The mass m, stiffness k, and natural frequency of an undamped SDF system are unknown. These prop

step 1 In this problem there are two cases arises. In case 1, given that the value of f is equal to 4 H

The mass m, stiffness k, and natural frequency of an...

The mass $m$, stiffness $k$, and natural frequency $\omega_n$ of an undamped SDF system are unknown. These properties are to be determined by harmonic excitation tests. At an excitation frequency of $4 \mathrm{~Hz}$, the response tends to increase without bound (i.e., a resonant condition). Next, a weight $\Delta w=5 \mathrm{lb}$ is attached to the mass $m$ and the resonance test is repeated. This time resonance occurs at $f=3 \mathrm{~Hz}$. Determine the mass and the stiffness of the system.

Key Concepts

Resonance in Single-Degree-of-Freedom Systems

Resonance in a single-degree-of-freedom (SDF) system occurs when the frequency of an applied periodic force strikes the system’s natural frequency. In an ideal undamped system, this results in an unbounded (theoretically infinite) amplitude. This phenomenon is important because it indicates that the applied frequency exactly matches the system’s inherent oscillatory tendency, making it possible to identify the natural frequency from experimental observations.

Natural Frequency Relationship

The natural frequency of an undamped SDF system is determined by the relation ?? = ?(k/m), where k represents the system’s stiffness and m its mass. This fundamental relationship is used to connect the dynamic behavior of the system to its physical parameters, allowing for parameter estimation when the natural frequency is observed in tests.

Effect of Added Mass on Natural Frequency

Introducing additional mass to the system changes its natural frequency. Since the natural frequency is inversely proportional to the square root of the mass, adding mass will decrease the natural frequency. Observing this change by comparing resonance tests with and without the added mass provides a way to form equations that relate the unknown mass and stiffness.

Parameter Identification via Multiple Tests

By performing experiments at different conditions—initially at one resonance frequency and then at a new resonance frequency after adding a known amount of mass—two equations can be set up using the natural frequency relation. Solving these coupled equations allows one to determine both the original mass and the stiffness of the SDF system. This method is a common process in experimental modal analysis and system identification.

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