency response functions for power allow one to predict power variables at a point in a vibratory system given the power input at a source position. In these papers both fluctuating and mean power were considered; fluctuating power does not contribute to energy flow, whereas mean power is associated with energy flow. Power input at a source point is given by the product of the input force and the driving point velocity; the response ‘‘power variable’’ is proportional to velocity squared: e.g., a damping coefficient multiplied by velocity squared gives dissipated power. Also, use of the frequency response function for power concept allows one to determine vibration severity as a ratio to the input power. Thus, by using the frequency response function for power and damping values the power dissipation in a system can be computed. The inverse of the frequency response function for power gives the apparent damping at a point. Only mean power is considered here. The purposes of this paper are fourfold. The first is to illustrate the use of connectivity theory for mobility functions [4–7] for the calculation of frequency response functions for power; the second is the development of connectivity theory by using frequency response functions for power; the third is the presentation of frequency response functions for power for simple and complex structures and is based upon mobility functions given in references [8, 9]: the fourth is the development of general power relationships. Connectivity theory 709 0022–460X/95/140709 + 17 $08.00/0
7 1995 Academic Press Limited
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is used to predict power distribution in a complex structure where the structure is considered to be built up of several structural elements. Mobility is a complex ratio of velocity at a point in a vibratory system to an applied force. The frequency response function for power is a real quantity and is a ratio of the velocity squared to the input power. 2. CONNECTIVITY THEORY
2.1. As the development of connectivity theory can be found in references [4–7], only a summary will be presented here. For the purpose of presentation the terms aij , bij , cij , etc., refer to mobility functions at a given frequency v; the frequency dependence will not be shown explicitly. Definitions of principal symbols are given in the Appendix. Shown in Figure 1 are two subsystems, b and c, to be attached to each other at point 2, where a force input (as well as power input) exists. The boundary conditions at point 2 are that the velocities of each of the attached bodies are the same and the force input is shared by both bodies. If b22 and c22 are the point mobilities at point 2 for body b and body c respectively, then the driving point mobility function for the combined system, a22 , is given by 1/a22 = 1/b22 + 1/c22 .
(1)
From Table 1 in reference [3], the frequency response function for power at zero frequency Hpvv (0), the ratio of mean square velocity to average power at point 2 for a sine wave force input, is given by Hpvv22 (0) = =a22 =2/R[a22 ],
(2)
where R represents the real value of the enclosed mobility function, p is input power, and vv is velocity squared. The first numerical subscript indicates a response point and the second subscript an input point. Shown in Figure 2 is a different situation, in which two structures are connected at point 2, force and power are input at point 1 on structure b, and the response at point 3 on structure c is required. The mobility function for this case is given by a31 = b21 c32/(b22 + c22 ),
(3)
where a31 is the transfer mobility from force at point 1 on body b to velocity at point 3 on body c. The frequency response function for power for this case is given by Hpvv31 (0) = =a31 =2/R[a11 ].
(4)
For a three-body system (see Figure 3), the transfer mobility from point 1 on structure b to point 4 on structure d, where structure b is connected to structure c at point 2 and structure c is connected to structure d at point 3, is given by a41 = b21 c32 d43 /[(d33 + c33 )(b22 + c22 ) − (c23 )2],
(5)
where dij represents the mobility functions for structure d. Equation (4) can then be employed to calculate Hpvv (0), with a41 used in equation (4) instead of a31 .
Figure 1. (a) Bodies b and c connected at point 2. (b) The free body diagram of bodies b and c.
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Figure 2. Bodies b and c are connected at point 2; input is at point 1 on body b and the response is required at point 3 on body c.
Thus, frequency response functions for power can be obtained directly from available point and transfer mobility functions of individual components. The mobility functions can be obtained from tests, from analytical procedures, from finite element analysis, or from available literature. The above equations have several restrictions. First, they are applicable to a sine wave input force; second, they apply for the case where the structures are connected at a single attachment point, except for equations (2) and (4) which apply to any mobility function; third, linear behaviour is assumed. 2.2. : The question arises as to whether frequency response functions for power for structural components can be combined in the same manner as for the case of connectivity for mobility functions. As will now be shown, the frequency response functions for power can be combined as per mobility connectivity theory to give an overall function for certain situations. Conservation of power at point 2 in Figure 1 gives the relation p1 = p2b + p2c ,
(6)
where p1 is input power and where p2b and p2c are the powers going into structures b and c, respectively. Equation (6) is now divided by v22 , where v2b = v2c = v2 , and, by using the definition that v22 /p gives a frequency response function for power, one obtains 1/Hpvv2 (0) = 1/Hpvv2b (0) + 1/Hpvv2c (0),
(7)
where Hpvv2 (0) is the overall frequency response function for power at point 2 of the connected structure, and Hpvv2b (0) and Hpvv2c (0) are the driving point frequency response functions for power of components b and c when isolated from each other, respectively. Equation (7) is similar in format to that of equation (1) but also applies to the case where the force consists of a band of noise, as the definition of the frequency response function for power for this case (driving point) is given by (see Table 1 in reference [3]) Hpvv11 (0) =
g
v2
R[a11 (h)]=a11 (h)=2 dh
v1
>g
v2
{R[a11 (h)]}2 dh,
(8)
v1
where v1 and v2 are the lower and upper frequency limits of the band of the random force input, and h is a dummy frequency variable. Equation (7) is valid for all types of force inputs and is independent of modal density.
2
Figure 3. The three-body system where input is at point 1 on structure b and the response is required at point 4 on structure d.
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The frequency response function for power equivalence of equation (3) is Hpvv31 = Hpvvb21 Hpvvc32 /(Hpvvb22 + Hpvvc22 ),
(9)
where the (0) has been left out from the frequency response functions for power for brevity. Equation (9) is applicable to cases where the velocities induced at the attachment point from the input force and from body c are uncorrelated. Otherwise, a term of the form fv2b v2c dt will appear in the evaluation of mean square velocity and the integral becomes zero when the two velocities are uncorrelated. This will correspond to situations of random force inputs and high modal density.
3. EXAMPLES
3.1. Consider the case of a pure stiffness element with spring constant k. The mobility function, a11 , of the spring is a11 = iv/k,
(10)
where i = z−1. Using the defintion of frequency response function for power, Hpvv (0) = =a11 =2/R[a11 ],
(11)
and equation (10) gives the following result for the stiffness element: Hpvv (0) = 1/0 = a.
(12)
Similarly, for a pure mass the frequency response function for power is infinite. Listed in Table 1 are mobility functions, frequency response functions and apparent damping for six fundamental structural vibration elements; apparent damping is the inverse of the frequency response function for power. As an example of the use of equation (7) i.e., connectivity for frequency response functions for power), consider a mass attached to a spring/dashpot system. The frequency response functions for power for a mass and spring with a dashpot in parallel are, from Table 1, a and 1/c, respectively. Employing equation (7) yields the overall frequency function for power as 1/Hpvv (0) = 1/(1/c) + 1/a,
(13)
T 1 Mobility functions and frequency response functions for power (Frfp) for simple elements
Element 1. 2. 3. 4. 5. 6.
Spring Mass Dashpot Spring and dashpot k, m and c Base input, k, m and c
Mobility ((Ns)−1 m)
Frfp† ((Ns)−1 m)
iv/k a −i/mv a 1/c 1/c iv/(k + ivc) 1/c iv/(k − mv 2 + ivc) 1/c iv[1/(k + c) − 1/(mv 2)] [1 − 2k/(mv 2) + (k + (vc)2)/(mv 2)2]/c
Note that k is the spring stiffness, m is the mass and c is the damping constant. † Frfp is the frequency response function for power.
