Primary Resonance in a Weakly Forced Oscillator With Both Parametric Damping and Stiffness

Ardister, Jamal; Afzali, Fatemeh; Feeny, Brian F.

doi:10.1115/1.4065037

Abstract

We study the primary resonance of a parametrically damped Mathieu equation with direct excitation. Potential applications include wind-turbine blade vibration with cyclic stiffening and aeroelastic effects, which may induce parametric damping, and devices with designed cyclic damping for resonance manipulation. The parametric stiffness, parametric damping, and the direct forcing all have the same excitation frequency, with phase parameters between these excitation sources. The parametric amplification at primary resonance is examined by applying the second-order method of multiple scales. With parametric stiffness and direct excitation, it is known that there is a primary parametric resonance that is an amplifier under most excitation phases, but can be a slight suppressor in a small range of phases. The parametric damping is shown to interact with the parametric stiffness to further amplify, or suppress, the resonance amplitude relative to the resonance under parametric stiffness. The effect of parametric damping without parametric stiffness is to shift the resonant frequency slightly, while inducing less significant resonance amplification. The phase of the parametric damping excitation, relative to the parametric stiffness, has a strong influence on the amplification or suppression characteristics. There are optimal phases of both the direct excitation and the parametric damping for amplifying or suppressing the resonance. The effect of the strength of parametric damping is also studied. Numerical simulations validate the perturbation analysis.

Graphical Abstract Figure

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1 Introduction

We study the responses of a system with both parametric stiffness and damping, while also under direct excitation. Our interest is in the resonance behavior of this dynamical system. Parametric stiffness is well known to occur in physical systems, for example a base-excited pendulum, a string with cyclic tension, and ship roll. Parametric damping can occur in aeroelastic oscillators or can be specifically designed into a system.

To illustrate parametric damping from aerodynamic affects, we can consider an airfoil with transverse displacement and velocity $x$ and $\dot{x}$ ⁠, in an ambient flow of speed $u$ with a relative cyclic direction, $α_{0} (t) \approx a_{0} + a_{1} \cos Ω t$ ⁠, as shown in Fig. 1, where $a_{0}$ and $a_{1}$ are the mean and fluctuation of the cyclic relative angle. This can happen, for example, in wind-turbine blades, which rotate in the flow field. Then the relative flow velocity is $v = u \cos α_{0} (t) i + (u \sin$ $α_{0} (t) - \dot{x})) j = v_{1} i + v_{2} j$ ⁠, where $i$ and $j$ are unit vectors in the directions in-line and transverse to the airfoil’s chord. The angle of attack is then $α = \tan^{- 1} v_{2} / v_{1} \approx α_{0} (t) - \dot{x} / u$ ⁠, and the flow speed squared is $v^{2} = u^{2} - 2 u \dot{x} \sin α + {\dot{x}}^{2} \approx u^{2} - 2 u α_{0} (t) \dot{x}$ ⁠, for small $α$ and $\dot{x}$ ⁠. The lift force is then $F_{L} = \frac{1}{2} C_{L} (α) ρ c v^{2} \approx (c_{L 0} + c_{L 1} α) v^{2},$ on an airfoil of chord length $c$ in a fluid of density $ρ$ ⁠, which takes the form $F_{L} \approx (c_{L 0} + c_{L 1} a_{0}) u^{2} + c_{L 1} u^{2} a_{1} \cos Ω t - 2 c_{L 0} u a_{0} \dot{x} - 2 c_{L 1} u a_{1} \cos Ω t \dot{x} .$ The first two terms are direct loading and the last term promotes cyclic damping. (This approximation neglects possible hysteresis in the lift force under oscillation [1].) Relative freestream oscillation can happen in idealized vertical-axis wind turbines [2], which rotate in the face of oncoming wind, and horizontal-axis wind turbines, which rotate in a wind-shear profile that can change in magnitude and direction with altitude. In addition, studies suggest the presence of parametric stiffness due to the effect of the gravitational field causing cyclic variations of tension and compression on the rotating blade, which may be significant when the turbines are very large [3–8].

Fig. 1

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(⁠ $a$ ⁠) The angle of attack of the relative velocity of wind $u$ on an airfoil depends on the cyclic angle of the wind and on the flatwise coordinate velocity $\dot{x}$ ⁠. (⁠ $b$ ⁠) An eddy-current damper can be designed for parametric damping if the magnet current $i (t)$ is cyclic.

Cyclic damping could be implemented by design using eddy-current damping [9–12], perhaps with electromagnets with cyclic input current, although such a device would be limited to positive net damping. An example schematic is shown in Fig. 1(b). Such a device could be relevant to the design of parametric amplification, which has been done in micro electromechanical systems resonators with combined forcing and parametric stiffness [13].

Combining cyclic parametric damping, parametric stiffness, and direct excitation results in an equation as follows:

\ddot{x} + c (t) \dot{x} + k (t) x = f (t)

(1)

where

c (t), k (t),

and

f (t)

are periodic with frequency

Ω

⁠. Equation is linear and has a homogeneous and particular solution,

x_{h}

and

x_{p}

⁠, respectively, where

x_{h}

satisfies the equation with

f (t) = 0

⁠. Floquet-based methods are often applied in analysis of the homogeneous case. In some special cases, the homogeneous equation is a reduction of Ince’s equation [14]. When

c (t)

is constant and

f (t) = 0

⁠, a special case is the damped Mathieu equation,

\ddot{x} + 2 ζ ω \dot{x} + (ω^{2} + ε \cos Ω t) x = 0

⁠, which has been thoroughly investigated for its stability [14–16] and response characteristics [17]. The case where

k (t)

is constant and

f (t) = 0

⁠, that of parametric damping, was studied for its stability (with application to rain-wind-induced vibration) [18] and response characteristics [19]. The case of combined parametric damping and stiffness with

f (t) = 0

was also studied [20]. Forced (⁠

f (t) \neq 0

⁠) parametrically excited linear and nonlinear systems have been studied mostly pertaining to parametric stiffness. The forced Mathieu equation (⁠

c (t)

is constant) with and without nonlinearity has been studied for parametric amplification [13,21–25] and other applications [26–31], with extensions to autoparametric systems [32–35], such as vibration absorbers, rotor dynamics, and energy harvesting. Studies on nonlinear generalizations include Refs. [36–45]. Nonlinear systems with parametric damping particularly have also been studied [46,47].

