On the Analysis of Certain Vibrograms

V. K. Nechaev

The character of the curves of free damped oscillations of mechanical systems depends on the character and laws of the damping forces present in the system.

In a number of cases, a vibrogram of free damped oscillations, obtained experimentally—in the usual coordinates: displacement \(x\) versus time \(t\)—makes it possible to reveal the resistances present in the system, knowledge of which is extremely necessary in the investigation of a number of dynamical problems. By appropriate processing of the vibrogram, the character and magnitude of these resistances can be established.

Below such a method is set forth for processing vibrograms, applicable primarily to the simplest mechanical oscillatory system with one degree of freedom (Fig. 1).

It is assumed here that the system contains only the forces of dry friction \(R_c\) (Coulomb friction) and fluid friction \(R_k\) (proportional to the velocity of the oscillatory motion of the mass \(m\) of the system under consideration), which give the total damping force \(R\), the cause of the damping of the free oscillations. Thus here:

Fig. 1

\[ \begin{aligned} R &= R_c + R_k,\\ R_c &= \mathrm{const}, \end{aligned} \tag{1} \]

\[ R_k = -k\frac{dx}{dt}, \tag{2} \]

where \(k\) is the so-called coefficient of velocity damping.

As shown below, the quantities \(R_c\) and \(k\) can be determined unambiguously by the appropriate processing of the vibrogram of free oscillations.

The differential equation of motion of the mass \(m\) of the system under consideration has (under the conditions of a linear characteristic of the spring, possessing a constant stiffness \(c\) kg/cm) the usual form:

\[ m \frac{d^2 x}{dt^2} - R + cx = 0, \]

where, according to (1) and (2):

\[ R = - k \frac{dx}{dt} \pm R_c, \]

with the sign \((+)\) referring to periods of motion of the mass downward, and the sign \((-)\) to periods of motion upward; the direction of the force \(R_k\) is taken into account automatically by the sign of the derivative \(\frac{dx}{dt}\).

Now, for periods of motion of the mass downward:

\[ m \frac{d^2 x}{dt^2} + k \frac{dx}{dt} - R_c + cx = 0 \]

or

\[ m \frac{d^2 x}{dt^2} + k \frac{dx}{dt} + c(x-a)=0, \tag{3} \]

where it is denoted:

\[ a = \frac{R_c}{c}. \tag{4} \]

Similarly, for periods of motion of the mass upward:

\[ m \frac{d^2 x}{dt^2} + k \frac{dx}{dt} + c(x+a)=0. \tag{5} \]

Equation (3), by the substitution

\[ x-a=y \]

is reduced to the elementary form

\[ m \frac{d^2 y}{dt^2} + k \frac{dy}{dx} + cy = 0 \]

and has the general solution:

\[ y = e^{-pt}(A \sin \omega t + B \cos \omega t), \]

where

\[ p = \frac{k}{2m} \tag{6} \]

\[ \omega = \sqrt{\frac{c}{m} - p^2} \tag{7} \]

The angular frequency \(\omega\) entering here is related by the relation:

\[ \omega = \frac{2\pi}{T} \tag{8} \]

to the period \(T\) of the free oscillations, which can be found directly from the vibrogram when the time scale along its abscissa axis is known.

Now for the downward motion of the mass:

\[ x=a+e^{-pt}(A\sin\omega t+B\cos\omega t) \tag{9} \]

and analogously for the upward motion of the mass:

\[ x=-a+e^{-pt}(A\sin\omega t+B\cos\omega t). \tag{10} \]

The constants of integration \(A\) and \(B\) are different for each successive half-period of motion of the oscillating mass (from one extreme position \(X_n\) to the other extreme position \(X_{n+1}\)) and must be determined each time separately from the initial conditions for the given half-period.

The following notation is adopted below:
\(X_0, X_1, X_2\ldots X_n, X_{n+1}\ldots\)—successive amplitudes of oscillation of the mass \(m\), taken with their sign (that is, \(X_0, X_2, X_4\ldots\) are positive, \(X_1, X_3\ldots\) are negative);
\(t\)—time measured from the beginning of the oscillatory process (point \(O\) in Fig. 1);
\(\tau\)—time measured from the instant of the preceding amplitude deviation of the mass \(\left(\tau=0\div \dfrac{T}{2}\right)\).