Apparent damping (Ns m−1) 0 0 c c c —
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or Hpvv (0) = 1/c,
(14)
which is the result given in references [2, 3] and for element 5 in Table 1. 3.2. A case is now presented in which the free end of a cantilever beam, body b, is attached to a fixed–fixed beam, body c, at a point 0·4 of the distance from one of its fixed ends to the other; see Figure 4. The frequency response function for power between point 1, input, and point 3, response, is required. The function will be calculated by two methods, first by using mobility functions and equations (3) and (11), and second by calculating the input power at point 1, and the output power variable, velocity squared, at point 3 and taking the appropriate ratio. Both beams being examined are made of steel and have dimensions of 1 m in length, 0·01 m in width and 0·05 m in height, with bending occurring in the direction of the height. The Mathworks ‘‘Matlab’’ routine was employed for the calculation of mobility functions and frequency response functions for power. Mobility functions are given by the equation (see Chapter 2 in reference [7] for forms of mobility functions, for a proportionally damped system) ajk , bjk
or
cjk = siv rCj rCk /[(kr − v 2mr ) + ivcr ],
(15)
where r is the mode number, mr , kr and cr are the modal mass, stiffness and damping in that order, and r Cj or k is the mode shape for position j or k for mode r. Modal data has been calculated from formulas and data given in reference [10] and is presented in Table 2. Mobility functions for the cantilever beam and the fixed–fixed beam in the unconnected state are given in Figures 5 and 6, respectively, for a frequency resolution of P radians/second. At point 1, the point mobility function for the connected condition is obtained from 2 a11 = b11 − b12 /(b22 + c33 ).
(16)
A comparison between the unconnected beam and the connected beam’s mobility function at point 1 is shown in Figure 7. A comparison between the transfer mobility and the frequency response function for power between points 1 and 3 is given in Figure 8. As can be observed from this figure, the forms of these functions are not the same, and thus the transfer mobility alone does not give the character of the power transfer from
Figure 4. The double beam problem. Input is at point 1 and the response is required at point 3; beams are attached at point 2. See Table 2 for beam properties.
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T 2 Properties of beam components A. General properties of both beams: (1) mass per unit length, 3·85 kg m−1; (2) product of modulus of elasticity and area moment, 21·875e3 Nm2 B. Cantilever beam modal properties: (1) assumed damping ratio for all modes, 0·001; (2) mode shape data are as follows: Mode
vr (rad s−1)
cr (Ns m−1)
kr (Nm−1)
1 2 3
265 1661 4651
2·04 12·79 35·81
2·70e5 1·06e7 8·33e7
C (point 2) 2 −2 2
C (point 1) 0·68 1·42 0·04
C. Fixed–fixed beam modal properties: (1) assumed damping ratio for all modes, 0·001; (2) mode shape data are as follows: Mode
vr (rad s−1)
cr (Ns m−1)
kr (Nm−1)
C (point 2)
C (point 3)
1 2 3
1686 4649 9114
12·98 35·73 70·18
1·09e7 8·29e7 3·20e8
1·46 1·04 −0·63
1·00 −1·51 1·11
D. Dimensions: (1) both beams are made of steel and have the same dimensions; (2) length 1 m, width 0·01 m and height 0·05 m point 1 to point 3. The frequency response function for power does not have as sharp peaks as the mobility function and is of higher amplitude. Computation of the frequency response function for power by both methods mentioned above give the same result. A comparison of the point mobility function and the frequency response function for power at point 1 is shown in Figure 9. At resonances, the two functions have the same magnitudes, and thus at resonance frequencies the power input and the frequency response functions for power can be calculated from the point mobility function.
Figure 5. The mobility functions of beam b prior to connection being made. ——, b22 ; – – –, b21 , - - -, b11 . Frequency resolution of p rad s−1. This resolution applies for Figures 5–10 inclusive.
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Figure 6. The mobility functions of beam c prior to connection being made. ——, c22 ; – – –, c32 .
Frequency response functions for power between the driving point, point 1, and points 1, 2 and 3 are shown in Figure 10; they all are positive and real, and they vary smoothly with frequency. Apparent damping is given by the inverses of these functions: e.g., at lower frequencies Hpvv11 (0) has an amplitude of 10·3, the inverse of which is 3·33 N sm−1 which can be compared to the damping of the first mode of the cantilever beam of 2·041 N sm−1. The highest magnitude frequency response function for power is at the driving point followed by the transfer to point 2, and then that by the transfer to point 3; apparent damping increases in the same order.
Figure 7. A comparison of mobility functions at point 1. – – –, b11 prior to connection; ——, a11 after connection of beams.
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Figure 8. A comparison of the frequency response function for power Hpvv13 (0) (——) with the mobility function a13 (– – –), after beams have been connected.
4. MODAL RELATIONSHIPS
The response of the combined structure of Figure 4 is given by equation (15), and the combined frequency response function for power is given by equation (4). If the assumption is made that the response near resonance is mainly due to the resonance, then the combination of equation (4) and equation (15) leads to the frequency response function for power Hpvvjk (0) = r Cj2 /cr ,
(17)
for mode r, where j is the response point and k is the driving point. Equation (17) gives the power transfer ratio; this ratio is inversely proportional to damping and directly
Figure 9. A comparison of the frequency response function for power Hpvv11 (0) (——) with the mobility function a11 (——), for a frequency of p rad s−1: case of connected beams.
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Figure 10. A comparison of the frequency response functions for power for the connected beam. ——, Hpvv11 (0); – – –, Hpvv12 (0); - - -, Hpvv13 (0).
proportional to the square of the response point mode shape only. For a single-degree-offreedom system, equation (17) gives the inverse of the apparent coefficient. The apparent damping is apparent damping = cr /r Cj2 .
(18)
Thus, if the response point mode shape is zero then the apparent damping is infinite. Many structures have a nearly sinusoidal mode shape distribution, especially for simply supported structures, or modes at high frequencies. For these types of situations a mean or average frequency response function for power can be calculated based upon the mean 2 /2, where Cmax is the maximum value of the mode shape function. square mode shape, r Cmax 2 The ratio of r Cmax /2cr is independent of modal scaling. The average frequency response function for power thus becomes 2 /2cr . Hpvvkj (0) = rCmax
(19)
A space averaged mean square velocity can also be used to determine a mean square mode shape. Equations (17) and (19) are of interest for determining power transfer. First, they depend only upon the response point mode shape or average mode shape; second, only modal damping is needed and this can be estimated from decay tests or a point mobility function. These two equations state that once power is injected into a mode the frequency response function for power depends only upon the modal damping and the modal displacement and not upon the point at which power is injected. 4.1. An example of structural damping will now be examined (see Chapter 2 in reference [7] for forms of mobility functions for structural damping). For the case of hysteretic (structural) proportional damping the frequency response function for power is Hpvvkj (0) = vr Cj2 /hr kr ,
(20)
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where hr is the hysteretic loss factor for mode r and kr is the modal stiffness for mode r. The frequency response function for power for general structural damping is Hpvvkj (0) = =rFj =2/vr hr ,
(21) where r Fj is the mass normalized mode shape for position j. If a point mobility function is represented by equation (15), and a resonance frequency is chosen a result similar to equation (17) is obtained. However, equation (17) is applicable to all points on a structure, not just a driving point. An example of general structural damping for the system shown in Figure 11 is now given (see page 51 of reference [7]). The frequency response function for power is calculated from equation (2) and by using the following form of the mobility function ajk = s i v rFj rFk /(vr2 − v 2 + ihr vr2 ).
