On the "Rigid" and "Elastic" Nonuniformity of Crankshaft Rotation
Abstract Generated abstract
This paper examines the distinction between rigid and elastic nonuniformity in crankshaft rotation, arguing that the rigid component is often neglected in torsional vibration calculations and torsiographic interpretation despite its possible importance in installations with small flywheel inertia or appreciable damping. Using simplified monoharmonic models, first a rigidly connected multimass system reduced to one inertia and then a two-mass elastic shaft system, the author derives expressions for forced vibration amplitudes, phase relations, work against damping, and degrees of rotational nonuniformity. The analysis shows that elastic vibration amplitudes are proportional to the rigid oscillation amplitude and depend strongly on proximity to resonance and on the location of the exciting torque relative to the node of free vibration. It concludes that excitation near the node can veil resonance effects, making resonance-peak sharpness an unreliable indicator of damping in some crankshaft torsiography results.
Full Text
On the “Rigid” and “Elastic” Nonuniformity of Crankshaft Rotation
V. K. Nechaev
In ordinary calculations, and also in experimental studies of torsional vibrations (by means of torsiographing), as a rule, due attention is not paid to the so-called “rigid” nonuniformity of rotation and to the “rigid” vibrations of the shaft of a motor installation.
However, “rigid” nonuniformity may, under certain conditions, prove to be quite substantial in motor installations with small flywheel masses and large damping. Neglecting it sometimes leads to false conclusions both in calculations and in the processing of experimental results.
The “rigid” and “elastic” nonuniformities of shaft rotation arise due to the presence, respectively, of “rigid” and “elastic” vibrations of this shaft.
These vibrations are caused and conditioned by the action on the shaft of the entire spectrum of harmonic moments created by the pressure of gases in the cylinders (“gas” harmonics), by the inertia forces of the elements of the engine crank mechanisms (“inertia” harmonics), and by the harmonic components of the moment of the external load on the engine.
The role and significance of each of these harmonic moments in the occurrence and development of “rigid” and “elastic” vibrations of the shaft are different and depend on a whole series of circumstances. A detailed analysis of this question in general form, for a multimass torsional system, presents considerable difficulties and is associated with cumbersome calculations.
Below only the simplest case is considered—the case of monoharmonic excitation, as applied to systems with a small number of masses.
1. “Rigid” vibrations of the shaft of a motor installation
In a motor installation with a very rigid (in the limit—absolutely rigid in torsion) shaft, the development of “rigid” vibrations of this shaft is possible and, correspondingly, the appearance of “rigid” nonuniformity of its rotation. Here the entire shaft, together with all the masses fastened to it, vibrates as one rigid whole.
Such rigid1 vibrations will arise under the action on a shaft that is rigid (in torsion) of some external harmonic exciting moment. These vibrations will exist despite the absence in the system under consideration of any elastic elements.
We have a multimass torsional system engine–flywheel (Fig. 1). External harmonic exciting moments of some one definite order \(h\) act on the masses of the engine. In addition, on
—the masses of the engine are acted upon by damping moments proportional to the instantaneous angular velocity of the oscillatory motion of the corresponding mass.
With a very high stiffness of the installation shaft \((c=\infty)\), all masses of the shafting can be replaced by a single concentrated total mass with moment of inertia \(\sum \Theta\), and all damping in the system, which is assumed to have the character of viscous friction, is taken into account by a certain total damping coefficient \(k_c\). Then the total damping moment in the installation is determined by the expression:
\[ M_d=-k_c\frac{d\varphi}{dt}, \]
where \(\varphi\) is the instantaneous angular deviation of each of the masses from the position it would occupy at the same instant if the shaft of the engine installation rotated perfectly uniformly \((\eta=\text{const})\).
