Journal of Sound and Vibration (1992) 155(l), 55-73. FREQUENCY ... types of force input and degrees of freedom of a system. Also, power transfer is associated.
Journal of Sound and Vibration (1992) 155(l),
55-73
FREQUENCY RESPONSE FUNCTIONS FOR POWER, POWER TRANSFER RATIO AND COHERENCE WANG Xu Department
AND
L. L. Koss
of Mechanical Engineering, Monash University,
(Received
12 November
Clayton,
Victoria 3 168, Australia
1990, and in revised form 20 March 1991)
The concept of frequency response functions for power variables was proposed in reference [3], where a convolution integral formula in the frequency domain was used to relate fluctuating power variables and to give the frequency response function for power. This convolution integral is simplified in the present paper for several cases of varying bandwidth random force inputs and cosine wave force input to an oscillator, which results in frequency response functions for power which are dependent on the type of force input. Coherence functions for power variables are also determined from theory. Theoretically derived results compare well to those obtained from experiments, and these results are used to demonstrate the bandwidth sensitivity of coherence between power and velocity squared. The zero frequency or dc. value of the frequency response function for power is shown to be related to the system damping, and system damping can be determined from an examination of this term. Several formulas are given for the calculation of system damping for different types of force input and degrees of freedom of a system. Also, power transfer is associated with the zero frequency term of the frequency response function for power, and other frequencies are associated with power fluctuations. Given the mobility functions of a system, the power transfer and effective damping can be determined for any force bandwidth by the formulas given in this paper.
I. INTRODUCTION
The concept of correlation of fluctuating power variables has been examined in three previous papers [l-3]. In these publications emphasis was placed on derivations relating power fluctuations at different points in a system. Several questions were raised regarding coherence of power fluctuating quantities and transfer of power from an input point to another point in a vibratory system. For example, under what conditions of force input will the coherence between power quantities be high, i.e., near or equal to unity, and which part of the frequency response function for power can be used for power transfer from a driving point to another point in the system? Derivations and experimental results given here treat and answer these queries for some basic mechanical systems and several types of force inputs. In the previous papers, derivations and experiments showed that for a sine wave force input coherence between fluctuating power variables is unity whereas for wide band force inputs coherence is low. A question arose as to what is the relationship between coherence and bandwidth and to the linearity of the frequency response function for power variables. These questions are examined and possibly answered for the case of a single-degree-offreedom vibratory system. In two other papers [4,5], power transfer to a vibratory system at a driving point was shown to be associated with the zero frequency or d.c. quantity of the frequency response function for power, and not the real part of the frequency response function for power at non-zero frequency as originally postulated in reference [3]. The ratio of mean square 55 0022-460X/92/100055 + 19 %03.00/O
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WANG
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1.. L. KOSS
velocity to power input was shown in references [4] and [5] to be equal to the inverse of the system damping for a single-degree-of-freedom system. Derivations of frequency response functions for power variables for different bandwidth force inputs are given in this paper along with experimental results which treat both d.c. and non-dc. frequencies. Two types of frequency response functions are used. The first is the mobility function, velocity over force, and the acronym FRF is used for this function. The second type of frequency response function is that which relates power variables, velocity squared over power, and this is given the acronym FRFP to distinguish it from the former FRF. Complex derivations are given in the Appendices and basic derivations are given in the text.
2. DERIVATIONS Derivations relating to different bandwidth force inputs are given in what follows here, and they are based upon general integral equations presented in reference [3]. In reference [3] a FRFP and power variable spectral density functions were derived for a random Gaussian force input to a linear system, these quantities being given by
SP”2(W)=-
1
W
K s -‘x
$r(n)%r(w
- t7)%(n)&(w
- q)W..“(w - dl* drl,
(3)
uz s”2v”(w)=;
%(n)%(w s --a3
-
rlWdrl)l*1f!.d~
- t7)12drl.