Perturbation methods can be used on the forced equation to obtain the resonances and instabilities. Examples include linear and nonlinear generalizations of the forced Mathieu equation [5,6,13,22,37,44,48–50].

In this work, we show the details of the analysis when Eq. takes the specific form of

\begin{aligned} \ddot{x} + 2 ε (μ_{0} + μ_{1} \cos (Ω t + ϕ)) \dot{x} + (ω^{2} + ε γ \cos Ω t) x = ε F \sin (Ω t + θ) \end{aligned}

(2)

where

ε

is a small parameter,

ω

is the natural frequency,

Ω

is the excitation frequency,

μ_{0}

is the mean damping coefficient,

μ_{1}

and

γ

are the scaled strength of variation of the damping and stiffness, respectively,

ϕ

is a phase of cyclic damping relative to the parametric stiffness,

ε F

is the amplitude of weak cyclic direct excitation, and

θ

is the direct excitation phase relative to the parametric stiffness. The damping

2 ε μ_{0} = 2 ζ ω

indicates that

ζ = ε μ_{0} / ω

is the damping ratio of the reference oscillator.

Equation can be recast by letting

τ = ω t

and

x = (ε F / ω^{2}) y

to obtain

\begin{aligned} y^{″} + (2 ζ + 2 ε ζ_{1} \cos (r τ + ϕ)) y^{'} + (1 + ε δ \cos r τ) y = \sin (r τ + θ) \end{aligned}

(3)

where

ζ_{1} = μ_{1} / ω, δ = γ / ω^{2},

and

r = Ω / ω

⁠, which reveals six independent parameters as

ζ, ε ζ_{1}

⁠,

ε δ

⁠,

r

⁠,

ϕ

⁠, and

θ

⁠. We accommodate this nondimensional form by analyzing Eq. , which is closer to the familiar form of the standard oscillator, and treat it as nondimensional with

ω = 1

and parameters

μ_{0}, μ_{1}, γ, ϕ, θ

⁠, and

Ω

(and with

F

producing only scaling effects by linearity) and use the bookkeeping parameter

ε

for the perturbation expansion.

This study follows up on the previous work with parametric stiffness and direct excitation [50]. In that study, the parametric and direct excitation had the same frequency, and parametric amplification was observed at primary resonance. The current study investigates how the addition of parametric damping may influence the resonance.

2 Method of Multiple Scales: Primary Resonance

A second-order method of multiple scales analysis [15,51] is used to expose the resonance conditions when weak harmonic forcing is applied. Second-order perturbation analyses have been useful in problems with parametric excitation [15,29,31,50,52–54]. In this study, we are only considering primary resonance. Secondary resonances may be present, but will not be explored at this time.

A second-order method of multiple scales analysis starts by letting

\begin{aligned} x = x_{0} (T_{0}, T_{1}, T_{2}) + ε x_{1} (T_{0}, T_{1}, T_{2}) + ε^{2} x_{2} (T_{0}, T_{1}, T_{2}) + O (ε^{3}) \end{aligned}

(4)

using three time scales,

T_{0}, T_{1},

and

T_{2}

⁠, where

T_{i} = ε^{i} t

⁠. The time derivative is of the form

\frac{d}{d t} = D_{0} + ε D_{1} + ε^{2} D_{2} + \dots

⁠, where

D_{i} = \frac{\partial}{\partial T_{i}}

⁠. Inserting these expansions into Eq. yields the following set of equations after collecting the coefficients of

ε^{0}, ε^{1}

⁠, and

ε^{2}

⁠:

O (ε^{0}) : {D_{0}}^{2} x_{0} + ω^{2} x_{0} = 0

(5)

\begin{aligned} O (ε^{1}) : {D_{0}}^{2} x_{1} + ω^{2} x_{1} = - 2 D_{0} D_{1} x_{0} - x_{0} γ \cos (Ω T_{0}) \\ - 2 (μ_{0} + μ_{1} \cos (Ω T_{0} + ϕ)) (D_{0} x_{0}) + F \sin (θ + Ω T_{0}) \end{aligned}

(6)

\begin{aligned} O (ε^{2}) : {D_{0}}^{2} x_{2} + ω^{2} x_{2} = - 2 D_{0} D_{1} x_{1} - 2 D_{0} D_{2} x_{0} - {D_{1}}^{2} x_{0} \\ - 2 (D_{0} x_{1} + D_{1} x_{0}) (μ_{0} + μ_{1} \cos (Ω T_{0} + ϕ)) - γ x_{1} \cos (Ω T_{0}) \end{aligned}

(7)

Solving the

O (ε^{0})

ordinary differential equation gives the solution to an undamped free vibration shown in the complex exponential form as follows:

x_{0} = A (T_{1}, T_{2}) e^{i ω T_{0}} + c . c

(8)

where

A (T_{1}, T_{2})

is complex and

c . c .

denotes the complex conjugate(s) of the previous term(s).

