On the Analysis of Curves of Free Damped Oscillations

V. K. Nechaev

As was shown earlier¹, the successive amplitudes \(X\) of free damped oscillations of a linear mechanical system (with one degree of freedom), in the presence in it of forces of viscous and dry friction, are determined by the expressions:

\[ X_{n+1}=-e^{-\beta}X_n+(1+e^{-\beta})\cdot a, \tag{1} \]

\[ X_{n+2}=-e^{-\beta}X_{n+1}-(1+e^{-\beta})\cdot a, \tag{2} \]

\[ X_{n+3}=-e^{-\beta}X_{n+2}+(1+e^{-\beta})\cdot a, \tag{3} \]

\[ X_{n+4}=-e^{-\beta}X_{n+3}-(1+e^{-\beta})\cdot a, \tag{4} \]

\[ \cdots \]

where

\[ \beta=\frac{kT}{4m}, \]

\[ a=\frac{R_c}{c}, \]

\(n\)—the number of an even (that is, positive) amplitude \(X_n\);

\(k\)—the so-called coefficient of velocity damping, entering into the expression

\[ R_k=-k\frac{dx}{dt} \]

for the force of viscous (fluid) friction \(R_k\) in the system, reduced to that point of the system whose oscillations are recorded on the vibrogram being analyzed;

\(R_c\)—the force of dry friction in the system, reduced to this same point;

\(T\)—the period of the free damped oscillations;

\(m\) and \(c\)—respectively the mass and stiffness of the oscillating system, reduced to the same point.

For \(q\) even and satisfying the condition \(q>n\), the mean values of the coefficient \(k\) and the force \(R_c\) over the interval of oscillations from \(X_n\) to \(X_q\) can be determined from the expressions:

\[ k=\frac{4m}{T}\ln\left(-\frac{X_n-X_q}{X_{n+1}-X_{q+1}}\right), \tag{5} \]

\[ R_c=c\frac{X_nX_{q+1}-X_{n+1}X_q}{(X_n-X_q)-(X_{n+1}-X_{q+1})}. \tag{6} \]

Here all even amplitudes are considered positive \((X_n, X_{n+2}, \ldots > 0)\), and all odd ones negative. But application of these formulas is possible only when an accurately drawn mean (zero) line is present on the vibrogram; obtaining such a line, however, is not always possible under experimental conditions. This may lead to substantial errors in the results of determining the quantities \(k\) and \(R_c\) from formulas (5) and (6).

Below is shown a method for determining the sought quantities \(k\) and \(R_c\) from the vibrogram without using the mean (zero) line, on the basis of appropriate processing of the measurement results on the vibrogram of consecutive “total” deviations (or ranges) of the oscillating point on both sides \(Z_{n,n+1}, Z_{n+1,n+2}, \ldots, Z_{q,q+1}\) (Fig. 1). Here all \(Z\) are already considered positive, that is,

Fig. 1

\[ X_n-X_{n+1}=|X_n|+|X_{n+1}|=Z_{n,n+1}, \tag{7} \]

\[ X_{n+1}-X_{n+2}=-|X_{n+1}|-|X_{n+2}|=-Z_{n+1,n+2} \tag{8} \]

\[ \cdots\cdots\cdots\cdots\cdots \]

From expressions (1) and (2)

\[ X_{n+2}-X_{n+1}=e^{-\beta}(X_{n+1}-X_n)-2(1+e^{-\beta})\,a \]

or, on the basis of relations (7) and (8):

\[ Z_{n+1,n+2}=e^{-\beta}Z_{n,n+1}-2(1+e^{-\beta})\,a. \tag{9} \]

In an analogous manner, from expressions (3) and (4)

\[ X_{n+2}-X_{n+3}=e^{-\beta}(X_{n+1}-X_{n+2})-2(1+e^{-\beta})\,a \]

or, taking into account that

\[ X_{n+2}-X_{n+3}=Z_{n+2,n+3}>0, \]

\[ X_{n+2}-X_{n+1}=Z_{n+1,n+2}>0, \]

we have

\[ Z_{n+2,n+3}=e^{-\beta}Z_{n+1,n+2}-2(1+e^{-\beta})\,a. \tag{10} \]