(22)
The frequency response function for power at a driving point, j = k = 1, and that for a transfer of power, j = 2 and k = 1, will be given. The resonance frequencies for this system are at 31·72, 62·78 and 63·77 rad/s, the last two resonances are closely spaced. A comparison of the point frequency response function for power and the mobility function is given in Figure 12. At the resonance frequencies the two functions are of equal values, the frequency response function for power increases monotonically, and the mobility function has two closely spaced resonances. The form of the frequency response function for power for general structural damping is different from that for proportional viscous damping, as can be seen by comparing the functions given in Figures 9 and 12. The transfer point functions for structural damping are plotted in Figure 13; for this case the functions have an equal amplitude only at the lower resonance frequency. Frequency response functions for power are different from mobility functions, although they have the same units. Frequency response functions for power are real, and have only positive values. The two functions have the same magnitudes only at a driving point and for resonance frequencies. Peaks in amplitude for the frequency response function for power do not necessarily occur at resonance and this is associated with the apparent damping concept. The effect of damping on response is greatest at resonance and off resonance its effect is less. The frequency response function for power as dimensions of inverse damping and thus at off resonance frequencies the function can have a greater amplitude than at resonance. 4.2. Equations (17)–(21) give the ratio of velocity squared to power for several cases of damping; however, the distribution of power from the driving point to other points along the structure is not described by these equations. Such a distribution can be determined by examining a power balance of a one-dimensional structure as shown in Figure 14, where x is distance along the structure, A is the constant cross-sectional area, p is power flow, ce is damping per unit volume of the structure at a point, vj is velocity at position j, and
Figure 11. A non-proportionally structurally damped system, as given on page 51 of reference [7]. m1 = 0·5 kg, m2 = 0·95 kg, m3 = 1·05 kg, all spring constants are equal to 1·0 × 103 N/m, h1 = 0·3k1 , and h2 , to, h6 = 0, where hi is the structural damping constant associated with spring ki .
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Figure 12. A comparison of the frequency response function for power (——) and the mobility function (– – – –) at position 1, which is the driving point for the system shown in Figure 11.
power dissipated in the control volume is ce vj2 A dx. By neglecting terms of order dx 2 and performing a power balance on the control volume one obtains dp(x)/dx = −ce Avj2 ,
(23) 2 j
2 j
where p(x) is the power at position x. Upon substituting pC /cr for v , the derivative of p(x) with respect to x becomes dp(x)/dx = −ce ApCj2 /cr .
(24)
Since p is a constant input power, it can be denoted by p0 , and dp(x) is then given by dp(x) = −P0 ce ACj2 dx/cr .
(25)
Figure 13. A comparison of the transfer frequency response function for power (——) and the transfer mobility function, (– – – – – – –) between points 1 and 2 for the system shown in Figure 11.
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Figure 14. Power transfer into and out of elemental volume. A is the cross-sectional area and dx is the volume length.
Integrating equation (25) to obtain p(x) gives p(x) = −p0
g
ce ACj2 dx/cr + pc .
(26)
where pc is a constant of integration and is associated with power input. If power is input at x = 0, then when x is at the other end of the structure, p(x) = 0 as all the input power will have been dissipated. In that case cr is given by
g
cr = ce ACj2 dx.
(27)
From equation (26) the power can be seen to decrease from the driving point at a rate dependent upon the integral of the mode shape squared with respect to distance, the ratios of the damping coefficients and the cross-sectional area. 4.3. In this section a simple relationship will be derived which relates the frequency response function for power and modal parameters. The space averaged mean square velocity for mode r, vr2 , is given by vr2 = svi2 n = p sCi2 /ncr ,
(28)
where n is the number of points used to describe the structure. If the mass distribution is isotropic and homogeneous then the modal mass, mr , is given by mi aCi2 , and upon also noting that the total mass is given by nmi , then the space averaged mean square velocity becomes vr2 = pmr /cr mt ,
(29)
where mt is the total mass of the structure. For many structures and modal scaling, e.g., beams, the modal mass, mr , is equal to the total mass, mt , and for these cases vr2 becomes vr2 = pr /cr ,
(30)
where pr is the power in mode r. The space averaged mean square velocity is thus the power in the mode divided by the modal damping. Next consider the case of a space averaged mean square velocity which is summed over a frequency band in which l modes exist. This variable is noted by v 2 and is given by l
v 2 = s pr /cr . r=1
(31)
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If the modal power is the same or approximately the same for all modes in the frequency band of interest, i.e., pr , then l
v 2 3 pr s 1/cr ,
(32)
r=1
and a frequency response function for power based upon equation (32) is given by v /pr , or 2
l
H pvv (0) = s 1/cr ,
(33)
r=1
where the hat implies space and frequency averaging. Equation (33) can be given in terms of the modal loss factor, hr , where cr equals hr mr vr , mr equals mt and vr is the natural frequency for mode r: l
H pvv (0) = 1/mt s 1/hr vr .
(34)
r=1
Frequency response functions for power can also be expressed in terms of a modal format for use in equations (7) and (9); thus, the effects of damping on power transfer can be directly computed from these two equations. A simple example can be given for the use of equation (8) in equation (7) for broadband input if the assumption is made that the majority of the response is close to resonance frequencies. For a viscous damping case, i.e., equation (15), and q modes in a frequency band, then the driving point frequency response function for power for an individual component is computed from equation (8) as q
0
q
1
Hpvvjj (0) = s rCj6 /cr3 s rCj4 /cr2 , r=1
r=1
(35)
where the integrals in equation (8) have been replaced by summation signs. The frequency response functions for power on the right side of equation (7) can be replaced by the appropriate version of equation (35). The engineer can then determine quite simply what the effects of changing damping would be on the frequency response function for power for the combined structure. 4.4. (9) An example pertaining to equation (9) and a further derivation regarding equation (9) will now be given. In section 2.2 the use of equation (9) is based upon the assumption that the analysis bandwidth and frequency content is such that the connected bodies act as independent oscillators. An attempt at testing this assumption will now be given by using the case of the double beam problem, as was the frequency response functions for power for a sine wave input which are given in Figure 10. The frequency response function for power Hpvv31 (0) was computed by using a bandwidth of 1571 rad s−1 and equation (9). The frequency response function for power for the sine wave input shown in Figure 10 was averaged over the same bandwidth. These results are shown in Figure 15 along with Hpvv31 (0) shown in Figure 10, where the circles represent the results of using equation (9) and the squares are the results of averaging the frequency response function for power obtained from the sine wave input (note that the circles and squares are at the end of the bandwidth and not at the centre frequencies). Both methods of averaging follow the general trend of the sine wave frequency response function for power. An examination of equation (9) will reveal that only the term Hpvvc32 (0) would change with position, point 3, along beam c; thus only this term in equation (9) will be averaged. The space and frequency averaged value of Hpvv32 (0) can be replaced by the result given
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Figure 15. A comparison of the frequency response function for power calculated over a bandwidth of 1571 rad s−1 and the sine wave result of Hpvv31 (0). ——, Sine wave result; w, equation (9), Q, bandwidth average of Hpvv13 (0).
