The study of pendulum oscillations is carried out using a setup, the diagram of which is shown in Fig. 5. The installation consists of a spring pendulum, a vibration recording system based on a piezoelectric sensor, a forced vibration excitation system, and an information processing system on a personal computer. The spring pendulum under study consists of a steel spring with a stiffness coefficient k and pendulum bodies m, in the center of which a permanent magnet is mounted. The movement of the pendulum occurs in a liquid and at low oscillation speeds the resulting friction force can be approximated with sufficient accuracy by a linear law, i.e.
Fig.5 Block diagram of the experimental setup
To increase the resistance force when moving in a liquid, the body of the pendulum is made in the form of a washer with holes. To record vibrations, a piezoelectric sensor is used, to which a pendulum spring is suspended. During the movement of the pendulum, the elastic force is proportional to the displacement X,
Since the EMF arising in the piezoelectric sensor is in turn proportional to the pressure force, the signal received from the sensor will be proportional to the displacement of the pendulum body from the equilibrium position.
Oscillations are excited using a magnetic field. The harmonic signal created by the PC is amplified and fed to an excitation coil located under the pendulum body. As a result of this coil, a magnetic field that is variable in time and non-uniform in space is formed. This field acts on a permanent magnet mounted in the body of the pendulum and creates an external periodic force. When a body moves, the driving force can be represented as a superposition of harmonic functions, and the oscillations of the pendulum will be a superposition of oscillations with frequencies mw. However, only the force component at the frequency will have a noticeable effect on the movement of the pendulum w, since it is closest to the resonant frequency. Therefore, the amplitudes of the components of the pendulum oscillations at frequencies mw will be small. That is, in the case of an arbitrary periodic influence, the oscillations with a high degree of accuracy can be considered harmonic at the frequency w.
The information processing system consists of an analog-to-digital converter and a personal computer. The analog signal from the piezoelectric sensor is represented in digital form using an analog-to-digital converter and fed to a personal computer.
Controlling the experimental setup using a computer
After turning on the computer and loading the program, the main menu appears on the monitor screen, the general appearance of which is shown in Fig. 5. Using the cursor keys , , , , you can select one of the menu items. After pressing the button ENTER the computer begins to execute the selected operating mode. The simplest hints on the selected operating mode are contained in the highlighted line at the bottom of the screen.
Let's consider the possible operating modes of the program:
Statics- this menu item is used to process the results of the first exercise (see Fig. 5) After pressing the button ENTER the computer requests the mass of the pendulum bob. After the next button press ENTER a new picture with a blinking cursor appears on the screen. Sequentially write down on the screen the mass of the load in grams and, after pressing the space bar, the amount of tension of the spring. Pressing ENTER go to a new line and again write down the mass of the load and the amount of tension of the spring. Data editing within the last line is allowed. To do this, press the key Backspace remove the incorrect mass or spring stretch value and write the new value. To change data in other lines, you must successively press Esc And ENTER, and then repeat the result set.
After entering the data, press the function key F2. The values of the spring stiffness coefficient and the frequency of free oscillations of the pendulum, calculated using the least squares method, appear on the screen. After clicking on ENTER A graph of the elastic force versus the amount of spring extension appears on the monitor screen. Return to the main menu occurs after pressing any key.
Experiment- this item has several sub-items (Fig. 6). Let's look at the features of each of them.
Frequency- in this mode, using the cursor keys, the frequency of the driving force is set. In the event that an experiment is carried out with free oscillations, then it is necessary to set the frequency value equal to 0
.
Start- in this mode after pressing the button ENTER the program begins to remove the experimental dependence of the pendulum's deviation on time. In the case when the frequency of the driving force is zero, a picture of damped oscillations appears on the screen. The values of the oscillation frequency and damping constant are recorded in a separate window. If the frequency of the exciting force is not zero, then along with the graphs of the dependences of the deviation of the pendulum and the driving force on time, the values of the frequency of the driving force and its amplitude, as well as the measured frequency and amplitude of the pendulum oscillations, are recorded on the screen in separate windows. Pressing a key Esc you can exit to the main menu.
