Faculty of engineering and ITElectrical and electronics department

General Physics 1Seminar:

Oscillations

Students: Armin Halilović (EEE) Mentor: prof. dr. Rajfa Musemić

Belmin Memišević (EEE)

Armin Dervić (IT)

Hikmet Muratović (EEE)

Sead Skopljak (EEE)

Ahmed Bajek (EEE)

09.01.2014.

A very special kind of motion occurs when the force acting on a body is proportional to the displacement of the body from some equilibrium position. If this force is always directed toward the equilibrium position, repetitive backand- forth motion occurs about this position. Such motion is called periodic motion, harmonic motion, oscillation, or vibration (the four terms are completely equivalent). You are most likely familiar with several examples of periodic motion, such as the oscillations of a block attached to a spring, the swinging of a child on a playground swing, the motion of a pendulum, and the vibrations of a stringed musical instrument. In addition to these everyday examples, numerous other systems exhibit periodic motion. For example, the molecules in a solid oscillate about their equilibrium positions; electromagnetic waves, such as light waves, radar, and radio waves, are characterized by oscillating electric and magnetic field vectors; and in alternating-current electrical circuits, voltage, current, and electrical charge vary periodically with time.

Simple harmonic motionIn mechanics and physics, simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement. It can serve as a mathematical model of a variety of motions, such as the oscillation of a spring. In addition, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum as well as molecular vibration. Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's Law. The motion is sinusoidal in time and demonstrates a single resonant frequency. In order for simple harmonic motion to take place, the net force of the object at the end of the pendulum must be proportional to the displacement.

Simple harmonic motion shown both in real space and phase space. The orbit is periodic. (Here the velocity and position axes have been reversed from the standard convention in order to align the two diagrams)

In the diagram a simple harmonic oscillator, comprising a mass attached to one end of a spring, is shown. The other end of the spring is connected to a rigid support such as a wall. If the system is left at rest at the equilibrium position then there is no net force acting on the mass. However, if the mass is displaced from the equilibrium position, a restoring elastic force which obeys Hooke's law is exerted by the spring.

Mathematically, the restoring force F is given by

where F is the restoring elastic force exerted by the spring (in SI units: N), k is the spring constant (N·m−1), and x is the displacement from the equilibrium position (in m).

For any simple harmonic oscillator:

When the system is displaced from its equilibrium position, a restoring force which resembles Hooke's law tends to restore the system to equilibrium.

Once the mass is displaced from its equilibrium position, it experiences a net restoring force. As a result, it accelerates and starts going back to the equilibrium position. When the mass moves closer to the equilibrium position, the restoring force decreases. At the equilibrium position, the net restoring force vanishes. However, at x = 0, the mass has momentum because of the impulse that the restoring force has imparted. Therefore, the mass continues past the equilibrium position, compressing the spring. A net restoring force then tends to slow it down, until its velocity vanishes, whereby it will attempt to reach equilibrium position again.

As long as the system has no energy loss, the mass will continue to oscillate. Thus, simple harmonic motion is a type of periodic motion.

Dynamics of simple harmonic motionFor one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, could be obtained by means of Newton's second law and Hooke's law.

where m is the inertial mass of the oscillating body, x is its displacement from the equilibrium (or mean) position, and k is the spring constant.

Therefore,

Solving the differential equation above, a solution which is a sinusoidal function is obtained.

where

In the solution, c1 and c2 are two constants determined by the initial conditions, and the origin is set to be the equilibrium position. Each of these constants carries a physical meaning of the motion: A is the amplitude (maximum displacement from the equilibrium position), ω = 2πf is the angular frequency, and φ is the phase.

Using the techniques of differential calculus, the velocity and acceleration as a function of time can be found:

Acceleration can also be expressed as a function of displacement:

Then since ω = 2πf,

and since T = 1/f where T is the time period,

These equations demonstrate that the simple harmonic motion is isochronous (the period and frequency are independent of the amplitude and the initial phase of the motion).

Time dependence of elongation x(t), speed v(t) and acceleration a(t) for simple harmonic motion, during one period of time T

Energy of simple harmonic motionThe kinetic energy K of the system at time t is:

and the potential energy is:

The total mechanical energy of the system therefore has the constant value

An undamped spring–mass system undergoes simple harmonic motion..

