Unique Loss Factor Images for Complex Dynamic Systems

Gregory McDaniel, J.; Liem, Alyssa; Kaminski, Allison

doi:10.1115/1.4054360

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 (N³), 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.

Researchers have been searching for ways to quantify and measure damping for at least a century. The 1956 bibliography by Demer [5] details early work, including one of the earliest in 1930 by Kimball [6]. Kimball considered the free vibration of a single degree-of-freedom system. Under the assumption of light damping, approximations were introduced that led to the key result:

Δ W = 2 W δ

(1)

where ΔW 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

δ \approx \frac{Δ y}{y}

(2)

where Δy 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].

Sometime between 1931 and the 1959 meeting of the American Society of Mechanical Engineers [10], attention shifted from free vibration to forced vibration. Attention also shifted to nonhomogenous structures. These shifts prompted a new damping metric, named the system loss factor and often given the symbol 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

η = \frac{D}{2 π W}

(3)

where 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

Begin with the classical definition of the system loss factor [11]:

η = \frac{D}{2 π W}

(4)

where 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:

W = \frac{1}{τ} \int_{0}^{τ} E (t) d t

(5)

where 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

E (t) = T (t) + U (t)

(6)

Substituting Eq. into Eq. gives

W = T_{ave} + U_{ave}

(7)

where T_ave is the average kinetic energy and U_ave is the average potential energy. The system loss factor defined by Eqs. and 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

Consider a finite element model of a viscously damped system. The equation of motion for the eth finite element is

M_{e} {\ddot{x}}_{e} (t) + α_{e} C_{e} {\dot{x}}_{e} (t) + K_{e} x_{e} (t) = F_{e} (t) for e = 1, 2, \dots, E

(8)

where 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

M \ddot{x} (t) + C \dot{x} (t) + K x (t) = F (t)

(9)

The force vector is assumed to be time-harmonic with a frequency ω and is therefore expressed as

F (t) = \tilde{F} (ω) \exp (j ω t)

(10)

where

\tilde{F} (ω)

is a vector of complex-valued force amplitudes that are independent of time. Seeking the steady-state response to this force, the displacement is expressed as

x (t) = \tilde{x} (ω) \exp (j ω t)

(11)

where

\tilde{x} (ω)

is a vector of complex-valued displacement amplitudes that are independent of time. Substitution of Eqs. and into Eq. gives

D (ω) \tilde{x} (ω) = \tilde{F} (ω)

(12)

where the dynamic stiffness matrix is

D (ω) = (j ω)^{2} M + (j ω) C + K

(13)

In many problems, the force vector is known and the displacement is found by a linear solve of Eq. . For the remainder of this section, the frequency dependence of

\tilde{F} (ω)

and

\tilde{x} (ω)

will be omitted.

The time-average of kinetic energy over a cycle is [17]

T_{ave} = (\frac{1}{4}) {\tilde{v}}^{H} M \tilde{v}

(14)

where the complex-valued velocity vector is

\tilde{v} = (j ω) \tilde{x}

(15)

The time-average of potential energy over a cycle is [17]

U_{ave} = (\frac{1}{4}) {\tilde{x}}^{H} K \tilde{x}

(16)

Substituting Eqs. and into Eq. gives

W = (\frac{1}{4}) {\tilde{v}}^{H} M \tilde{v} + (\frac{1}{4}) {\tilde{x}}^{H} K \tilde{x}

(17)

The energy dissipated over a cycle is [17]

D = (\frac{2 π}{ω}) (\frac{1}{2}) {\tilde{v}}^{H} C \tilde{v}

(18)

Substituting Eqs. and into Eq. gives

η = (\frac{2}{ω}) \frac{{\tilde{v}}^{H} C \tilde{v}}{{\tilde{v}}^{H} M \tilde{v} + {\tilde{x}}^{H} K \tilde{x}}

(19)

The physical interpretation of 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.

In order to quantify the contributions of each finite element to the energy dissipated over a cycle, Eq. is rewritten as a sum over E finite elements

η = \sum_{e = 1}^{E} α_{e} η_{e}

(20)

where the element loss factor is the loss factor contribution of the eth finite element:

η_{e} = (\frac{2}{ω}) \frac{{\tilde{v}}_{e}^{H} C_{e} {\tilde{v}}_{e}}{{\tilde{v}}^{H} M \tilde{v} + {\tilde{x}}^{H} K \tilde{x}}

(21)

The vector

{\tilde{v}}_{e}

holds the velocities associated with the eth 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.