For the first downward motion of the mass (from \(X_0\) to \(X_1\)), equation (9) is valid, which may be written in the form:

\[ x=a+e^{-pt}(A_{01}\sin\omega t+B_{01}\cos\omega t), \tag{11} \]

where at \(t=0\)

\[ x=X_0;\qquad \frac{dx}{dt}=0. \tag{12} \]

Under these initial conditions, from (11):

\[ B_{01}=X_0-a, \tag{13} \]

\[ A_{01}=\frac{p}{\omega}(X_0-a). \tag{14} \]

Thus the equation of the downward motion of the mass on the interval \(X_0 \div X_1\) is:

\[ x=a+e^{-pt}(X_0-a)\left(\frac{p}{\omega}\sin\omega t+\cos\omega t\right). \]

The deflection \(x\) reaches its maximum negative value:

\[ X_1=a+e^{-pt}(X_0-a)\left(\frac{p}{\omega}\sin\omega t_1+\cos\omega t_1\right) \]

at the instant \(t_1\), at which once again

\[ \frac{dx}{dt}=0. \]

But it is easy to show that

\[ e^{-pt}\left(\frac{p}{\omega}\sin\omega t+\cos\omega t\right)=\max \]

when

\[ t=t_1=\frac{\pi}{\omega}. \]

Therefore,

\[ X_1=a+e^{-p\frac{\pi}{\omega}}(X_0-a)\cos\pi \]

or

\[ X_1=-e^{-p\frac{\pi}{\omega}}X_0+\left(1+e^{-p\frac{\pi}{\omega}}\right)a. \tag{15} \]

For the second motion of the mass (upward, from \(X_1\) to \(X_2\)), according to (10), the following equation is valid:

\[ x=-a+e^{-pt}(A_{12}\sin\omega t+B_{12}\cos\omega t) \]

with the initial conditions:

\[ \text{at } t=t_1=\frac{\pi}{2} \begin{cases} x=X_1,\\[4pt] \dfrac{dx}{dt}=0. \end{cases} \]

This equation can obviously be rewritten in the form:

\[ x=-a+e^{-p\tau}(A_{12}\sin\omega\tau+B_{12}\cos\omega\tau) \tag{16} \]

with the initial conditions:

\[ \text{at } \tau=0 \begin{cases} x=X_1,\\[4pt] \dfrac{dx}{dt}=0. \end{cases} \]

Under these conditions, from (16) we obtain:

\[ B_{12}=X_1+a, \]

\[ A_{12}=\frac{p}{\omega}(X_1+a). \]

Consequently, for the upward motion of the mass (from \(X_1\) to \(X_2\)) the following equation will hold:

\[ x=-a+e^{-p\tau}(X_1+a)\left(\frac{p}{\omega}\sin\omega\tau+\cos\omega\tau\right). \]

But, as noted above, the expression

\[ e^{-p\tau}\left(\frac{p}{\omega}\sin\omega\tau+\cos\omega\tau\right) \]

attains its maximum at

\[ \tau=\frac{\pi}{\omega}, \]

therefore

\[ X_2=-a+e^{-p\frac{\pi}{\omega}}(X_1+a)\cos\pi \]

or

\[ X_2=-e^{-p\frac{\pi}{\omega}}X_1-\left(1+e^{-p\frac{\pi}{\omega}}\right)a. \tag{17} \]

Now, taking relation (8) into account, instead of (15) and (17) one may write:

\[ X_1=-e^{-p\frac{T}{2}}X_0+\left(1+e^{-p\frac{T}{2}}\right)a, \tag{18} \]

\[ X_2=-e^{-p\frac{T}{2}}X_1-\left(1+e^{-p\frac{T}{2}}\right)a \tag{19} \]

and similarly further:

\[ X_3=-e^{-p\frac{T}{2}}X_2+\left(1+e^{-p\frac{T}{2}}\right)a, \tag{20} \]

\[ X_4=-e^{-p\frac{T}{2}}X_3-\left(1+e^{-p\frac{T}{2}}\right)a. \]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

In the general case, for even \(n\) (that is, \(X_n\) is positive):

\[ \tag{21} \]

\[ X_n=-e^{-\beta}X_{n-1}-\left(1+e^{-\beta}\right)a, \tag{22} \]

\[ X_{n+1}=-e^{-\beta}X_n+\left(1+e^{-\beta}\right)a, \tag{23} \]

\[ X_{n+2}=-e^{-\beta}X_{n+1}-\left(1+e^{-\beta}\right)a, \tag{24} \]

\[ X_{n+3}=-e^{-\beta}X_{n+2}+\left(1-e^{-\beta}\right)a. \tag{24a} \]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where the following notation has been introduced:

\[ \beta=p\frac{T}{2}=\frac{kT}{4m}. \tag{25} \]

Relations (18)—(24) can also be used to determine the quantities \(k\) and \(R_c\) in the given oscillatory system from the recorded vibrogram of free damped oscillations.

