## Abstract

Over the past century, a number of scalar metrics have been proposed to measure the damping of a complex system. The present work explores these metrics in the context of finite element models. Perhaps the most common is the system loss factor, which is proportional to the ratio of energy dissipated over a cycle to the total energy of vibration. However, the total energy of vibration is difficult to define for a damped system because the total energy of vibration may vary considerably over the cycle. The present work addresses this ambiguity by uniquely defining the total energy of vibration as the sum of the kinetic and potential energies averaged over a cycle. Using the proposed definition, the system loss factor is analyzed for the cases of viscous and structural damping. For viscous damping, the system loss factor is found to be equal to twice the modal damping ratio when the system is excited at an undamped natural frequency and responds in the corresponding undamped mode shape. The energy dissipated over a cycle is expressed as a sum over finite elements so that the contribution of each finite element to the system loss factor is quantified. The visual representation of terms in the sum mapped to their spatial locations creates a loss factor image. Moreover, analysis provides an easily computed sensitivity of the loss factor with respect to the damping in one or more finite elements.

## Introduction

Engineers are often challenged by the need to increase damping in a complex system. To this end, recent work has investigated optimization of damping in automotive design [1], wind turbine design [2], and multiple tuned mass dampers [3]. There are two challenges, the first being location. Where in the system should the engineer introduce or change materials that provide damping? Most complex dynamic systems are highly coupled and the effect of a damping material can only be assessed after a model is constructed and solved. The second challenge is material selection. If materials are too flexible or too stiff relative to other parts of the system, their strain energies are relatively low and therefore their energy dissipation is low.

Since the cost of constructing and analyzing models of complex dynamic systems is high, most engineers cannot afford to optimize by iterating over a large number of designs. Direct assessment of a design often requires the very high price of solving a large linear system at each frequency of interest. For example, the number of floating-point operations for most direct solvers scales as (*N*^{3}), where *N* is the size of the system [4]. Therefore, the goal of the present work is to provide an approximate yet efficient method that indicates the locations and materials that optimize damping.

*W*is the energy dissipation over a cycle,

*W*is the total energy of vibration, and

*δ*is the damping factor. Kimball also showed that

*δ*is the decrement, approximated as

*y*is the reduction in displacement over a cycle of vibration and

*y*is the value of displacement at some time in that cycle. Kimball provided a table of measured values of

*δ*for 17 materials. The following year, von Heydekampf [7] published a similar paper, providing additional data and proposing a related quantity called the

*specific damping capacity*, which is twice the decrement. The measurement of specific damping received much attention since the publications in the 1930s, as evidenced by Maringer’s bibliography covering the period 1955–1965 [8] and the thesis by Chu [9].

*η*. Ungar et al. [11] cited three papers from that conference [12–14], as well as an additional one by Lazan [15], when describing the system loss factor in his 1962 paper. Ungar summarized that work by first writing the definition of the system loss factor as

*D*is the “energy dissipated per cycle (or the energy that must be supplied to the system to maintain steady-state conditions” and

*W*is the “total (kinetic plus potential) energy associated with the vibration.” Unless the damping is light, the definition of

*W*is ambiguous as the total energy varies during a cycle. The following five definitions for

*W*were summarized by Ungar from Refs. [12–15] and are quoted as follows:

total kinetic plus strain energy at any instant,

total kinetic energy at an instant of zero stress at a reference point,

total kinetic energy at an instant of zero strain at a reference point,

total strain energy at an instant of maximum stress at a reference point, and

total strain energy at an instant of maximum strain at a reference point.

Ungar found that these five definitions “…give neither unique nor equal results…” for computing the system loss factor.

This history places in context three new and significant contributions of the present work. The first contribution is to uniquely, clearly, and justifiably define the total energy of vibration used in the definition of the system loss factor. The second contribution is to quantify the spatial distribution of damping that contributes to the system loss factor, so that one may rank the damping effectiveness of various materials in the system. The present work proposes a loss factor image that is a visual representation of the contributions to the system loss factor. The third contribution is to quantify the effects of small design changes on the system loss factor by way of a sensitivity analysis.

Before proceeding with the analysis, conventions and notations are reviewed. A bold-faced variable denotes either a matrix or vector. Complex exponential representations [16, Chapter 2] are used for all time-dependent quantities, with a tilde representing a complex amplitude. A superscript *T* shall denote a transpose and a superscript *H* denote a Hermitian transpose.

