RU132913U1 - DEVICE FOR DEMONSTRATION AND RESEARCH OF FORCED OSCILLATIONS OF THE MECHANICAL SYSTEM - Google Patents

Abstract

1. A device for demonstration and study of forced vibrations of a mechanical system, containing a vibrational element in the form of a disk mounted on a common base, a lever with a shaft coupled to the disk through an elastic link, a drive motor connected to the lever by means of a gearbox and a transmission mechanism with a connecting rod, vibration damper a disk, a power supply and control unit, characterized in that it is equipped with sensors of the angles of rotation of the disk and the lever, kinematically connected to them, and a unit for recording and processing signals sensors, the disk is equipped with a central shaft mounted horizontally and coaxially with the lever shaft on the base, the elastic link is made in the form of a spiral spring, one end of which is fixed to the disk shaft and the other on the lever, the connecting rod of the transmission mechanism is directly connected to the lever, the damper is made in in the form of two electromagnets mounted coaxially with each other on two sides of the disk with a gap relative to it, while the disk is made of conductive non-magnetic material, and the power supply and control unit are electrically connected n with damper and sensors. 2. The device according to claim 1, characterized in that as a unit for recording and processing sensor signals, a personal computer with an analog-to-digital converter is used, electrically connected to the sensors of rotation angles of the disk and lever. The device according to claim 1, characterized in that the angle sensors of rotation of the disk and the lever are made in the form of DC resistors.

Description

Device for demonstration and study of forced oscillations of a mechanical system

The utility model relates to educational and research equipment in theoretical mechanics and can be used in higher technical schools to study the theory of free and forced vibrations of a mechanical system with kinematic excitation of vibrations through an elastic element.

State of the art

A device for demonstrating oscillatory processes, comprising an oscillating element in the form of a disk, a lever with a shaft, articulated with the disk through an elastic link, a drive motor connected to the lever by means of a gear and a transmission mechanism with a connecting rod, and a disk vibration damper, as well as a power and control unit (see USSR Author's Certificate No. 1707613, class G09B 23/06, 1989).

In the specified device, the recording and processing of experimental results is carried out according to visual observations, which allows to obtain only individual and approximate characteristics of the oscillatory process. In addition, in this device, the oscillating disk together with the damper is vertically suspended on an elastic element - a string, which does not provide a stable position of the axis of oscillations and the reliability of the damper, and the transmission mechanism connecting the gearbox with the connecting rod has a bulky design that does not provide sufficient accuracy for obtaining harmonic the law of change of driving force.

Utility Model Disclosure

The objective of the utility model is to increase the information content, reliability and accuracy of the demonstration and study of the characteristics of the oscillation

                                                         2 Appendix No. 2

mechanical system, including amplitude-frequency (AFC) and phase-frequency (PFC) characteristics.

The problem is solved in that a device for demonstrating and studying the forced oscillations of a mechanical system, containing a vibrational element in the form of a disk mounted on a common base, a lever with a shaft coupled to the disk through an elastic link, a drive motor connected to the lever through a gearbox and a transmission mechanism with a connecting rod , the vibration damper of the disk, the power supply and control unit, according to the utility model, is equipped with sensors of rotation angles of the disk and lever kinematically connected to them, and a recording unit and signal processing of sensors, the disk is equipped with a central shaft mounted horizontally and coaxially on the lever shaft, the elastic link is made in the form of a spiral spring, one end of which is fixed to the disk shaft and the other on the lever, the connecting rod of the transmission mechanism is directly connected to the lever, the damper is made in the form of two electromagnets mounted coaxially to each other on both sides of the disk with a gap relative to it, while the disk is made of conductive non-magnetic material, and the power supply and control unit connected to damper and sensors.

In addition, as a unit for recording and processing sensor signals, a personal computer with an analog-to-digital converter is used, electrically connected to the sensors of rotation angles of the disk and lever, and the sensors of rotation angles of the disk and lever are made in the form of DC resistors - potentiometers.

List of figures

Figure 1-3 presents a General view of the device.

Figure 4 shows a graph of linear damped oscillations.

Figure 5 shows a typical frequency response (λ = λ (z) of a mechanical system.

механической системы.Figure 6 shows a typical phase response

mechanical system.

Utility Model Implementation

The device diagram is shown in FIGS. 1-3 (here, the power supply unit and the recording and processing unit of sensor signals are shown conditionally in FIG. 1 and not shown in FIGS. 2, 3).

