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Contents of sinusoidal excitation of a resonator

- Introduction - contents

- Damping

- Driven oscillations

- Various looks of resonance curves

- Resonant peak properties

- The resonance peak - qualitatively

- More on the impact model - impulses in two directions

- Mathematical response to a very short impulse
- Summing impulses using calculus

- Buildup and decay of the envelope

- Q - a review and more

- Coupling of waves to a resonator

- Coupling strength

- Waveguide excited resonator with circulator

- Resonant scattering of waves - one dimensional cases

- Circuit analysis of the acoustical circuit in Fig. 35

- Electrical model of the acoustical circuit in Fig. 35 of posting 3.15

- Looking at the resonator part of the circuit

- Steady state response of the circuit: transmission line, coupling inductor and resonator

- Transmission line resonators

- Steady state response of a transmission line resonator via a Smith Chart

- Differential equation solution for transient response and SPICE simulation

- Summary of analysis of a transmission line excited resonator using a circulator

- Electrical model of the acoustical circuit in Fig. 35 of posting 3.15
- Circuit analysis of resonant scattering with two output channels
- Separating out the radiated and simply reflected components - mathematical
- Applying methods of 3.18 to the circuit of 3.17
- Derivation of wave properties for transmission lines
- Reflection and absorption coefficients for transmission lines with loads
- Comparison of series and parallel LRC resonant circuits

3.1 Sinusoidal excitation of a resonator

In this series of postings we will discuss sinusoidal excitation of a resonator, specifically directed towards mass-and-spring resonators. We chose this resonator because it makes for very visual animations. The math is the same for all simple resonators. Immediately below are animations of variations of the the mass and spring resonator. Mouse over each one to see the action. We will work with the variation shown in Fig. 1b, a mass hanging from a spring which is attached to a rigid support.

| Fig. 1b. Animation of the particular mass/spring resonator discussed next. |

The mechanical resonator

Fig. 2. Mass/spring resonator. This consists of a mass suspended by a spring from a strong fixed beam.

Figure 2 shows the simple mass and spring resonator. We will now derive an equation for motion of this resonator.

The spring exerts a force on the mass of:

*F*_{spring} = −*kx* , (1)

where *k* is the spring constant of the spring and *x* is the vertical displacement of the mass from its equilibrium position. Springs for which *k* is constant and does not vary with *x* are called "linear" because the graph of force versus *x* is a straight line as shown in Fig. 3. A stiff, i.e. strong, spring has a larg spring constant while a weak spring has a small spring constant.

Fig. 3. Force generated by a linear spring versus the distance the spring is stretched or compressed.

We combine (1) with Newton's second law of motion which governs the motion of all objects moving slowly compared to the speed of light. In this case we are considering the effect that the spring force has on the motion of the mass (i.e. the bob):

This equation relates the force on the bob to the acceleration the bob will experience because of the force. In (2) *m* is the mass of the bob and *t* is time. The operator *d*^{2}/*dt*^{2} performs a second time derivative on the displacement *x* which is a function of time. This second derivative operation yields the acceleration. The displacement *x* is the vertical distance the bob is displaced from its position when the spring is unstretched and uncompressed.

Effect of gravity

In addition to the spring, gravity also exerts a force on the bob. Gravity exerts a constant downward force of:

*F _{gravity}* = −

*mg*, (3)

where *g* = 9.8 m/s^{2} is the local acceleration of gravity for the surface of the Earth.

We add the force of the spring (equation 2) to that of gravity (equation 4) and substitute this total into (2) to have:

In general gravity will make the bob move down a little. Anticipating this, we assume that *x* will be of the form:

*x* = *x*_{0} + *x*_{1}(*t*) , (5)

where *x*_{0} is a constant shift (or sag) due to gravity and *x*_{1}(*t*) is the time varying part of the displacement. Substituting (5) into (4), we get:

Considering the case where the time varying part is zero, (6) becomes −*kx*_{0} − *mg* = 0. This can be solved for *x*_{0}, i.e.: *x*_{0} = −*mg*/*k* . This gives the static "sag" of the spring due to gravity. Substituting this value for *x*_{0} into (6) gives:

Equation 8 is basically the same as equation 5 without the complication of gravity. Another difference is that the time varying part *x*_{1} is the displacement of the bob from the "sagging" position. If we remember this modified displacement definition, then we are free to just ignore gravity. In this spirit, we rewrite (7) just using this new definition of *x*:

I might note that we could not separate out the effect of gravity so simply if the spring were not linear (as graphed in Fig. 3 above).

Fig. 4. Motion of a mass/string resonator as recorded by a strip chart recorder (similar to those used in EKG machines, seismographs and polygraphs). The particular solution shown is

*x*=

*A*sin

*ωt*. Note that on this type of recorder, time progresses towards the left, backwards of the normal convention. Mouse over the figure to see the action.

Solutions to the differential equation

We can rearrange (8) to be in the standard second order differential equation format as:

The standard solutions to the very simple second order differential equation as given by (9) are sines, cosines, and complex exponentials in the forms:

*x* = *A* sin *ωt*, *x* = *A* cos *ωt*, *x* = *A* e^{±iωt} (10)

and sums of these. All of these mathematically describe the oscillations that we associate with a mass dangling at the end of a spring.

Fig. 5. This animation shows how various parameters of this resonator vary versus time. The red arrow on the bob indicates the spring force on the bob, while the blue arrow indicates the bob's momentum. For discussion of the interactions of force, momentum, and energy see my earlier posting. Note that the total energy of the resonator is constant but varies in its form and location, between the form of potential energy in the spring and kinetic energy in the mass. Mouse over the figure to see the action. Click on it to restart.

The angular frequency *ω* (in radians per second) and regular frequency *f* (in Hz) of the oscillations are given by:

Equation 12 for *ω* can be had by substituting any of the standard solutions (10) back into (9) and solving for *ω*. The frequency *f* is related to *ω* by the definition of *ω*, i.e. by *ω* = 2*πf*. The amplitude *A* of the oscillation is arbitrary and is determined by the impulse that is given to the mass to set it into oscillations.

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