Fig. 1
The motion of the reduced single-mass system (Fig. 1b) is described by the elementary differential equation:
\[ (\sum \Theta)\frac{d^2\varphi}{dt^2}+k_c\frac{d\varphi}{dt} = M_h\sin(h\eta t-\varepsilon), \tag{1} \]
where \(M_h\) is the amplitude of the total harmonic exciting moment of the \(h\)-th order in the engine. For the principal orders \(h\), the quantity \(M_h\) is equal to the arithmetic sum of the amplitudes of the exciting harmonic moments of the given order in all cylinders of the engine; for secondary orders, \(M_h=0\) (in engines with uniform alternation of firing).
Integrating equation (1) and discarding the nonstationary part of the solution, we have:
\[ \varphi=A\sin(h\eta t-\varepsilon)+B\cos(h\eta t-\varepsilon), \]
where
\[ A=-M_h\frac{\sum \Theta}{h^2\eta^2(\sum \Theta)^2+k_c^2}, \]
\[ B=-M_h\frac{k_c}{h\eta\left[h^2\eta^2(\sum \Theta)^2+k_c^2\right]}. \tag{3} \]
Now the amplitude \(\Phi_{\mathrm{ж}}\) of the rigid oscillations of the shaft, caused by the action on the system in Fig. 1b of the external exciting harmonic torque \(M_h\), is determined by the expression:
\[ \Phi_{\mathrm{ж}}=\sqrt{A^2+B^2} = -\frac{M_h}{h^2\eta^2\Sigma\Theta}\cdot \frac{1}{\sqrt{1+\left(\frac{k_c}{h\eta\Sigma\Theta}\right)^2}} . \tag{4} \]
Therefore, instead of (2) one may write:
\[ \varphi=\Phi_{\mathrm{ж}}\sin\left[h\eta t-\varepsilon-(\pi-\delta)\right], \tag{5} \]
where
\[ \operatorname{tg}\delta=-\frac{B}{A}=\frac{k_c}{h\eta\Sigma\Theta}. \tag{6} \]
Thus, the forced rigid oscillations of the system under consideration in Fig. 1a lag behind the reduced (total) exciting torque \(M_h\) by the angle \(\pi-\delta\).
\[ M_{\mathrm{in}\,\max}=(\Sigma\Theta)\,\Phi_{\mathrm{ж}}\,(h\eta)^2 \]
\[ M_{a\,\max}=-k_c\Phi_{\mathrm{ж}}h\eta \]
Fig. 2
As follows from the vector diagram (Fig. 2), when viscous friction is present in the system, the component
\[ M_h''=M_h\sin\delta \]
of the reduced torque \(M_h\) will be expended in overcoming this friction. The second component of the torque \(M_h\),
\[ M_h'=M_h\cos\delta \]
will overcome the inertial resistance of the oscillating masses.
Naturally, the work \(L_{\text{ж}}\) of the moment \(M_h'\) over one cycle of oscillations is equal to zero:
\[ L_{\text{ж}}=\int_{0}^{2\pi/h\eta} M_h \cos\delta \cdot \cos(h\eta t-\varepsilon)\,d\varphi=0. \]
The moment \(M_h''\) will perform “useful” work, the magnitude of which over a cycle of oscillations is determined by the expression:
\[ L_c=\int_{0}^{2\pi/h\eta} M_h \sin\delta \cdot \sin(h\eta t-\varepsilon)\,d\varphi . \]
Hence, after substituting the value for \(\varphi\) from (5), we obtain:
\[ L_c=\pi M_h\Phi_{\text{ж}}\sin\delta . \tag{7} \]
In the absence of damping in the system \((k_c=0)\)
\[ \delta=0 \]
and
\[ L_c=0 . \]
It follows from formula (4) that, for “gas” exciting harmonic moments, the amplitude \(\Phi_{\text{ж}}\) of rigid oscillations at \(M_h=\text{const}\) varies, approximately, inversely proportional to the square of the angular speed of shaft rotation \(\eta\). In the case of “inertial” exciting moments, the magnitude \(M_h\) increases in proportion to \(\eta^2\); that is, the amplitude \(\Phi_{\text{ж}}\) almost (for small \(k_c\)) does not depend on \(\eta\).
