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Chapter 22: Problem 38

The machine has a mass \(m\) and is uniformly supported by four springs, each having a stiffness \(k\). Determine the natural period of vertical vibration.

Short Answer

Expert verified
The natural period of the machine's vertical vibration is \(\frac{1}{2}\sqrt{\frac{m}{k}}\).

Step by step solution

01

Calculate the overall spring stiffness

For a system with multiple springs, the effective stiffness is the sum of the stiffness of all the springs combined. Since there are four springs, each having stiffness \(k\), the total spring stiffness \(K\) is \(4k\).
02

Find the natural frequency

The natural frequency of a system is given by \(\sqrt{\frac{K}{m}}\), where \(K\) is the overall stiffness and \(m\) is the mass. For our problem here, \(K\) is \(4k\), so the natural frequency is \(\sqrt{\frac{4k}{m}}\).
03

Calculate the natural period

The natural period \(T\) of a system is the reciprocal of the natural frequency. In other words, \(T=\frac{1}{f}\). Using the natural frequency calculated in step 2, the natural period is \(\frac{1}{\sqrt{\frac{4k}{m}}}\). By the properties of fractions and square roots, this simplifies to \(T=\frac{1}{2}\sqrt{\frac{m}{k}}\).

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Most popular questions from this chapter

The block, having a weight of \(12 \mathrm{lb}\), is immersed in a liquid such that the damping force acting on the block has a magnitude of \(F=(0.7|v|) \mathrm{lb}\), where \(v\) is in \(\mathrm{ft} / \mathrm{s}\). If the block is pulled down \(0.62 \mathrm{ft}\) and released from rest, determine the position of the block as a function of time. The spring has a stiffness of \(k=53 \mathrm{lb} / \mathrm{ft}\). Assume that positive displacement is downward.

A 2-lb weight is suspended from a spring having a stiffness \(k=2 \mathrm{lb} / \mathrm{in}\). If the weight is pushed \(1 \mathrm{in}\). upward from its equilibrium position and then released from rest, determine the equation which describes the motion. What is the amplitude and the natural frequency of the vibration?

If the block-and-spring model is subjected to the periodic force \(F=F_{0} \cos \omega t\), show that the differential equation of motion is \(\ddot{x}+(k / m) x=\left(F_{0} / m\right) \cos \omega t\), where \(x\) is measured from the equilibrium position of the block. What is the general solution of this equation?

A 7-lb block is suspended from a spring having a stiffness of \(k=75 \mathrm{lb} / \mathrm{ft}\). The support to which the spring is attached is given simple harmonic motion which may be expressed as \(\delta=(0.15 \sin 2 t) \mathrm{ft}\), where \(t\) is in seconds. If the damping factor is \(c / c_{c}=0.8\), determine the phase angle \(\phi\) of forced vibration.

If the \(20-\mathrm{kg}\) wheel is displaced a small amount and released, determine the natural period of vibration. The radius of gyration of the wheel is \(k_{G}=0.36 \mathrm{~m}\). The wheel rolls without slipping.
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