Equations and may be used to assess the sensitivity of the system loss factor with respect to changes of the damping in a single element. This is done by evaluating the first derivative of η with respect to the scaling factor, α_e, which was previously introduced in the equation of motion of the eth finite element given in Eq. . This evaluation gives

\frac{\partial η}{\partial α_{e}} = η_{e}

(22)

Therefore, η_e is the sensitivity of the system loss factor with respect to changes in the damping of the eth 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

This section will explore the relationship of the system loss factor as defined here and the critical damping ratio for the case where there is light damping and the structure is vibrating in a single mode. The undamped eigenvalue properties used below may be found in Ref. [16]. The analysis begins with , repeated here for convenience:

M \ddot{x} (t) + C \dot{x} (t) + K x (t) = F (t)

(23)

The associated undamped eigenvalue problem is

(M - ω_{n}^{2} K) ϕ_{n} = 0

(24)

where ϕ_n is the nth undamped mode shape and ω_n is the nth 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:

\begin{aligned} ϕ_{m}^{T} M ϕ_{n} = {\begin{matrix} 0 for m \neq n \\ 1 for m = n \end{matrix} \end{aligned}

(25)

\begin{aligned} ϕ_{m}^{T} K ϕ_{n} = {\begin{matrix} 0 for m \neq n \\ ω_{n}^{2} for m = n \end{matrix} \end{aligned}

(26)

The “light damping approximation” presented in Ref. [16, Chapter 4] presents an approximation of the modal damping ratio given by

ζ_{n} = \frac{1}{2 ω_{n}} ϕ_{n}^{T} C ϕ_{n}

(27)

Note that this approximation is exact if certain conditions are satisfied [18], most notably proportional damping [16, Section 4.3].

Now assume that the structure is excited at its nth undamped natural frequency, ω_n, and is vibrating in its nth undamped mode shape, ϕ_n, so that

ω = ω_{n}

(28)

\tilde{x} = C_{n} ϕ_{n}

(29)

\tilde{v} = j ω_{n} C_{n} ϕ_{n}

(30)

Substituting Eqs. – and Eqs. – into the definition of system loss factor in Eq. gives

η = \frac{1}{ω_{n}} ϕ_{n}^{T} C ϕ_{n}

(31)

Comparing Eqs. and gives

η = 2 ζ_{n}

(32)

Under the light damping approximation, the system loss factor defined here is equal to twice the modal damping ratio in cases where the structure is vibrating in the mode shape and at the undamped natural frequency of one mode.

Applications to Systems With Structural Damping

The system loss factor will now be applied to a system with structural damping [19,20] represented by a complex-valued stiffness matrix. Some steps in the analysis are repeated from this section so this section presents a complete analysis. The force vector is assumed to be time-harmonic with a frequency ω and is therefore expressed as

F (t) = \tilde{F} (ω) \exp (j ω t)

(33)

where

\tilde{F} (ω)

is a vector of complex-valued force amplitudes that are independent of time. Seeking the steady-state response to this force, the displacement is expressed as

x (t) = \tilde{x} (ω) \exp (j ω t)

(34)

where

\tilde{x} (ω)

is a vector of complex-valued displacement amplitudes that are independent of time.

The equation of motion for one element with structural damping is

[(j ω)^{2} M_{e} + K_{e}] {\tilde{x}}_{e} (ω) = {\tilde{F}}_{e} (ω)

(35)

where 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

[(j ω)^{2} M + K] \tilde{x} (ω) = \tilde{F} (ω)

(36)

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

K = K_{r} + j K_{i}

(37)

The time-average of kinetic energy over a cycle is [17]

T_{ave} = (\frac{1}{4}) {\tilde{v}}^{H} M \tilde{v}

(38)

where the complex-valued velocity vector is

\tilde{v} = (j ω) \tilde{x}

(39)

The time-average of potential energy over a cycle is [17]

U_{ave} = (\frac{1}{4}) {\tilde{x}}^{H} K_{r} \tilde{x}

(40)

Substituting Eqs. and into Eq. gives

W = (\frac{1}{4}) {\tilde{v}}^{H} M \tilde{v} + (\frac{1}{4}) {\tilde{x}}^{H} K_{r} \tilde{x}

(41)

The dissipated power is given by

P (t) = F^{T} (t) v (t)

(42)

The energy dissipated over a cycle is

D = \int_{0}^{τ} P (t) d t

(43)

where τ is the period of vibration given by τ = 2π/ω. Substituting Eqs. – and into Eq. and performing the integration results in

D = π {\tilde{x}}^{H} K_{i} \tilde{x}

(44)

Substituting Eqs. and into Eq. gives the system loss factor

η = 2 \frac{{\tilde{x}}^{H} K_{i} \tilde{x}}{{\tilde{v}}^{H} M \tilde{v} + {\tilde{x}}^{H} K_{r} \tilde{x}}

(45)

In order to quantify the contributions of each finite element to the energy dissipated over a cycle, Eq. is rewritten as a sum over finite elements:

η = \sum_{e = 1}^{E} η_{e}

(46)

where the element loss factor is the loss factor contribution of the eth finite element:

η_{e} = 2 \frac{{\tilde{x}}_{e}^{H} Im {K_{e}} {\tilde{x}}_{e}}{{\tilde{v}}^{H} M \tilde{v} + {\tilde{x}}^{H} K_{r} \tilde{x}}

(47)

The vector

{\tilde{x}}_{e}

holds the displacements associated with the eth 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 0.5f₁ to 1.5f₁₀, where f₁ and f₁₀ 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].

