harmonic and periodic motions

7/29/2019 Harmonic and Periodic Motions

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Harmonic and periodic motions

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ne-dimensional harmonic motion of a particle can be described by a sine (or cosine) function of time form

u(t) = u0sin t+

here

u(t) is the displacement measured in [m],

0 is amplitude of motion [m],

is angular frequency [rad/s],

is time [s],

is an initial phase, phase constant or epoch [rad].

he angular frequency is related to the cyclic frequency fby

= 2 f

herefhas the dimension [1/s].

hen the harmonic motion is related to the rotary motion we can consider fas revolutions per second

d define

n = 60f [1/min]

R.P.M. or revolutions per minute.

we add Tto t, then the value ofu(t) must remain unaltered. This requires that the argument of the si

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s been increased by 2 . From it follows that T= 2 . The quantity Tis called period and is

T= [s].

he velocity and acceleration of a harmonic motion are given by derivatives of displacement with res

time

(t) = u0 cos t+ and (t) = - u0 sin t+

here u0 and u0 are amplitudes of velocity and acceleration respectively.

he motion is harmonic if it is described by a sine or cosine function in time.

he motion is periodic if it has the same pattern after a period Tor after its integer multiples. For the

nction describing it we can write u(t) = u(t + kT), for k = 0,1,2,3,...

ombination (superposition) of two harmonic motions being expressed by

u1 = u10sin t+ and u2 = u20sin t+

ves a simple harmonic (sinusoidal) wave if and only if their angular frequencies are the same, i.e.

. In such a case

u = u1 + u2 = u0sin t+

here



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u0 = u102 + u20

2 + 2 u10u20cos( - ) , = = ,

= arctan .

he sum of two harmonic motions with different angular frequencies is not harmonic, i.e. u = u1 + u2

sin t + . It could, however, be periodic if the ratio of frequencies is a rational number.

special case of interest is when the frequencies are slightly different. Let us consider the motion in trm

u = u1 + u2 = u10sin( t) + u10sin ( + )t . (5

mple trigonometric manipulation reveals that

u = Usin t= 2 u10cos t sin + t . (5

he resulting motion could be considered as a sine wave with angular frequency = + /2 and

ith varying amplitude U= 2 u10cos t/2 . Every time the amplitude reaches a maximum, there

id to be a beat. Beats are well pronounced if the difference of frequencies is small, i.e. if

ead carefully the following Matlab program VHMharpoh1a.m. Observe the results shown in Fig. 5.

g. 5.2 and Fig. 5.3. Observe what happens as the input parameters are varied.

ear u10=10; u20=20; om1=0.3; om2=1.05; fi1=0.1; fi2=0.4; t=0:pi/64:12*pi; u1=u10*sin(om1*t+fi1);

=u20*sin(om2*t+fi2); uv=u1+u2;



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gure(1) subplot(3,1,1); plot(t,u1,'k'); axis('off') subplot(3,1,2); plot(t,u2,'k'); axis('off') subplot(3,1,3); plot(t,

,'k'); title('sum of two periodic motions','fontsize',13); xlabel('does not generally give a periodic

otion','fontsize',13); print VHMharpohf1.eps -deps

Figure 5.1:Harmonic and periodic motions

m1=2; om2=4; u1=u10*sin(om1*t+fi1); u2=u20*sin(om2*t+fi2); uv=u1+u2;

gure(2) subplot(3,1,1); plot(t,u1,'k'); axis('off'); subplot(3,1,2); plot(t,u2,'k'); axis('off'); subplot(3,1,3); plot(t,

,'k') title('if the frequency ratio is an integer','fontsize',13); xlabel('then the resulting motion is

riodic','fontsize',13) print VHMharpohf2.eps -deps



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Figure 5.2:Periodic motions

wo wave motions with 'close' frequencies lead to the well-known phenomenon of beat frequencies.

m1=4.1; om2=4.8; u1=u10*sin(om1*t+fi1); u2=u10*sin(om2*t+fi2); uv=u1+u2;

gure(3) subplot(3,1,1); plot(t,u1,'k'); axis('off') subplot(3,1,2); plot(t,u2,'k'); axis('off') subplot(3,1,3); plot(t,

,'k') title('superposition of close frequencies - beats','fontsize',13) print VHMharpohf3.eps -deps



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Figure 5.3:Beats

a particle performs harmonic motions in two mutually perpendicular directions, say

u = u0sin t w = w0sin t+ (5

en as a resulting motion we get interesting patterns, known as Lissajouse figures. Their appearance

pends on the ratio of frequencies and the phase difference. Simple trigonometrical manipulation sho

at for the same frequencies we obtain a tilted ellipse, which in some cases could degenerate into a ci

a straight line.

ee program VHMlissa1.m showing how (see Fig. 5.4) this effect could be employed for the phase



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entification.

ear t=0:0.001:8; om1=2; om2=2; u0=1; w0=1; fi=pi/2; i=0; figure(1) for fi=0:pi/34:pi u=u0*sin(om1*t);

=w0*sin(om2*t+fi); i=i+1; subplot(6,6,i); lab=['fi = ' num2str(fi)]; plot(u,w,'k'); axis('square'); axis('off') end

11,-2,'Lissajouse with a phase shift','fontsize',13) print VHMlissa1.eps -deps

Figure 5.4:Lissajouse pattern - the influence of a shift

the ratio of frequencies is not a rational number we obtain curves that are generally not closed. An

ample of such a curve is generated by the program VHMlissa2.m - see Fig. 5.5.



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ear t=0:0.001:1; om1=102; om2=203; u0=1; w0=1; fi=0; figure(1) u=u0*sin(om1*t); w=w0*sin(om2*t+fi);

w,'k'); axis('off') print VHMlissa2.eps -deps

Figure 5.5:Lissajouse pattern - just for fun

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ext:Phase and group velocitiesUp:VibrationsPrevious:Introduction to vibration Contents

arcel 2001-08-14
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harmonic and periodic motions

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