Mechanical ad Electrical vibration EASY question ?

In summary, the general solution to the equation mu'' + ku = 0 is u = Acos(ωt) + Bsin(ωt), where ω^2 = k/m. This can be derived by taking the second derivative of both the cosine and sine function, showing that they satisfy the given equation. This is known as the general solution because it includes all possible solutions for this type of equation. The specific values of A and B can then be determined by initial conditions or boundary conditions.
  • #1
dwilmer
11
0
Mechanical ad Electrical vibration EASY question please help?

In Boyce and Diprima textbook it says that:

mu'' + ku = 0 .

Then it says the general solution of this is:

u = Acos (w^2)t + B sin (w^2)t , where w^2 = k/m


It provides no explanation how it arrives at this. Where does the cos and sin come from?

PS: it is supposed to be the greek letter w, whatever that is called and it also has a sub-0 on it, but i didnt include it because it looks confusing without the right fonts.

please help, thanks
 
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  • #2
Welcome to PF!

Hi dwilmer! Welcome to PF! :smile:

(have a little omega: ω and try using the X2 tag just above the Reply box :wink:)
dwilmer said:
mu'' + ku = 0 .

Then it says the general solution of this is:

u = Acos (w^2)t + B sin (w^2)t , where w^2 = k/m


It provides no explanation how it arrives at this. Where does the cos and sin come from?

Because [cos(ωt)]'' is obviously -ω2cos(ωt), and [sin(ωt)]'' is obviously -ω2sin(ωt) :smile:
 
  • #3

1. What is mechanical and electrical vibration?

Mechanical and electrical vibration refers to the back-and-forth motion of an object or system, caused by an external force or energy. In mechanical vibration, this can be seen in the movement of machines or structures, while electrical vibration occurs in electrical circuits due to changes in voltage and current.

2. What are the causes of mechanical and electrical vibration?

Mechanical vibration can be caused by various factors such as imbalances in rotating machinery, misalignments, worn bearings, or external forces like wind or earthquakes. Electrical vibration is often caused by fluctuations in power supply, electromagnetic interference, or faulty electrical components.

3. How is mechanical and electrical vibration measured?

Mechanical vibration is typically measured using accelerometers, which detect the acceleration of a vibrating object. Electrical vibration can be measured using oscilloscopes, which display the electrical signals generated by the vibration.

4. What are the effects of excessive mechanical and electrical vibration?

In machinery, excessive vibration can lead to increased wear and tear, decreased efficiency, and even failure of components. In electrical systems, excessive vibration can cause damage to sensitive components and disrupt the flow of electricity, leading to malfunctions or failures.

5. How can mechanical and electrical vibration be reduced or controlled?

To reduce mechanical vibration, techniques such as balancing, alignment, and damping can be used. In electrical systems, shielding, grounding, and proper wiring can help control vibration. Regular maintenance and monitoring can also help identify and address sources of vibration before they become a major issue.

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