Save- if the result of the experiment is satisfactory, then it can be saved by pressing the corresponding menu key.
New Series- this menu item is used if there is a need to abandon the data of the current experiment. After pressing the key ENTER in this mode, the results of all previous experiments are erased from the machine’s memory, and a new series of measurements can be started.
After the experiment, they switch to the mode Measurements. This menu item has several sub-items (Fig. 7)
Frequency response graph- this menu item is used after the end of the experiment to study forced oscillations. The amplitude-frequency characteristic of forced oscillations is plotted on the monitor screen.
FFC schedule- In this mode, after the end of the experiment to study forced oscillations, a phase-frequency characteristic is plotted on the monitor screen.
Table- this menu item allows you to display on the monitor screen the values of the amplitude and phase of oscillations depending on the frequency of the driving force. These data are copied into a notebook for the report on this work.
Computer menu item Exit- end of the program (see, for example, Fig. 7)
Exercise 1. Determination of the spring stiffness coefficient using the static method.
Measurements are carried out by determining the elongation of a spring under the action of loads with known masses. It is recommended to spend at least 7-10 measurements of spring elongation by gradually suspending weights and thereby changing the load from 20 before 150 d. Using the program operation menu item Statistics the results of these measurements are stored in the computer memory and the spring stiffness coefficient is determined using the least squares method. During the exercise, it is necessary to calculate the value of the natural frequency of oscillation of the pendulum
The operation of most mechanisms is based on the simplest laws of physics and mathematics. The concept of a spring pendulum has become quite widespread. Such a mechanism has become very widespread, since the spring provides the required functionality and can be an element of automatic devices. Let's take a closer look at such a device, its operating principle and many other points in more detail.
Definitions of a spring pendulum
As previously noted, the spring pendulum has become very widespread. Among the features are the following:
- The device is represented by a combination of a load and a spring, the mass of which may not be taken into account. A variety of objects can act as cargo. At the same time, it may be influenced by an external force. A common example is the creation of a safety valve that is installed in a pipeline system. The load is attached to the spring in a variety of ways. In this case, exclusively the classic screw version is used, which is the most widely used. The basic properties largely depend on the type of material used in manufacturing, the diameter of the coil, correct alignment and many other points. The outer turns are often made in such a way that they can withstand a large load during operation.
- Before deformation begins, there is no total mechanical energy. In this case, the body is not affected by elastic force. Each spring has an initial position, which it maintains over a long period. However, due to a certain rigidity, the body is fixed in the initial position. It matters how the force is applied. An example is that it should be directed along the axis of the spring, since otherwise there is a possibility of deformation and many other problems. Each spring has its own specific compression and extension limits. In this case, maximum compression is represented by the absence of a gap between individual turns; during tension, there is a moment when irreversible deformation of the product occurs. If the wire is elongated too much, a change in the basic properties occurs, after which the product does not return to its original position.
- In the case under consideration, vibrations occur due to the action of elastic force. It is characterized by quite a large number of features that must be taken into account. The effect of elasticity is achieved due to a certain arrangement of turns and the type of material used during manufacture. In this case, the elastic force can act in both directions. Most often, compression occurs, but stretching can also be carried out - it all depends on the characteristics of the particular case.
- The speed of movement of a body can vary over a fairly wide range, it all depends on the impact. For example, a spring pendulum can move a suspended load in a horizontal and vertical plane. The effect of the directed force largely depends on the vertical or horizontal installation.
In general, we can say that the definition of a spring pendulum is quite general. In this case, the speed of movement of the object depends on various parameters, for example, the magnitude of the applied force and other moments. Before the actual calculations, a diagram is created:
- The support to which the spring is attached is indicated. Often a line with back hatching is drawn to show it.
- The spring is shown schematically. It is often represented by a wavy line. In a schematic display, the length and diametrical indicator do not matter.
- The body is also depicted. It does not have to match the dimensions, but the location of direct attachment is important.
A diagram is required to schematically show all the forces that influence the device. Only in this case can we take into account everything that affects the speed of movement, inertia and many other aspects.