in equation (33); also, the other frequency response function for power terms in equation (9) can be replaced by the result given in equation (17). Making these substitutions gives
0
2 Hpvvc (0) = s rb C2b /crb s1/crc
1>0
1
2 s rb C2b /crb + s rc C2c2 /crc ,
(36)
where rb and rc refer to the modes of beams b and c respectively, and 2b and 2c refer to position 2 for beams b and c respectively. In a given bandwidth each beam may have a different number of modes, so the summations are over the number of modes for the given bandwidth. 4.5. 3 Listed in Table 3 are mobility functions, frequency response functions for power and apparent damping for a number of commercially used structural elements. The mobility functions were obtained from references [8] and [9], and the frequency response functions for power were obtained from reference [11]. The apparent damping is the inverse of the frequency response function for power. 5. CONCLUSIONS
Several different concepts related to connectivity theory and frequency response functions for power have been developed in this paper. Two different methods have been provided for the calculation of frequency response functions for connected structures. The first is to use connectivity theory and mobility functions for the unconnected structures to obtain the mobility function for the connected structure and then to employ equations (2) or (4) to obtain the frequency response function for power for the connected structure. The other method employs (a) the use of equation (7) and the individual frequency response functions for power for the unconnected components when structures are connected at a driving point, or (b) equation (9) or similar equations for the case of random vibrations when a bandwidth analysis is used. Comparisons of mobility functions and frequency response functions for power show that they do not give the same information, nor are they of the same form. Mobility functions identify the transfer of force at a driving point to velocity; frequency response
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T 3 Driving point mobility functions and frequency response functions for power (Frfp) for complex elements (see also accompanying sketches of elements (a)–(j)) Element (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Element
Definition Infinite beam with longitudinal wave motion Infinite beam with torsional wave motion Infinite beam with flexural wave motion—force excitation Infinite beam with flexural wave motion—torque excitation Infinite plate with flexural wave motion—normal force excitation Ring with normal force Corrugated plate with normal force input Ribbed plate with normal force input Plain cylinder with normal force input Ring stiffened cylinder with normal force input Mobility
Frfp
(a) (b) (c) (d) (e)
Apparent damping
1/(2Az(rE)) 1/(2Az(rE)) 2Az(rE) 1/(2Jz(rG)) 1/(2Jz(rG)) 2Jz(rG) (1 − i)k/4mv k/2mv 2mv/k (1 + i)v/4Bk v/2Bk 2Bk/v 1/(8z(B'm)) 1/(8z(B'm)) 8z(B'm) (note that ka Q p/2) (f) (1 − i)k/4mv k/2mv 2mv/k (g) 1/8z(m(Bx B')0·5 ) 1/8(zm(Bx B')0·5 ) 8z(m(Bx B')0·5 ) (h) 1/8z(m(Bx1 By1 )0·5 ) 1/8z(m(Bx1 By1 )0·5 ) 8z(m(Bx1 By1 )0·5 ) (i) VW1 0·306 zV(1 + i)/mch 0·612zV/mch mch/0·612zV Vq1 1/8z(Bm) 1/8z(Bm) 8z(Bm) (j) VW1 0·306zV(1 + i)/mchr 0·612zV/mchr mchr /0·612zV Vq1 1/(8 z(m2 (Bx1 By )0·5 )) 1/(8z(m2 (Bx1 By )0·5 )) 8z(mBx By )0·5 Notation: c is longitudinal sound speed, V = vR/c, m = r(h + (th1 + bh2 )/s), Ar = th1 + bh2 , hr = z12((Ir + sh 3/12)/(Ar + sh))0·5, Ir = (bh23 + th1 )/12 + (bh2 (2h1 + h2 + h)2 + th1 (h + h1 )2 )/4, Bx = Eh 3/12, By = E(bh23 + th13 + sh 3 + 3bh 2 (h + 2h1 + h2 )2 + 3th1 (h + h1 )2 )/12
F
h1
h
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functions for power identify the transfer of power to velocity squared. The two functions have the same amplitudes at a driving point for resonance frequencies only. Two examples of structures with different types of damping have been given: the first is that of the connection of two beams, each viscously damped, and the second is that of a spring mass system with structural damping. These examples demonstrate the equality as noted in the two sentences above. Relationships between modal properties and power are given in section 4. The frequency response function for power for a given point in the structure is given by equation (17) for the case of viscous damping. This equation demonstrates that the power transfer is independent of the driving point mode shape and is given by the mode shape value squared at the point divided by the modal damping factor. A similar result is obtained for structural damping, as given by equations (20) and (21). Equation (26) shows that power decreases linearly with respect to the integral of the mode shape squared, the ratio of the volumetric damping to the modal damping and the cross-sectional area. The space averaged mean square velocity is given by equation (30), which is the ratio of power dissipated in a mode to the modal damping factor. The frequency response function for power averaged over space and frequency is given by equations (31)–(34). These equations demonstrate that the damping values at the different modal frequencies are added up on an inverse basis, as for resistors in parallel. The relationships given in section 4 allow an engineer to estimate the effect of adding or diminishing damping or mass on power transfer from a driving point to another point on the structure. ACKNOWLEDGMENTS
The author wishes to thank the Australian Research Grants Committee for their support of this investigation, and Mr Wang Xu for his assistance. REFERENCES 1. L. L. K 1988 Journal of Sound and Vibration 125, 511–522. Power variables—frequency domain. 2. L. L. K and W X 1991 Proceedings of the Fourth International Conference on Recent Advances in Structural Dynamics, ISVR, U.K., July 1991. Power transfer to a structure using frequency response functions. 3. W X and L. L. K 1992 Journal of Sound and Vibration 155, 55–73. Frequency response functions for power, power transfer ratio and coherence. 4. R. E. D. B and D. C. J 1960 The Mechanics of Vibration. Cambridge: Cambridge University Press. 5. W. S 1981 Vibrations of Shells and Plates. New York: Marcel Dekker. 6. B. J. S 1990 The Benefits of a Systems Approach to Solving Vibration Problems. Department of Mechanical Engineering, University of Western Australia. 7. D. J. E 1989 Modal Testing: Theory and Practice. Somerset, England: Research Studies Press. 8. 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 1: introduction and approximate analysis of beam and plate-like foundations. 9. R. J. P 1988 ISVR Technical Report 162. Approximate mobilities of built-up structures. Southampton, England: University of Southampton. 10. W. T. T 1965 Vibration Theory and Applications. Englewood Cliffs, New Jersey: Prentice-Hall. 11. W. Z, X. W and L. L. K 1991 Mobility Functions, Frequency Response Functions for Power and Power Transfer, and Effective Damping for Built-up Structures. Department of Mechanical Engineering, Monash University (to be published).
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APPENDIX: DEFINITIONS OF TERMS cross-sectional area Young’s modulus moment of area about neutral axis density area moment of inertia shear modulus =!−1 mass per unit length of beam or mass per unit area of plate k wavenumber: elements c and d, v 0·5 (m/ B)0·25, elements e and f, same as c and d except that B = B'
A E I r J G i m
B =EI h thickness n Poisson ratio B' = Eh 3/(12(1 − n 2 )) u distance between ribs t width of rib H depth of rib as measured from top of plate m1 = r[hu + t(H − h)]/u Bx = Bx1 − Eh 3/12 By = B By1 = Euh 3/{12(u + t[(h/H)3 − 1)]}
FREQUENCY RESPONSE FUNCTIONS FOR POWER AND CONNECTIVITY L. L. K Department of Mechanical Engineering, Monash University, Clayton, Victoria 3168, Australia (Received 9 March 1993, and in final form 31 January 1994) Methods of calculating vibratory power transfer for complex structures by using frequency response functions for power are developed. Connectivity theory for mobility functions is employed to determine frequency response functions for power for connected structures and to develop connectivity relations for power transfer by employing frequency response functions for power. Mobility functions and frequency response functions are computed and compared for two complex structures with different types of damping. The comparisons show that both the mobility method and the frequency response function for power method yield identical results. An exact relationship for power sharing for all types of force input, equation (7), and a relationship which applies to cases of high frequencies and random force inputs, equation (9), are derived and employed for selected structural elements. General formulas relating power transfer and modal properties are developed by using the frequency response function for power concept. These formulas give further insight into the role of damping in power transfer relations; see, e.g., equation (36). Frequency response functions for power for many different structural elements are presented in Tables 1 and 3.