Total energy of a body which makes free harmonic motion is the sum of it’s kinetic energy and it’s elastic potential energy

Simple harmonic oscillatorA simple harmonic oscillator is an oscillator that is neither driven nor damped. It consists of a mass m, which experiences a single force, F, which pulls the mass in the direction of the point x=0 and depends only on the mass's position x and a constant k. Balance of forces (Newton's second law) for the system is

Solving this differential equation, we find that the motion is described by the function

where

The motion is periodic, repeating itself in a sinusoidal fashion with constant amplitude, A. In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its period T, the time for a single oscillation or its frequency f = 1⁄T, the number of cycles per unit time. The position at a given time t also depends on the phase, φ, which determines the starting point on the sine wave. The period and frequency are determined by the size of the mass m and the force constant k, while the amplitude and phase are determined by the starting position and velocity.

Momentum of harmonic oscillator:

By making the sum we get:

Graphically momentum of oscillator as the function of displacement x, gives the elipse.(Area=Total Energy) :

The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position but with shifted phases. The velocity is maximum for zero displacement, while the acceleration is in the opposite direction as the displacement.

The potential energy stored in a simple harmonic oscillator at position x is

x2

A2 + p2

m2 A2ω2 =1

p=mv=−mA ωsin(ωt +ϕ )

E=f⋅S= fπA2 mω= f∮ pdx

Pendulum (mathematics)

Mathematical pendulum makes the bob with mass m hanging on light rope (thread) of length l.

The pendulum makes simple harmonic motion as it swings back and forth provided that the arc through which the pendulum bob moves is a fairly small one (small amplitude) not more than about 5◦ on either side of the vertical

.

The figure shows a pendulum of length l whose bob has a mass m , together with a diagram of the forces acting on the bob.

The weight of the bob, mg, which acts vertically downward, may be resolved into two forces Ft and FN, which act perpendicular to and parallel to the supporting string l, that is:

Ft + FN = mg

The period of swing of a simple gravity pendulum depends on its length, the local strength of gravity, and to a small extent on the maximum angle that the pendulum swings away from vertical, θ0, called the amplitude. It is independent of the mass of the bob. If the amplitude is limited to small swings, the period T of a simple pendulum, the time taken for a complete cycle, is:

where L is the length of the pendulum and g is the local acceleration of gravity.

For small swings the period of swing is approximately the same for different size swings: that is, the period is independent of amplitude. This property, called isochronism, is the reason

pendulums are so useful for timekeeping. Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.

For larger amplitudes, the period increases gradually with amplitude so it is longer than given by equation (1). For example, at an amplitude of θ0 = 23° it is 1% larger than given by (1). The period increases asymptotically (to infinity) as θ0 approaches 180°, because the value θ0 = 180° is an unstable equilibrium point for the pendulum. The true period of an ideal simple gravity pendulum can be written in several different forms (see Pendulum (mathematics) ), one example being the infinite series:

The difference between this true period and the period for small swings (1) above is called the circular error. In the case of a longcase clock whose pendulum is about one metre in length and whose amplitude is ±0.1 radians, the θ2 term adds a correction to equation (1) that is equivalent to 54 seconds per day and the θ4 term a correction equivalent to a further 0.03 seconds per day.

For small swings the pendulum approximates a harmonic oscillator, and its motion as a function of time, t, is approximately simple harmonic motion:

For real pendulums, corrections to the period may be needed to take into account the presence of air, the mass of the string, the size and shape of the bob and how it is attached to the string, flexibility and stretching of the string, motion of the support, and local gravitational gradients.

Compound or physical pendulum

Physical pendulum is a solid body which can make harmonic oscillations about fixed horizontal axis due to gravitational forces.If we resolve the weight force vector into two components, then only the component F makes angular momentum which can return the body to the equilibrium position

The force moment is:

m - mass, s - distance from the fixed hanging point and the center of the mass of the body

(On the picture s is signed by L).

For small q we have

and M=-mgs q

As we know the II Newton law for rotation is

α is angular acceleration= second derivation of angle θ;

Differential equation of pendulum is:

Coefficient next θ represents angular frequence so,the equation of motion for pendulum can be written as:

It is identical as the differential equation of harmonic oscillator.

For small oscillations physical pendulum executes harmonic motion, which can be described by equation:

The Period is

(I – moment of inertia)

The length L of the ideal simple pendulum above, used for calculating the period, is the distance from the pivot point to the center of mass of the bob. A pendulum consisting of any swinging rigid body, which is free to rotate about a fixed horizontal axis is called a compound pendulum or physical pendulum. For these pendulums the appropriate equivalent length is the distance from the pivot point to a point in the pendulum called the center of oscillation. This is located under the center of mass, at a distance called the radius of gyration, that depends on the mass distribution along the pendulum. However, for any pendulum in which most of the mass is concentrated in the bob, the center of oscillation is close to the center of mass.