The solution for the $O (ε^{0})$ equation is then inserted into the $O (ε^{1})$ equation. That is, we insert Eq. (8) into Eq. (6). For primary resonance, we consider $Ω \approx ω$ ⁠. Specifically, we let $Ω = ω + ε σ$ ⁠, where $σ$ is a detuning parameter. In this article, we will only present the results of primary resonance. Therefore, we will not explore the nonresonant, subharmonic, and superharmonic cases here.

We obtain the solvability condition for the

O (ε^{1})

by equating the coefficients of terms that produce secular terms to zero, namely,

2 ω D_{1} A + 2 μ_{0} ω A + \frac{1}{2} F e^{i θ} = 0

(9)

This equation is used in a later stage of the second-order multiple scales analysis. By examining the secular terms in this order, we can see that they do not contain the effect of the parametric damping, as shown by the absence of

μ_{1}

⁠. After removing the secular terms, we solve the

O (ε^{1})

equation in Eq. for

x_{1}

to obtain its particular solution as

x_{1} = K A e^{i T_{0} (Ω + ω)} + L \bar{A} e^{i T_{0} (Ω - ω)} + c . c

(10)

where

K = \frac{γ + 2 i μ_{1} ω e^{i ϕ}}{2 (Ω^{2} + 2 ω Ω)}, L = \frac{γ - 2 i μ_{1} ω e^{i ϕ}}{2 (Ω^{2} - 2 ω Ω)}

(11)

Inserting Eq. into Eq. , letting

Ω = ω + ε σ

⁠, and eliminating the secular terms, the solvability condition of

O (ε^{2})

is obtained as follows:

\begin{aligned} 2 i ω D_{2} A + 2 μ_{0} D_{1} A + D_{1}^{2} A + \frac{1}{2} γ K A \\ + \frac{1}{2} γ \bar{L} A + 2 i μ_{1} Ω K e^{- i ϕ} A + \frac{1}{2} γ L \bar{A} e^{2 i σ T_{1}} = 0 \end{aligned}

(12)

Unlike the first-order solvability condition , this second-order solvability condition does include the effect of the parametric damping, shown by the presence of

μ_{1}

⁠, directly apparent and also through

K

and

L

⁠. Considering

K

and

L

as defined in Eq. and appearing in Eq. , there is one term in Eq. that involves

μ_{1}

independently from

γ

⁠, while the other

μ_{1}

terms show up as

μ_{1} γ

⁠, suggesting that much of the effect of parametric damping occurs in combination with parametric stiffness.

Next, we use the

O (ε^{1})

and

O (ε^{2})

solvability conditions to reconstitute

A

back into the original time scale [51]. We first obtain expressions for

D_{1} A

and

D_{2} A

⁠. From Eq. , we obtain

D_{1} A = - μ_{0} A - \frac{F e^{i θ}}{4 ω} e^{i σ T_{1}}

(13)

Differentiating Eq. yields

D_{1}^{2} A = - μ_{0} D_{1} A - i σ \frac{F e^{i θ}}{4 ω} e^{i σ T_{1}}

(14)

Applying Eq. to the middle term in Eq. leads to

D_{1}^{2} A = {μ_{0}}^{2} A + \frac{μ_{0} F e^{i θ}}{4 ω} e^{i σ T_{1}} - \frac{i σ F e^{i θ}}{4 ω} e^{i σ T_{1}}

(15)

We then insert Eqs. and into Eq. to obtain

\begin{aligned} D_{2} A & = \frac{1}{4 ω} (2 i (2 μ_{0} D_{1} A + D_{1}^{2} A) - A (2 μ_{1} (K e^{- i ϕ} (ω + Ω) \\ + \bar{L} e^{i ϕ} (ω - Ω)) - i γ (K + \bar{L})) + L e^{2 i σ T_{1}} \bar{A} (i γ + 2 μ_{1} e^{i ϕ} (ω - Ω))) \end{aligned}

(16)

By using Eqs. and , we consider the multiple-scale derivative and write

\frac{d A}{d t} = ε D_{1} A + ε^{2} D_{2} A

⁠. This results in

\begin{aligned} \frac{d A}{d t} & = - \frac{ε}{4 ω} (F e^{i (θ + σ ε t)} + 4 μ_{0} ω A) \\ + \frac{ε^{2}}{8 ω^{2}} (F (σ - i μ_{0}) e^{i (θ + σ ε t)} + 2 i ω γ K A \\ + 2 i ω γ \bar{L} A - 4 i μ_{0}^{2} ω A - 8 μ_{1} ω^{2} K e^{- i ϕ} A + 2 i ω γ L \bar{A} e^{2 i σ ε t}) \end{aligned}

(17)

Note that in this process of reconstitution, we have substituted

T_{1} = ε t

⁠, and

A (T_{1}, T_{2})

is now expressed as

A (t)

⁠.

Here, we can see that the parametric damping affects the resonant amplitude at a very slow time scale indicated by the

μ_{1}

terms in Eq. being of

O (ε^{2})

after reconstitution of

A

⁠. We let

A = (X + i Y) e^{i σ ε t},

where

X

and

Y

are real and dependent on

t

(equivalently, dependent on

T_{1}

and

T_{2}

prior to reconstitution). Inserting this representation of

A

into Eq. , applying Eq. for

K

and

L

using

Ω = ω

and then separating the real and imaginary parts, expressions involving

\frac{d X}{d t}

and

\frac{d Y}{d t}

are obtained:

\begin{aligned} \frac{d X}{d t} + (ε μ_{0} + \frac{ε^{2} γ μ_{1} \cos ϕ}{4 ω^{2}}) X \\ - (ε σ - \frac{ε^{2} γ^{2}}{24 ω^{3}} + \frac{ε^{2} μ_{0}^{2}}{2 ω} + \frac{ε^{2} μ_{1}^{2}}{3 ω} - \frac{ε^{2} γ μ_{1} \sin ϕ}{12 ω^{2}}) Y = b_{1} \end{aligned}