                                                          2a

The device comprises a disk 1 and a lever 2 of the drive of the disk in oscillatory motion, which through the shafts 3 and 4 are installed in the supports 5 and 6, mounted on the base 7, while the shafts 3 and 4 are mounted coaxially to each other. The disk and the lever are interconnected by means of a spiral spring 8, one end of which is fixed on the shaft 3 of the disk, and the other on the lever 2. The lever is by means of a transmission mechanism including a connecting rod 9, which is articulated with the axles 10 and 11 with the lever and crank 12, is connected to the gearbox 13, while the length of the crank can be adjusted by moving the axis 11 in the groove of the crank, and the amplitude of rotation of the lever will change. The crank 12 is rigidly fixed to the output shaft of the gearbox connected to the electric motor 14. On both sides of the disk 1 with gaps relative to its side surfaces, permanent electromagnets 15 of the braking device — a disk vibration damper — are mounted coaxially to each other. In this case, the disk is made of a non-magnetic conductive material, for example, an aluminum alloy. The gearbox 13, the motor 14 and the damper electromagnets are fixedly mounted on the base 7. On the supports 5 and 6 are mounted sensors 16 and 17 of the rotation angles of the disk and the lever, kinematically, for example by means of a gear transmission, connected to their shafts 3 and 4. The base is also fixed graduated scale 18, and the disk and lever are equipped with arrows 19 and 20 - indicators of the angles of rotation relative to the scale 18.

The device is also equipped with a power and control unit 21 and a unit 22 for recording and processing sensor signals (these units are shown in FIG. 1 conditionally, but not shown in FIGS. 2 and 3). The power and control unit is electrically connected to an electric motor, damper and sensors. As a unit for recording and processing sensor signals, this device uses a personal computer that is electrically connected to sensors 16 and 17 through an analog-to-digital converter (ADC).

This device works as follows.

First, when you turn on the block 21, the power supply is supplied to the electromagnets 15 of the damper and the sensor 16 of the angle of rotation of the disk 1, while the engine remains off and the lever 2 is stationary. Turning the crank 12 (see figure 1), set on a scale of 18, the lever and the disk to zero positions. Then, the disc is deflected by a predetermined angle by hand, twisting the coil spring 8, i.e. tell the disk the initial deviation from the equilibrium position and then release it, for example, without an initial speed. As a result, the disk comes into free rotational vibrational motion, while the sensor signal 16 enters block 22, in which it is processed, recorded, displayed visually and printed using the appropriate software, including, for example, in real time. Oscillation damping occurs due to resistance to the movement of viscous friction forces in the environment, in the sensor and in the bearings of the disk shaft, as well as electromagnetic forces of the interaction of the eddy currents that arise in the body of the disk during its movement in the magnetic field of the damper, with this field itself. The damping rate of oscillations of the disk in this device is regulated by changing the supply voltage of the electromagnets 15 of the damper.

A typical graph of linear damped oscillations is shown in figure 4. Here:

φ is the angle of deviation of the disk from the equilibrium position,

A _i and A _{i + 1} are the values of two adjacent maxima of this deviation,

t is the oscillation time,

t _i and t _{i + 1} - time moments corresponding to adjacent maximums of the disc deviations,

T ₁ - the conditional period of free damped oscillations of the disk.

According to the recording data of this graph, the conditional period T _{1 of the} damped oscillations of the disk is determined and the main parameters of these oscillations are calculated by the following formulas:

;

;

;

;

;

;

;

;

where ω is the circular frequency of the damped oscillations;

η is the logarithmic decrement of oscillations;

n is the damping coefficient of oscillations (generalized coefficient of resistance);

ω is the natural frequency of the disk without resistance

Q is the quality factor of the system.

According to the parameters thus obtained, the calculated (theoretical) frequency response and phase response of this system are determined by the following ratios:

;

;

where λ is the coefficient of dynamism;

z = p / ω is the detuning coefficient;

p is the frequency of forced oscillations of the system;

γ = ψ _in -ψ is the delay in phase oscillations, i.e. the phase difference between the disturbance phase ψ _in and the phase of the forced oscillations ψ.

As an example, typical frequency response (λ = λ (z)) and phase response (γ = γ (z)) are shown in FIGS. 5 and 6, respectively. In Fig. 5, positions 23, 24, and 25 indicate the dependences λ = λ (z) for small, medium, and large resistance (here, λ _max1 , λ _max2 λ _max3 are the corresponding maximum values of the dynamic coefficient). 6, positions 26, 27, and 28 mark the dependences γ = γ (z) for small, medium, and large resistance, respectively.

Then, the engine 14 is turned on (see Figs. 1-3) and, by means of a reducer 13, a crank 12 and a connecting rod 9, the lever 2 is brought into oscillatory motion (rotation around the axis of the shaft 4), and this movement of the lever occurs according to a law close to harmonic:

,

,

where φ _in - the current value of the angle of rotation of the lever;

- амплитуда угла поворота рычага;

- the amplitude of the angle of rotation of the lever;

β is the initial phase of the perturbation.