According to (4), (6), (7), rigid oscillations are damped in the same way as elastic oscillations, and require for their maintenance the expenditure of a certain work \(L_c\), the greater the larger the angle \(\delta\), that is, the larger the damping coefficient \(k_c\) and the smaller the total moment of inertia of the masses of the installation. However, for moderate values of \(k_c\) and not too small values of \(\Sigma\Theta\) and \(h\eta\), the influence of damping on the magnitude \(k_c\) is very small. Practically, here the moment \(M_h\) works as if on a lossless load. Therefore, in most cases the following expression will be valid:
\[ \Phi_{\text{ж}}=-\frac{M_h}{(h\eta)^2\Sigma\Theta}, \tag{8} \]
which is obtained from (3) and (4) for \(k_c=0\).
The rigid oscillations considered here, with amplitude \(\Phi_{\text{ж}}\), cause the appearance of the “rigid” nonuniformity of shaft rotation, the numerical characteristic of which is the degree of nonuniformity\(^1\):
\[ \delta_{\text{ж}}=2\Phi_{\text{ж}}h . \]
But in actual engine installations with a comparatively elastic, pliable shaft, the picture is different. Here, in addition to rigid oscillations, elastic oscillations of the shaft also arise and, correspondingly, an “elastic” nonuniformity of its rotation.
\(^1\) V. K. Nechaev. Theoretical torsiograms for the shaft of an internal-combustion engine. Izvestiya of the Tomsk Industrial Institute, vol. 58, no. 2, 1937.
2. Relations between the rigid and elastic vibrations of the shaft
Let us consider, as an example, the system shown in Fig. 3a. Here the external harmonic torque \(M_h \sin h\eta t\) acts on the middle mass \(\Theta_0\). There are no exciting harmonic torques on the other masses of the system. Damping in the system is absent.
Fig. 3
Assuming that the quantity \(\Theta_0\) is small in comparison with \(\Theta_1\) and \(\Theta_2\), we shall take, in order to simplify the subsequent calculations,
\[ \Theta_0 = 0. \]
Then we obtain the system shown in Fig. 3b, for which the differential equations of motion may be written in the form:
\[ \left. \begin{aligned} \Theta_1 \frac{d^2\varphi_1}{dt^2} + c_{01}(\varphi_1 - \varphi_0) &= 0,\\ \Theta_2 \frac{d^2\varphi_2}{dt^2} + c_{02}(\varphi_2 - \varphi_0) &= 0,\\ c_{01}(\varphi_0 - \varphi_1) + c_{02}(\varphi_0 - \varphi_2) &= M_h \sin h\eta t. \end{aligned} \right\} \tag{9} \]
Here:
\(\varphi_0, \varphi_1, \varphi_2\) are, respectively, the instantaneous angles of deflection of section 0 (the point of application of the torque \(M_h\)), of the left mass 1, and of the right mass 2; these deflections are measured from those angular positions of section 0 and of masses 1 and 2 which section 0 and masses 1 and 2 would occupy at the given instant of time under uniform rotation of the shaft, that is, in the absence of any rigid or elastic vibrations of it;
\(c_{01}, c_{02}\) are, respectively, the stiffnesses of the shaft portions 01 and 02. The combined
the solution of equations (9) gives the desired expressions for the forced oscillations of points 0, 1, and 2 of the system under consideration