(18)

\begin{aligned} \frac{d Y}{d t} + (ε σ + \frac{5 ε^{2} γ^{2}}{24 ω^{3}} + \frac{ε^{2} μ_{0}^{2}}{2 ω} + \frac{ε^{2} μ_{1}^{2}}{3 ω} + \frac{5 ε^{2} γ μ_{1} \sin ϕ}{12 ω^{2}}) X \\ + (ε μ_{0} - \frac{ε^{2} γ μ_{1} \cos ϕ}{4 ω^{2}}) Y = b_{2} \end{aligned}

(19)

where

b_{1} = \frac{ε^{2} F μ_{0} \sin (θ)}{8 ω^{2}} + \frac{ε^{2} F σ \cos (θ)}{8 ω^{2}} - \frac{ε F \cos (θ)}{4 ω}

(20)

b_{2} = - \frac{ε^{2} F μ_{0} \cos (θ)}{8 ω^{2}} + \frac{ε^{2} F σ \sin (θ)}{8 ω^{2}} - \frac{ε F \sin (θ)}{4 ω}

(21)

Equations and take the matrix form

\dot{x} + A x = b,

where

x = [X Y]^{T}

⁠,

b = [b_{1} b_{2}]^{T}

is the forcing vector, and

A

is the matrix of coefficients (these are constants, not to be confused with complex scalar amplitude

A

⁠). The steady-state solution is given by

x = A^{- 1} b .

The

A

matrix also holds useful information about stability. In this case, the determinant of

A

changes sign as the parameters are varied through a stability transition. Since the determinant is in the denominator of the steady-state solution, instability occurs as the homogeneous system crosses the boundary of a region of instability, and simultaneously the steady-state forced amplitude becomes unbounded. This pattern also occurs in cases of parametric stiffness only [13,23,24,50]. The effect of parameters on stability of the current system was examined in Ref. [20].

We solve for

X

and

Y

and seek the steady-state response amplitude,

a

⁠, defined using

A = \frac{1}{2} a e^{i β} = (X + i Y) e^{i ε σ t}

=

B e^{i ε σ t}

⁠. If

B = \frac{1}{2} a e^{- i ψ}

and

Ω = ω + ε σ

⁠, then the leading-order solution in Eq. becomes

x_{0} = A e^{i ω T_{0}} + c . c . = B e^{i Ω T_{0}} + c . c . = a \cos (Ω T_{0} - ψ)

(22)

This indicates that the response occurs at the excitation frequency, with a phase lag

ψ

relative to the parametric excitation. Note that

| X + i Y | = a / 2

⁠. Therefore, the response amplitude is

a = 2 \sqrt{X^{2} + Y^{2}}

⁠. As the parametric damping amplitude,

μ_{1}

⁠, and phase,

ϕ

⁠, are involved in the matrix

A

⁠, we expect them to influence the response amplitude and phase,

a

and

ψ

⁠, respectively.

The solution to

x_{1}

contributes to the next order of approximation,

x \approx x_{0} + ε x_{1}

⁠. From Eq. , for the case of

Ω = ω + ε σ

⁠, we have

x_{1} = K A e^{i (2 Ω - ε σ) T_{0}} + L \bar{A} e^{i σ T_{1}} + c . c .

Using

A = B e^{i ε σ t} = \frac{1}{2} a e^{- i ψ} e^{i σ T_{1}}

⁠, then

x_{1} = \frac{K a}{2} e^{i (2 Ω T_{0} - ψ)} + \frac{L a}{2} e^{i ψ} + c . c

(23)

The complex

K

and

L

can be further simplified using

Ω \approx ω

⁠, and

x_{1}

can be put into real form. The point is that the first term contributes a harmonic of frequency

2 Ω

⁠, and the second term is a constant term. Both terms are scaled at order

ε a

in the asymptotic approximation.

3 Behavior of Primary Resonance

In this section, we display features of the response amplitudes and phases of the leading-order term $x_{0}$ for various parameter cases. Our interest is in the general behavior of this dynamical system and not the specific behavior of a device. Thus, parameters are chosen to illustrate possible system behavior and expand on the case presented for a forced system with only parametric stiffness [50].

To this end, we will use the values of $θ = 3 π / 4$ and $π / 4$ ⁠, and $ϕ = π / 2$ and $- π / 2$ ⁠, as it turns out that they show the extreme effects of parametric stiffness and damping, respectively. We have also focused on values of $γ$ near $γ = 3.5$ to illustrate distinguishable features in the dynamics presented, without going unstable in the ranges of damping used. Such values of $γ$ are easily achieved in some mechanical systems. For example, a linearized pendulum with vertical base-excitation displacement $a \cos (Ω t)$ has $ε γ = Ω^{2} a / g$ ⁠, and a first-mode approximation of a string stretched by $δ_{0} + δ_{1} \cos (Ω t)$ will achieve a parametric variation $ε γ / ω^{2} = δ_{1} / δ_{0}$ ⁠. The parametric effect of a horizontal-axis wind turbine is more speculative, but seems to become significant as the blades get large (say 80–100 m or more) [4,55,56] depending on the geometry. The bookkeeping parameter is chosen as $ε = 0.1$ throughout, such that other parameters are roughly in the next order of magnitude. For the lower values of $γ$ the effects are still present, but are not as strong.

As noted, the effect of $μ_{1}$ on primary resonance is strongest in collaboration with parametric stiffness $γ$ ⁠. Therefore, we first look at the behavior of these combined effects and then briefly look at the resonance behavior when $γ = 0$ and $μ_{1} \neq 0$ ⁠. For all of the cases presented, $ω = 1$ ⁠, and the parameter values given are nondimensional.

