EG260 Harmonic motion

EG260 Harmonic motion - A.K Slone EG-260 Dynamics(1 EG-260...

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A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2006 1 of 16 EG-260 DYNAMICS I – Harmonic motion Introduction . ..................................................................... 2 1. Revision - Motion in a Circle ..................................... 2 2. Circular Motion and the Sine Function .................... 4 3. Simple Harmonic Motion (SHM) .............................. 5 3.1. Solution of the SHM equation . ............................. 6 3.2. Motion of a single spring-mass system . ............... 7 3.3. Properties of SHM . .............................................. 11 3.4. SHM & circular motion . ..................................... 14 4. Solutions for Harmonic motion . ......................... 15

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A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2006 2 of 16 Introduction Oscillatory motion may repeat at regular time intervals as in the balance wheel of a clock or be highly irregular as in an earthquake. When the motion repeats itself in equal time intervals it is called periodic motion The simplest form of periodic motion is harmonic motion and this is most simply illustrated by the projection of the motion of a point that is moving in a circle with constant speed onto a straight line. 1. Revision - Motion in a Circle Consider a particle P moving in an anti-clockwise direction around the circumference of a circle. Initially it is at position Q, and after a time it is at R. The angle between the position vectors Q and R is θ . The angular velocity of P is given by ω, which is defined as the rate of increase in the angle θ . If θ is increasing at a constant rate then angular velocity, ω, is uniform. Angles are measured in radians, where 2 π radians = 360 o . The units of angular velocity are one radian per second, 1 rad/s. O θ R Q ω Figure 1 Circular Motion
A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2006 3 of 16 If the angle θ has been travelled in time t , at a rate of ω rad/s then θ = ω t . Let the time to travel one complete revolution, i.e. 2 π rads, be T, then using the relationship: distance = velocity x time ω π 2 2 = = T T T is called the period of the oscillation , the time for one complete cycle and its reciprocal is the frequency , f , which is measured in Hertz, Hz, which is cycles /second.

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A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2006 4 of 16 2. Circular Motion and the Sine Function Consider the particle P moving around the circle as above, the projection of the motion of P onto a straight line is shown in Figure 2 Sine Function x θ π/2 π 3π/2 A A -A -A θ =π/2 π 3π/2 x P P θ X(t) Figure 2 Relationship between the Sine Function and Circular Motion From our knowledge of the graphs of trigonometrical relationships, we recognise this above graph as that of the sine function, hence: () ( ) ( ) t A A t x ω θ sin sin = =
A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2006 5 of 16 3. Simple Harmonic Motion (SHM)

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EG260 Harmonic motion - A.K Slone EG-260 Dynamics(1 EG-260...

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