In the same way we obtain

\[ Z_{n+3,\,n+4}=e^{-\beta}Z_{n+2,\,n+3}-2(1+e^{-\beta})a \tag{11} \]

\[ \cdots\cdots\cdots\cdots\cdots \]

\[ \cdots\cdots\cdots\cdots\cdots \]

\[ Z_{q-1,\,q}=e^{-\beta}Z_{q-2,\,q-1}-2(1+e^{-\beta})a \tag{12a} \]

\[ Z_{q,\,q+1}=e^{-\beta}Z_{q-1,\,q}-2(1+e^{-\beta})a . \tag{12} \]

From formulas (12) and (9):

\[ Z_{n+1,\,n+2}-Z_{q,\,q+1}=e^{-\beta}\left(Z_{n,\,n+1}-Z_{q-1,\,q}\right). \]

Whence

\[ e^{-\beta}=\frac{Z_{n+1,\,n+2}-Z_{q,\,q+1}}{Z_{n,\,n+1}-Z_{q-1,\,q}}, \tag{13} \]

\[ \beta=\ln\left(\frac{Z_{n,\,n+1}-Z_{q-1,\,q}}{Z_{n+1,\,n+2}-Z_{q,\,q+1}}\right) \tag{14} \]

and the sought coefficient of viscous friction is

\[ k=\frac{4\beta m}{T}=\frac{4m}{T}\ln\left(\frac{Z_{n,\,n+1}-Z_{q-1,\,q}}{Z_{n+1,\,n+2}-Z_{q,\,q+1}}\right). \tag{15} \]

For simplicity of notation, taking

\[ n=0 \]

(that is, denoting by \(Z_{01}\) the first swing of the oscillations on the investigated portion of the vibrogram), we finally obtain:

\[ k=\frac{4m}{T}\ln\left(\frac{Z_{01}-Z_{q-1,\,q}}{Z_{12}-Z_{q,\,q+1}}\right), \tag{16} \]

where \(q\) is even, and all \(Z\)’s are positive and are determined by simple measurement on the vibrogram, without using the zero line.

In the general case formula (16) gives a certain mean value of the coefficient \(k\) over an interval of \(\dfrac{q+1}{2}\) cycles of oscillation. But this formula can also be extended to an integral number of cycles. Indeed, subtracting expression (12a) from expression (9), we obtain (with \(q\), as before, even):

\[ Z_{n+1,\,n+2}-Z_{q-1,\,q}=e^{-\beta}\left(Z_{n,\,n+1}-Z_{q-2,\,q-1}\right). \]

Whence

\[ e^{-\beta}=\frac{Z_{n+1,\,n+2}-Z_{q-1,\,q}}{Z_{n,\,n+1}-Z_{q-2,\,q-1}} \]

and

\[ k=\frac{4m}{T}\ln\left(\frac{Z_{n,\,n+1}-Z_{q-2,\,q-1}}{Z_{n+1,\,n+2}-Z_{q-1,\,q}}\right) \]

or, for \(n=0\), finally:

\[ k=\frac{4m}{T}\ln\left(\frac{Z_{01}-Z_{q-2,\,q-1}}{Z_{12}-Z_{q-1,\,q}}\right). \tag{17} \]

Formula (17) gives the mean value of the coefficient \(k\) over an interval of \(\dfrac{q}{2}\) cy-

periods of free oscillations. Thus, for example, for \(q=4\) (two periods of oscillations):

\[ k=\frac{4m}{T}\ln\left(\frac{Z_{01}-Z_{23}}{Z_{12}-Z_{34}}\right). \]

In an analogous way, the mean values (over the interval being processed) of the dry-friction force \(R_c\) in the oscillating system under consideration can also be found. To obtain the necessary computational expressions, we sum relations (9) and (12):

\[ Z_{n+1,n+2}+Z_{q,q+1}=e^{-s}(Z_{n,n+1}+Z_{q-1,q})-4(1+e^{-s})a. \]

Hence

\[ 4a(1+e^{-s})=e^{-s}(Z_{n,n+1}+Z_{q-1,q})-(Z_{n+1,n+2}+Z_{q,q+1}) \]

or

\[ a=\frac{1}{4(1+e^{-s})}\left[e^{-s}(Z_{n,n+1}+Z_{q-1,q})-(Z_{n+1,n+2}+Z_{q,q+1})\right]. \tag{18} \]