3.1 Combined Parametric Damping and Stiffness.

Figure 2 shows the effect of $μ_{1}$ for $ϕ = π / 2$ ⁠, $θ = 3 π / 4$ ⁠, $γ = 3.5$ ⁠, $μ_{0} = 0.25,$ and $F = 1$ with $μ_{1}$ varying from 0 to 0.5. Figure 2(a) plots the real and imaginary parts of the complex response coefficients $X + i Y$ for various values of $μ_{1}$ ⁠. This plot packages the response amplitude and phase together and shows that the resonance is amplified with $μ_{1}$ ⁠. The smallest ellipse represents $μ_{1} = 0$ and the largest represents $μ_{1} = 0.5$ ⁠, showing parametric damping resonance amplification for $ϕ = π / 2$ and $θ = 3 π / 4$ radians. The maximum amplitude occurs on or close to the imaginary axis, implying that the response phase $ψ$ of the resonance peak is $π / 2$ for all $μ_{1}$ shown.

Fig. 2

Resonance characteristics for ϕ=π/2, θ=3π/4, μ0=0.25, ε=0.1, γ=3.5, 0≤μ1≤0.5 in increments of 0.1, Ω=ω+εσ, and F=1. (a) Half complex amplitudes (X+iY) parameterized by −1≤σ≤1. Amplitudes increase as μ1 increases. (b) Response amplitude, a, versus Ω. The circles and triangles show numerical simulation results for μ1=0.5 and μ1=0.3, respectively. (c) Response phase lag α=ψ+θ versus Ω.

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Resonance characteristics for $ϕ = π / 2$ ⁠, $θ = 3 π / 4$ ⁠, $μ_{0} = 0.25$ ⁠, $ε = 0.1$ ⁠, $γ = 3.5$ ⁠, $0 \leq μ_{1} \leq 0.5$ in increments of 0.1, $Ω = ω + ε σ$ ⁠, and $F = 1$ ⁠. (⁠ $a$ ⁠) Half complex amplitudes (⁠ $X + i Y$ ⁠) parameterized by $- 1 \leq σ \leq 1$ ⁠. Amplitudes increase as $μ_{1}$ increases. (⁠ $b$ ⁠) Response amplitude, $a,$ versus $Ω$ ⁠. The circles and triangles show numerical simulation results for $μ_{1} = 0.5$ and $μ_{1} = 0.3$ ⁠, respectively. (⁠ $c$ ⁠) Response phase lag $α = ψ + θ$ versus $Ω$ ⁠.

Figure 2(b) depicts the resonance peak, plotting the amplitude as a function of excitation frequency, $Ω$ ⁠. When $μ_{1} = 0$ ⁠, there is already an amplified resonance due to the presence of parametric stiffness with $γ = 3.5$ ⁠, for which the resonance peak is increased and shifted to the left relative to the standard (nonparametric) forced oscillator. With $ϕ = π / 2,$ the peak increases with increasing $μ_{1}$ ⁠, shown in the range of $0 \leq μ_{1} \leq 0.5$ ⁠. For example, the case of $μ_{1} = 0.5$ shows significant amplification relative to the case without parametric damping. This might be interpreted as a reduced damping effect with the introduction of parametric damping at $ϕ = π / 2$ ⁠. Also increasing $μ_{1}$ pushes the peak further to the left, indicating a decrease in the resonance frequency.

Figure 2(c) plots the response phase lag as $α = ψ + θ$ (i.e., here, $α$ is relative to the direct excitation) as a function of $Ω$ ⁠, and shows a minimal effect of parametric damping, as the curves are virtually indistinguishable from one another in the range of $μ_{1}$ presented. At resonance, the response lags the excitation by approximately $π / 2$ ⁠, like a standard forced oscillator.

The symbols shown in Fig. 2(b) are numerical validations. The circles were obtained for $μ_{1} = 0.5$ ⁠, and the triangles were obtained for $μ_{1} = 0.3$ ⁠, both using the other parameters listed for the figure. The equation of motion Eq. (2) was numerically solved with matlab function ode45 over many periods to achieve steady state. The fast Fourier transform (FFT) was applied to a period of response data, and the peak value at the excitation frequency was used to extract $a$ ⁠. The sampling was conducted at an exact integer number of samples per period (in this case 100 samples per period) to prevent leakage effects. The numerical solutions also included small mean and $2 Ω$ components as expected from the analysis of $x_{1}$ ⁠.

Figure 3 shows the effect of $μ_{1}$ for $ϕ = - π / 2$ ⁠, $θ = 3 π / 4$ ⁠, $γ = 3.5$ ⁠, $μ_{0} = 0.25,$ and $F = 1$ as $μ_{1}$ varies from 0 to 0.5. Figure 3(a) shows the half complex amplitudes (⁠ $X + i Y$ ⁠). The smallest ellipse represents $μ_{1} = 0.5$ and the largest ellipse represents $μ_{1} = 0$ ⁠. Thus, the ellipses decrease as the parametric damping increases. This indicates that the parametric damping with $ϕ = - π / 2$ has become a parametric suppressor as opposed to an amplifier as seen with $ϕ = π / 2$ in Fig. 2. Like Fig. 2, Fig. 3 also shows some amount of vertical symmetry, hinting that $μ_{1}$ still has very little effect on the phase at resonance. Figure 3(b) further shows the suppressing effect as $μ_{1}$ is increased for $ϕ = - π / 2$ ⁠. The case of $μ_{1} = 0$ shows the highest peak. Numerically simulated amplitudes are shown for the case of $μ_{1} = 0.5$ ⁠. The effects of suppression are not as strong as the effects of amplification shown in Fig. 2. In contrast to the case of amplification in Fig. 2(b), Fig. 3(b) shows that the resonance peaks shift slightly to the right with increasing $μ_{1}$ ⁠. Figure 3(c) displays the phase lag $α = ψ + θ$ for the case when $ϕ = - π / 2$ ⁠, again lagging the excitation by approximately $π / 2$ at resonance.

