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0Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 2, pp. 187-200. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd230202
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 70E55
Singularities of the Dynamics of Some Bar Systems with Unilateral Constraints
G. M. Rozenblat, V. T. Grishakin
This paper deals with a formulation and a solution of problems of the dynamics of mechanical systems for which solutions that do not take into account the unilateral nature of the constraints imposed on the objects under study have been obtained before. The motive force in all the cases considered is the gravity force applied to the center of mass of each body of the mechanical system. Since unilateral constraints are imposed on all systems of bodies considered in the above-mentioned problems, their correct solution requires taking into account the unilateral action of the constraint reaction forces applied to the bodies of the systems under study. A detailed analysis of the motion of the systems after zeroing out the constraint reaction forces is carried out. Results of numerical experiments are presented which are used to construct motion patterns of the systems of bodies illustrating the motions of the above-mentioned systems after they lose contact with the supporting surfaces.
Keywords: unilateral constraints, normal reactions, kinetic energy, free fall
1. Introduction
Problems whose correct solution is only possible if the unilateral nature of the constraints imposed on mechanical systems is taken into account were addressed simultaneously with the development of analytical methods in mechanics. The increasing interest in research in this area in our country and abroad in the second half of the 20th century led to the emergence of the theory of calculation of mechanical systems with unilateral constraints.
Received May 24, 2022 Accepted December 21, 2022
Grigory M. Rozenblat gr51@mail.ru
Moscow Automobile and Road Construction State Leningradsky prosp. 64, Moscow, 125319 Russia
Vitaly T. Grishakin Grichacin@yandex.ru
Moscow Automobile and Road Construction State Leningradsky prosp. 64, Moscow, 125319 Russia Moscow Aviation Institute (National Research University), Volokolamskoe sh. 4, Moscow, 125993 Russia
Technical University (MADI),
Technical University (MADI),
The application of bar systems to machine-building practice can be observed everywhere: in general machine-building, highway engineering and industrial engineering, extractive and processing industries, and medicine [1]. For example, mathematical models and studies of modern external skeletons can be found in [2-4]. A detailed survey of the possibilities of military application of the above-mentioned bar systems is presented in [5].
However, the problems of dynamical calculation of systems with unilateral constraints and problems concerning the motion of the above-mentioned systems after loss of contact with supporting surfaces still remain underexplored.
The motivation for the study of such problems with unilateral constraints stems from a widespread use of robotic devices moving so that their bar supports are in contact with various surfaces. Situations involving losses of contact of such supports with surfaces must undoubtedly be taken into consideration in designing and operating such devices.
In addition, the study of problems concerning losses of contact with the surface with unilateral constraints should be actively introduced into the scientific and educational practice when working with students in order to get the latter involved in active research in the area of mechanics and mechatronics. Finally, in our opinion, detection and correction of any inaccuracies and errors made in well-known and popular books of problems is a useful and progressive activity.
It may be regarded as indisputable that the solution to any problem of mechanics can be obtained only by idealizing some processes taking place in the case at hand. The validity of some assumptions is obvious and does not lead to considerable errors in the resulting solutions. The application of such an approach is justified in quite a number of cases, also from an economic point of view [6]. However, in some problems the assumptions made during solution may entail substantial changes in results.
In this paper we address two problems of the dynamics of bar systems which have been solved before under the assumption that the constraints imposed on the bodies are not unilateral [7]. Cessation of the action of the above-mentioned constraints during the motion of the bar systems increases the degrees of mobility of the systems, and taking them into account leads to a considerable discrepancy between the results obtained and those presented in [7].
2. Formulations of the problems
Computational schemes of the problems considered are shown in Fig. 1.
From the computational schemes shown in Fig. 1 it can be seen that all mechanical systems are conservative and are acted upon by the gravity forces and reaction forces of the supporting surfaces. The motion of the mechanical system in the former case occurs as a result of the loss of stability, and in the latter case, as a consequence of breakage of the thread BD.
A
B
R
D
h
(a)
(b)
Fig. 1. Formulations of the problems considered
3. Computational models and derivation of equations of motion
The solutions to the problems posed have been obtained using the theorem of the motion of the center of mass and the law of conservation of the total mechanical energy and presented in [8]. Below we present the main dependences derived for the problems under consideration in [8], mathematical models describing the behavior of mechanical systems with the support reactions zeroed out, and the results of numerical experiments conducted on the basis of the dependences revealed in [8] and obtained in this work.