Spring pendulums are used not only in calculations or solving various problems, but also in practice. However, not all properties of such a mechanism are applicable.
An example is the case when oscillatory movements are not required:
- Creation of locking elements.
- Spring mechanisms associated with the transportation of various materials and objects.
Calculations of the spring pendulum allow you to select the most suitable body weight, as well as the type of spring. It is characterized by the following features:
- Diameter of turns. It can be very different. The diameter largely determines how much material is required for production. The diameter of the coils also determines how much force must be applied to achieve full compression or partial extension. However, increasing the size can create significant difficulties with the installation of the product.
- The diameter of the wire. Another important parameter is the diametrical size of the wire. It can vary over a wide range, depending on the strength and degree of elasticity.
- Length of the product. This indicator determines how much force is required for complete compression, as well as what elasticity the product can have.
- The type of material used also determines the basic properties. Most often, the spring is made using a special alloy that has the appropriate properties.
In mathematical calculations, many points are not taken into account. The elastic force and many other indicators are determined by calculation.
Types of spring pendulum
There are several different types of spring pendulum. It is worth considering that classification can be carried out according to the type of spring installed. Among the features we note:
- Vertical vibrations have become quite widespread, since in this case there is no frictional force or other influence on the load. When the load is positioned vertically, the degree of influence of gravity increases significantly. This execution option is common when carrying out a wide variety of calculations. Due to the force of gravity, there is a possibility that the body at the starting point will perform a large number of inertial movements. This is also facilitated by the elasticity and inertia of the body at the end of the stroke.
- A horizontal spring pendulum is also used. In this case, the load is on the supporting surface and friction also occurs at the time of movement. When positioned horizontally, gravity works somewhat differently. The horizontal position of the body has become widespread in various tasks.
The movement of a spring pendulum can be calculated using a sufficiently large number of different formulas, which must take into account the influence of all forces. In most cases, a classic spring is installed. Among the features we note the following:
- The classic coiled compression spring has become very widespread today. In this case, there is a space between the turns, which is called a pitch. The compression spring can stretch, but often it is not installed for this. A distinctive feature is that the last turns are made in the form of a plane, which ensures uniform distribution of force.
- A stretch version can be installed. It is designed for installation in cases where the applied force causes an increase in length. For fastening, hooks are placed.
The result is an oscillation that can last for a long period. The above formula allows you to carry out a calculation taking into account all the points.
Formulas for the period and frequency of oscillation of a spring pendulum
When designing and calculating the main indicators, quite a lot of attention is also paid to the frequency and period of oscillation. Cosine is a periodic function that uses a value that does not change after a certain period of time. This indicator is called the period of oscillation of a spring pendulum. The letter T is used to denote this indicator; the concept characterizing the value inverse to the oscillation period (v) is also often used. In most cases, the formula T=1/v is used in calculations.
The period of oscillation is calculated using a somewhat complicated formula. It is as follows: T=2п√m/k. To determine the oscillation frequency, the formula is used: v=1/2п√k/m.
The considered cyclic frequency of oscillation of a spring pendulum depends on the following points:
- The mass of a load that is attached to a spring. This indicator is considered the most important, as it affects a variety of parameters. The force of inertia, speed and many other indicators depend on the mass. In addition, the mass of the cargo is a quantity whose measurement does not pose any problems due to the presence of special measuring equipment.
- Elasticity coefficient. For each spring this indicator is significantly different. The elasticity coefficient is indicated to determine the main parameters of the spring. This parameter depends on the number of turns, the length of the product, the distance between the turns, their diameter and much more. It is determined in a variety of ways, often using special equipment.
Do not forget that when the spring is strongly stretched, Hooke's law ceases to apply. In this case, the period of spring oscillation begins to depend on the amplitude.
The universal time unit, in most cases seconds, is used to measure the period. In most cases, the amplitude of oscillations is calculated when solving a variety of problems. To simplify the process, a simplified diagram is constructed that displays the main forces.
Formulas for the amplitude and initial phase of a spring pendulum
Having decided on the features of the processes involved and knowing the equation of oscillation of the spring pendulum, as well as the initial values, you can calculate the amplitude and initial phase of the spring pendulum. The value of f is used to determine the initial phase, and the amplitude is indicated by the symbol A.