Fig. 3

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Resonance characteristics for $ϕ = - π / 2$ ⁠, $θ = 3 π / 4$ ⁠, $μ_{0} = 0.25$ ⁠, $ε = 0.1$ ⁠, $γ = 3.5$ ⁠, $0 \leq μ_{1} \leq 0.5$ in increments of 0.1, $Ω = ω + ε σ$ ⁠, and $F = 1$ ⁠. (⁠ $a$ ⁠) Half complex amplitudes (⁠ $X + i Y$ ⁠) parameterized by $- 1 \leq σ \leq 1$ ⁠. Amplitudes decrease as $μ_{1}$ increases. (⁠ $b$ ⁠) Response amplitude, $a$ ⁠, versus $Ω$ ⁠. The circles show numerical simulation results for $μ_{1} = 0.5$ ⁠. (⁠ $c$ ⁠) Response phase $ψ + θ$ versus $Ω$ ⁠.

Figure 4 indicates the effect of $μ_{1}$ for $ϕ = π / 2$ ⁠, $θ = π / 4$ ⁠, $γ = 3.5$ ⁠, $μ_{0} = 0.25,$ and $F = 1$ with $μ_{1}$ varying from 0 to 0.5. This value of $θ$ produces a twin-peak response as observed in Ref. [50]. Figure 4(a) shows how the twin-peak resonance can be interpreted as the (half) complex amplitude with a horizontal major axis, such that the local maximum amplitudes occur at two points (associated with frequencies in the parameterization) on the ellipse. With increasing $μ_{1}$ ⁠, the ellipses become shorter and wider, thus accentuating the local maximum (twin peaks of the resonance). The ellipse corresponding to $μ_{1} = 0.5$ is the widest with the largest major axis, and the ellipse of $μ_{1} = 0$ has the smallest major axis. The vertical symmetry suggests here that the phase lag when the amplitude is at the local minimum is close to $π / 2.$ Figure 4(b) shows the twin peaks on a magnified frequency axis to help visualize the features. The circles indicate numerical simulations for the case of $μ_{1} = 0.5$ ⁠, showing good agreement. The implication is that a frequency sweep may detect two resonant frequencies in the vicinity of primary resonance in this single-degree-of-freedom oscillator. Figure 4(c) plots the phase lag varying between 0 and $π$ as in a nonparametric forced oscillator, but with some nuanced features.

Fig. 4

Resonance characteristics for ϕ=π/2, θ=π/4, μ0=0.25, ε=0.1, γ=3.5, 0≤μ1≤0.5 in increments of 0.1, Ω=ω+εσ, and F=1. (a) Half complex amplitudes (X+iY) parameterized by −1≤σ≤1. The curves become wider as μ1 increases. (b) Response amplitude, a, versus Ω. The circles indicate numerical simulations for the case of μ1=0.5. (c) Response phase ψ+θ versus Ω.

View large Download slide

Resonance characteristics for $ϕ = π / 2$ ⁠, $θ = π / 4$ ⁠, $μ_{0} = 0.25$ ⁠, $ε = 0.1$ ⁠, $γ = 3.5$ ⁠, $0 \leq μ_{1} \leq 0.5$ in increments of 0.1, $Ω = ω + ε σ$ ⁠, and $F = 1$ ⁠. (⁠ $a$ ⁠) Half complex amplitudes (⁠ $X + i Y$ ⁠) parameterized by $- 1 \leq σ \leq 1$ ⁠. The curves become wider as $μ_{1}$ increases. (⁠ $b$ ⁠) Response amplitude, $a,$ versus $Ω$ ⁠. The circles indicate numerical simulations for the case of $μ_{1} = 0.5$ ⁠. (⁠ $c$ ⁠) Response phase $ψ + θ$ versus $Ω$ ⁠.

We have established that parametric damping can act as both an amplifier and a suppressor relative to the parametric stiffness case, which itself has gain relative to the standard nonparametric oscillator. For design applications of resonators or harvesters, the gain is of interest. The gain is defined as $G = a_{p} / a_{0}$ ⁠, where $a_{p}$ is the peak amplitude of the parametric resonance and $a_{0}$ is the resonant amplitude of the standard oscillator [13,22].

Figure 5(a) shows the gain as a function of the parametric damping phase angle $ϕ$ ⁠. Each curve has a different value of $θ$ ⁠. Solid and dashed curves are used only to help visually distinguish intersecting curves. As $θ$ increases through a $π$ cycle, the solid curves show the phases for which the gain increases with $θ$ ⁠, while the dashed curves show the phases for which the gain decreases. Except when $θ = π / 4$ (the case of twin peaks in Fig. 4), the plotted amplitude gains reach a maximum when $ϕ$ is somewhat near $π / 2$ ⁠. Conversely, for $ϕ$ near $3 π / 2$ (same as $- π / 2$ due to the circular nature of the parameter), we see minima. This is consistent with Figs. 2 and 3.

Fig. 5

Primary resonance gain versus ϕ for parameter values, ε=0.1, μ0=0.25, γ=3.5. (a) The set of curves for θ={0,π8,2π8,…,7π8}, and μ1=0.25. (b) The set of curves for 0≤μ1≤0.5, with θ=3π/4, with numerical simulations (circles) for the case of μ1=0.5.