Using the parallel axis theorem, the radius of gyration L of a rigid pendulum can be shown to be

M = I 0 α= I 0d2θdt2

d2 θdt2 + mgs

I 0θ= 0I 0

d2 θdt2 + mgs θ= 0

ω02

d2 θdt 2 + ω0

2 θ= 0

Substituting this into (1) above, the period T of a rigid-body compound pendulum for small angles is given by

where I is the moment of inertia of the pendulum about the pivot point, m is the mass of the pendulum, and R is the distance between the pivot point and the center of mass of the pendulum.

For example, for a pendulum made of a rigid uniform rod of length L pivoted at its end, I = (1/3)mL2. The center of mass is located in the center of the rod, so R = L/2. Substituting these values into the above equation gives T = 2π√2L/3g. This shows that a rigid rod pendulum has the same period as a simple pendulum of 2/3 its length.

Damped oscillations

Damped oscillations – when the motion of an oscillator is reduced by an external force we call that damped osc.

The damping force which resists to the motion is proportional to velocity

Designation: ω0 own frequency non-damped oscillators; δ= b/2m factor of damping

Using II Newton’s law we can write

02 202

2

xdtdx

dtxd

mk

0

ma = Fel + Ftr

Solution:

Damped oscillations – graph

The Amplitude decreases upon exponential way with respect to time; as the damped factor is greater, the amplitude decreases faster. Energy is going on to overcome the friction forces .(β instead δ). The damping decreases the frequency of oscillating

Damped harmonic oscillator

A damped harmonic oscillator, which slows down due to friction

In real oscillators, friction, or damping, slows the motion of the system. Due to frictional force, the velocity decreases proportional to the acting frictional force. Whereas Simple harmonic motion oscillates with only the restoring force acting on the system, Damped Harmonic motion experiences friction. In many vibrating systems the frictional force Ff can be modeled as being proportional to the velocity v of the object: Ff = −cv, where c is called the viscous damping coefficient.

)sin()cos(2)sin()( 222

2

teAteAteAdt

xdta ttt

)cos()sin()( teAteAdtdxtv tt

220

)sin()( tAetx t

Balance of forces (Newton's second law) for damped harmonic oscillators is then

This is rewritten into the form

Where:

is called the 'undamped angular frequency of the oscillator' and

is called the 'damping ratio'.

Step-response of a damped harmonic oscillator; curves are plotted for three values of μ = ω1 = ω0√1−ζ2. Time is in units of the decay time τ = 1/(ζω0).

The value of the damping ratio ζ critically determines the behavior of the system. A damped harmonic oscillator can be:

Overdamped (ζ > 1): The system returns (exponentially decays) to steady state without oscillating. Larger values of the damping ratio ζ return to equilibrium slower.

Critically damped (ζ = 1): The system returns to steady state as quickly as possible without oscillating. This is often desired for the damping of systems such as doors.

Underdamped (ζ < 1): The system oscillates (with a slightly different frequency than the undamped case) with the amplitude gradually decreasing to zero. The angular frequency of the underdamped harmonic oscillator is given by

The Q factor of a damped oscillator is defined as

Q is related to the damping ratio by the equation

Forced oscillations

Each oscillator has own frequency when it oscillates without outher effects (by oneself). Then it makes own or free oscillations.However, an outher periodic force makes force to the oscillator to oscillate with the frequency of the periodic force, besides with own, so it means that oscillator executes complex motion.

Forced oscillations - equations

m d2 sdt 2 =−ks−b ds

dt+F0sin ωt

d2 sdt 2 +2 δ ds

dt+ω0

2 s=A0sin ωt

F=F0 sin ωt , A0=F0

m , δ=b2m , ω0

2=km

s( t )=A (ω )sin (ωt−ϕ )

d2 sdt 2 +2 δ ds

dt+ω0

2 s=A0 sin ωt

A(ω )=A0

√(ω02−ω2 )2+4 δ 2ω2

, tgϕ=2 δωω0

2−ω

ωr=√ω02−2 δ2

Resonant frequency is smaller then own, and difference will be smaller when the damping is smaller. In limit case (without any resistance) r=0.

In an ideal case, at the resonance the amplitude would be indefinite.

Forced oscillations

The greatest amplitude will be reached when the frequency of outher force is equal to the own frequency of free harmonic motion

This is resonant frequency of the system .

The amplitude of forced vibration decreases as the damping is greater.