\[ \varphi_0=\Phi_0\sin h\eta t, \]
\[ \varphi_1=\Phi_1\sin h\eta t, \]
\[ \varphi_2=\Phi_2\sin h\eta t, \]
where
\[ \begin{aligned} \Phi_0&=-\frac{M_h}{h^2\eta^2(\Theta_1+\Theta_2)}\cdot \frac{\left[1-\gamma^2(1-a_1)\right]\left[1-\gamma^2(1+a_2)\right]}{1-\gamma^2},\\[6pt] \Phi_1&=-\frac{M_h}{h^2\eta^2(\Theta_1+\Theta_2)}\cdot \frac{1-\gamma^2(1+a_2)}{1-\gamma^2},\\[6pt] \Phi_2&=-\frac{M_h}{h^2\eta^2(\Theta_1+\Theta_2)}\cdot \frac{1-\gamma^2(1-a_1)}{1-\gamma^2}. \end{aligned} \tag{10} \]
Here:
\[ \gamma \]
is a coefficient determined by the expression
\[ \gamma=\frac{h\eta}{\omega_0}, \]
\[ \omega_0 \]
is the angular frequency of the natural free oscillations of the two-mass system under consideration in Fig. 3b:
\[ \omega_0=\sqrt{\frac{c(\Theta_1+\Theta_2)}{\Theta_1\Theta_2}}, \]
\[ a_1,a_2 \]
are coefficients characterizing the position of point 0:
\[ a_1=\frac{x}{l_1};\qquad a_2=\frac{x}{l_2}, \]
\(x\) is the distance from point 0, at which the moment \(M_h\) is applied, to the node of the free oscillations of the system (point \(Z\));
\(l_1,l_2\) are the distances from this node, respectively, to points 1 and 2.
But, analogously to (8), for \(k_c=0\):
\[ -\frac{M_h}{h^2\eta^2(\Theta_1+\Theta_2)}=\Phi_{\text{ж}}. \tag{11} \]
Now, instead of (10), we have:
\[ \Phi_0=\Phi_{\text{ж}}\, \frac{\left[(1-\gamma^2)+\gamma^2a_2\right]\left[(1-\gamma^2)-\gamma^2a_1\right]}{1-\gamma^2}, \tag{12} \]
\[ \Phi_1=\Phi_{\text{ж}}\, \frac{(1-\gamma^2)-\gamma^2a_1}{1-\gamma^2}, \tag{13} \]
\[ \Phi_2=\Phi_{\text{ж}}\, \frac{(1-\gamma^2)+\gamma^2a_1}{1-\gamma^2}. \tag{14} \]
Expression (14) may be represented in the form:
\[ \Phi_2=\Phi_{\text{ж}}+\Phi_{2y}, \tag{15} \]
where the notation adopted is:
\[ \Phi_{2y}=\Phi_{\text{ж}}\frac{\gamma^2a_1}{1-\gamma^2}. \tag{16} \]
The quantity \(\Phi_{2y}\) entering here should be regarded as the amplitude of the elastic oscillations of the second mass \((\Theta_2)\) relative to the rigid transport oscillatory motion characterized by the first term \((\Phi_{\mathrm{r}})\) on the right-hand side of formula (15). In the absence of damping in the system, these elastic oscillations coincide in phase with the rigid oscillations \(\Phi_{\mathrm{r}}\) or are opposite to them (see below).
Analogously to (15), from expressions (12) and (13) we have:
\[ \Phi_0=\Phi_{\mathrm{r}}+\Phi_{0y}, \tag{17} \]
\[ \Phi_1=\Phi_{\mathrm{r}}+\Phi_{1y}, \tag{18} \]
where
\[ \Phi_{0y}=-\Phi_{\mathrm{r}}\gamma^2 \left( 1+\frac{\gamma^2 a_1 a_2}{1-\gamma^2}+a_2-a_1 \right), \tag{19} \]
\[ \Phi_{1y}=-\Phi_{\mathrm{r}}\frac{\gamma^2 a_2}{1-\gamma^2}. \tag{20} \]
Here \(\Phi_{0y}\) and \(\Phi_{1y}\) are, respectively, the amplitudes of the elastic oscillations of section 0 and of mass 1.
The amplitude \(\Phi_{\mathrm{r}}\) of the rigid oscillations does not depend on \(\gamma\) (that is, for a given \(\eta\), on the value of \(\omega_0\)) and is retained for any, even infinitely large, stiffness of the shaft of the installation.