View large Download slide

Primary resonance gain versus $ϕ$ for parameter values, $ε = 0.1$ ⁠, $μ_{0} = 0.25$ ⁠, $γ = 3.5.$ (⁠ $a$ ⁠) The set of curves for $θ = {0, \frac{π}{8}, \frac{2 π}{8}, \dots, \frac{7 π}{8}}$ ⁠, and $μ_{1} = 0.25$ ⁠. (⁠ $b$ ⁠) The set of curves for $0 \leq μ_{1} \leq 0.5$ ⁠, with $θ = 3 π / 4$ ⁠, with numerical simulations (circles) for the case of $μ_{1} = 0.5.$

Figure 5(b) shows the gain versus $ϕ$ with varying $μ_{1}$ ⁠, while the parametric stiffness phase is held at $θ = 3 π / 4$ ⁠. The horizontal line at an amplitude of about 2.6 is the reference, showing the system without the parametric damping term (⁠ $μ_{1} = 0$ ⁠), but with $γ = 3.5$ ⁠. The addition of parametric damping induces fluctuation in the gain with phase $ϕ$ ⁠, which increases with increasing $μ_{1}$ ⁠. The plot shows that parametric damping has a substantial effect on amplifier gain. The potential for increased amplification with parametric damping is about twice as strong as the potential for suppression. At about $ϕ \approx 0$ and $ϕ \approx π$ ⁠, the curves shift between an amplifier and a suppressor. Numerical simulations (circles) are shown for the case of $μ_{1} = 0.5$ ⁠.

The resonance amplitude can become unbounded as $μ_{1}$ is increased. The approach to a vertical asymptote coincides with an approach to a region of instability in the parameter space. ( The effects of $μ_{1}$ ⁠, $γ$ ⁠, and $ϕ$ on stability and free responses were examined in Ref. [20].) Figure 6 shows the peak of amplitude $a$ versus the strength of the parametric damping, $μ_{1}$ ⁠. Figure 6(a) is for $θ = 3 π / 4$ for a set of values of $ϕ$ ranging from $0$ through a cycle of $2 π$ at increments of $π / 4$ ⁠, with $γ = 3.5$ ⁠. The three intermediate curves involve nearly overlapping curves, as labeled in the figure. The circles on the plot represent numerical simulations for the cases of $ϕ = 0, π / 4,$ and $π / 2$ ⁠. The simulations were conducted at the peak frequency, as determined from a numerical maximum in the asymptotic amplitude expressions (obtained from fixed points of $X$ and $Y$ in Eqs. (18) and (19)), and were allowed to approach steady state over 200–800 periods, the latter being necessary when close to the instability asymptote, at which the real part of the eigenvalues of the system of Eqs. (18) and (19) approach zero. The triangles show selected simulations at $ϕ = 3 π / 4$ and $ϕ = π$ ⁠. While the asymptotic analysis produces nearly indistinguishable curves, the simulation deviation suggests that the true curves may deviate slightly and become distinguishable as $μ_{1}$ gets close to a vertical asymptote.

Fig. 6

Amplitude versus μ1 and γ=3.5, with a set of curves for values of ϕ through a cycle of 2π in increments of π/4. The ∘ symbols are numerical simulations at ϕ=0,π/4, and π/2, while △ symbols are simulations at ϕ=3π/4 and π. (a) θ=3π/4. (b) θ=π/4. Note the difference in the vertical axes of the two plots.

View large Download slide

Amplitude versus $μ_{1}$ and $γ = 3.5$ ⁠, with a set of curves for values of $ϕ$ through a cycle of $2 π$ in increments of $π / 4$ ⁠. The $\circ$ symbols are numerical simulations at $ϕ = 0, π / 4$ ⁠, and $π / 2$ ⁠, while $△$ symbols are simulations at $ϕ = 3 π / 4$ and $π$ ⁠. (⁠ $a$ ⁠) $θ = 3 π / 4 .$ (⁠ $b$ ⁠) $θ = π / 4 .$ Note the difference in the vertical axes of the two plots.

Figure 6(b) is for the case of $θ = π / 4 .$ In both cases shown, as $ϕ$ gets closer to $π / 2$ the amplitudes become unbounded with the lowest critical values of $μ_{1}$ ⁠. As $ϕ$ ranges through a cycle of $2 π$ radians, the stability boundary varies cyclically in the $γ$ -versus- $ω$ space (not plotted), with larger $μ_{1}$ promoting greater variation. The parameter $θ$ shows little effect on the critical values of $μ_{1}$ ⁠, but does influence the stable response amplitudes.

To see the combined effect of the parametric damping and parametric stiffness, Fig. 7 plots half complex amplitudes parameterized in excitation frequency $Ω$ for various $θ$ and $ϕ$ ⁠. The set of values $θ = {0, \frac{π}{8}, \frac{2 π}{8}, \dots, \frac{7 π}{8}}$ and the set of values $ϕ = {0, \frac{π}{8}, \frac{2 π}{8}, \dots, \frac{15 π}{8}}$ ⁠, with $μ_{0} = 0.25$ and $μ_{1} = 0.25$ ⁠, are applied in Fig. 7(a) with $γ = 3$ ⁠, Fig. 7(b) with $γ = 3.5$ ⁠, and Fig. 7(⁠ $c$ ⁠) with $γ = 4$ ⁠. As $γ$ is increased, we can start to see “petals” open up and become more defined. Looking at the plot as collection of “petals on a lotus” helps to distinguish the effects of $θ$ and $ϕ$ ⁠. Each petal is related to a particular $θ$ value, and the group of curves within each petal is made up of different values of $ϕ$ ⁠. The plot gives insight about the resonance amplitude and phase given the parameter values, and how they depend on $θ$ and $ϕ$ ⁠. The resonance peak is at the tip of a curve within a petal, and the resonance phase is determined by the angle of this peak value in the complex plane. The excitation phase $θ$ primarily influences the orientation of a petal and hence the response phase at resonance.