## System Loss Factor

*D*is the energy dissipated per cycle. The definition of

*W*proposed here is that

*W*is the time-averaged total energy,

*E*(

*t*), over a cycle of vibration:

*τ*is the period of vibration. The total energy of a dynamic system is the sum of kinetic,

*T*(

*t*), and potential,

*U*(

*t*), energies, so

*T*

_{ave}is the average kinetic energy and

*U*

_{ave}is the average potential energy. The system loss factor defined by Eqs. (4) and (7) may be greater than unity, as it is possible to dissipate more energy over a cycle than the average total energy.

The energies used in the system loss factor may be found from computed responses, eigenvectors, or operating deflection shapes. The model used may be linear or nonlinear, as long as the excitation is periodic. In many cases, computing the response of a system is significantly faster than computing the eigenvectors. In these cases, the system loss factor will have a considerable advantage over the modal loss factor or modal damping ratio.

## Application to Viscously Damped Systems

*e*th finite element is

**M**

_{e}is the mass matrix,

**C**

_{e}is the damping matrix,

**K**

_{e}is the stiffness matrix,

**x**

_{e}(

*t*) is the displacement vector,

*E*is the number of elements, and

**F**

_{e}(

*t*) is the force vector. The variable

*α*

_{e}is a factor that scales the damping in element

*e*. Varying

*α*

_{e}represents a physical change in the damping of the material being modeled by element

*e*. It will be used later to examine the sensitivity of the loss factor with respect to scaling the damping matrix of element

*e*. A finite element model constructed by enforcing boundary conditions between elements has

*N*degrees-of-freedom governed by

*ω*and is therefore expressed as

*η*is the fraction of energy dissipated per cycle of vibration, where energy is interpreted as the sum of kinetic and potential energies that are time-averaged over a cycle of vibration.

*E*finite elements

*e*th finite element:

*e*th finite element. The visual representation of the quantities

*η*

_{e}at the associated locations of the finite elements shall be described here as a

*loss factor image*. The purpose of the loss factor image is to reveal elements of low or high contribution to the system loss factor. The sum of the loss factor image is equal to the system loss factor.

*η*with respect to the scaling factor,

*α*

_{e}, which was previously introduced in the equation of motion of the

*e*th finite element given in Eq. (8). This evaluation gives

*η*

_{e}is the sensitivity of the system loss factor with respect to changes in the damping of the

*e*th finite element. Using the element loss factor, one can approximate the effects of small changes in the damping of any element or group of elements on the system loss factor.

## Relationship to the Critical Damping Ratio for a Mode

*ϕ*_{n}is the

*n*th undamped mode shape and

*ω*

_{n}is the

*n*th undamped natural frequency. The mode shape

*ϕ*_{n}and natural frequency

*ω*

_{n}are real-valued. If the mode shapes are normalized appropriately, the following identities result:

*n*th undamped natural frequency,

*ω*

_{n}, and is vibrating in its

*n*th undamped mode shape,

*ϕ*_{n}, so that

## Applications to Systems With Structural Damping

*ω*and is therefore expressed as

**M**

_{e}is the mass matrix and

**K**

_{e}is a complex-valued stiffness matrix of the element. A finite element model constructed by enforcing boundary conditions between elements has

*N*degrees-of-freedom governed by

**K**is a complex-valued stiffness matrix which may be written as

*τ*is the period of vibration given by

*τ*= 2

*π*/

*ω*. Substituting Eqs. (33)–(34) and (42) into Eq. (43) and performing the integration results in

*e*th finite element:

*e*th finite element.

## Numerical Examples

The first example is a steel beam in flexure connected to ground by dashpots, as shown in the schematic of Fig. 1. There is no damping in the beam. The beam is excited by a transverse force at *x* = *L*/3. Properties of the beam and dashpots are given in Table 1. Frequency of excitation is varied in 1000 equal steps from 0.5*f*_{1} to 1.5*f*_{10}, where *f*_{1} and *f*_{10} are the first and tenth nonzero natural frequencies of the beam with no dashpots. A finite element model is constructed of this system, using 500 beam flexure elements which may be found in Ref. [21, Chapter 12].

