A 400 kg machine is placed at the midspan of a 3-m-long, 200-kg simply supported beam. The beam

step 1 So assume the beam as a rigid bar, ABC, hinged at A to find the equivalent stiffness of the spri

A 400 kg machine is placed at the midspan of a 3-m-long,...

A $400 \mathrm{~kg}$ machine is placed at the midspan of a $3-\mathrm{m}$-long, 200-kg simply supported beam. The beam is made of a material of elastic modulus $200 \times 10^9 \mathrm{~N} / \mathrm{m}^2$ and has a cross-sectional moment of inertia of $1.4 \times 10^{-5} \mathrm{~m}^4$. Use a three degree-offreedom model to approximate the system's lowest natural frequency.

Key Concepts

Natural Frequency

Natural frequency is the rate at which a system oscillates when set into motion without external forces. It is determined by the balance between the system's stiffness and its mass distribution and is critical in avoiding resonance in structural dynamics.

Degree-of-Freedom Modeling

Degree-of-freedom modeling involves reducing a continuous system to a few discrete points where the dynamic behavior is captured. In this context, a three degree-of-freedom model approximates the distributed mass and stiffness of the beam into three key points to simplify the analysis of its vibrational characteristics.

Simply Supported Beam

A simply supported beam is one that rests on supports at its ends, allowing it to freely rotate at the supports while resisting vertical loads. The boundary conditions associated with a simply supported beam significantly influence its deflection profile and modal shapes in vibration analysis.

Bending Stiffness

Bending stiffness, calculated as the product of the elastic modulus and the second moment of area (moment of inertia), quantifies a beam's resistance to bending. It is a fundamental property that governs the deflection and natural frequencies of beam structures under loading.

Mass Distribution

Mass distribution refers to how mass is spread throughout the structure, including both the beam's inherent mass and any additional point loads. Accurate representation of mass distribution is essential for predicting the dynamic response and natural frequencies of the system.

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