Mathematics
Vertical Oscillation
Vertical oscillation refers to the up and down movement of an object or system around a fixed point or equilibrium position. It is a measure of the amplitude of the oscillation and is often used in the study of vibrations and waves. In mathematics, vertical oscillation can be described using functions such as sine and cosine.
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4 Key excerpts on "Vertical Oscillation"
- Andrei D. Polyanin, Alexei Chernoutsan(Authors)
- 2010(Publication Date)
- CRC Press(Publisher)
Chapter P4 Oscillations and Waves P4.1. Oscillations ◮ General definitions. Oscillations are repetitive variations in the state of a system such that the state parameters vary in time according to a periodic or almost periodic law. If oscillations occur without external action due to system deviation from a stable equilibrium, the oscillations are said to be free or natural . If the oscillations occur under the action of an external periodic force, then they are said to be forced . The oscillations are characterized by their period T and frequency ν = 1 /T (measured in hertzs : 1 Hz = 1 s – 1 ). The term vibrations is often used in a narrower sense to mean mechanical oscillations, but sometimes is used synonymously with oscillations. P4.1.1. Harmonic Oscillations. Composition of Oscillations ◮ Simple harmonic oscillations. An oscillation of a quantity x is said to be a simple harmonic oscillation (or simple harmonic motion ) if x varies in time t by the law x = A cos( ωt + ϕ 0 ), (P4. 1 . 1 . 1 ) where A is the amplitude , ϕ = ωt + ϕ 0 is the phase , ϕ 0 is the initial phase , and ω = 2 π/T is the angular or circular frequency of the oscillation. The first and second time-derivatives of the quantity x , ˙ x = – Aω sin( ωt + ϕ 0 ) = Aω cos ( ωt + ϕ 0 + π/ 2 ) , ¨ x = – Aω 2 cos( ωt + ϕ 0 ) = Aω 2 cos( ωt + ϕ 0 + π ), (P4. 1 . 1 . 2 ) oscillate harmonically with the same frequency but with amplitudes ωA and ω 2 A and with the phase shifts π/ 2 and π , respectively. Example. If the initial values (at t = 0 ) of the quantity x and its derivative, x ( 0 ) = x 0 and ˙ x ( 0 ) = v 0 , are known, then the amplitude and the initial phase of the oscillation can be determined. The equations x 0 = A cos ϕ 0 and v 0 = – ωA sin ϕ 0 allow one to find A = radicalbig x 2 0 + ( v 0 /ω ) 2 and tan ϕ 0 = – v 0 / ( ωx 0 ).
- Christof M. Aegerter(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
67 Periodic Motions: Oscillations ✲ t ✻ t E v(t) v 0 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ✠ v 0 − a 0 t t 0 t ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ↓ Max. height Figure 3.5 Velocity profile for a constant acceleration. The dashed area underneath the curve corresponds the the distance traveled between time t 0 and time t . t E marks the end of the motion, which for the armadillo in Figure 3.1 is when it reaches ground again. The jumping and falling heights are the same, which implies that the areas of the triangles above and below the axis are equal. determine the integral graphically. As Figure 3.5 shows, the area underneath the curve for the velocity corresponds to the travelled path. 3.2 Periodic Motions: Oscillations An oscillation is a very special form of movement, namely one which repeats itself after a certain time. This time after which the motion repeats itself is called the period of the oscillation and is the most important variable that describes an oscillation. For some types of oscillation, harmonic oscillations, this period, together with the amplitude, i.e., the distance that the object moves maximally, is sufficient for a complete description of the movement. In general, we speak of an oscillation when a physical quantity changes in time around an average, resting value. These general oscillations can also have increasing or decaying (damped) amplitudes in time; see Section 3.8.1. In that case, they are no longer strictly periodic. Oscillations can run by themselves or be forced externally, which we will discuss in depth in Section 4.2. Further examples of oscillating systems are found in the central and inner ear, where the eardrum, the hammer, and the basilar membranes begin to vibrate according to the 68 Motions and Oscillations arriving sound waves, thus allowing the sensors in the ear to send appropriate signals to the brain.
eBook - PDFWave Optics
Basic Concepts and Contemporary Trends
- Subhasish Dutta Gupta, Nirmalya Ghosh, Ayan Banerjee(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
25 Oscillations and vibrations constitute one of the major areas of study in physics. Most systems can oscillate freely. Generally, heavier ones have low oscillation frequency while the lighter ones have large frequencies. The vari-ety of phenomena exhibiting repetitive motions have been discussed nicely by French [1]: “Systems can vibrate freely in a large variety of ways. Broadly speaking, the predominant natural vibrations of 1 2 Wave Optics: Basic Concepts and Contemporary Trends small objects are likely to be rapid, and those of large ob-jects are likely to be slow. A mosquito’s wings, for example, vibrate hundreds of times per second and produce an audible note. The whole earth, after being jolted by an earthquake, may continue to vibrate at the rate of about one oscilla-tion per hour. The human body itself is a treasure-house of vibratory phenomena.” All the above phenomena have one thing in common, i.e., repetitive motion or periodicity. The same pattern of displacement is repeated over and over again. It can be simple or complicated. Irrespective of the nature of oscillations, the pattern is generally represented by plots where the horizontal axis represents the steady progress of time. Such pictures make it easy to recognize one cycle or one period of oscillation, which keeps on repeating. 1.1 Sinusoidal oscillations Sinusoidal oscillations take place in a vast majority of mechanical systems. This is due to the fact that in most cases, the restoring force is proportional to the displacement. Such motion is always possible if the displacement is small enough. In general the restoring force F can have the following dependence on the displacement x : F ( x ) = − ( k 1 x + k 2 x 2 + k 3 x 3 + ... ) . (1.1) For small displacements we can ignore the terms proportional to x 2 , x 3 and other higher-order terms. This leads to an equation of motion m d 2 x dt 2 = − k 1 x, (1.2) which has a solution of the form x = A sin( ωt + φ 0 ) , ω = radicalbigg k 1 m .
eBook - PDF- Md Nazoor Khan, Simanchala Panigrahi(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
1 Oscillations and Waves 1.1 Introduction Objects subjected to restoring forces when displaced from their normal positions and released, perform to and from or vibrating motions. They move back and forth along a path, repeating over and over again, a series of motions. Such motion of constant frequency is called periodic motion or harmonic motion and objects performing such type of motion are called harmonic oscillators. In this book, it is tacitly assumed that there is a linear relationship between force and displacement; frequency remains constant throughout the motion. In real systems however, the linear behavior, implicit in simple harmonic motion, is rarely obeyed. If the frequency of the oscillatory system is not constant, then it is called anharmonic motion – its study is beyond the scope of this book due to its mathematical complexities. In oscillatory systems, it is not necessarily the bodies themselves who execute oscillations; bodies may be at rest. If the physical properties of a system undergo changes in an oscillatory manner, the system will also be called an oscillatory system. The electromagnetic energy transfer between the capacitor and inductor in a tank circuit used in electronic gadgets, variation of pressure in air due to propagation of sound waves, vibration of the diaphragm of a speaker in sound systems, flow of alternating current, variation of electric and magnetic vectors during propagation of electromagnetic waves, etc., are examples of oscillatory systems. 1.1.1 Parameters of an oscillatory system i. Mean position The position of the oscillating body when there is no oscillation is called the mean position or equilibrium position. This is the rest position of the oscillating body. 2 Principles of Engineering Physics 1 ii. Amplitude ( r ) It is the absolute value of the maximum displacement of the oscillating particle from its mean position or equilibrium position.
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