But, according to relation (13):

\[ e^{-s}(Z_{n,n+1}+Z_{q-1,q})-(Z_{n+1,n+2}+Z_{q,q+1})= \]

\[ =2\,\frac{Z_{n+1,n+2}\cdot Z_{q-1,q}-Z_{n,n+1}\cdot Z_{q,q+1}}{Z_{n,n+1}-Z_{q-1,q}} \]

and, moreover,

\[ 1+e^{-s}= \frac{(Z_{n,n+1}+Z_{n+1,n+2})-(Z_{q-1,q}+Z_{q,q+1})}{Z_{n,n+1}-Z_{q-1,q}}. \]

Now, instead of expression (18), we obtain:

\[ a=\frac{1}{2}\, \frac{Z_{n+1,n+2}\cdot Z_{q-1,q}-Z_{n,n+1}\cdot Z_{q,q+1}} {(Z_{n,n+1}+Z_{n+1,n+2})-(Z_{q-1,q}+Z_{q,q+1})}. \]

Hence the magnitude of the dry-friction force is:

\[ R_c=ca=\frac{1}{2}\,c\, \frac{Z_{n+1,n+2}\cdot Z_{q-1,q}-Z_{n,n+1}\cdot Z_{q,q+1}} {(Z_{n,n+1}+Z_{n+1,n+2})-(Z_{q-1,q}+Z_{q,q+1})}. \tag{19} \]

Similarly, summing (9) and (12a), one can obtain:

\[ R_c=\frac{1}{2}\,c\, \frac{Z_{n+1,n+2}\cdot Z_{q-2,q-1}-Z_{n,n+1}\cdot Z_{q,q+1}} {(Z_{n,n+1}+Z_{n+1,n+2})-(Z_{q-2,q-1}+Z_{q-1,q})}. \tag{20} \]

In formulas (19) and (20), as before, \(q\) is an even number. Formula (19) gives the mean value of the reduced dry-friction force \(R_c\) over the interval of \(\frac{q+1}{2}\) periods (cycles) of oscillations, while formula (20) gives it over the interval of \(\frac{q}{2}\) complete cycles.

Taking \(n=0\), instead of expressions (19) and (20) we obtain, respectively:

\[ R_c=\frac{c}{2}\, \frac{Z_{12}\cdot Z_{q-1,q}-Z_{01}\cdot Z_{q,q+1}} {(Z_{01}+Z_{12})-(Z_{q-1,q}+Z_{q,q+1})}, \tag{21} \]

\[ R_c=\frac{c}{2}\, \frac{Z_{12}\cdot Z_{q-2,q-1}-Z_{01}\cdot Z_{q-1,q}} {(Z_{01}+Z_{12})-(Z_{q-2,q-1}+Z_{q-1,q})}. \tag{22} \]

Now, for example, for the interval of \(2T\), with \(q=4\):

\[ R_c=\frac{c}{2}\, \frac{(Z_{12}Z_{23}-Z_{01}Z_{34})} {(Z_{01}+Z_{12})-(Z_{23}+Z_{34})}. \]

Thus, formulas (15), (16), (20), (21) make it possible to determine the mean values of the coefficient \(k\) and of the dry-friction force \(R_c\) for any number (beginning with three) of half-periods of the free damped oscillations of the system under consideration. Obviously, before applying these formulas to a complex oscillating system, it is necessary to carry out the corresponding computational work of reducing the actual system to a simple system with one degree of freedom. The entire mass \(m\) of this reduced system is concentrated at the point whose oscillations are recorded on the vibrogram being analyzed; the mass \(m\) is connected with a fictitious spring whose stiffness \(c\) is equal to the stiffness, reduced to the same point, of all elements of the actual system. The fundamental basis for such a reduction is the condition that, at every instant of time, the potential and kinetic energies of the actual system be respectively equal to the potential and kinetic energies of the reduced system.

1. V. K. Nechaev, “On the analysis of certain vibrograms,” Proceedings of the Tomsk Polytechnic Institute named after S. M. Kirov, vol. 68, issue 1, 1951. ↩