To determine the amplitude, the formula can be used: A = √x 2 +v 2 /w 2. The initial phase is calculated using the formula: tgf=-v/xw.
Using these formulas, you can determine the main parameters that are used in the calculations.
Vibration energy of a spring pendulum
When considering the oscillation of a load on a spring, one must take into account the fact that the movement of the pendulum can be described by two points, that is, it is rectilinear in nature. This moment determines the fulfillment of the conditions relating to the force in question. We can say that the total energy is potential.
It is possible to calculate the oscillation energy of a spring pendulum by taking into account all the features. The main points are the following:
- Oscillations can take place in the horizontal and vertical plane.
- Zero potential energy is chosen as the equilibrium position. It is in this place that the origin of coordinates is established. As a rule, in this position the spring retains its shape provided there is no deforming force.
- In the case under consideration, the calculated energy of the spring pendulum does not take into account the friction force. When the load is vertical, the friction force is insignificant; when the load is horizontal, the body is on the surface and friction may occur during movement.
- To calculate the vibration energy, the following formula is used: E=-dF/dx.
The above information indicates that the law of conservation of energy is as follows: mx 2 /2+mw 2 x 2 /2=const. The formula used says the following:
It is possible to determine the oscillation energy of a spring pendulum when solving a variety of problems.
Free oscillations of a spring pendulum
When considering what causes the free vibrations of a spring pendulum, attention should be paid to the action of internal forces. They begin to form almost immediately after movement has been transferred to the body. Features of harmonic oscillations include the following points:
- Other types of forces of an influencing nature may also arise, which satisfy all the norms of the law, called quasi-elastic.
- The main reasons for the action of the law may be internal forces that are formed immediately at the moment of a change in the position of the body in space. In this case, the load has a certain mass, the force is created by fixing one end to a stationary object with sufficient strength, the second to the load itself. In the absence of friction, the body can perform oscillatory movements. In this case, the fixed load is called linear.
We should not forget that there are simply a huge number of different types of systems in which oscillatory motion occurs. Elastic deformation also occurs in them, which becomes the reason for their use for performing any work.
Definition
Spring pendulum called a system that consists of an elastic spring to which a load is attached.
Let us assume that the mass of the load is $m$ and the elasticity coefficient of the spring is $k$. The mass of the spring in such a pendulum is usually not taken into account. If we consider the vertical movements of the load (Fig. 1), then it moves under the influence of gravity and elastic force if the system is taken out of equilibrium and left to its own devices.
Equations of oscillations of a spring pendulum
A spring pendulum that oscillates freely is an example of a harmonic oscillator. Let us assume that the pendulum oscillates along the X axis. If the oscillations are small, Hooke’s law is satisfied, then the equation of motion of the load has the form:
\[\ddot(x)+(\omega )^2_0x=0\left(1\right),\]
where $(нu)^2_0=\frac(k)(m)$ is the cyclic frequency of oscillations of the spring pendulum. The solution to equation (1) is the function:
where $(\omega )_0=\sqrt(\frac(k)(m))>0$ is the cyclic frequency of oscillations of the pendulum, $A$ is the amplitude of oscillations; $((\omega )_0t+\varphi)$ - oscillation phase; $\varphi $ and $(\varphi )_1$ are the initial phases of oscillations.
In exponential form, the oscillations of a spring pendulum can be written as:
Formulas for the period and frequency of oscillation of a spring pendulum
If Hooke’s law is satisfied in elastic vibrations, then the period of oscillation of a spring pendulum is calculated using the formula:
Since the oscillation frequency ($\nu $) is the reciprocal of the period, then:
\[\nu =\frac(1)(T)=\frac(1)(2\pi )\sqrt(\frac(k)(m))\left(5\right).\]
Formulas for the amplitude and initial phase of a spring pendulum
Knowing the equation of oscillations of a spring pendulum (1 or 2) and the initial conditions, one can completely describe the harmonic oscillations of a spring pendulum. The initial conditions are determined by the amplitude ($A$) and the initial phase of oscillations ($\varphi $).