Problem of Fig. 1a. To solve this problem, we introduce a moving left Cartesian coordinate system with origin at point D in which the bars are connected by a joint to form a rotational kinematic couple. The positive direction of the ordinate axis coincides with the direction of motion of point D. As a generalized coordinate we take the angle p made by the bar with the horizontal.
The expression for the kinetic energy of the bar of length l = AD = BD can be written as
l2 /4 1 2 \ 2 T=— - + -:---3.1
8 \3 cos Lp COS Lp J y '
In deriving the formula (3.1), the following assumptions were taken into account: yCi =
= Vc2=V = l^g V ~ sin xc2 = ~xc1 = 1cos <P, ™ =
Applying the law of conservation of the total mechanical energy, in which the potential energy of the system is n = -gy, we obtain an expression for the square of the generalized velocity
2 4(1 — COS Lp) t.g Lp
V =(7~l-1-2 ' (3J)
3 cos4 ip cos ip
where a = | is the parameter of the length of the bar.
The second derivative of the generalized coordinate with respect to time, the physical meaning of which is the angular acceleration of the bar, can be found using the identity (p = This yields
<£ = '2(T-^^--^----^--3.3
cos2 Lp U ,__1___2
± I____
3 cos4 (f cos f
It should be noted that the formulae (3.2) and (3.3) hold only in the case where the reaction force of the supporting surface on the bars is not equal to zero. This reaction force is directed perpendicularly to the bars, and its absolute value can be found using the theorem of the motion of the center of mass:
2y = 2g - 2N cos p,
whence
N = —> 0, (3.4)
cos p
where y = isint^ + ^ J ^ + i ^--cos^ j ^ is the projection of the acceleration of the
cos3 f
center of mass of the bar onto the ordinate axis.
In [8] it is shown that, when points A and B (the ends of the bars) coincide with the tops of the shoulders which are the points of contact of the bars with the supporting surface, one
has y = > 9- Hence, at this instant of motion, N < 0 and the bars lose contact with the supporting surface.
Analysis of the function N(p) whose graph is presented in Fig. 2 has shown that the support reaction disappears when p1 = 0.815 rad, or 46.7°. It has also become possible to establish that the value of p1 does not depend on the mass and length of the bars.
N(ip) 5
0—
0 0.262 0.524 0.785 1.047
Fig. 2. Dependence N(p)
Figure 3 depicts the position of the system at the instant when the bars lose contact with the supporting surface and shows the linear and angular dimensions of the characteristic quantities. The angular dimensions are shown in degrees, and the linear dimensions are shown in fractions of the bars' length l. Figure 3 also shows the position of the instantaneous velocity center of bar 1 — point P1. If one knows the coordinate of this point, it is easy to find the velocity of any point of the bar in the above-mentioned position.
Fig. 3. The position of the system for p = p1
After zeroing out the support reaction N the mechanical system becomes a system with two degrees of freedom. As generalized coordinates we take the quantities p and y, which have been used before. After the bars lose contact with the supporting surface, they move with a variable angular velocity, and their motion can be investigated, for example, using the Lagrange equations of the second kind [10, 11].
In this case, the kinetic energy of the bars takes the form
T = ^l2<p2 + (y2 - yip cos ip),
(3.5)
and the generalized forces can be found in the form Q^ = —gl cos p, Qy = 2g.
Then the differential equations of motion of the mechanical system take the following form:
l2Cp — yl cos p = -gl cos p,
2y — l(p cos p + lip2 sin p = 2g.
(3.6)
Eliminating from the equations the vertical acceleration of the centers of the bars, y, we obtain a differential equation of rotational motion of a bar about its center of mass:
2 1
1
c/3^---cos ipj+-ip sin cos = 0, from which the angular velocity can be found by the formula
(3.7)
. . /1 - \ cos2 ip, <P(<P) = <P(<Pl -1 ..„9 » <P><Pl-
1 2
3 — ^ cos^ ip
(3.8)
Figure 4 shows graphs of dependences of the angular velocity w(p) = <p(p) and the angular acceleration e(p) = <p(p) of the bars when they fall freely. Figure 5 shows the corresponding dependences for the functions vD(p) = y(p) and aD(p) = y(p), i. e., the velocity and acceleration of point D.
2.11
-1
-2
-2.576
-3
<Pi
1.2
<P
1.4
Fig. 4. Dependences <p(p) and <p(p)
To carry out a detailed analysis of the motions of the bars of the mechanical system under study, we show the motion pattern of the bars before and after they lose contact with the supporting surface in Fig. 6.
