Question

How did shortening the length of the spring affect the resonant frequencies? How does this confirm the relationship v = ? f when velocity is constant? And how musical instruments create higher and lower tones. Use a string instrument as an example.

I can't explain above question well....When the wavelength becomes short, it increases its frequency.

Answer 1

Resonating systems will be described in terms of a "spring system." From a practical point of view, any machine, structural part or combination of assembled parts that can be deflected by a force and then returned to their original positions when the force is removed, can be treated as a spring or "spring system."

A shaft or spindle supported between two bearings may be regarded as an individual spring; so also may the span of concrete floor as well as the columns supporting the floor. A motor armature, its frames, and pulleys on shafts may also be considered spring systems. However, parts that "deflect" by pivoting about a friction joint, such as a door hinge, are not spring systems. To be considered a spring, the part itself must flex and be able to return.

Each spring or spring system has its own natural frequency, which -- when matched by a vibration frequency -- will resonate. Pipes and tubes in process plants are long springs whose resonant frequencies are determined not only by their length and weight but also by the spacing between supports and the rigidity added by elbows and support hangers. Fluid in a pipe adds weight which lowers the pipe's resonant frequency as well as increases vibration damping.

The more flexible a part, the lower its natural frequency; the more rigid, the higher the natural frequency. Adding weight to a spring system will also lower the natural frequency. The natural frequency can also be changed by moving the location of the weight. To get a general idea of what affects the natural frequency or resonance frequency of a spring system, refer to Fig. 1 which illustrates a cantilevered flat beam that is sagging slightly (static deflection) due to its own weight.

Although engineers can use equations found in college vibration textbooks that describe the relationship between the different variables composing a spring system, it's enough for this course to simply note that whatever changes the static deflection changes the resulting natural frequency (resonance frequency). For example, in Fig. 1 there is static deflection due just to the weight of the flat spring and its resulting resonance frequency. If the flat spring was shortened as shown in Fig. 2, or cantilevered less due to adding a brace as illustrated in Fig. 3, the static deflection would decrease. The smaller the deflection, the higher the resonance frequency. The greater the static deflection, the lower the resonance frequency. For a non-cantilevered spring system such as a span of beam supported by two columns, the same principle applies. The shorter the beam, the more rigid, the higher its resonance frequency. The longer the span, the less rigid (or more flexible), the lower its resonance frequency.

Assuming the same rigidity and same length but now supporting a weight as shown in Fig. 4, the deflection is greater. The greater the deflection, the lower the resonance frequency. Moving the weight to a place that results in greater deflection, further lowers the resonant frequency. Moving it to a place that results in less deflection results in a higher resonant frequency.

The "real life" spring systems in the plant such as beams, lengths of pipe, columns, sections of bases and pedestals, and so on, combine the effects of rigidity, length and weight for a net result of more or less static deflection. More deflection -- lower frequency. Less deflection -- higher frequency.

Analogies, including this one, are never perfect. The error in the above analogy assumes that weight is affected by gravity, thereby resulting in a static deflection. However, the same spring system could be repositioned as seen in Fig. 5, so that the effect of gravity doesn't result in the same static deflection. The resonant frequency would remain essentially the same as described before. The analogy using static deflection is only intended to help the analyst visualize how to adjust the resonant frequency of a single part or a whole system.