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Chapter 9: Problem 52

A single-degree-of-freedom system is used to represent an automobile, of mass \(m\), damping constant \(c\), and stiffness \(k\), which travels on a rough road that is in the form of a sinusoidal surface with an amplitude \(Y\) and wavelength \(l\). If the automobile travels at a velocity \(v\), derive an expression for the transmissibility of the vertical motion of the automobile mass \((m)\).

Short Answer

Expert verified
The short answer for the transmissibility of the vertical motion of the automobile mass (\(m\)) is given by the expression: \( T = \frac{1}{\sqrt{ (2\zeta \omega/\omega_n)^2 + (1-\omega/\omega_n^2)^2 }} \), where \(\zeta = \frac{c}{2\sqrt{mk}}\) is the damping ratio, \(\omega_n = \sqrt{\frac{k}{m}}\) is the natural frequency of the system, and \(\omega = \frac{2 \pi v}{l}\) is the angular frequency related to the road's sinusoidal surface.

Step by step solution

01

Equations of Motion of the System

First, let's write down the equations of motion for the system: \[ m\ddot{x} + c\dot{x} + kx = F(t) \] where \(x\) is the displacement of the mass from its equilibrium position, \(\ddot{x}\) is the acceleration, \(\dot{x}\) is the velocity, and \(F(t)\) is the external force acting on the system. Since the road is a sinusoidal surface, we can describe the road profile as: \[ h(t) = Y \sin(\omega t) \] where \(Y\) is amplitude, \(\omega\) is angular frequency, and \(\omega = \frac{2 \pi v}{l}\) with \(v\) being the velocity of the automobile and \(l\) being the wavelength of the sinusoidal surface.
02

Determine the External Force

The external force \(F(t)\) acting on the system is due to the road irregularity, which acts on the mass in the vertical direction. Therefore, we can write the force as: \[ F(t) = -ky(t) \] where \(y(t) = x + h(t)\) is the relative displacement between the mass and the road. Taking the time derivative of \(y(t)\), we find the relative velocity: \[ \dot{y}(t) = \dot{x} + \dot{h}(t) = \dot{x} + Y\omega \cos(\omega t) \]
03

Substitute the External Force into the Equation of Motion

Now, substitute the external force into the equation of motion: \[ m\ddot{x} + c\dot{x} + kx = -kY \sin(\omega t) \]
04

Solve the Equation of Motion

We are only interested in the steady-state solution of the equation of motion. So, let's assume a solution of the form: \[ x(t) = X \sin(\omega t + \phi) \] where \(X\) is the amplitude and \(\phi\) is the phase angle. Now, substitute this solution into the equation of motion and solve for \(X\) and \(\phi\). After solving the equation, we obtain the amplitude of displacement as: \[ X = \frac{Y}{\sqrt{ (2\zeta \omega/\omega_n)^2 + (1-\omega/\omega_n^2)^2 }} \] where \(\zeta = \frac{c}{2\sqrt{mk}}\) is the damping ratio, and \(\omega_n = \sqrt{\frac{k}{m}}\) is the natural frequency of the system.
05

Transmissibility

Finally, the transmissibility of vertical motion is defined as the ratio between the amplitude of the mass motion and the amplitude of the base motion: \[ T = \frac{X}{Y} = \frac{1}{\sqrt{ (2\zeta \omega/\omega_n)^2 + (1-\omega/\omega_n^2)^2 }} \] This expression represents the transmissibility of the vertical motion of the automobile mass \(m\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding a Single-Degree-of-Freedom System
Think of a single-degree-of-freedom (SDOF) system like a simple mechanical system that can move in only one way. For instance, a weight hanging from a spring. This system can bounce up and down, which is just one motion鈥攈ence the term 'single-degree'.

In our textbook exercise, we consider an automobile as a SDOF system. This means we are only looking at one type of motion of the car, which is the up-and-down movement due to a bumpy road. Here, the mass (\(m\)), stiffness (\(k\)), and damping constant (\(c\)) of the vehicle's suspension system define this motion.

When we calculate the transmissibility of vertical motion, we're effectively looking for how much of the road's bumpiness transfers to the car's body. Students need to remember that in an SDOF system, understanding these three key properties (mass, stiffness, damping) is crucial to analyze the system's dynamic behavior.
Damping Constant and its Significance
A damping constant (\(c\)) in the context of mechanical systems refers to the resistive force that opposes the motion鈥攍ike shock absorbers in a car. Imagine pushing a child on a swing; if the swing has no resistance, it would keep going indefinitely. However, air resistance and friction at the hinge provide damping, so it gradually slows down.

In our exercise, damping helps us understand how the vehicle's suspension absorbs the shocks from the road. A high damping constant means that the car would absorb most of the road shock, leading to a smoother ride with less motion transferred to the passengers. On the other hand, too much damping can make the ride feel stiff and uncomfortable.

Students should keep in mind the damping ratio (\( \frac{c}{2\root{mk}} \) is derived from the damping constant. It plays a vital role in determining the system's response to vibrations鈥攂asically telling us if the car will be bouncy, firm, or just right.
Equations of Motion: The Foundation of Dynamics
The equations of motion are the fundamental principles that govern the behavior of a dynamic system. Just like a recipe tells you what ingredients you need and in what order to add them, equations of motion tell us how forces and movement are related.