1. INTRODUCTION
In several previous papers [1–3], frequency response functions for power were defined and related to mobility functions. Frequency response functions for power allow one to predict power variables at a point in a vibratory system given the power input at a source position. In these papers both fluctuating and mean power were considered; fluctuating power does not contribute to energy flow, whereas mean power is associated with energy flow. Power input at a source point is given by the product of the input force and the driving point velocity; the response ‘‘power variable’’ is proportional to velocity squared: e.g., a damping coefficient multiplied by velocity squared gives dissipated power. Also, use of the frequency response function for power concept allows one to determine vibration severity as a ratio to the input power. Thus, by using the frequency response function for power and damping values the power dissipation in a system can be computed. The inverse of the frequency response function for power gives the apparent damping at a point. Only mean power is considered here. The purposes of this paper are fourfold. The first is to illustrate the use of connectivity theory for mobility functions [4–7] for the calculation of frequency response functions for power; the second is the development of connectivity theory by using frequency response functions for power; the third is the presentation of frequency response functions for power for simple and complex structures and is based upon mobility functions given in references [8, 9]: the fourth is the development of general power relationships. Connectivity theory 709 0022–460X/95/140709 + 17 $08.00/0
7 1995 Academic Press Limited
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is used to predict power distribution in a complex structure where the structure is considered to be built up of several structural elements. Mobility is a complex ratio of velocity at a point in a vibratory system to an applied force. The frequency response function for power is a real quantity and is a ratio of the velocity squared to the input power. 2. CONNECTIVITY THEORY
2.1. As the development of connectivity theory can be found in references [4–7], only a summary will be presented here. For the purpose of presentation the terms aij , bij , cij , etc., refer to mobility functions at a given frequency v; the frequency dependence will not be shown explicitly. Definitions of principal symbols are given in the Appendix. Shown in Figure 1 are two subsystems, b and c, to be attached to each other at point 2, where a force input (as well as power input) exists. The boundary conditions at point 2 are that the velocities of each of the attached bodies are the same and the force input is shared by both bodies. If b22 and c22 are the point mobilities at point 2 for body b and body c respectively, then the driving point mobility function for the combined system, a22 , is given by 1/a22 = 1/b22 + 1/c22 .
(1)
From Table 1 in reference [3], the frequency response function for power at zero frequency Hpvv (0), the ratio of mean square velocity to average power at point 2 for a sine wave force input, is given by Hpvv22 (0) = =a22 =2/R[a22 ],
(2)
where R represents the real value of the enclosed mobility function, p is input power, and vv is velocity squared. The first numerical subscript indicates a response point and the second subscript an input point. Shown in Figure 2 is a different situation, in which two structures are connected at point 2, force and power are input at point 1 on structure b, and the response at point 3 on structure c is required. The mobility function for this case is given by a31 = b21 c32/(b22 + c22 ),
(3)
where a31 is the transfer mobility from force at point 1 on body b to velocity at point 3 on body c. The frequency response function for power for this case is given by Hpvv31 (0) = =a31 =2/R[a11 ].
(4)
For a three-body system (see Figure 3), the transfer mobility from point 1 on structure b to point 4 on structure d, where structure b is connected to structure c at point 2 and structure c is connected to structure d at point 3, is given by a41 = b21 c32 d43 /[(d33 + c33 )(b22 + c22 ) − (c23 )2],
(5)
where dij represents the mobility functions for structure d. Equation (4) can then be employed to calculate Hpvv (0), with a41 used in equation (4) instead of a31 .
Figure 1. (a) Bodies b and c connected at point 2. (b) The free body diagram of bodies b and c.
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Figure 2. Bodies b and c are connected at point 2; input is at point 1 on body b and the response is required at point 3 on body c.
Thus, frequency response functions for power can be obtained directly from available point and transfer mobility functions of individual components. The mobility functions can be obtained from tests, from analytical procedures, from finite element analysis, or from available literature. The above equations have several restrictions. First, they are applicable to a sine wave input force; second, they apply for the case where the structures are connected at a single attachment point, except for equations (2) and (4) which apply to any mobility function; third, linear behaviour is assumed. 2.2. : The question arises as to whether frequency response functions for power for structural components can be combined in the same manner as for the case of connectivity for mobility functions. As will now be shown, the frequency response functions for power can be combined as per mobility connectivity theory to give an overall function for certain situations. Conservation of power at point 2 in Figure 1 gives the relation p1 = p2b + p2c ,
(6)
where p1 is input power and where p2b and p2c are the powers going into structures b and c, respectively. Equation (6) is now divided by v22 , where v2b = v2c = v2 , and, by using the definition that v22 /p gives a frequency response function for power, one obtains 1/Hpvv2 (0) = 1/Hpvv2b (0) + 1/Hpvv2c (0),
(7)
where Hpvv2 (0) is the overall frequency response function for power at point 2 of the connected structure, and Hpvv2b (0) and Hpvv2c (0) are the driving point frequency response functions for power of components b and c when isolated from each other, respectively. Equation (7) is similar in format to that of equation (1) but also applies to the case where the force consists of a band of noise, as the definition of the frequency response function for power for this case (driving point) is given by (see Table 1 in reference [3]) Hpvv11 (0) =
g
v2
R[a11 (h)]=a11 (h)=2 dh
v1
>g
v2
{R[a11 (h)]}2 dh,
(8)
v1
where v1 and v2 are the lower and upper frequency limits of the band of the random force input, and h is a dummy frequency variable. Equation (7) is valid for all types of force inputs and is independent of modal density.
2
Figure 3. The three-body system where input is at point 1 on structure b and the response is required at point 4 on structure d.
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The frequency response function for power equivalence of equation (3) is Hpvv31 = Hpvvb21 Hpvvc32 /(Hpvvb22 + Hpvvc22 ),
(9)
where the (0) has been left out from the frequency response functions for power for brevity. Equation (9) is applicable to cases where the velocities induced at the attachment point from the input force and from body c are uncorrelated. Otherwise, a term of the form fv2b v2c dt will appear in the evaluation of mean square velocity and the integral becomes zero when the two velocities are uncorrelated. This will correspond to situations of random force inputs and high modal density.
3. EXAMPLES
3.1. Consider the case of a pure stiffness element with spring constant k. The mobility function, a11 , of the spring is a11 = iv/k,
(10)
where i = z−1. Using the defintion of frequency response function for power, Hpvv (0) = =a11 =2/R[a11 ],
(11)
and equation (10) gives the following result for the stiffness element: Hpvv (0) = 1/0 = a.
(12)
Similarly, for a pure mass the frequency response function for power is infinite. Listed in Table 1 are mobility functions, frequency response functions and apparent damping for six fundamental structural vibration elements; apparent damping is the inverse of the frequency response function for power. As an example of the use of equation (7) i.e., connectivity for frequency response functions for power), consider a mass attached to a spring/dashpot system. The frequency response functions for power for a mass and spring with a dashpot in parallel are, from Table 1, a and 1/c, respectively. Employing equation (7) yields the overall frequency function for power as 1/Hpvv (0) = 1/(1/c) + 1/a,
(13)
T 1 Mobility functions and frequency response functions for power (Frfp) for simple elements
Element 1. 2. 3. 4. 5. 6.