The amplitude can be found as:
the initial phase in this case:
where $v_0$ is the speed of the load at $t=0\ c$, when the coordinate of the load is $x_0$.
Vibration energy of a spring pendulum
In the one-dimensional motion of a spring pendulum, there is only one path between two points of its motion, therefore, the condition of force potentiality is satisfied (any force can be considered potential if it depends only on the coordinates). Since the forces acting on a spring pendulum are potential, we can talk about potential energy.
Let the spring pendulum oscillate in the horizontal plane (Fig. 2). Let us take the position of its equilibrium as the zero potential energy of the pendulum, where we place the origin of coordinates. We do not take into account friction forces. Using the formula relating potential force and potential energy for the one-dimensional case:
taking into account that for a spring pendulum $F=-kx$,
then the potential energy ($E_p$) of the spring pendulum is equal to:
We write the law of conservation of energy for a spring pendulum as:
\[\frac(m(\dot(x))^2)(2)+\frac(m((\omega )_0)^2x^2)(2)=const\ \left(10\right), \]
where $\dot(x)=v$ is the speed of the load; $E_k=\frac(m(\dot(x))^2)(2)$ is the kinetic energy of the pendulum.
From formula (10) the following conclusions can be drawn:
- The maximum kinetic energy of a pendulum is equal to its maximum potential energy.
- The time-average kinetic energy of the oscillator is equal to its time-average potential energy.
Examples of problems with solutions
Example 1
Exercise. A small ball with a mass of $m=0.36$ kg is attached to a horizontal spring, the elasticity coefficient of which is equal to $k=1600\ \frac(N)(m)$. What was the initial displacement of the ball from the equilibrium position ($x_0$), if it oscillates through it with a speed of $v=1\ \frac(m)(s)$?
Solution. Let's make a drawing.
According to the law of conservation of mechanical energy (since we assume that there are no friction forces), we write:
where $E_(pmax)$ is the potential energy of the ball at its maximum displacement from the equilibrium position; $E_(kmax\ )$ is the kinetic energy of the ball at the moment of passing the equilibrium position.
Potential energy is equal to:
In accordance with (1.1), we equate the right-hand sides of (1.2) and (1.3), we have:
\[\frac(mv^2)(2)=\frac(k(x_0)^2)(2)\left(1.4\right).\]
From (1.4) we express the required value:
Let's calculate the initial (maximum) displacement of the load from the equilibrium position:
Answer.$x_0=1.5$ mm
Example 2
Exercise. A spring pendulum oscillates according to the law: $x=A(\cos \left(\omega t\right),\ \ )\ $where $A$ and $\omega $ are constants. When the restoring force first reaches $F_0,$ the potential energy of the load is $E_(p0)$. At what point in time will this happen?
Solution. The restoring force for a spring pendulum is the elastic force equal to:
We find the potential energy of vibration of the load as:
At the moment of time that should be found $F=F_0$; $E_p=E_(p0)$, means:
\[\frac(E_(p0))(F_0)=-\frac(A)(2)(\cos \left(\omega t\right)\ )\to t=\frac(1)(\omega ) \arc(\cos \left(-\frac(2E_(p0))(AF_0)\right)\ ).\]
Answer.$t=\frac(1)(\omega )\ arc(\cos \left(-\frac(2E_(p0))(AF_0)\right)\ )$
Let's consider the simplest system in which mechanical vibrations can be realized. Let us assume that a load of mass $m$ is suspended on an elastic spring whose stiffness is $k,$. The load moves under the influence of gravity and elasticity if the system is taken out of equilibrium and left to its own devices. We consider the mass of the spring to be small in comparison with the mass of the load.
The equation for the movement of the load during such oscillations has the form:
\[\ddot(x)+(\omega )^2_0x=0\left(1\right),\]
where $(\omega )^2_0=\frac(k)(m)$ is the cyclic frequency of oscillations of the spring pendulum. The solution to equation (1) is the function:
where $(\omega )_0=\sqrt(\frac(k)(m))>0$ is the cyclic frequency of oscillations of the pendulum, $A$ and $B$ are the amplitude of oscillations; $((\omega )_0t+\varphi)$ - oscillation phase; $\varphi $ and $(\varphi )_1$ are the initial phases of oscillations.