Fig. 1

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Schematic of a beam in flexure attached to ground by three dashpots

Table 1

Properties and dimensions for systems shown in Figs. 1 and 5

Variable name	Variable symbol	Value
Beam Young’s modulus	E	2.0 × 10¹¹ Pa
Beam Poisson’s ratio	ν	0.3
Beam mass density	ρ	7800 kg/m³
Beam length	L	1 m
Beam base	b	0.01 m
Beam height	h	0.01 m
Dashpot 1	c₁	100 N s/m
Dashpot 2	c₂	200 N s/m
Dashpot 3	c₃	300 N s/m
Spring 1	k₁	(1 + 0.1j)(1 × 10⁵) N/m
Spring 2	k₂	(1 + 0.1j)(2 × 10⁵) N/m
Spring 3	k₃	(1 + 0.1j)(3 × 10⁵) N/m

Variable name	Variable symbol	Value
Beam Young’s modulus	E	2.0 × 10¹¹ Pa
Beam Poisson’s ratio	ν	0.3
Beam mass density	ρ	7800 kg/m³
Beam length	L	1 m
Beam base	b	0.01 m
Beam height	h	0.01 m
Dashpot 1	c₁	100 N s/m
Dashpot 2	c₂	200 N s/m
Dashpot 3	c₃	300 N s/m
Spring 1	k₁	(1 + 0.1j)(1 × 10⁵) N/m
Spring 2	k₂	(1 + 0.1j)(2 × 10⁵) N/m
Spring 3	k₃	(1 + 0.1j)(3 × 10⁵) 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.

Fig. 2

Color contour of velocity amplitude, |v~(ω)|, versus location, x, and frequency, f. The velocity amplitude is plotted in dB re 1 m/s.

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Color contour of velocity amplitude, $| \tilde{v} (ω) |$ ⁠, versus location, x, and frequency, f. The velocity amplitude is plotted in dB re 1 m/s.

Fig. 3

Velocity amplitude, |v~(ω)|, at locations where dashpots are attached: x1 = 0, x2 = L/2, and x3 = L. The velocity amplitude is plotted in dB re 1 m/s.

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Velocity amplitude, $| \tilde{v} (ω) |$ ⁠, at locations where dashpots are attached: x₁ = 0, x₂ = L/2, and x₃ = L. The velocity amplitude is plotted in dB re 1 m/s.

Fig. 4

Logarithm of system loss factor, η, and logarithm of element loss factor, ηe, for attached dashpots c1–c3 attached at x1 = 0, x2 = L/2, and x3 = L. The system loss factor was computed from Eq. (19) and the element loss factor was computed from Eq. (21).

View large Download slide

Logarithm of system loss factor, η, and logarithm of element loss factor, η_e, for attached dashpots c₁–c₃ attached at x₁ = 0, x₂ = L/2, and x₃ = L. The system loss factor was computed from Eq. (19) and the element loss factor was computed from Eq. (21).

Inspection of Figs. 3 and 4 reveals the utility of writing the system loss factor as a sum over elements. While dashpot c₁ 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 c₃ 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 k₃ is smaller than the displacements of the other two damped springs over most of the frequency range, Fig. 8 shows that the contribution of k₃ 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.

Fig. 5

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Schematic of a beam in flexure attached to ground by three damped springs

Fig. 6

Plot of displacement amplitude, |x~(ω)|, versus location, x, and frequency, f. The displacement amplitude is plotted in dB re 1 m.

View large Download slide

Plot of displacement amplitude, $| \tilde{x} (ω) |$ ⁠, versus location, x, and frequency, f. The displacement amplitude is plotted in dB re 1 m.

Fig. 7

Displacement amplitude, |x~(ω)|, at locations where springs are attached: x1 = 0, x2 = L/2, and x3 = L. The displacement amplitude is plotted in dB re 1 m.

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Displacement amplitude, $| \tilde{x} (ω) |$ ⁠, at locations where springs are attached: x₁ = 0, x₂ = L/2, and x₃ = L. The displacement amplitude is plotted in dB re 1 m.

Fig. 8

Logarithm of system loss factor, η, and logarithm of element loss factor, ηe, for attached damped springs k1–k3 attached at x1 = 0, x2 = L/2, and x3 = L. The system loss factor was computed from Eq. (45) and the element loss factor was computed from Eq. (47).

View large Download slide

Logarithm of system loss factor, η, and logarithm of element loss factor, η_e, for attached damped springs k₁–k₃ attached at x₁ = 0, x₂ = L/2, and x₃ = L. The system loss factor was computed from Eq. (45) and the element loss factor was computed from Eq. (47).

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.

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Unique Loss Factor Images for Complex Dynamic Systems

Abstract

Introduction

System Loss Factor

Application to Viscously Damped Systems

Relationship to the Critical Damping Ratio for a Mode

Applications to Systems With Structural Damping

Numerical Examples

Conclusions

Acknowledgment

Conflict of Interest

Data Availability Statement

References

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Unique Loss Factor Images for Complex Dynamic Systems

Abstract

Introduction

System Loss Factor

Application to Viscously Damped Systems

Relationship to the Critical Damping Ratio for a Mode

Applications to Systems With Structural Damping

Numerical Examples

Conclusions

Acknowledgment

Conflict of Interest

Data Availability Statement

References

Get Email Alerts

Cited By

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