Our exercise outlines a basic equation of motion for a SDOF system (\(m\textcolor{#016936}{\textstyle \textstyle \bfseries \textbackslash ddot\textbraceleft x\textbraceright}} + c\textcolor{#016936}{\textstyle \textstyle \bfseries \textbackslash dot\textbraceleft x\textbraceright}} + kx = F(t)\) which incorporates mass, velocity, and displacement against an external force. This equation helps predict how the car's body will react to the sinusoidal road. By solving the equation for our given conditions, such as the sinusoidal force caused by the road (\(F(t) = -kY \textcolor{#016936}{\textstyle \textstyle \textbackslash sin}\textcolor{#0070C1}{\textstyle (\textcolor{#0070C1}{\textstyle \textcolor{#0070C1}{\textstyle \textomega t}})}\) we can determine the car's steadiness, comfort, and overall performance on such surfaces.

Students should grasp that these equations are not only tools for solving problems but also are fundamental in understanding the behavior of a system and predicting how it will react under different forces and movements.

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Most popular questions from this chapter

Derive an expression for the displacement transmissibility of a damped single- degree-offreedom system whose base is subjected to a general periodic displacement.

Consider a single-degree-of-freedom system with Coulomb damping (which offers a constant friction force, \(F_{c}\) ). Derive an expression for the force transmissibility when the mass is subjected to a harmonic force, \(F(t)=F_{0} \sin \omega t\).

An internal combustion engine has a rotating unbalance of \(1.0 \mathrm{~kg}-\mathrm{m}\) and operates between 800 rpm and 2000 rpm. When attached directly to the floor, it transmitted a force of \(7018 \mathrm{~N}\) at \(800 \mathrm{rpm}\) and \(43,865 \mathrm{~N}\) at \(2000 \mathrm{rpm} .\) Find the stiffness of the isolator that is necessary to reduce the force transmitted to the floor to \(6000 \mathrm{~N}\) over the operating-speed range of the engine. Assume that the damping ratio of the isolator is \(0.08,\) and the mass of the engine is \(200 \mathrm{~kg}\).

Design the suspension of a car such that the maximum vertical acceleration felt by the driver is less than \(2 g\) at all speeds between \(70 \mathrm{~km} / \mathrm{h}\) and \(140 \mathrm{~km} / \mathrm{h}\) while traveling on a road whose surface varies sinusoidally as \(y(u)=0.5 \sin 2 u \mathrm{~m},\) where \(u\) is the horizontal distance in meters. The weight of the car, with the driver, is \(700 \mathrm{~kg}\) and the damping ratio of the suspension is to be \(0.05 .\) Use a single-degree-of-freedom model for the car.

Ground vibrations from a crane operation, a forging press, and an air compressor are transmitted to a nearby milling machine and are found to be detrimental to achieving specified accuracies during precision milling operations. The ground vibrations at the locations of the crane, forging press, and air compressor are given by \(x_{c}(t)=A_{c} e^{-\omega_{c} \zeta_{c} t} \sin \omega_{c} t, x_{f}(t)=A_{f}\) \(\sin \omega_{f} t, \quad\) and respectively, where \(A_{c}=20 \mu \mathrm{m}, A_{f}=30 \mu \mathrm{m}\) \(A_{a}=25 \mu \mathrm{m}, \omega_{c}=10 \mathrm{~Hz}, \omega_{f}=15 \mathrm{~Hz}, \omega_{a}=20 \mathrm{~Hz},\) and \(\zeta_{c}=0.1 .\) The ground vibrations travel at the shear wave velocity of the soil, which is equal to \(300 \mathrm{~m} / \mathrm{s}\), and the amplitudes attenuate according to the relation \(A_{r}=A_{0} e^{-0.005 r}\), where \(A_{0}\) is the amplitude at the source and \(A_{r}\) is the amplitude at a distance of \(r \mathrm{ft}\) from the source. The crane, forging press, and air compressor are located at a distance of \(18 \mathrm{~m}, 24 \mathrm{~m},\) and \(12 \mathrm{~m},\) respectively, from the milling machine. The equivalent mass, stiffness, and damping ratio of the machine tool head in vertical vibration (at the location of the cutter) are experimentally determined to be \(500 \mathrm{~kg}\), \(480 \mathrm{kN} / \mathrm{m},\) and \(0.15,\) respectively. The equivalent mass of the machine tool base is \(1000 \mathrm{~kg}\). It is proposed that an isolator for the machine tool be used, as shown in Fig. \(9.55,\) to improve the cutting accuracies [9.2]. Design a suitable vibration isolator, consisting of a mass, spring, and damper, as shown in Fig. \(9.55(\mathrm{~b}),\) for the milling machine such that the maximum vertical displacement of the milling cutter, relative to the horizontal surface being machined, due to ground vibration from all the three sources does not exceed \(5 \mu \mathrm{m}\) peak-to-peak.
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