(4)
Here p is the product of force f and velocity U, &(w) is the autospectral density of force, H,“(o) is the frequency response function between force and a transfer point velocity, H_,(w) is the frequency response function between force and the driving point velocity, H,,(w) is the frequency response function between the driving point velocity u and the transfer point velocity v; o is a radian frequency, 17is a dummy variable and an asterisk represents complex conjugation; H,J(w) is the frequency response function between input power p and response velocity squared u*, i.e., FRFP, S,,(w) is the autospectral density of the input power p, Q(w) is the cross-spectral density between input power p, and response velocity squared v*, and L&(w) is the autospectral density of response velocity squared v*. 2.1. COSINE WAVE FORCE INPUT Consider the case of very narrow band force input which has an autospectral density function of an impulse in the frequency domain and is defined by the following: &r(w) =6(w),
s~(w-w~)=6(0--w0)=0, no $K(o-wO) dw= I, s -0X
o#wo,
(5,6) (7)
FREQUENCY
RESPONSE FUNCTIONS
a, s --m
FOR POWER
57
m S/f(w -ho)&
dw =
$A% ~o)=m?l-~oh
&(a - WOW/,(W) dw = &(oo), s --03
&Id@- t7,oo) = Wlw - VI- 00).
(8)
(9,lO)
Here w. is any given radian frequency of interest. The above single frequency force input is equivalent to a cosine force input in the time domain. The spectral densities and frequency response functions for power (FRFP) for a singledegree-of-freedom spring-mass-dashpot system, as shown in Figure 1, with stiffness k, mass m and damping c, have been evaluated for all frequencies except w equal to zero by complex integration of equations (l)-(4) and the use of delta function definitions to give Spp(o)=402/7r[(4k-mmW2)2+4C2m2],
Q&o)
Spuz(o) =
= ~~co~/K[(~~-wKD~)~+~c~w~]~, 8i03
(12) (13,14)
rm2(4k - mu2 + i 2cw) ’
KYd(@)=
(11)
i2w 4k-mo2+i2co’
(15)
In these equations, o is radian frequency and &J(W) represents the coherence function for power variables p and u2, and i is fi. The cross-spectra Sp.2(o) and HP&o) are complex quantities, whereas the autospectral quantities are real only. For this situation the coherence is 1 at all frequencies, and corresponds to a previous derivation for normalized correlation function of one in reference [ 1] for sine wave input; the result for zero frequency is given in section 3.
v(f)
F(fl
Figure I. Schematic of a single-degree-of-freedom oscillator; mass m, stiffness k and dashpot c. The differential equation of motion is m dv/dt + CD+ k l v dr = F(r), where v is the velocity.
The spectral densities and frequency response functions for power for a multi-degreeof-freedom system are derived in Appendix II and are
where H,“(o) is the transfer point FRF and firU(w) is the driving point FRF. Equations (16) and (17) show that frequencies for FRFP are double those of the corresponding FRF and, by the use of known FRF, FRFP may be evaluated for a single frequency force input.
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2.2. FINITE BANDWIDTH RANDOM FORCE INPUT A finite bandwidth random force input, to a linear mechanical system, is represented by a unity amplitude autospectral density function as
(18) (19) where w, and o2 are respectively the lower and upper frequency limits of the input random force. With the bandwidth do =02-a1 compared with the arithmetic average central frequency o,= (w, + w2)/2, three types of bandwidths are used to evaluate the integrals in equations (l)-(4). They are (a) w2-WI (0, + 02)/2, where w2 - wI = (or + w2)/2 is defined as the critical bandwidth. For finite bandwidths the integrals are evaluated numerically in the common range of w, < 1~1~ a2 and w, G jw - 71~ w2. Emphasis on the three frequency ranges will be made with respect to coherence and linearity of HP&w) and this will be discussed in section 4. 2.3. INFINITE BANDWIDTH RANDOM FORCE INPUT A wide-band Gaussian random force is represented by 5°C n) = 1 and .SR(W- n) = 1. A derivation in Appendix I for the single-degree-of-freedom system shown in Figure 1 gives the following results : S”V( w ) =
2(m204 - w2(3km - c’) + 4k2) mc(m2ei2 + c2)(m204 - 8kmw’ + 4c2w2 + 16k2) ’
SpV2(w)= (mu2 - 2k - icw)/mc(mo
(2% 21)
- ic)(mo2 - 4k - 2icw)i,
(22)
HP&o) = 2(mw2 - 2k - icw)/(mw - ic)(mo2 -4k - 2icw)i.
(23j
y&(o) = (m’w” + 4k2 - 4kmu2 + c202)/(m2u4 + 4k2 - 3kmo’ .‘ciic--‘j.