From the superposed positions of the system shown in Fig. 6, where at the stage of free fall each new position corresponds to a change in the angle <p by the quantity A¡p = 0.1 (§ — = = 4.33°, it can be seen that the length of the vertical motion of joint D until the instant when
Fig. 5. Dependences y(p) and y(p)
the bars mutually touch each other is 2.141. The ends of the bars (points A and B) move along trajectories whose height is 1.371.
Analysis of the region where the bars are located (Fig. 6) suggests that the mechanical system does not come again into contact with the supporting surface, i.e., when p = p1 the bars leave the supporting surface without a repeated contact (impact) with it. This fact makes it possible to model the process in the entire interval of variation of the generalized coordinate without having to investigate the impact interaction of the bars and the supporting surface.
Problem of Fig. 1b. As above, to solve this problem, we take as a generalized coordinate the angle p made by the bar CD with the horizontal (Fig. 1), with origin at point C, which by virtue of the symmetry of the bar system will be the projection of the center of mass of the mechanical system onto the horizontal axis Cx, and, since no external forces act on the system in the direction of the above-mentioned axis, it will have no horizontal component of the motion. Consequently, until point C loses contact with the supporting surface, the internal bars BC and CD will perform rotational motions and the external bars AB and DE will perform plane-parallel motion.
Without violating this assumption, we will assume that the masses of the internal and external bars are equal to m1 and m2, respectively. The ratio of the masses of the external bars and the mass of the entire system is taken into account by means of the coefficient A = = 1].
Let us consider the motion of the bar DE. By virtue of the homogeneity of the bars of the system, its center of mass (point K) is at the center of the bar and has the coordinates
3l h l . ,
xk = J cos <P, VK = ~ g = ~ 2 Sm ^ '
where l is the length of the bar.
We find the projections of the velocity of point K by differentiating its coordinates with respect to time:
3l l
xK = -— sm<p ■ <p, yK = -2 cos <P-<P>
whence the square of the velocity of point K is
l2
V2K = XK + VK = J i1 + 8 sin2 <fi) V2- (3-10)
The kinetic energy of the bar DE can be calculated by Konig's theorem, and the kinetic energy of the internal bar CD can be found as the energy of rotation about the fixed axis:
T l-
1de- 2
m2/2 + m^(i + 8sin2^
•2 rp
V , J-CD = 2
m
l2
12
p2.
12 4
Hence, the kinetic energy of the mechanical system takes the form
T = 2TCD + 2TDE = (roj + m2) Q + 2Asin2 ipj i2up2. (3.11)
According to the law of conservation of the total mechanical energy, the kinetic energy of the system is equal to the work of the gravity forces, which can be calculated by the formula
A(mg) = 2(m.i + m2)fif^(siny?(0) — siny?), (3.12)
where p(0) is the initial value of the angle of inclination of the bars to the horizontal.
Equating (3.11) and (3.12), we obtain the following expression for the square of the generalized velocity:
2 siny(O) -siny
p = a—j---~-, (3.13)
| + 2Asin p
where a = f is the parameter of the length of the bar.
We find the expression for the angular acceleration of the bars by differentiating (3.13) with respect to the generalized coordinate p as we did in solving the preceding problem. As a result, we obtain
(h + 2A sin y>(0) sin p — A sin2 p) cos (fi
p = —a—-¡j---. (3.14)
+ 2Asin2 p)
The magnitude of the normal reaction N of the supporting surface on the bars of the system can be determined by applying the theorem of the motion of the center of mass projected onto the vertical axis Cy to the entire mechanical system:
2(m.i + )~ (p2 sint£> — pcosp) = 2(m.i + m2)g — N, (3.15)
whence
N = 2(m1 + m2)g
1 — 7— (p2 sin Lp — Lf COS Lp) 2a
> 0. (3.16)
It should be noted that (3.16) is a necessary condition for all the mathematical calculations presented above to remain valid. If this condition is not fulfilled, the bar system becomes a system with two degrees of freedom as was the case when the problem of Fig. 1a was considered.
By analyzing the expression (3.16), one can find the condition under which the motion of the bars of the system occurs without loss of contact with the supporting surface. This condition takes the form
Lp2 sin p — (^cos p< 2a, (3.17)
or
siny?(0) - siny? . + 2Asint/?(0) sint/? - Asin2t/?) cos2c/?
