The 450 -kg trailer is pulled with a constant speed over the...

The 450 -kg trailer is pulled with a constant speed over the surface of a bumpy road, which may be approximated by a cosine curve having an amplitude of $50 \mathrm{mm}$ and wave length of $4 \mathrm{m}$. If the two springs $s$ which support the trailer each have a stiffness of $800 \mathrm{N} / \mathrm{m},$ determine the speed $v$ which will cause the greatest vibration (resonance) of the trailer. Neglect the weight of the wheels.

Key Concepts

Resonance

Resonance is the phenomenon that occurs when a system is driven at a frequency equal to its natural frequency, causing the amplitude of oscillations to increase significantly. This concept is critical in understanding why certain speeds or forcing frequencies can lead to large vibrations in mechanical systems.

Natural Frequency

Natural frequency is the frequency at which a system tends to oscillate when not subject to a continuous or repeated external force. It is determined by the system's intrinsic properties, such as mass and stiffness, and serves as a baseline for predicting dynamic responses in vibrational analyses.

Mass-Spring System Dynamics

Mass-spring system dynamics involves analyzing how a mass attached to a spring responds to displacements and external forces. The system’s behavior, characterized by its natural frequency, is fundamental in studying oscillatory motion and forms the basis for understanding more complex mechanical vibrations.

Forced Vibration

Forced vibration refers to oscillations induced by an external periodic force acting on a system. When the frequency of this external force matches the system's natural frequency, resonance occurs, leading to amplified vibrations that are central to analyses in mechanical and structural engineering.

Equivalent Stiffness in Parallel

When multiple springs are arranged in parallel supporting a mass, their individual stiffnesses combine to form an equivalent stiffness. This concept is vital in accurately modeling the dynamic behavior of systems that rely on several supporting elements, as it directly affects the system's natural frequency.

Excitation Frequency from Periodic Inputs

In scenarios where a system is influenced by a spatially periodic input—such as a periodic bump or cosine-shaped road profile—the frequency of excitation experienced by the system is the product of the moving speed and the spatial frequency of the input. This relationship bridges the gap between spatial variations and temporal forcing, enabling the prediction of resonant conditions.

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