Spring Mass Dashpot Spring and dashpot k, m and c Base input, k, m and c
Mobility ((Ns)−1 m)
Frfp† ((Ns)−1 m)
iv/k a −i/mv a 1/c 1/c iv/(k + ivc) 1/c iv/(k − mv 2 + ivc) 1/c iv[1/(k + c) − 1/(mv 2)] [1 − 2k/(mv 2) + (k + (vc)2)/(mv 2)2]/c
Note that k is the spring stiffness, m is the mass and c is the damping constant. † Frfp is the frequency response function for power.
Apparent damping (Ns m−1) 0 0 c c c —
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or Hpvv (0) = 1/c,
(14)
which is the result given in references [2, 3] and for element 5 in Table 1. 3.2. A case is now presented in which the free end of a cantilever beam, body b, is attached to a fixed–fixed beam, body c, at a point 0·4 of the distance from one of its fixed ends to the other; see Figure 4. The frequency response function for power between point 1, input, and point 3, response, is required. The function will be calculated by two methods, first by using mobility functions and equations (3) and (11), and second by calculating the input power at point 1, and the output power variable, velocity squared, at point 3 and taking the appropriate ratio. Both beams being examined are made of steel and have dimensions of 1 m in length, 0·01 m in width and 0·05 m in height, with bending occurring in the direction of the height. The Mathworks ‘‘Matlab’’ routine was employed for the calculation of mobility functions and frequency response functions for power. Mobility functions are given by the equation (see Chapter 2 in reference [7] for forms of mobility functions, for a proportionally damped system) ajk , bjk
or
cjk = siv rCj rCk /[(kr − v 2mr ) + ivcr ],
(15)
where r is the mode number, mr , kr and cr are the modal mass, stiffness and damping in that order, and r Cj or k is the mode shape for position j or k for mode r. Modal data has been calculated from formulas and data given in reference [10] and is presented in Table 2. Mobility functions for the cantilever beam and the fixed–fixed beam in the unconnected state are given in Figures 5 and 6, respectively, for a frequency resolution of P radians/second. At point 1, the point mobility function for the connected condition is obtained from 2 a11 = b11 − b12 /(b22 + c33 ).
(16)
A comparison between the unconnected beam and the connected beam’s mobility function at point 1 is shown in Figure 7. A comparison between the transfer mobility and the frequency response function for power between points 1 and 3 is given in Figure 8. As can be observed from this figure, the forms of these functions are not the same, and thus the transfer mobility alone does not give the character of the power transfer from
Figure 4. The double beam problem. Input is at point 1 and the response is required at point 3; beams are attached at point 2. See Table 2 for beam properties.
. .
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T 2 Properties of beam components A. General properties of both beams: (1) mass per unit length, 3·85 kg m−1; (2) product of modulus of elasticity and area moment, 21·875e3 Nm2 B. Cantilever beam modal properties: (1) assumed damping ratio for all modes, 0·001; (2) mode shape data are as follows: Mode
vr (rad s−1)
cr (Ns m−1)
kr (Nm−1)
1 2 3
265 1661 4651
2·04 12·79 35·81
2·70e5 1·06e7 8·33e7
C (point 2) 2 −2 2
C (point 1) 0·68 1·42 0·04
C. Fixed–fixed beam modal properties: (1) assumed damping ratio for all modes, 0·001; (2) mode shape data are as follows: Mode
vr (rad s−1)
cr (Ns m−1)
kr (Nm−1)
C (point 2)
C (point 3)
1 2 3
1686 4649 9114
12·98 35·73 70·18
1·09e7 8·29e7 3·20e8
1·46 1·04 −0·63
1·00 −1·51 1·11
D. Dimensions: (1) both beams are made of steel and have the same dimensions; (2) length 1 m, width 0·01 m and height 0·05 m point 1 to point 3. The frequency response function for power does not have as sharp peaks as the mobility function and is of higher amplitude. Computation of the frequency response function for power by both methods mentioned above give the same result. A comparison of the point mobility function and the frequency response function for power at point 1 is shown in Figure 9. At resonances, the two functions have the same magnitudes, and thus at resonance frequencies the power input and the frequency response functions for power can be calculated from the point mobility function.
Figure 5. The mobility functions of beam b prior to connection being made. ——, b22 ; – – –, b21 , - - -, b11 . Frequency resolution of p rad s−1. This resolution applies for Figures 5–10 inclusive.
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Figure 6. The mobility functions of beam c prior to connection being made. ——, c22 ; – – –, c32 .
Frequency response functions for power between the driving point, point 1, and points 1, 2 and 3 are shown in Figure 10; they all are positive and real, and they vary smoothly with frequency. Apparent damping is given by the inverses of these functions: e.g., at lower frequencies Hpvv11 (0) has an amplitude of 10·3, the inverse of which is 3·33 N sm−1 which can be compared to the damping of the first mode of the cantilever beam of 2·041 N sm−1. The highest magnitude frequency response function for power is at the driving point followed by the transfer to point 2, and then that by the transfer to point 3; apparent damping increases in the same order.
Figure 7. A comparison of mobility functions at point 1. – – –, b11 prior to connection; ——, a11 after connection of beams.
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Figure 8. A comparison of the frequency response function for power Hpvv13 (0) (——) with the mobility function a13 (– – –), after beams have been connected.
4. MODAL RELATIONSHIPS
The response of the combined structure of Figure 4 is given by equation (15), and the combined frequency response function for power is given by equation (4). If the assumption is made that the response near resonance is mainly due to the resonance, then the combination of equation (4) and equation (15) leads to the frequency response function for power Hpvvjk (0) = r Cj2 /cr ,
(17)
for mode r, where j is the response point and k is the driving point. Equation (17) gives the power transfer ratio; this ratio is inversely proportional to damping and directly
Figure 9. A comparison of the frequency response function for power Hpvv11 (0) (——) with the mobility function a11 (——), for a frequency of p rad s−1: case of connected beams.
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Figure 10. A comparison of the frequency response functions for power for the connected beam. ——, Hpvv11 (0); – – –, Hpvv12 (0); - - -, Hpvv13 (0).
proportional to the square of the response point mode shape only. For a single-degree-offreedom system, equation (17) gives the inverse of the apparent coefficient. The apparent damping is apparent damping = cr /r Cj2 .
(18)
Thus, if the response point mode shape is zero then the apparent damping is infinite. Many structures have a nearly sinusoidal mode shape distribution, especially for simply supported structures, or modes at high frequencies. For these types of situations a mean or average frequency response function for power can be calculated based upon the mean 2 /2, where Cmax is the maximum value of the mode shape function. square mode shape, r Cmax 2 The ratio of r Cmax /2cr is independent of modal scaling. The average frequency response function for power thus becomes 2 /2cr . Hpvvkj (0) = rCmax
(19)
A space averaged mean square velocity can also be used to determine a mean square mode shape. Equations (17) and (19) are of interest for determining power transfer. First, they depend only upon the response point mode shape or average mode shape; second, only modal damping is needed and this can be estimated from decay tests or a point mobility function. These two equations state that once power is injected into a mode the frequency response function for power depends only upon the modal damping and the modal displacement and not upon the point at which power is injected. 4.1. An example of structural damping will now be examined (see Chapter 2 in reference [7] for forms of mobility functions for structural damping). For the case of hysteretic (structural) proportional damping the frequency response function for power is Hpvvkj (0) = vr Cj2 /hr kr ,
(20)
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where hr is the hysteretic loss factor for mode r and kr is the modal stiffness for mode r. The frequency response function for power for general structural damping is Hpvvkj (0) = =rFj =2/vr hr ,
(21) where r Fj is the mass normalized mode shape for position j. If a point mobility function is represented by equation (15), and a resonance frequency is chosen a result similar to equation (17) is obtained. However, equation (17) is applicable to all points on a structure, not just a driving point. An example of general structural damping for the system shown in Figure 11 is now given (see page 51 of reference [7]). The frequency response function for power is calculated from equation (2) and by using the following form of the mobility function ajk = s i v rFj rFk /(vr2 − v 2 + ihr vr2 ).