—¡—-——-j— sin <p H — — 72 < 2- (3.18)
3
+ 2Asin tp (i + 2Asin2pY
The left-hand side of inequality (3.18) is a function of the generalized coordinate p and the parameter A, and depends on the initial angle of inclination of the bars, p(0). It is obvious that the largest angular velocities of the bars can arise when they move from the vertical initial position in which t£>(0) = Then condition (3.18) takes the form
1-sinw (I + 2Asint^ - A sin2 ip) cos2 p
M<P, A) = 1-L0. .2 sin ip + A6---H < 2. (3.19)
| + 2Asm p + 2Asin p)
Analysis of the dependence fi(p, A) suggests that in the range of variation of the generalized coordinate p G , 0] this function has, for any value of A e [0, 1], one extremum arising at p = = p*. In particular, the dependence fi(p, 0) = 3(1 — sin t/?) sin t^ + |cos2(/? has a maximum at p* = arcsin|. The maximal values of the function max//(<£?, A) = n(<p*, A), which arise at various values of the parameter A, are presented in Table 1.
It can be seen from Table 1 that the only value of the parameter A for which n(p, A) = 2, corresponds to A = 0. In this case, condition (3.19) is not fulfilled only when p = p*, i.e., the
Table 1. Values of the angles <*, <, <2 and the function ¡1 (<*, X)
A 0 0.25 0.5 0.75 1
ip*, rad 0.34 0.25 0.2 0.19 0.17
ip1, rad 0.34 0.47 0.45 0.42 0.41
<p2, rad 0.34 -0.022 -0.034 -0.039 -0.043
2.0 2.46 2.82 3.12 3.39
contact between the bar system and the supporting surface is broken only for an infinitely small time interval. In this case, the angle of separation from the supporting surface and the angle of fall of the bars are the same: <1 = <2. However, such a value of the parameter X is only possible for m2 = 0, i.e., in the absence of the outermost bars, which leads to consideration of a problem that is somewhat different in its condition. For other values of the parameter X, condition (3.19) is not fulfilled either, and hence, when < = <*, the bar system loses contact with the supporting surface no matter what the masses of the bars are.
Figure 7 shows graphs of the dependence of 2{rn'+m )g ^ e [§' ' which allows a qualitative estimation of the behavior of the normal reaction of the supporting surface on the bars of the system which have been constructed for two cases of practical significance: X = 0 and X = = 0.5. We note that the analysis of the case X = 1 has no sense because at this value of the parameter X the mass of the internal bars is zeroed out, which is impossible for the mechanical system considered.
0.6
2(m1+m2)g
0.4
0.6
N(v>)
2(m1+m2)g
0.4
(a) A = 0
(b) A = 0.5
Fig. 7. Dependence ^„f^ jg for A = 0 (a) and A = 0.5 (b)
As can be seen from Fig. 7, the supporting reaction N(<) is zeroed out at different angles of <1, which are presented in Table 1 for different values of the parameter X. Table 1 also presents values of the angles <2 at which the bars come again into contact with the supporting surface. We note that the calculation of the angles of fall, <2, is only possible by considering the motion of the system in the absence of the supporting reaction N.
The motion of the mechanical system with N(<) = 0 can be studied using the approaches described in the corresponding part of the solution of the problem of Fig. 1a, i.e., using the Lagrange equation of the second kind [10, 11]. By virtue of the symmetry of the mechanical
system (Fig. 1b), the free motion of the bars can be given by two generalized coordinates p, y, where p is the angle made by the bars with the horizontal x, and y is the vertical coordinate of point C, i.e., the central connecting joint measured downward along the vertical. The kinetic energies of the internal and external bars are, respectively,
2T1 = ml 2T2 = m2
l
l
y--pcospj + l--psmp
y--pcospj +I-J0sin<£>
l2 2
l2 2 + ™2-^P ■
The kinetic energy of the system in this case takes the form
T = (mj + m2)
o /1 O \ O o
y — ypl cos <£+(-+ 2A sin (p ) I (p
(3.20)
and the generalized forces can be found by the formulae: = -m0gl cos p, Qy = 2m0g.
The system of differential equations of motion of the bar system under study with NN = 0 takes the form
—y cos tp + 2 (pi ^ + 2A sin2 p^j + 4p2l\ sin p cos p = —g cos (p,
y = g+l- (~(p2 sinp + pcosp) . Eliminating y from the last two equations, we obtain the following equation for p:
(3.21)
(p ( i + 2Asin2 ip j = i cos tp (ipcos (p)' — 2Xp2 sinocos (p.
