MPEI, the innovative program
28 problems
Short condition of a problem -
  1. Dynamics of a point 5 problems about rectilinear movement of a point
    1 2 3

    The Maple-program
  2. Dynamics of a point The car in weight m brakes, moving on a horizontal straight line. Force of resistance of air depends on speed R=kv, friction factor f. For what time speed of the car will decrease with v0 to v1?
    1
  3. Theorems of dynamics of a pointOn a rectilinear site of a way the washer is dispersed during time t by the variable force F directed under a corner to moving. On a curvilinear site of an axis constant force of resistance operates. a decision Example.

    1 2 3
  4. The theorem of movement of the centre of weights The mechanism consisting of cargo, the block and the cylinder, is established on a prism which is on a horizontal plane. The friction between a prism and a plane is absent. Cargo receives moving concerning a prism along its surface to the left or (in those variants where it hangs) on a vertical downwards. Where and on what distance the prism will move?
    1 2 3 4 5
  5. Dynamic reactions of a shaft On an axis rotating in bearings under the influence of the constant moment, the rotor consisting of the cylinder and a rigid weightless core with dot weight on the end is fixed. The cylinder axis makes a small corner with a rotation axis. To find dynamic components of reactions of bearings
    1 2 3 4 5 6 7 8 9 10
  6. Kinetic energy. The resulted weightThe decision in integers the Example 1. An example 2.
    1 2 3 Decision in integers
  7. The theorem of change of kinetic energy of system (1) To find speed of one of system bodies.
    1 2 3 4 5 6 7 8 9 10
  8. Dynamic calculation of the mechanism with unknown parametre. The theorem of change of kinetic energy (2) Using the theorem of change of kinetic energy to find one of system parametres on its set movement.
    1 2 3 4 5 6 7 8 9 10
  9. The theorem of change of kinetic energy of system (3) The mechanism consisting of cargo, the block and the cylinder, is established on a prism fixed on a plane. Under the influence of a gravity from a condition of rest the mechanism has come to movement. What speed was developed by cargo, having moved on distance S?
    1 2 3 4 5
    The simplified variants (without a friction):
    6 7  8 9  10  

  10. The analytical mechanics
  11. Calculation of number of degrees of freedom of mechanical systemTo use formula W=3-2Ш-. a decision Example. (LaTeX)

    1 2 3 4 5 6 7 8 9 10
  12. The general equation of dynamics for system with one degree of freedom The flat sharnirno-rod mechanism with one degree of freedom moves in a vertical plane under the influence of a gravity and moment M which rotates link OA with constant angular speed. In knots A, B, C and in centre E of link AB material points are located. On axes of motionless hinges O and D there is a friction with constant moment Mfr. Force of resistance to movement л - Ffr, other communications ideal. Neglecting weights of cores to define size of moment M.
    1 2 3 4 5 6 7 8 9 10
  13. Principle of possible speeds (Definition of reactions of support) To find reactions of support of a compound design
    1 2 3 4 5 6 7 8 910
  14. Principle of possible speeds. The mechanism with a disk. The decision in integers To find force F
    1 2 3 Example 1. An example 2. The Maple-program for example 2.
  15. Dynamics of a side scene To make and integrate equation г. To use Maple 6,7,8,9
    1 2 3 4 5 6 7 8 9 10

    Animation 1 Animation 2 Animation 3


  16. Example
  17. Equation (two degrees of freedom) The mechanical system from two homogeneous cylinders 1 and 2 and б 3 with ideal stationary communications has two degrees of freedom and moves under the influence of force F. A friction to neglect. Weights are given in kgs, force - in ьюх. To find acceleration б, sliding on a smooth surface.
    1 2 3 4 Decision in integers
    Example. Stanislav Zajtseva's decision (MPEI)
  18. Equation г of 2nd sort (two degrees of freedom) The mechanical system with ideal stationary communications has two degrees of freedom and moves under the influence of a gravity.
          Three elements of the mechanism
          are allocated by weights, multiple to some weight m. A friction to neglect. Mobile and motionless blocks to consider as homogeneous cylinders. To find acceleration And cargo or the centre of cylinder A.
    1 2 3 4 5 6 7 8 9 10

    Example
  19. Lagrange equation of the second kind (two degrees of freedom) The mechanical system with ideal stationary communications has two degrees of freedom and consists of five bodies. The Block (or the homogeneous cylinder) D slides without льыя on a motionless horizontal plane or on the mobile cart. Cargoes A, B and an axis of homogeneous cylinder E move vertically under the influence of a gravity. To find acceleration of cargo A.
    1 2 3 4 5 6 7 8 9 10
  20. Lagrange equation of the second kind for conservative systems The conservative mechanical system with ideal stationary communications has two degrees of freedom and represents the mechanism consisting of cargo, the block and the cylinder. The mechanism is established on a prism fixed on axes of two homogeneous cylinders. The constant is enclosed to a prism on size horizontal force. Using equation г of 2nd sort for conservative systems to find prism acceleration.
    1 2 3 4 5 6 7 8 9 10
  21. Lagrange equation of the second kind - an examination problem To work out equation г of mechanical system. Two bodies of system have weights, cylinders slide without resistance.
         For the decision at first to make kinematic columns, then to find kinetic energy, then - the generalised force. A problem for 20-45 minutes.
    1 2 3 4 5 6 7 8 9 10
  22. 11
    Illustrations
    120 problems on the same theme see in All book except answers and without the listing right
  23. Lagrange equation of the second kind - definition of accelerations on the set kinetic energy and the generalised force Equation г
    1 2 3 4 5 6 7 8 9 10
  24. Function of Hamilton H
    1 2 3
  25. The equations of Hamilton H
    1

    The Theory of Oscillations
  26. Oscillations of system with two degrees of freedom (1) The system with 2 degrees of freedom consists of three bodies allocated in weights, and one spring. To find frequency of fluctuations.
    1 2 3 4 5 6 7 8 9 10
  27. Oscillations of system with two degrees of freedom (2). The frequency analysisTo find rigidity of one of system springs at which the difference of own frequencies is minimum. a decision Example

    1 2 3 4 5 6 7 8 9 10
  28. Oscillations of system with two degrees of freedom (3). Limiting frequencies To find limiting frequencies of system with 2 degrees of freedom. (Limiting frequency is equal own weight at unlimited growth)
    1 2 3 4 5 6 7 8 9 10
  29. Oscillations of system with two degrees of freedom (4). Cylinders The mechanical system with two degrees of freedom consists of two homogeneous cylinders and several linearly elastic springs with identical rigidity c. Cylinders go for a drive without льыя and resistance on a horizontal surface, springs in position of balance have no preliminary pressure. Weight of springs to neglect. To define frequencies of own fluctuations of system.the Example
    1 2 3 4 5 6 7 8 9 10
  30. Oscillations of knot of a truss  In one of hinges of a flat farm (in drawing it is allocated) there is a point with weight m. Farm cores elastic. Rigidity of cores EF; l=1. The farm is located in a horizontal plane. Neglecting weight of cores to define frequencies of own small fluctuations of the hinge fermy./
    1 2 3 4 5 6 7 8 9 10
  31. Problems for a practical training at the rate "Robototeñhnique systems and complexes". Pankrateva G. V, Zatsepin M. F.


Solver. The theoretical mechanics. A Fig. 129.
Moscow.  <Hermitage>