Thus, for example, let us add (23) and (24):

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

Hence

\[ \beta=\frac{kT}{4m}=\ln\left(-\frac{X_n+X_{n+1}}{X_{n+1}+X_{n+2}}\right) \tag{26} \]

and the desired coefficient of velocity damping is:

\[ k=\frac{4m}{T}\ln\left(-\frac{X_n+X_{n+1}}{X_{n+1}+X_{n+2}}\right). \tag{27} \]

Further, from (23):

\[ a=\frac{X_{n+1}+e^{-\beta}X_n}{1+e^{-\beta}} \]

or, on the basis of (26):

\[ a=\frac{X_{n+1}(X_n+X_{n+1})-X_n(X_{n+1}+X_{n+2})}{X_n-X_{n+2}}. \]

Thus, according to (4):

\[ \frac{R_c}{c}=a=\frac{X_{n+1}^2-X_nX_{n+2}}{X_n-X_{n+2}}, \]

and hence the magnitude of the sought dry-friction force \(R_c\):

\[ R_c=c\,\frac{X_{n+1}^{2}-X_nX_{n+2}}{X_n-X_{n+2}}. \tag{28} \]

By formulas (27) and (28), the problem posed above of determining the characteristics of the damping forces in the system from a vibrogram of decaying free oscillations is, in essence, completely solved. To determine the quantities \(R_c\) and \(k\), it is sufficient to know three neighboring amplitudes of the free oscillations, \(X_n\), \(X_{n+1}\), and \(X_{n+2}\) (the first and third of which are positive).

If a vibrogram is available with a sufficiently long record of the process of decay of the oscillations (that is, with a sufficiently large number of recorded oscillation cycles), the method of determining the quantities \(R_c\) and \(k\) can be made more accurate by expanding the interval of the cycles of decaying oscillations covered by the calculation.

Thus, for example, subtracting (24a) from (23), we obtain (for even \(n\)):

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

whence

\[ k=\frac{4m}{T}\ln\left(-\,\frac{X_n-X_{n+2}}{X_{n+1}-X_{n+3}}\right). \tag{29} \]

Summing (23) and (24a):

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

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

or, on the basis of (26):

\[ a=\frac{X_nX_{n+3}-X_{n+1}X_{n+2}}{(X_n-X_{n+2})-(X_{n+1}-X_{n+3})}, \]

whence

\[ R=c\,\frac{X_nX_{n+3}-X_{n+1}X_{n+2}}{(X_n-X_{n+2})-(X_{n+1}-X_{n+3})}. \tag{30} \]

The limits of the calculation interval (which, for formulas (29) and (30), extend over \(1\frac{1}{2}\) periods of oscillation) may be expanded still further.

In the general case, when using on the vibrogram the interval of amplitudes from \(X_n\) to \(X_q\) (where \(n\) and \(q\) are even, that is, \(X_n\) and \(X_q\) are positive, and \(q>n\)) and, correspondingly, of negative amplitudes from \(X_{n+1}\) to \(X_{q+1}\), analogously to (22)—(24), one may write:

\[ X_{n+1}=-e^{-\beta}X_n+(1+e^{-\beta})a, \tag{31} \]

\[ X_{q+1}=-e^{-\beta}X_q+(1+e^{-\beta})a. \tag{32} \]

Subtracting (31) from (32) gives:

\[ e^{-\beta}=-\,\frac{X_{n+1}-X_{q+1}}{X_n-X_q}, \tag{33} \]

whence

\[ k=\frac{4m}{T}\ln\left(-\,\frac{X_n-X_q}{X_{n+1}-X_{q+1}}\right). \tag{34} \]

Summing (31) and (32):

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

hence

$$ 2a=\frac{(X_{n+1}+X_{q+1})+e^{-\beta}(X_n+X_q)}{1+e^{-\beta}} $$

and on the basis of (33) and (4):

$$ R_c=c\,\frac{X_nX_{q+1}-X_{n+1}X_q}{(X_n-X_q)-(X_{n+1}-X_{q+1})}. \tag{35} $$

Formulas (34) and (35) evidently give certain mean values of the quantities \(k\) and \(R_c\) over the entire interval from \(n\) to \(q\). If the fundamental assumption adopted above is valid for the oscillating system under consideration—namely, that only dry and fluid (“viscous”) resistances are present in the system—then the values of \(k\) and \(R_c\) found from (34) and (35) will be independent of the magnitude and position, on the vibrogram, of the chosen computational interval \(X_n \div X_q\). A change in the quantities \(k\) and \(R\) under another choice of the numbers \(n\) and \(q\) (for example, when this interval is narrowed) will indicate the presence in the system of damping forces (resistances) of a more complex character than was assumed in formulas (1) and (2).

Finally, it is clear that the method set forth above may also be used for processing curves of free damped oscillations of oscillatory systems more complex than was assumed above, on p. 253. In these cases the quantities \(m\) and \(c\) will denote, respectively, the mass and stiffness of the oscillatory system reduced to that point of the system whose oscillations are recorded on the vibrogram being studied. The quantities \(k\) and \(R_c\), calculated by formulas (34) and (35), will also be reduced to this same point.