Variable name | Variable symbol | Value |
---|---|---|

Beam Young’s modulus | E | 2.0 × 10^{11} Pa |

Beam Poisson’s ratio | ν | 0.3 |

Beam mass density | ρ | 7800 kg/m^{3} |

Beam length | L | 1 m |

Beam base | b | 0.01 m |

Beam height | h | 0.01 m |

Dashpot 1 | c_{1} | 100 N s/m |

Dashpot 2 | c_{2} | 200 N s/m |

Dashpot 3 | c_{3} | 300 N s/m |

Spring 1 | k_{1} | (1 + 0.1j)(1 × 10^{5}) N/m |

Spring 2 | k_{2} | (1 + 0.1j)(2 × 10^{5}) N/m |

Spring 3 | k_{3} | (1 + 0.1j)(3 × 10^{5}) N/m |

Variable name | Variable symbol | Value |
---|---|---|

Beam Young’s modulus | E | 2.0 × 10^{11} Pa |

Beam Poisson’s ratio | ν | 0.3 |

Beam mass density | ρ | 7800 kg/m^{3} |

Beam length | L | 1 m |

Beam base | b | 0.01 m |

Beam height | h | 0.01 m |

Dashpot 1 | c_{1} | 100 N s/m |

Dashpot 2 | c_{2} | 200 N s/m |

Dashpot 3 | c_{3} | 300 N s/m |

Spring 1 | k_{1} | (1 + 0.1j)(1 × 10^{5}) N/m |

Spring 2 | k_{2} | (1 + 0.1j)(2 × 10^{5}) N/m |

Spring 3 | k_{3} | (1 + 0.1j)(3 × 10^{5}) N/m |

Figure 2 shows a color contour of the velocity amplitude versus location and frequency while Fig. 3 shows the velocity amplitude versus frequency at the locations where the dashpots are attached. The effects of the dashpots grow with frequency, as expected from classical analysis of viscously damped systems. Figure 4 shows the element loss factors for the three dashpots as well as the system loss factor that is the sum of the three element loss factors.

Inspection of Figs. 3 and 4 reveals the utility of writing the system loss factor as a sum over elements. While dashpot *c*_{1} experiences the largest velocity over most of the frequency range, its contribution to the system loss factor is only largest over a much smaller frequency range. While *c*_{3} is the largest dashpot constant, its contribution to the system loss factor is only greatest over a very small frequency range. This example shows that the decomposition of the system loss factor is essential to understand the contribution of each damped element to the system loss factor, as it cannot be guessed from the dashpot velocity or the dashpot constant.

Figure 5 contains a schematic of the second example, a beam in flexure connected to ground by springs. This example is identical to the first one, however the dashpots have been replaced by damped springs that provide damping through a complex-valued stiffnesses as given in Table 1. Figure 6 shows a color contour of the displacement amplitude versus location and frequency while Fig. 7 shows the displacement amplitude versus frequency at the locations where the damped springs are attached. Figure 8 shows the element loss factors for the three springs as well as the system loss factor that is the sum of the three element loss factors. While the displacement of *k*_{3} is smaller than the displacements of the other two damped springs over most of the frequency range, Fig. 8 shows that the contribution of *k*_{3} to the system loss factor is highest over most of the frequency range. Again, the element loss factors provide a clear picture of the relative importance of elements in the model.

## Conclusions

In this work, the system loss factor was made unique by defining the total energy of a dynamic system undergoing periodic motion to be equal to the average of the total energy over a cycle. This uniqueness of the system loss factor allows comparisons of damping effectiveness in complex systems, regardless of the damping model used. Analysis of the system loss factor is illustrated for models using viscous damping and structural damping. The system loss factor was written as a sum over the element loss factors, providing for a graphic representation of element loss factors with a proposed name of *loss factor image*. The loss factor image illustrates the spatial distribution of contributions to the system loss factor. For systems with light viscous damping, it was shown that the system loss factor is equal to twice the modal damping ratio when the system is excited at an undamped natural frequency and vibrates in the corresponding undamped mode shape. Examples were presented for a beam with viscous dashpots and damped springs, illustrating the generality of the definition as well as the power of the loss factor image in ranking the contributions of individual elements to the system loss factor.

## Acknowledgment

This work was supported by ONR under Award Number N00014-19-1-2100.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

## References

*Proceedings of the Annual Meeting*

*Papers Presented at a Colloquium on Structural Damping Held at the ASME Annual Meeting*

*Structural Damping: Papers Presented at a Colloquium on Structural Damping Held at the ASME Annual Meeting*

*Structural Damping: Papers Presented at a Colloquium on Structural Damping Held at the ASME Annual Meeting*

*Structural Damping: Papers Presented at a Colloquium on Structural Damping Held at the ASME Annual Meeting*