Making straightforward transformations, we obtain the following differential equation for the function p from the last equation:
<P
1
1
-+l- + 2A]sinV
+ ( i + 2A ) p¿ sin ip cos (p = 0.
Introducing the notation u = (p2, we obtain the exact differential equation
du dp
l + Q + 2A)sin2p
+ u
d_
dp
12 + (l +2A ' sin(^c0s(^
0.
Integration of the last equation gives us the following dependences:
u = w2(p) = (p2 = h
Q + 2A)sin2p
j
w(p) = p = —
l+Q + 2A)sin2p
J/2
e(p) = p = —h ( - + 2A ) sin p cos p
l + Q+2A)sin2p
2
h = (p2 (0)
¿ + Q + 2A)sin2p1
= const.
2
2
2
We note that the law of variation of the angle p does not depend on the acceleration of gravity g (in particular, this law is valid also in the absence of gravity, i.e., in weightlessness). We take the minus sign in the expression for the angular velocity because, due to the specificity of the problem, the angle p obviously decreases monotonically up to a new collision of the bar system with the supporting surface.
Using the formulae obtained above, it is easy to write the dependence of the acceleration of point C along the vertical axis y on the angle p:
icy = y = g
| + 2A) sinp
[i + (I + 2A) sin2p]:
(3.22)
Assuming that the initial conditions for y are zero conditions (the coordinate and the velocity are zero for p = ), we obtain the following formula for the function y(p):
yiv) = -(sinp — sinp1) --yjufa) COSP! J
Vi
dtp g 2
Vi
d^
\
/
Here we need to take as the function u(^) the square of the angular velocity as a function of the variable which was calculated above. The integrals appearing in the last formula are elliptic. Therefore, further investigations are numerical. We note that similar formulae can be written for the coordinates of other points of the bar system as well.
Figure 8 shows graphs of the dependences of the angular velocity w(p) = pp(p) and the angular acceleration e(p) = p(p) of the freely falling bars for the case A = 0.5.
63.798
100
50
10 u(<p)
T^ 0
-50
-50.618
-100
<Pi
0.3 0.2
y
0.1
<P 2
Fig. 8. Dependences ^p(p) and <p(p)
2
Figure 9 shows the corresponding dependences for the functions vC(p) = y(p) and aC (p) = = y(p), i.e., the velocity and acceleration of point C for p £ [p1, p2]. To make the graphs of the angular velocity and the velocity of point C more illustrative, the values of the ordinates of the quantities presented in Figs. 8 and 9 were increased tenfold.
The difference of the acceleration of point C from zero at time instants corresponding to the angles p2 can be explained by the fact that the mechanical system comes again into contact with the supporting surface at points B and D (Fig. 10). For a better illustration, we present
20.321
lOtfcfc) 10 ac{ip)
Fig. 9. Dependences y(p>) and y(p>)
V/////X
V77777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777\
V77777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777\
V77777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777\
Fig. 10. Motion pattern of the mechanical system of Fig. 1a
in Fig. 10 the motion pattern of the bar system under study — the sequence of the positions occupied by the bar system at p £ [p],, p2] and constructed for the case A = 0.5 on the basis of the above-mentioned dependences.
We note that for the case A = 0.5 represented in Fig. 10, at the instant when all bars take a horizontal position, i. e., when p = 0, the lengths of vertical motions of all points of the system are the same, being equal to 0.031. We also note that the maximal lengths of vertical motions of points A, C and E are the same, being equal to 0.0331.
Conclusion
The analysis carried out in this work has revealed that the unilateral nature of the constraints imposed on the mechanical systems shown in Fig. 1 has a considerable influence on the motion of the systems considered. We have presented the values of the angular coordinates of the bars at which the above-mentioned constraints are violated. In the first case (Fig. 1a) we established that the contact of the bodies with the supporting surface is not resumed. In the second case (Fig. 1b) we showed at which angular coordinates and at which points the mechanical system under study will collide with the supporting surface. We analyzed the influence of the ratio of the masses of the bars on the processes under study.
Acknowledgments
The authors express their gratitude to Academician V. F. Zhuravlev from the Russian Academy of Sciences for discussion of the results obtained and for his support.
Conflict of interest
The authors declare that they have no conflicts of interest.
References
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