Dynamics of machines - Applied Mechanics

Machine dynamics

Of great importance in the design of machines and mechanisms is the calculation of the dynamics of the structure and its elements. When there are dynamic processes on the mechanical components and parts there are additional loads. They can cause fracture, which has a fatigue character. The simplest design model of oscillations of structural elements with one degree of freedom for force excitation with frequency and amplitude F 0 is described by the equation

where n is the damping coefficient, is the natural oscillation frequency; С - coefficient of rigidity; m is the weight of the cargo.

The solution of the equation has the form


where is the attenuation decrement.

Dynamic factor/СД1Ш equals the ratio of the amplitude of the oscillations to the static displacement бс under the action of the force F0 and determines the increase in the load on the structure. For different values ​​of with force excitation in Fig. 4.10, a shows the dependencies from , and in Fig. 4.10, b - the dependence of the phase φ on η.

In calculating the strength, you need to take into account the dynamism factor of KD1SH and set the force . At resonance, when the oscillation frequency coincides with the natural frequency of the system ). the maximum value of .

So, with the decrement of oscillations the maximum value of

The loads are significantly reduced when the excitation frequencies ω and the natural oscillations of the structure are spaced apart. To do this, the exciting frequency must be either in the pre-resonance zone , or in the resonance frequency

During the operation of products, forced oscillations are most often encountered. Thus, when the drive of its transmission is operated, an external dynamic load is experienced (uneven engine shaft rotation, force action at the output, etc.) and an internal additional dynamic load when the teeth interact in engagement.

Fig. 4.10

This is due to impact when entering and leaving the teeth from the gear and manufacturing error. The unevenness of the acting forces is observed in the lever and cam mechanisms, which also leads to an additional loading of their links. Reduction of uneven movement can be achieved by means of balancing or flywheel. Parametric oscillations of transmissions are also possible because of the change in rigidity, since different numbers of teeth, for example, one or two, are involved in engagement. To identify and eliminate parametric oscillations, a stability study is carried out. With a rapid change in the load, the static component adds an oscillatory component. It can increase the dynamic factor to a value of in a system with one degree of freedom with stepwise loading (figure 4.11, a).

If the drive is started, a more gradual increase in the exciting force 1 is characteristic (Figure 4.11, b). With an increase in time t , the vibrational component 2 will decrease and will tend from 2 to 1 (with ). In the calculations of torsional and bending vibrations of shafts, the dynamic drive model consisting of rigid wheels and elastic shafts, couplings and supports is most often used. Own oscillations of the shafts often have a low frequency and affect loads, which can lead to increased contact and flexural stresses in the teeth. The increase in the speeds of machines also leads to an increase in dynamic loads. Let's consider the simplest cases.

The torsional oscillations of a shaft with two wheels are described by the equation

where is the natural oscillation frequency, Hz; - coefficient of rigidity of the shaft; - rigidity of the shaft; - the length of the shaft; - moments of inertia of the wheels.

Fig. 4.11

Consider the free oscillations of two shafts with wheels (Figure 4.12). For generalized coordinates, we take the rotation angles of the wheels (a system with three degrees of freedom). The rotation angle of the wheel 3 is determined by the angle of rotation wheels 2: (- gear ratio).

Having determined the kinetic and potential energy and using the Lagrange equation of the second kind, we obtain


where is the moment of inertia of the wheels 2 and 3, reduced to the shaft 1; - the moments of inertia of the gears 1,2 and 3; C12, C 34 are the stiffness coefficients of the shafts I and I.

Solving the system of equations (4.2), we define two natural frequencies of torsional oscillations of the shafts. There is a zero, which is caused by the fact that the system has one degree of mobility. Similarly, one can obtain a system of equations for more shafts and determine their own frequencies. To determine the loads, forced oscillations are considered for a given perturbation. In this case, it is necessary to take into account the damping forces of energy dissipation in gearing, clutches and bearings.

In high-speed reversing drives, the speed at acceleration and deceleration is important, which is achieved by reducing the moment of inertia of the entire transmission, brought to the motor shaft. This is achieved by the fact that when dividing by stages, large values ​​of the gear ratio are set at the last stages, therefore wheels with a larger mass are installed on the slow shafts.

If there is a danger of significant influence of bending vibrations of shafts, one must study and evaluate their influence on the loads. The dangerous dynamic loads that arise during the operation of the machines need to be reduced or limited by the scope of their action.

The presence of elastic couplings in the drive, which can reduce the load on the transmission, can lead to a decrease in the number of couplings.

Fig. 4.12

frequencies of torsional vibrations and there is a danger of resonances.

Vibroactivity and vibration protection. During the operation of machines, the following mechanical effects occur: overload, vibration and shock loads.

The vibrational effect can be force and kinematic. For example, a force action - from an unbalanced motor rotor, and kinematic - when considering the dynamic effect of base oscillations on devices.

The ability of mechanisms to not break down when subjected to mechanical shocks from vibration is called vibration resistance , and normal functioning is vibration resistance (no opening of electrical circuits in vibration switches, screwing of threaded connections, etc.).

Serious attention should be paid to the harmful effects of vibration on maintenance personnel, as new high-performance machines based on vibratory and vibro-impact processes are used in industry and construction.

In the oscillation of machines involved kinematic chain, consisting of shafts with concentrated masses of gears. Calculations of the natural oscillations of the shafts are performed and in the case of their coincidence with the perturbing forces, measures are taken to eliminate the resonances. The most dangerous is the fundamental tone of the oscillations. When the frequencies of the intrinsic bending vibrations of the shaft coincide with the frequency of their rotation from the action of the centrifugal forces, a disbalance of the wheels causes a resonance. This speed is called critical . The source of excitation is unbalanced details. The mechanism should not operate at a speed close to critical.

Sources of vibroactivity: excitement from the strength of technological resistance and the inertia force of the portable movement of the working organs. Dynamic loads in gearing gears can occur due to manufacturing and mounting errors (kinematic accuracy, smooth running, side clearance and contact quality), uneven rotation of the rotor (step and contactless DC motors), as well as uneven movement of the working mechanisms of the mechanism. To reduce the harmful effect of vibration on the mechanisms and bases used vibration protection.

Methods of vibration protection. Vibration protection uses:

• change the design of the object to change its natural frequencies and increase dissipation (dissipation) of energy;

• Introduction of vibration isolation.

Active and passive vibration protection devices are used to protect objects from vibration. Active devices have an independent energy source and have elements that give out forces that compensate for the load from the source of vibration. Passive devices consist of inertial, elastic and dissipative elements.

The objectives of vibration protection. With force excitation the objectives of vibration protection are:

• a decrease in the amplitude of the reaction , transmitted to a stationary base (estimated by the vibration protection coefficient , where - the amplitudes of the reaction and the excitation force ω is the circular frequency of the exciting oscillations;

• decrease in the amplitude A of the forced oscillations of the load from the action of the exciting force (estimated by the dynamic coefficient , where C is the stiffness coefficient of the suspension nodes).

With kinematic excitation the objectives of vibration protection are:

• a decrease in the amplitude of the absolute acceleration (overload) of the cargo (it is estimated by the coefficient of vibration isolation , where A and - amplitudes of the absolute vibro-displacement of the load and the base);

• a decrease in the amplitude of the relative vibro-displacement A, of the load relative to the ground (estimated by the dynamic coefficient ).

Carrying out design measures for vibration protection has a significant positive effect on the dynamics of structural elements (eliminating dangerous fluctuations, reducing dynamic loads).

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