The amplitudes \(\Phi_{2y}\), \(\Phi_{1y}\), and \(\Phi_{0y}\) of the elastic oscillations are directly proportional to the amplitude \(\Phi_{\mathrm{r}}\). In addition, they depend on \(\gamma\) (that is, on the degree of closeness of the forced oscillatory process to resonance, \(\gamma=1\)) and on the quantities \(a_1\) and \(a_2\), that is, on the distance \(x\) of the point of application of the exciting moment \(M_h\) from the node \(Z\) of the free oscillations. This last dependence is now the most essential for us.
The forced oscillations, for example, of mass 2 with amplitude \(\Phi_2\), lead to the appearance of a certain degree of nonuniformity of rotation of the extreme right-hand (in Fig. 3) section of the shaft. The magnitude of this degree of nonuniformity is determined by the expression:
\[ \delta_2= \frac{ \left(\dfrac{d\varphi_2}{dt}\right)_{\max} - \left(\dfrac{d\varphi_2}{dt}\right)_{\min} }{\eta} \]
or, taking into account that \(\varphi_2=\Phi_2\sin h\eta t\):
\[ \delta_2=2\Phi_2 h. \]
But on the basis of (15) one may write:
\[ \delta_2=2\Phi_{\mathrm{r}}h+2\Phi_{2y}h=\delta_{\mathrm{r}}+\delta_y, \]
where \(\delta_{\mathrm{r}}\) and \(\delta_y\) are the degrees of nonuniformity due, respectively, to the rigid and elastic oscillations of section 2 of the shaft.
Taking into account (16), (19), and (20), we introduce the notation:
\[ D_1=\frac{\Phi_{1y}}{\Phi_{\mathrm{r}}} = -\frac{\gamma^2 a_2}{1-\gamma^2}, \tag{21} \]
\[ D_2=\frac{\Phi_{2y}}{\Phi_{\mathrm{r}}} = +\frac{\gamma^2 a_1}{1-\gamma^2}, \tag{22} \]
\[ D_0=\frac{\Phi_{0y}}{\Phi_{\mathrm{r}}} = -\gamma^2 \left( 1+\frac{\gamma^2 a_1 a_2}{1-\gamma^2}+a_2-a_1 \right) \tag{23} \]
for the ratios of the amplitudes of the elastic and rigid components of the oscillations of sections 1, 2, and 0 of the shaft of the installation according to Fig. 3b.
It is obvious that these ratios, as well as the ratios \(\delta_y/\delta_{\text{rig}}\), decrease in proportion to the decrease of the coefficients \(a_1\) and \(a_2\), that is, as the point \(0\) of application of the exciting moment \(M_h\) approaches the node of the free vibrations of the elastic system. These ratios attain maxima when \(a_1\) or \(a_2\) is equal to unity, that is, when the moment \(M_h\) acts on the left- or right-hand mass of the system.
The quantities \(D_1\), \(D_2\), and \(D_0\) are determined by the specific values of the coefficients \(\gamma\) and \(a\). Thus, for example, if the node of the free vibrations of the system in Fig. 3 is located between points \(0\) and \(2\), that is, lies to the right of the point of application of the moment \(M_h\), then \(a_1 > 0\) and \(a_2 > 0\); therefore here:
a) in the subresonance region \((\gamma < 1)\):
\[ D_2 > 0;\qquad D_1 < 0 \]
and the vibrations \(\Phi_{2y}\) and \(\Phi_{\text{rig}}\) have the same phase and are added, whereas the vibrations \(\Phi_{1y}\) and \(\Phi_{\text{rig}}\) have opposite phases, that is, they partly cancel one another; when \(D_1 = -1\), that is, according to (21) when
\[ \gamma = \sqrt{\frac{1}{1 + a_2}} \]
we have
\[ \Phi_1 = \Phi_{\text{rig}} + \Phi_{1y} = 0, \]
therefore, the left-hand mass takes no part in the vibrations of the system and rotates uniformly;
b) in the superresonance region \((\gamma > 1)\):
\[ D_2 < 0;\qquad D_1 > 0 \]
and the vibrations \(\Phi_{2y}\) and \(\Phi_{\text{rig}}\) are opposite in phase and partly cancel one another; when \(D_2 = -1\), that is when
\[ \gamma = \sqrt{\frac{1}{1 - a_1}} \]
the vibrations of the right-hand mass are absent; now the left-hand mass \((\Theta_1)\) takes upon itself the entire action of the moment \(M_h\) and, with respect to the right-hand mass \((\Theta_2)\), acts as an ideal dynamic damper.