(22)
The frequency response function for power at a driving point, j = k = 1, and that for a transfer of power, j = 2 and k = 1, will be given. The resonance frequencies for this system are at 31·72, 62·78 and 63·77 rad/s, the last two resonances are closely spaced. A comparison of the point frequency response function for power and the mobility function is given in Figure 12. At the resonance frequencies the two functions are of equal values, the frequency response function for power increases monotonically, and the mobility function has two closely spaced resonances. The form of the frequency response function for power for general structural damping is different from that for proportional viscous damping, as can be seen by comparing the functions given in Figures 9 and 12. The transfer point functions for structural damping are plotted in Figure 13; for this case the functions have an equal amplitude only at the lower resonance frequency. Frequency response functions for power are different from mobility functions, although they have the same units. Frequency response functions for power are real, and have only positive values. The two functions have the same magnitudes only at a driving point and for resonance frequencies. Peaks in amplitude for the frequency response function for power do not necessarily occur at resonance and this is associated with the apparent damping concept. The effect of damping on response is greatest at resonance and off resonance its effect is less. The frequency response function for power as dimensions of inverse damping and thus at off resonance frequencies the function can have a greater amplitude than at resonance. 4.2. Equations (17)–(21) give the ratio of velocity squared to power for several cases of damping; however, the distribution of power from the driving point to other points along the structure is not described by these equations. Such a distribution can be determined by examining a power balance of a one-dimensional structure as shown in Figure 14, where x is distance along the structure, A is the constant cross-sectional area, p is power flow, ce is damping per unit volume of the structure at a point, vj is velocity at position j, and
Figure 11. A non-proportionally structurally damped system, as given on page 51 of reference [7]. m1 = 0·5 kg, m2 = 0·95 kg, m3 = 1·05 kg, all spring constants are equal to 1·0 × 103 N/m, h1 = 0·3k1 , and h2 , to, h6 = 0, where hi is the structural damping constant associated with spring ki .
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Figure 12. A comparison of the frequency response function for power (——) and the mobility function (– – – –) at position 1, which is the driving point for the system shown in Figure 11.
power dissipated in the control volume is ce vj2 A dx. By neglecting terms of order dx 2 and performing a power balance on the control volume one obtains dp(x)/dx = −ce Avj2 ,
(23) 2 j
2 j
where p(x) is the power at position x. Upon substituting pC /cr for v , the derivative of p(x) with respect to x becomes dp(x)/dx = −ce ApCj2 /cr .
(24)
Since p is a constant input power, it can be denoted by p0 , and dp(x) is then given by dp(x) = −P0 ce ACj2 dx/cr .
(25)
Figure 13. A comparison of the transfer frequency response function for power (——) and the transfer mobility function, (– – – – – – –) between points 1 and 2 for the system shown in Figure 11.
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Figure 14. Power transfer into and out of elemental volume. A is the cross-sectional area and dx is the volume length.
Integrating equation (25) to obtain p(x) gives p(x) = −p0
g
ce ACj2 dx/cr + pc .
(26)
where pc is a constant of integration and is associated with power input. If power is input at x = 0, then when x is at the other end of the structure, p(x) = 0 as all the input power will have been dissipated. In that case cr is given by
g
cr = ce ACj2 dx.
(27)
From equation (26) the power can be seen to decrease from the driving point at a rate dependent upon the integral of the mode shape squared with respect to distance, the ratios of the damping coefficients and the cross-sectional area. 4.3. In this section a simple relationship will be derived which relates the frequency response function for power and modal parameters. The space averaged mean square velocity for mode r, vr2 , is given by vr2 = svi2 n = p sCi2 /ncr ,
(28)
where n is the number of points used to describe the structure. If the mass distribution is isotropic and homogeneous then the modal mass, mr , is given by mi aCi2 , and upon also noting that the total mass is given by nmi , then the space averaged mean square velocity becomes vr2 = pmr /cr mt ,
(29)
where mt is the total mass of the structure. For many structures and modal scaling, e.g., beams, the modal mass, mr , is equal to the total mass, mt , and for these cases vr2 becomes vr2 = pr /cr ,
(30)
where pr is the power in mode r. The space averaged mean square velocity is thus the power in the mode divided by the modal damping. Next consider the case of a space averaged mean square velocity which is summed over a frequency band in which l modes exist. This variable is noted by v 2 and is given by l
v 2 = s pr /cr . r=1
(31)
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If the modal power is the same or approximately the same for all modes in the frequency band of interest, i.e., pr , then l
v 2 3 pr s 1/cr ,
(32)
r=1
and a frequency response function for power based upon equation (32) is given by v /pr , or 2
l
H pvv (0) = s 1/cr ,
(33)
r=1
where the hat implies space and frequency averaging. Equation (33) can be given in terms of the modal loss factor, hr , where cr equals hr mr vr , mr equals mt and vr is the natural frequency for mode r: l
H pvv (0) = 1/mt s 1/hr vr .
(34)
r=1
Frequency response functions for power can also be expressed in terms of a modal format for use in equations (7) and (9); thus, the effects of damping on power transfer can be directly computed from these two equations. A simple example can be given for the use of equation (8) in equation (7) for broadband input if the assumption is made that the majority of the response is close to resonance frequencies. For a viscous damping case, i.e., equation (15), and q modes in a frequency band, then the driving point frequency response function for power for an individual component is computed from equation (8) as q
0
q
1
Hpvvjj (0) = s rCj6 /cr3 s rCj4 /cr2 , r=1
r=1
(35)
where the integrals in equation (8) have been replaced by summation signs. The frequency response functions for power on the right side of equation (7) can be replaced by the appropriate version of equation (35). The engineer can then determine quite simply what the effects of changing damping would be on the frequency response function for power for the combined structure. 4.4. (9) An example pertaining to equation (9) and a further derivation regarding equation (9) will now be given. In section 2.2 the use of equation (9) is based upon the assumption that the analysis bandwidth and frequency content is such that the connected bodies act as independent oscillators. An attempt at testing this assumption will now be given by using the case of the double beam problem, as was the frequency response functions for power for a sine wave input which are given in Figure 10. The frequency response function for power Hpvv31 (0) was computed by using a bandwidth of 1571 rad s−1 and equation (9). The frequency response function for power for the sine wave input shown in Figure 10 was averaged over the same bandwidth. These results are shown in Figure 15 along with Hpvv31 (0) shown in Figure 10, where the circles represent the results of using equation (9) and the squares are the results of averaging the frequency response function for power obtained from the sine wave input (note that the circles and squares are at the end of the bandwidth and not at the centre frequencies). Both methods of averaging follow the general trend of the sine wave frequency response function for power. An examination of equation (9) will reveal that only the term Hpvvc32 (0) would change with position, point 3, along beam c; thus only this term in equation (9) will be averaged. The space and frequency averaged value of Hpvv32 (0) can be replaced by the result given
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Figure 15. A comparison of the frequency response function for power calculated over a bandwidth of 1571 rad s−1 and the sine wave result of Hpvv31 (0). ——, Sine wave result; w, equation (9), Q, bandwidth average of Hpvv13 (0).