Frequency and period of oscillation of a spring pendulum
Cosine (sine) is a periodic function, the displacement $x$ will take the same values at certain equal intervals of time, which are called the oscillation period. The period is designated by the letter T.
Another quantity characterizing oscillations is the reciprocal of the oscillation period, it is called frequency ($\nu $):
The period is related to the cyclic frequency of oscillations as:
Knowing that for a spring pendulum $(\omega )_0=\sqrt(\frac(k)(m))$, we define its oscillation period as:
From expression (5) we see that the period of oscillation of a spring pendulum depends on the mass of the load located on the spring and the elasticity coefficient of the spring, but does not depend on the amplitude of oscillations (A). This property of oscillations is called isochrony. Isochrony holds as long as Hooke's law holds. At large stretches of the spring, Hooke's law is violated, and a dependence of the oscillations on the amplitude appears. Note that formula (5) for calculating the period of oscillation of a spring pendulum is valid for small oscillations.
The unit of measure for a period is time, in the International System of Units it is seconds:
\[\left=с.\]
Examples of problems for the period of oscillation of a spring pendulum
Example 1
Exercise. A small load was attached to an elastic spring, and the spring stretched by $\Delta x$=0.09 m. What will be the period of oscillation of this spring pendulum if it is thrown out of balance?
Solution. Let's make a drawing.
Let us consider the equilibrium state of a spring pendulum. The weight is attached, after which the spring is stretched by the amount $\Delta x$, the pendulum is in a state of equilibrium. There are two forces acting on the load: gravity and elastic force. Let's write down Newton's second law for the equilibrium state of the load:
Let us write the projection of equation (1.1) onto the Y axis:
Since the load according to the conditions of the problem is small, the spring did not stretch much, therefore Hooke’s law is satisfied, we find the magnitude of the elastic force as:
Using expressions (1.2) and (1.3) we find the ratio $\frac(m)(k)$:
The period of oscillation of a spring pendulum for small oscillations can be found using the expression:
Replacing the ratio of the load mass to the spring stiffness with the right side of expression (1.4), we obtain:
Let's calculate the period of oscillation of our pendulum if $g=9.8\ \frac(m)(s^2)$:
Answer.$T$=0.6 s
Example 2
Exercise. Two springs with stiffnesses $k_1$ and $k_2$ are connected in series (Fig. 2), a load of mass $m$ is attached to the end of the second spring. What is the period of oscillation of this spring pendulum, if the masses of the springs can be neglected, the elastic force acting on the load is obeys Hooke's law.
Solution. The period of oscillation of a spring pendulum is equal to:
If two springs are connected in series, then their resulting stiffness ($k$) is found as:
\[\frac(1)(k)=\frac(1)(k_1)+\frac(1)(k_2)\to k=\frac(k_1k_2)(k_1(+k)_2)\left(2.2\ right).\]
Instead of $k$ in the formula for calculating the period of a spring pendulum, we substitute the right side of expression (2.2), we have:
Answer.$T=2\pi \sqrt(\frac(m(k_1(+k)_2))(k_1k_2))$
), one end of which is rigidly fixed, and on the other there is a load of mass m.
When an elastic force acts on a massive body, returning it to an equilibrium position, it oscillates around this position. Such a body is called a spring pendulum. Oscillations occur under the influence of an external force. Oscillations that continue after the external force has ceased to act are called free. Oscillations caused by the action of an external force are called forced. In this case, the force itself is called forcing.
In the simplest case, a spring pendulum is a rigid body moving along a horizontal plane, attached by a spring to a wall.
Newton's second law for such a system, provided there are no external forces and friction forces, has the form:
If the system is influenced by external forces, then the vibration equation will be rewritten as follows:
, Where f(x)- this is the resultant of external forces related to a unit mass of the load.In the case of attenuation proportional to the oscillation speed with the coefficient c:
see also
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