For \(x = 0\) (the moment \(M_h\) is applied at the node of the free vibrations, \(a_1 = 0\), \(a_2 = 0\)) from (21) and (22) we have:
\[ D_1 = 0;\qquad D_2 = 0 \]
or
\[ \Phi_{1y} = 0;\qquad \Phi_{2y} = 0 \]
and
\[ \Phi_1 = \Phi_2 = \Phi_{\text{rig}}. \]
Consequently, for \(x = 0\) masses 1 and 2 vibrate as they would vibrate if the entire shaft of the system were absolutely rigid. Here, for any (large or small) torsional stiffness of the shaft and at any frequency \(h\eta\) of the moment \(M_h\), both masses of the system will have only a “rigid” nonuniformity of rotation.
According to formula (11), for \(M_h = \text{const}\) the amplitude \(\Phi_{\text{rig}}\) of these rigid vibrations is inversely proportional to the total moment of inertia \((\Theta_1 + \Theta_2)\) of the masses of the system and decreases monotonically with increasing rotational speed of the shaft \(\eta\) (Fig. 4). Consequently, under the action of the moment \(M_h\) at node \(Z\), torsiographing sections 1 or 2 of the shaft of the system according to Fig. 3, carried out over any wide interval of values of \(\eta\), would not reveal in the system any
resonance phenomena. The amplitudes of the torsiogram here would decrease continuously as the number of revolutions of the crankshaft increased (provided, of course, that the amplitude of the moment \(M_h\) remains unchanged).
Fig. 4
But at \(x=0\), according to (17) and (23):
\[ D_0=-\gamma^2 \]
and
\[ \Phi_0=\Phi_{\text{ж}}(1-\gamma^2), \]
that is, the amplitude of the vibrations of point 0 (which at \(x=0\) coincides with point \(z\)) depends strongly on the magnitude \(\gamma\). At resonance (\(\gamma=1\)) the elastic and rigid vibrations of this point cancel each other out:
\[ \Phi_0=0 \]
and the elastic deformation curve of the shaft under resonant vibrations assumes the peculiar form shown in Fig. 5. It is easy to see that this curve is completely different from the curve of the form of free vibrations (line III in Fig. 3), which is usually adopted as the basis for calculating resonant torsional vibrations.
Fig. 5
The character of the dependence of the amplitudes \(\Phi_2\), \(\Phi_1\), and \(\Phi_0\) on \(x\), expressed analytically by relations (15)—(20), is presented graphically in Fig. 6 for one particular installation according to Fig. 3 at
\[ \frac{\Theta_1}{\Theta_2}=2 \]
and \(\gamma=0.8\)
(analogous graphs could also be constructed for degrees of nonuniformity—
...of the nonuniformity of rotation \(\delta\) of each of the masses). Here the ordinates of the straight lines II, I and of the curve 0 give the full amplitudes of the oscillations \(\Phi_2\), \(\Phi_1\) and \(\Phi_0\), respectively, of masses 2, 1 and section 0, obtained under the action of the exciting moment \(M_h\) at that point on the shaft axis (on the abscissa axis of the diagram) at which these ordinates are measured1.