(2.4)
These equations give explicit results for the spectral quantities, including the coherence between power and velocity squared. Frequency response function for power and cohcrence given in equations (23) and (24) for the system shown in Figure i are treated in section 4.
3. THE D.C. TERM OF THE FREQUENCY RESPONSE FUNCTION FOR POWER The d.c. term (i.e., the zero frequency term) of the FRFP represents a steady situation, and is the ratio of average response velocity squared over average input power. Several cases of frequency response function values at zero frequency will be evaluated by using various force input spectral densities. The frequency response function for power at zero frequency can be derived from equation (1) with o set equal to zero,
(25) where 2 is the time-averaged velocity squared and jj is the time-averaged input power.
FREQUENCY
RESPONSE FUNCTIONS
59
FOR POWER
For the case of a single-degree-of-freedom system, as shown in Figure 1, a single frequency force input into the system gives H&O), upon using equation (25),
2 j-“, ~2(lrll-~o)~(~7)I~(-t1)12 drl jzrn62((#-w,)[H(~)H*(-q)+(H(-rj)12] d,=t’
(26)
Hpu*(o)=
A finite bandwidth force input into the system yields
2 ]I:: fWWW-r7)12 dv+2 j:: WrlW(-~)12dtl HP”*= -a,, I_,, [IH(-r1)12+H(tl)H*(-rl)l drl+j:: [IW-rl 12+WW*(-t7)1
=‘. drl c
(27)
The d.c. term of FRFP for a unity amplitude white noise broadband force input into the system shown in Figure 1 is obtained by using the Cauchy residue theory, and is given by
03) The conclusion is that the d.c. term of FRFP for various force inputs into a single-degreeof-freedom system is equal to one over the system damping coefficient, and this is a very important result. As noted in references [4] and [5] the d.c. value of FRF for a force input displacement output single-degree-of-freedom spring-mass-dashpot system is one over the system stiffness. For the case of FRFP the d.c. value is one over the damping ratio, and this implies that a constant power gives a constant mean square velocity. An FRFP can be used either to predict or to measure damping. For the case of a multi-degree-of-freedom system, and a unity amplitude cosine wave force input at frequency wo, the FRFP at zero frequency is given by 2 jTrn 62(lrll - oo)H/u(t7)Huu(-~)IHf,(-tl)12 Hpu*(o)=
Equation
{-“, S’(lql-
wo)[H,u(rl)Hf;(-17)
dtl
+ lH,u(-~)121
W_4~oV+v(--~o)
dq = H,u(-00)
+ &(mo)’
(29) can be simplified further by using the identities Hfv(wo) = H);(-oo) in equation (29) to give
(29)
and
Hf,(wo) = Hj,(-oo)
HP&9
= I~/v~~o~l’/~~~,~~~o~1
(30)
where 9Z[Hf,,(wo)] is the real part of HfM(wo). For generality w can be used instead of w. in equation (30). Equation (30) can be obtained from equation (4) of reference [6] for a sine wave input. However, equation (30) allows for a transfer point calculation, whereas equation (4) in reference [6] is valid only for a driving point situation. The d.c. value of FRFP for a single-degree-of-freedom system can be obtained from equation (2.1.22) of reference [7], which is one over the system damping. A finite band force input into the multi-degree-of-freedom system with lower and upper frequency limits ml, o2 is calculated from
Hpoz(O) =
2j~:H/,(tl)H~“(-rl)[H/t(-rl)+Hlt(t7)ldrl j:: If!,u(-rl)
+ WuW12
drl
’
(31)
WANG XU AND L. L. KOSS
60
For viscous damping or structural damping the identities for frequency response functions noted above can be substituted into equation (31) to give
Equations (30) and (32) tions. The value Hj,&O) quency bandwidth, and calculation of d.c. value
are much easier to use than the more complicated integral equadepends upon the type of force input into the system; i.e., frecentre frequency. Listed in Table 1 are the formulas for the of FRFP as given in the above equations. TABLE 1
Formulas for calculation of d.c. value of FRFP D.c. value of FRFP at the driving point (u= u) and transfer point Force input type in frequency domain
Single-degree-offreedom system
Multi-degree-of-freedom system
An impulse in frequency domain (a sine wave of frequency o. in time domain) A finite bandwidth force input with the frequency limits [w,, wz] Ij,@)= in the positive frequency axis
l/c
H,,*(t)) = lH/.(@)lZ
wffu(~o)l
(30)
j:: ~[H,.(rl)llH/,(t1)12 dtl 5”z1alH,,(tl)]j2 dtl o)I
(32)
l/c
An infinite bandwidth white noise H (o) = I-“, I%(tl)l*H/.(~) dv force input PO2 I-“,
W!_(WM~)
l/c
dq
Note: q is a dummy frequency variable. The inverse of H&(O) is the effective damping.