. Здесь c - коэффициент жесткости пружины, µ - коэффициент сопротивления системы. Электромагнитный тормоз данной экспериментальной установки создает момент торможения, практически пропорциональный угловой скорости поворота диска. Величина момента торможения может регулироваться с помощью блока управления, что позволяет изменять коэффициент затухания n и соответственно добротность Q системы.The second end of the spring is connected to the disk and when the lever is deflected by an angle φ _{, the} disk is rotated by an angle φ from the neutral position. At the same time, the moment of elastic spring deformation forces x (φ-φ _in ) and the total moment of resistance forces, including the moment of resistance of the damper, act on it -

. Here c is the coefficient of spring stiffness, μ is the coefficient of resistance of the system. The electromagnetic brake of this experimental setup creates a braking torque that is practically proportional to the angular velocity of the disk. The magnitude of the braking torque can be adjusted using the control unit, which allows you to change the attenuation coefficient n and, accordingly, the quality factor Q of the system.

With each change in p and µ, in addition to forced oscillations, free oscillations arise (see Fig. 4), which decay after a certain period of time - the transition process t *, which is determined by the formula:

t * = 3τ ₀ = 3 / n,

where τ ₀ = 1 / n is the time constant of the damped oscillations.

It is from the moment of the end of the transition process that steady-state forced oscillations occur and the motion parameters of the disk and the lever are recorded. The equation of forced oscillations of the disk has the following form:

φ = φ ₀ sin (pt + β-γ),

where φ ₀ is the amplitude of the forced oscillations.

) и запаздывание у колебаний по фазе при фиксированных значениях коэффициента расстройки z определяются по формулам:According to the recorded parameters of the oscillations of the disk and the lever, the experimental values of the dynamic coefficient λ (the ratio of the amplitude of the forced oscillations of the disk to its static displacement under the action of a constant moment equal in magnitude

) and the delay in phase oscillations at fixed values of the detuning coefficient z are determined by the formulas:

и

,

and

,

- амплитудное отклонение рычага относительно его нулевого положения, равное в данном случае статическому смещению диска от положения равновесия;Where

- the amplitude deviation of the lever relative to its zero position, equal in this case to the static displacement of the disk from the equilibrium position;

- моменты реального времени, соответствующие ближайшим по времени амплитудам отклонений диска и рычага от нулевого положения.t ₀ and

- real-time moments corresponding to the nearest in time amplitudes of deviations of the disk and the lever from the zero position.

, отображаемых на соответствующих расчетных графиках (см. фиг.5 и 6). Это позволяет сопоставить результаты экспериментов и расчетов, полученных с использованием линейной математической модели.Each oscillation period corresponds to a pair of experimental points (z _i , λ _i ),

displayed on the corresponding calculated graphs (see figure 5 and 6). This allows us to compare the results of experiments and calculations obtained using a linear mathematical model.

Thus, the proposed device for demonstration and study of forced vibrations of a mechanical system can improve the reliability, accuracy and information content of studies of vibrations of a mechanical system. Moreover, the implementation of a braking device in the form of an electromagnetic damper simplifies and increases the accuracy of regulation of the process of damping oscillations, and supplying the device with measuring sensors and a unit for recording and processing sensor signals provides a record of these processes and their quantitative study, as well as the construction and study of theoretical and experimental amplitude frequency and phase-frequency characteristics of a mechanical system. For these reasons, this device can significantly improve the quality of the study of the theory of free and forced oscillations of mechanical systems in higher technical educational institutions.

Claims (3)

1. A device for demonstration and study of forced vibrations of a mechanical system, containing a vibrational element in the form of a disk mounted on a common base, a lever with a shaft coupled to the disk through an elastic link, a drive motor connected to the lever by means of a gearbox and a transmission mechanism with a connecting rod, vibration damper a disk, a power supply and control unit, characterized in that it is equipped with sensors of the angles of rotation of the disk and the lever, kinematically connected to them, and a unit for recording and processing signals sensors, the disk is equipped with a central shaft mounted horizontally and coaxially with the lever shaft on the base, the elastic link is made in the form of a spiral spring, one end of which is fixed to the disk shaft and the other on the lever, the connecting rod of the transmission mechanism is directly connected to the lever, the damper is made in in the form of two electromagnets mounted coaxially with each other on two sides of the disk with a gap relative to it, while the disk is made of conductive non-magnetic material, and the power supply and control unit are electrically connected n with damper and sensors. 2. The device according to claim 1, characterized in that as a unit for recording and processing sensor signals, a personal computer with an analog-to-digital converter is used, electrically connected to the sensors of rotation angles of the disk and lever. 3. The device according to claim 1, characterized in that the angle sensors of rotation of the disk and the lever are made in the form of DC resistors.

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