Fig. 5
As the coefficient \(\gamma\) approaches unity, the ratios \(D_2\), \(D_1\) and \(D_0\) increase rapidly. Thus, for example, from (22), as \(\gamma \to 1\):
\[ D_2=\frac{\Phi_{2y}}{\Phi_{\mathrm{zh}}}\to\infty, \]
since here
\[ \frac{\gamma^2}{1-\gamma^2}\to\infty. \]
However, in reality, when damping is present in the system, the ratio \(D_2\), as well as the ratios \(D_1\) and \(D_0\), remain bounded at resonance \((\gamma=1)\), and the greater this damping is, the smaller they are. With increasing damping, the height of the resonance peak (for example, the quantity \(\Phi_{2\max}\)) decreases, while the relative weight of the rigid oscillations in the full (total) oscillations increases.
The factor
\[ \beta=\frac{1}{1-\gamma^2} \]
in formulas (21)—(23) is analogous to the dynamic coefficient \(\beta\) in a system with one degree of freedom without damping.
In the presence of damping in our two-mass system, this coefficient will obviously have a more complicated form. Without undertaking a detailed analysis of this question here (which will constitute the subject of the author’s next article), let us note that, in the system of Fig. 3, when viscous friction is present in it, formulas (16) and (20) for the case \(\gamma = 1\) (resonance) may approximately be replaced by others1:
\[ \Phi_{2y}=\beta_{\mathrm{res}}\Phi_{\text{ж}}a_1, \]
\[ \Phi_{1y}=-\beta_{\mathrm{res}}\Phi_{\text{ж}}a_2, \]
where the dynamic coefficient \(\beta_{\mathrm{res}}\) has a finite value, the smaller the greater the damping in the system.
Fig. 7
Labels in the figure: \(\Phi_2\); \(\gamma\); \(M_h=\mathrm{const.}\); \(x=\dfrac{a}{l_1}\); \(a_1=1.0\) \((\beta_{\mathrm{res}}=10)\); \(a_1=0.5\); \(a_1=0.2\); \(\Phi_{\text{ж}}\).
But these latter expressions, together with expressions (16) and (20), show that the height of the entire resonance curve (for example, for \(\Phi_2\)) depends not only on the intensity of damping in the system, but also on the magnitudes \(a_1\) and \(a_2\), which characterize the distance of the point of application of the exciting moment \(M_h\) from the node of the system’s free vibrations.
In Fig. 7, by way of example, resonance curves are presented for \(\Phi_2\) as a function of \(\gamma\) for a number of values of \(x\) (within the limits from \(x=0\) to \(x=l_1\)), calcu-
calculated by formulas (11), (15), and (16) under the assumption that the amplitude \(M_h\) of the exciting moment does not depend on the shaft rotational speed;
\[ M_h=\mathrm{const}. \]
The peaks of the resonance curves in the region \(\gamma=1\) are bounded by dashed lines corresponding to the value \(\beta_{\mathrm{rez}}=10\) adopted for the example. Here the curve \(\Phi_{\mathrm{ж}}\), which for \(M_h=\mathrm{const}\) is a hyperbola, is shown separately.
It is seen from Fig. 7 that already at \(a_1=0.5\) (that is, at \(x=0.5\,l_1\)) the resonance peak is rather weakly expressed, while at \(a_1=0.2\) (that is, at \(x=0.2\,l_1\)) this resonance peak almost disappears.
Thus, bringing the point of application of the moment \(M_h\) closer to the node of the system’s free vibrations leads, as it were, to an apparent increase of damping in the system. It is now evident that the sharpness (and height) of the resonance peak in systems with small values of \(a_1\) and \(a_2\) cannot serve as a reliable characteristic for estimating damping in this system. The presence of rigid shaft vibrations here leads to a peculiar effect of veiling the resonance vibrations.
As will be shown in the author’s next article, these circumstances must also be taken into account when processing the results of engine torsiography.