4. COMPUTATIONAL RESULTS AND EXPERIMENTAL VERIFICATION In this section computed results based upon the previously given equations will be compared to those obtained from experiments on a cantilever beam, with its first mode initially considered as a single-degree-of-freedom system. Frequency response functions for power, coherence and the d.c. value of HP&o) will be examined for cosine wave input, broadband input and the three categories of finite-band force inputs defined in section 2.2. The first mode properties of the cantilever beam used in the previous equation are mass m equal to 0.464 kg, stiffness k equal to 17935.15 N/m and damping c equal to 12.62 Ns/m. The first mode natural frequency and damping ratio for this vibrator are 31.29 Hz and O-069 respectively. The beam has dimensions of 0.508 m length, O-0254 m width and 0.0127 m height. The vibration is in the direction of the smallest dimension. Shown in Figure 2 is the experimental set-up used to measure FRFP and coherence. In the experimental results to be presented, measured data for FRFP and coherence is noisier than that of the calculated data and is due to noise generated in the analog multiplier. 4.1. FREQUENCYRESPONSEFUNCTIONSFOR POWERAND
COHERENCE
4.1.1. Cosine wave force input Results for cosine wave force input into the single-degree-of-freedom system computed by using equation (15) are given in Figure 3 ; for this situation the coherence is unity from theory. Peak frequency for FRFP is double that of FRF, of equation (16), and shows up
FREQUENCY
Measurement
RESPONSE FUNCTIONS
61
FOR POWER
points Force transducer
Veloaty -signal
Power slgnal
Velouty squared sngnal
Figure 2. Experimental set-up for measurement of frequency response functions for power and coherence functions for power.
0
--
t ---------
L -_
2
i
5 5
--_ '\ \
-20 ()-_
\
DC. value
-
130 s
m -40
. .._ _.... ~_~_________________.. __._ _.._.__.....
0
$ _z a
\ ‘\ ‘.-__
----_____ -
-100
L 0
% & 3
I
I
I
L
20
40
60
80
-100
-180 100
Frequency (Hz)
Figure 3. Frequency response function for power; computed results for cosine wave force input into a singledegree-of-freedom system. -, FRFP amplitude; - - -, phase.
62.58 Hz. The FRFP also has a continuous phase change from a starting value of 180 to a final value of -90”. Experimental results for comparison with computed data for a sine sweep between 15 and 80 Hz are shown in Figure 4. The measured coherence is unity over the frequency range of twice the sine sweep. The experimental peak frequency for FRFP has doubled as per equation (16), and the computed and experimental amplitude of the FRFP match quite well. Computed and experimentally measured phase have similar shapes with respect to frequency but are out by a constant value of approximately 180”, which is due to inversion of phase in the accelerometer preamplifier. at
62
WANG
XU AND
Frequency
L. L. KOSS
(Hz)
Figure 4. Experimentally measured frequency response function for power and-phase (a), and coherence of power variables (b), for a sine force sweep between IS and 80 Hz input into a single-degree-of-freedom system. Twenty spectral averages, Hanning window, 50 percent overlap processing, and a 0.25 Hz resolution bandwidth.
41.2. Finite bandwidth force input In this section the three cases of finite bandwidth force inputs, given in section 2.2, will be examined. For the first case the bandwidth is less than the arithmetic average of the force excitation band limits; this situation was simulated and measured by using a onethird octave band force input centred at 31.5 Hz. The computed result is given in Figure 5 for amplitude and phase of the FRFP. Due to the nature of the computation certain frequency regions have no calculable results, in this case FRFP data is available only in two frequency regimes: i.e., between 0 Hz and 7.3 Hz corresponding to d.c. and the difference between the band limits, and between 56.4 Hz and 71 Hz corresponding to twice the lower frequency and twice the upper frequency of the one-third octave band. The calculated
0
No doto calculable 2 T
4 -20
,‘\
I
\
