Final answer:
Natural frequencies of a system can be determined in rad/sec and then converted to Hz. They describe how often a system vibrates naturally, and the mode shape indicates the pattern of these vibrations. Determining both requires knowledge of the system's properties and involves a conversion between radians and Hertz for practical applications.
Step-by-step explanation:
To find the natural frequencies of a system and the mode shapes, we need to consider different types of vibrations. Depending on whether the system is free or forced to vibrate, and whether there is damping involved, the natural frequencies may vary.
Natural Frequency in Rad/sec
The natural frequency in radians per second (rad/sec) is the frequency at which a system oscillates during free or forced vibration. This value can be calculated based on the properties of the system, such as mass and stiffness in the case of mechanical systems.
Natural Frequency in Hz
To convert the natural frequency from rad/sec to Hz, divide by 2π. The formula f = ω/(2π) reveals the frequency in Hertz (Hz), illustrating the number of oscillations per second.
Mode Shape
The mode shape describes the spatial distribution of vibrations within the system. For a string, the normal modes correspond to standing wave patterns, with the fundamental frequency being the first harmonic and any additional frequencies being overtones.
Summary of Steps
- Determine the properties of the system (e.g., mass, stiffness).
- Use these properties to calculate the natural frequency in rad/sec.
- Convert the natural frequency to Hz for practical applications.
- Identify the mode shape by analyzing the spatial distribution of vibrations.