comparing the amplitudes of the imposed force |F| with the force transmitted to the base |FB| gives
Equation (2.21) is plotted in Figure 2.10. The ratio |FB|/| F| is sometimes called the force transmissibility TF. The force amplitude transmitted to the machine support base, FB, is seen to be much greater than 1, if the exciting frequency is at the system resonance frequency. The results in Eq. (2.21) and Figure 2.10 have important applications to machinery noise problems that will be discussed again in detail in Chapter 9 of this book.
Figure 2.10 Force transmissibility, TF, for a damped simple system.
Briefly, we can observe that these results can be utilized in designing vibration isolators for a machine. The natural frequency ωn of a machine of mass M resting on its isolators of stiffness K and damping constant R must be made much less than the forcing frequency ω. Otherwise, large force amplitudes will be transmitted to the machine base. Transmitted forces will excite vibrations in machine supports and floors and walls of buildings, and the like, giving rise to additional noise radiation from these other areas.
Chapter 9 of this book gives a more complete discussion on vibration isolation.
Example 2.4
What is the maximum stiffness of an undamped isolator to provide 80% isolation for a 300‐kg machine operating at 1000 rpm?
Solution
The excitation frequency is f = 1000/60 = 16.7 Hz, or ω = 1000 × (2π/60) = 104.7 rad/s. For 80% isolation the maximum force transmissibility is 0.2.
Using Eq. (2.21) with δ = 0 and noting that isolation only occurs when
2.4 Multi‐Degree of Freedom Systems
The simple mass‐spring‐damper system excited by a harmonic force was discussed in the preceding sections assuming a single mass which could move in one axis only. This single‐degree‐of‐freedom system idealization is reasonable when the mass is fairly rigid, the springs are lightweight and its motion can be described by means of one variable. For simple systems vibrating at low frequencies, it is also often possible to represent continuous systems with discrete or lumped parameter models. However, real systems have more than just one degree of freedom and, consequently, more than one natural frequency of vibration. For example, systems with more than one mass or systems in which a mass has considerable translation or rotation in more than one direction need to be modeled as multi‐degree of freedom systems. In multi‐degree of freedom systems, we have to consider the relationship between the motions of the various masses, i.e. their relative motion.
The general form of the equation that governs the forced vibration of an n‐degree‐of‐freedom linear system with viscous damping can be written in matrix form as
where [M] is the n × n mass matrix, [R] is the n × n damping matrix (that incorporates viscous damping terms in the matrix formulation), [K] is the n × n stiffness matrix, q is the n‐dimensional column vector of time‐dependent displacements, and f(t) is the n‐dimensional column vector of dynamic forces that act on the system. Therefore, the system governed by Eq. (2.22) exhibits motion which is governed by a set of n simultaneously second‐order differential equations. These equations can be derived using either Newton's laws for free body diagrams or energy methods. In particular, it can be shown that the mass and stiffness matrix are symmetric. This fact is assured if energy methods are used to derive the differential equations. However, symmetric mass and stiffness matrices can also be obtained after algebraic manipulation of the equations. In general, damping matrices are not symmetric unless the system is proportionally damped, i.e. the damping matrix is a linear combination of the mass matrix and stiffness matrix.
The algebraic complexity of the solution grows exponentially with the number of degrees of freedom and the general solution of Eq. (2.22) can be difficult to obtain for systems with a large number of degrees of freedom. Therefore, approximate and numerical approaches are often required to obtain the vibration properties and system response of a multi‐degree of freedom system.
2.4.1 Free Vibration – Undamped
By free vibration, we mean that the system is set into motion by some forces which then cease (at t = 0) and the system is then allowed to vibrate freely for t > 0 with no external forces applied. First we will consider a free undamped multi‐degree of freedom system, i.e. [R] = [0] and f(t) = 0. Therefore, Eq. (2.22) now becomes
Similarly to the case of the single‐degree‐of‐freedom system discussed in Section 2.3, we assume harmonic solutions in the form
where A is the vector of amplitudes. Substituting