in equation (33); also, the other frequency response function for power terms in equation (9) can be replaced by the result given in equation (17). Making these substitutions gives
0
2 Hpvvc (0) = s rb C2b /crb s1/crc
1>0
1
2 s rb C2b /crb + s rc C2c2 /crc ,
(36)
where rb and rc refer to the modes of beams b and c respectively, and 2b and 2c refer to position 2 for beams b and c respectively. In a given bandwidth each beam may have a different number of modes, so the summations are over the number of modes for the given bandwidth. 4.5. 3 Listed in Table 3 are mobility functions, frequency response functions for power and apparent damping for a number of commercially used structural elements. The mobility functions were obtained from references [8] and [9], and the frequency response functions for power were obtained from reference [11]. The apparent damping is the inverse of the frequency response function for power. 5. CONCLUSIONS
Several different concepts related to connectivity theory and frequency response functions for power have been developed in this paper. Two different methods have been provided for the calculation of frequency response functions for connected structures. The first is to use connectivity theory and mobility functions for the unconnected structures to obtain the mobility function for the connected structure and then to employ equations (2) or (4) to obtain the frequency response function for power for the connected structure. The other method employs (a) the use of equation (7) and the individual frequency response functions for power for the unconnected components when structures are connected at a driving point, or (b) equation (9) or similar equations for the case of random vibrations when a bandwidth analysis is used. Comparisons of mobility functions and frequency response functions for power show that they do not give the same information, nor are they of the same form. Mobility functions identify the transfer of force at a driving point to velocity; frequency response
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T 3 Driving point mobility functions and frequency response functions for power (Frfp) for complex elements (see also accompanying sketches of elements (a)–(j)) Element (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Element
Definition Infinite beam with longitudinal wave motion Infinite beam with torsional wave motion Infinite beam with flexural wave motion—force excitation Infinite beam with flexural wave motion—torque excitation Infinite plate with flexural wave motion—normal force excitation Ring with normal force Corrugated plate with normal force input Ribbed plate with normal force input Plain cylinder with normal force input Ring stiffened cylinder with normal force input Mobility
Frfp
(a) (b) (c) (d) (e)
Apparent damping
1/(2Az(rE)) 1/(2Az(rE)) 2Az(rE) 1/(2Jz(rG)) 1/(2Jz(rG)) 2Jz(rG) (1 − i)k/4mv k/2mv 2mv/k (1 + i)v/4Bk v/2Bk 2Bk/v 1/(8z(B'm)) 1/(8z(B'm)) 8z(B'm) (note that ka Q p/2) (f) (1 − i)k/4mv k/2mv 2mv/k (g) 1/8z(m(Bx B')0·5 ) 1/8(zm(Bx B')0·5 ) 8z(m(Bx B')0·5 ) (h) 1/8z(m(Bx1 By1 )0·5 ) 1/8z(m(Bx1 By1 )0·5 ) 8z(m(Bx1 By1 )0·5 ) (i) VW1 0·306 zV(1 + i)/mch 0·612zV/mch mch/0·612zV Vq1 1/8z(Bm) 1/8z(Bm) 8z(Bm) (j) VW1 0·306zV(1 + i)/mchr 0·612zV/mchr mchr /0·612zV Vq1 1/(8 z(m2 (Bx1 By )0·5 )) 1/(8z(m2 (Bx1 By )0·5 )) 8z(mBx By )0·5 Notation: c is longitudinal sound speed, V = vR/c, m = r(h + (th1 + bh2 )/s), Ar = th1 + bh2 , hr = z12((Ir + sh 3/12)/(Ar + sh))0·5, Ir = (bh23 + th1 )/12 + (bh2 (2h1 + h2 + h)2 + th1 (h + h1 )2 )/4, Bx = Eh 3/12, By = E(bh23 + th13 + sh 3 + 3bh 2 (h + 2h1 + h2 )2 + 3th1 (h + h1 )2 )/12
F
h1
h
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. .
functions for power identify the transfer of power to velocity squared. The two functions have the same amplitudes at a driving point for resonance frequencies only. Two examples of structures with different types of damping have been given: the first is that of the connection of two beams, each viscously damped, and the second is that of a spring mass system with structural damping. These examples demonstrate the equality as noted in the two sentences above. Relationships between modal properties and power are given in section 4. The frequency response function for power for a given point in the structure is given by equation (17) for the case of viscous damping. This equation demonstrates that the power transfer is independent of the driving point mode shape and is given by the mode shape value squared at the point divided by the modal damping factor. A similar result is obtained for structural damping, as given by equations (20) and (21). Equation (26) shows that power decreases linearly with respect to the integral of the mode shape squared, the ratio of the volumetric damping to the modal damping and the cross-sectional area. The space averaged mean square velocity is given by equation (30), which is the ratio of power dissipated in a mode to the modal damping factor. The frequency response function for power averaged over space and frequency is given by equations (31)–(34). These equations demonstrate that the damping values at the different modal frequencies are added up on an inverse basis, as for resistors in parallel. The relationships given in section 4 allow an engineer to estimate the effect of adding or diminishing damping or mass on power transfer from a driving point to another point on the structure. ACKNOWLEDGMENTS
The author wishes to thank the Australian Research Grants Committee for their support of this investigation, and Mr Wang Xu for his assistance. REFERENCES 1. L. L. K 1988 Journal of Sound and Vibration 125, 511–522. Power variables—frequency domain. 2. L. L. K and W X 1991 Proceedings of the Fourth International Conference on Recent Advances in Structural Dynamics, ISVR, U.K., July 1991. Power transfer to a structure using frequency response functions. 3. W X and L. L. K 1992 Journal of Sound and Vibration 155, 55–73. Frequency response functions for power, power transfer ratio and coherence. 4. R. E. D. B and D. C. J 1960 The Mechanics of Vibration. Cambridge: Cambridge University Press. 5. W. S 1981 Vibrations of Shells and Plates. New York: Marcel Dekker. 6. B. J. S 1990 The Benefits of a Systems Approach to Solving Vibration Problems. Department of Mechanical Engineering, University of Western Australia. 7. D. J. E 1989 Modal Testing: Theory and Practice. Somerset, England: Research Studies Press. 8. 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 1: introduction and approximate analysis of beam and plate-like foundations. 9. R. J. P 1988 ISVR Technical Report 162. Approximate mobilities of built-up structures. Southampton, England: University of Southampton. 10. W. T. T 1965 Vibration Theory and Applications. Englewood Cliffs, New Jersey: Prentice-Hall. 11. W. Z, X. W and L. L. K 1991 Mobility Functions, Frequency Response Functions for Power and Power Transfer, and Effective Damping for Built-up Structures. Department of Mechanical Engineering, Monash University (to be published).
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APPENDIX: DEFINITIONS OF TERMS cross-sectional area Young’s modulus moment of area about neutral axis density area moment of inertia shear modulus =!−1 mass per unit length of beam or mass per unit area of plate k wavenumber: elements c and d, v 0·5 (m/ B)0·25, elements e and f, same as c and d except that B = B'
A E I r J G i m
B =EI h thickness n Poisson ratio B' = Eh 3/(12(1 − n 2 )) u distance between ribs t width of rib H depth of rib as measured from top of plate m1 = r[hu + t(H − h)]/u Bx = Bx1 − Eh 3/12 By = B By1 = Euh 3/{12(u + t[(h/H)3 − 1)]}