How Do You Find Amplitude of Resonance?

To find the amplitude of resonance, you first determine the steady-state amplitude of oscillation for a given system and then identify its maximum possible value as the driving frequency is varied. This maximum amplitude is the resonance amplitude.

Understanding Steady-State Amplitude

When a system capable of oscillation (like a mass on a spring with damping or an RLC circuit) is subjected to a continuous external force or signal, it eventually settles into a steady-state oscillation at the same frequency as the driving force. For a driving force described as F0 cos(ωt), the steady-state response can often be expressed in the form A cos(ωt) + B sin(ωt).

The steady-state amplitude (denoted as C) of this oscillation represents the overall magnitude of the response. As defined in the provided reference, this amplitude is calculated from the coefficients A and B using the formula:

C = √A² + B²

Here, A and B are determined by the system's properties (like mass, damping, stiffness, resistance, inductance, capacitance) and the driving frequency ω. Consequently, the steady-state amplitude C is also a function of the driving frequency ω.

What is Resonance Amplitude?

The amplitude of resonance, often denoted as C*, is the maximum value that the steady-state amplitude C can achieve as the driving frequency ω is changed. This peak amplitude occurs at a specific driving frequency known as the practical resonance frequency (denoted as ω*).

Resonance is a phenomenon where a relatively small driving force can produce large amplitude oscillations when the driving frequency is close to the system's natural frequency. The resonance amplitude C* quantifies how large this maximum response can be.

Steps to Find Resonance Amplitude (C*)

Finding the exact value of the resonance amplitude C* involves calculating the steady-state amplitude C as a function of the driving frequency ω and then determining the maximum value of that function.

Here are the general steps:

1. Determine the Steady-State Amplitude Function C(ω):

   • Analyze the system to find the coefficients A and B of the steady-state response A cos(ωt) + B sin(ωt) when driven by F0 cos(ωt). These coefficients will depend on the system parameters (e.g., mass m, damping coefficient c, stiffness k for a mechanical system) and the driving frequency ω.
   • Calculate the steady-state amplitude C using the formula: C(ω) = √A(ω)² + B(ω)². This gives you the amplitude as a function of the driving frequency.

2. Find the Practical Resonance Frequency ω*:

   • The resonance amplitude occurs at the frequency ω* that maximizes the function C(ω). To find this frequency, you typically use calculus: find the derivative of C(ω) with respect to ω, set it equal to zero, and solve for ω. The value(s) of ω that yield a maximum are the resonance frequency (or frequencies, though often there is one primary practical resonance).
   • The reference mentions the tuned practical resonance frequency ω* where the resonant amplitude C* occurs.

3. Calculate the Resonance Amplitude C*:

   • Once you have found the practical resonance frequency ω*, substitute this value back into the steady-state amplitude function C(ω) calculated in Step 1.
   • The result is the resonance amplitude: C* = C(ω*).

The reference highlights the relationship between the amplitude at any frequency C and the resonant amplitude C* by mentioning the ratio 100C/C*, which can be used to express the current amplitude as a percentage of the maximum possible amplitude.

In summary, finding the amplitude of resonance C* requires calculating the general steady-state amplitude C as a function of frequency ω using C = √A² + B² and then determining the peak value of this function.