A small trailer and its load have a total mass of 250 kg. The trailer is supported by two spring

step 1 To solve this problem, let us write the given data first. Given that the value of m is equal to

A small trailer and its load have a total mass of 250 kg....

A small trailer and its load have a total mass of $250 \mathrm{kg}$. The trailer is supported by two springs, each of constant $10 \mathrm{kN} / \mathrm{m},$ and is pulled over a road, the surface of which can be approximated by a sine curve with an amplitude of $40 \mathrm{mm}$ and a wavelength of $5 \mathrm{m}$ (i.e., the distance between successive crests is $5 \mathrm{m}$ and the vertical distance from crest to trough is $80 \mathrm{mm}$ ). Determine (a) the speed at which resonance will occur, (b) the amplitude of the vibration of the trailer at a speed of $50 \mathrm{km} / \mathrm{h}$.

Key Concepts

Forced Vibration

In a forced vibration analysis, an external periodic force drives the system. The response of the system depends on the relationship between the excitation frequency and the system's natural frequency. When these frequencies are close or match, resonance occurs, leading to significantly amplified vibrations. This is crucial for understanding how road irregularities can induce vibrations in vehicles.

Relationship between Road Profile and Excitation Frequency

When a vehicle moves over a spatially periodic road surface, such as a sine wave profile, the fixed geometric wavelength of the road can be converted into a temporal excitation frequency via the vehicle's speed. This means that the speed of the vehicle essentially determines the frequency at which the vehicle experiences the periodic input from the road, and resonance can be induced if this frequency matches the natural frequency of the vehicle’s suspension system.

Mass-Spring System Dynamics

This concept involves modeling the trailer as a rigid mass supported by springs. In such systems, the dynamics are determined by both the mass and the combined stiffness of the springs. When springs are arranged in parallel, as in this case with two springs, their stiffnesses add to provide an effective spring constant that governs the system's response to dynamic loads.

Natural Frequency

The natural frequency is the frequency at which a mass-spring system will oscillate when disturbed from its equilibrium position, without any external forcing or damping. It is determined by the square root of the ratio of the effective spring constant to the mass. This intrinsic property of the system plays a critical role in resonance phenomena, where external excitations at this frequency can cause large amplitude vibrations.

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