Mechanical Systems with One Degree of Freedom

Abstract

In noise reducing engineering, the consequences of changes made to a system must be understood. Questions posed could be on the effects of changes to the mass, stiffness, or losses of the system and how these changes can influence the vibration of or noise radiation from some structures. Real constructions certainly have many or in fact infinite modes of vibration. However, to a certain extent, each mode can often be modeled as a simple vibratory system. The most simple vibratory system can be described by means of a rigid mass, mounted on a vertical mass less spring, which in turn is fastened to an infinitely stiff foundation. If the mass can only move in the vertical direction along the axis of the spring, the system has one degree of freedom (1-DOF). This is a vibratory system never actually encountered in practice. However, certain characteristics of systems with many degrees of freedom, or rather, continuous systems with an infinite degree of freedom, can be demonstrated by means of the very simple 1-DOF model. For this reason, the basic mass–spring system is used in this chapter to illustrate some of the basic concepts concerning free vibrations, transient, harmonic, and other types of forced excitation. Kinetic and potential energies are discussed as well as their dependence on the input power to the system and its losses.

Problems

1.1

Determine the energy dissipated over one period for a simple mass–spring system if the losses are (a) viscous and (b) hysteretic. Assume that the displacement of the mass is described by $x(t) = x_0 \sin (\omega t)$.

1.2

The displacement of the mass of a simple mass–spring system is given by $x(t) = x_0 \sin (\omega t)$. Determine the force required to maintain this motion if the damping force is due to (i) viscous losses and (ii) frictional losses. In a diagram, show the force as function of displacement. Make some appropriate assumption concerning the magnitude of the properties m, $k_0$, c and $F_d$.

1.3

The mass in Fig. 1.20 is excited and is thereafter left to oscillate freely. Determine the displacement as function of time if the losses are assumed to be frictional. Assume that the displacement is $x_0$ and the velocity zero at time $t = 0$.

1.4

Show that for a critically damped system, the displacement can be zero for time t being finite and that this can only happen at one instance.

1.5

The mass of a simple mass–spring system is excited by an impulse I at time intervals T. Determine the response of the mass. Consider only harmonic solutions, i.e., assume that the excitation process was started at $t = - \infty$. The system is lightly damped.

1.6

A mass–spring system is at rest for $t < 0$. The mass is excited by a force F(t) at $t = 0$. The force is given by $F(t) = F_0$ for $0 \leqslant t \leqslant T$; $F(t) = 0$ for $t < 0$ and $t > T$.

Fig. 1.20

Mass-spring system with frictional losses

Determine the response of the mass. In particular, consider the cases for which the product $\omega _r T$ is equal to $\pi / 2$, $\pi$ and $2\pi$ with $\omega _r$ defined in Eq. (1.14). Assume that $\beta T\ll 1$. For definitions, see Sect. 1.2.

1.7

For the problem described in Example 1.6 determine the maximum amplitude as function of T.

1.8

A function x(t) is expanded in a Fourier series as

$$ x(t) = \dfrac{a_0 }{2} + \displaystyle \sum \limits _{n = 1}^\infty {(a_n \cos \omega _n t + } b_n \sin \omega _n t); \quad \omega _n = 2\pi n / T;\quad n = 1,2,\ldots $$

Show that the coefficients $a_n$ and $b_n$ are

$$ a_n = \dfrac{2}{T}\displaystyle \int _0^T {x(t)\cos (\omega _n t)\mathrm{d}t};\quad b_n = \dfrac{2}{T}\displaystyle \int _0^T {x(t)\sin (\omega _n t)\mathrm{d}t} $$

1.9

A harmonic force F(t) with the period T is exciting the mass of a simple 1-DOF system. Determine the displacement of the mass if

$$ F(t) = F(t + T) = G_0 / 2 + \displaystyle \sum \limits _{n = 1}^\infty {G_n \cos (\omega _n t) + } \displaystyle \sum \limits _{n = 1}^\infty {H_n \sin (\omega _n t)};\quad \omega _n = 2\pi n / T $$

Assume the losses to be viscous.

1.10

Solve Problem 1.5 by expanding the force and response in Fourier series.

1.11

A 1-DOF system is excited by a force $F(t) = F_0 \cdot e^{i\omega t}$. Determine the time averages of kinetic and potential energies as well as the time average of the input power to the system. Assume that the equation governing the motion of the system is

$$ m\ddot{x} + kx = F;\quad k = k_0 (1 + i\delta ) $$

According to Eq. (1.81), $\delta = 2\omega m\beta / k_0$. Since $\beta = c / (2m)\, \delta$ is written $\delta = c\omega / k_0$. Discuss the difference between viscous and structural damping.

1.12

A 1-DOF system is governed by the equation $m\ddot{x} + c\dot{x} + k_0 x = F(t)$. A function $h(t - \tau )$ satisfies the equation $m\ddot{h} + c\dot{h} + k_0 h = \delta (t - \tau )$ show that x(t) is given by

$$ x(t) = \displaystyle \int _{ - \infty }^t\mathrm{d}\tau F(\tau )h(t - \tau ). $$

1.13

The displacement of a 1-DOF system can be described in two different ways as

• (i) $m\ddot{x} + kx = F$; $k = k_0 (1 + i\delta )$

• (ii) $m\ddot{x} + c\dot{x} + k_0 x = F$

Assume $F = F_0 \cdot e^{i\omega t}$ and $x = x_0 \cdot e^{i\omega t}$ and derive the input power to the system for both cases. Show in the first case that the input power is proportional to the potential energy of the system and in the second case to the kinetic energy.