Fig. 7

Half complex amplitudes (X+iY) for the sets θ={0,π8,2π8,…,7π8} and ϕ={0,π8,2π8,…,15π8}, with μ0=0.25, μ1=0.25, and (a) γ=3, (b) γ=3.5, and (c) γ=4. (d) A group of half complex amplitude curves for θ=3π/4 and a set of eight values of ϕ within a 2π cycle, with γ=3.5.

View large Download slide

Half complex amplitudes (⁠ $X + i Y$ ⁠) for the sets $θ = {0, \frac{π}{8}, \frac{2 π}{8}, \dots, \frac{7 π}{8}}$ and $ϕ = {0, \frac{π}{8}, \frac{2 π}{8}, \dots, \frac{15 π}{8}}$ ⁠, with $μ_{0} = 0.25$ ⁠, $μ_{1} = 0.25$ ⁠, and (⁠ $a$ ⁠) $γ = 3$ ⁠, (⁠ $b$ ⁠) $γ = 3.5$ ⁠, and (⁠ $c$ ⁠) $γ = 4$ ⁠. (⁠ $d$ ⁠) A group of half complex amplitude curves for $θ = 3 π / 4$ and a set of eight values of $ϕ$ within a $2 π$ cycle, with $γ = 3.5.$

We can crosscheck this plot with those shown earlier, for example, by looking at the petal corresponding to $θ = 3 π / 4$ (the same value used for most of the other plots), which is shown in Fig. 7(d) for $γ = 3.5$ ⁠. Figure 7(d) also clarifies the effect of $ϕ$ in the vertically oriented resonance petal; the pattern of this variation carries over to the petals with other values of $θ$ ⁠, as shown in Figs. 7(a)–7(c). When $ϕ$ varies through a $2 π$ -cycle of values, the ellipses within the petal grow (solid lines) and shrink (dashed lines) and return to the original curve after one cycle. (The solid and dashed curves are used merely to help visually distinguish intersecting ovular curves.) The ellipse size is maximum near $ϕ = π / 2$ ⁠, and thus, this value of $ϕ$ gives the greatest parametric amplification. The ellipse size is smallest near $ϕ = 3 π / 2$ (equivalenty $ϕ = - π / 2$ ⁠), corresponding in fact to resonance suppression. These observations are consistent with the amplification and suppression observed in Figs. 2 and 3. During this $ϕ$ cycle, we can see that $ϕ$ causes as slight variation of the response phase of the resonance peak within each petal. That is, resonance occurs when the curve within a petal reaches its maximum distance from the origin, with a phase lag $ψ$ defined by the angle in the complex plane. Depending on $ϕ$ ⁠, the curves within a petal depict different maximum amplitudes accompanied by slightly different phases $ψ$ ⁠.

Thus, Fig. 7 illustrates that, given the excitation levels, the phase $θ$ of direct excitation dictates the orientation and the range of sizes of the “petals.” The orientation roughly indicates the resonance response phase lag, while the size reveals the range of resonance amplitudes. Then, given the excitation phase $θ$ (i.e., a specific “petal”), the phase $ϕ$ influences whether parametric damping will further amplify or suppress the resonance, and with what phase lag. When $γ$ is larger, the effect of $μ_{1}$ and $ϕ$ is stronger.

Summarizing this section, we see amplification and suppression effects of parametric damping when added to a system with parametric stiffness and direct forcing, all of the same frequency. The system has six independent parameters, and the various figures in this section sort out the effects of these parameters. Figures 2–4 indicate the effect of $Ω$ and $μ_{1}$ on resonance, for specific cases of excitation phases, which show that these phases determine if parametric damping serves to amplify or suppress the resonance. Figure 5 fills in the phase effects on the gain, while Fig. 6 shows that for many cases of $ϕ$ and $θ$ ⁠, responses can be unbounded as the parametric damping increases. Figure 7 displays how $θ$ and $ϕ$ coordinate in their effect on both resonance amplitude and phase. The indicated behaviors of these figures are consistent. For most phases (⁠ $θ$ ⁠) of direct excitation relative to parametric stiffness, parametric amplification takes place. Addition of parametric damping with phase $ϕ$ enhances the parametric amplification for $0 < ϕ < π$ (approximately) and suppresses responses for $π < ϕ < 2 π$ (approximately). The amplification effect of the phase $θ$ of direct excitation cycles through $π$ ⁠, while the effect of the phase $ϕ$ of parametric damping cycles through $2 π$ ⁠. The response tends to lag the direct excitation phase (⁠ $θ$ ⁠) in a transition from 0 to $π$ as the excitation frequency goes through resonance, regardless of the relative phases between the parametric and direct excitation terms. The parametric damping amplitude and phase influence the instability of the parametric oscillator, and resonant responses approach infinity as the parametric instability is approached. Thus, if both parametric damping and stiffness are present (by design or due to ambient effects) in a forced system, the implication is that resonances may be enhanced or suppressed, depending on relative phases.

3.2 Parametric Damping Without Parametric Stiffness.

If we set

γ = 0

in Eqs. and and obtain a steady-state solution for

X

and

Y

⁠, we can then establish the amplitude as

a = 2 \sqrt{X^{2} + Y^{2}}

⁠. We obtain

a^{2} = {(\frac{F}{4 ω^{2}})}^{2} \frac{(ε μ_{0})^{2} + (2 ω - ε σ)^{2}}{μ_{0}^{2} + (σ + ε μ)^{2}}

(24)

where

μ = \frac{μ_{0}^{2}}{2 ω} + \frac{μ_{1}^{2}}{3 ω} .

In some cases, we can approximate the maximum

a^{2}

to occur near the minimum of the denominator. The denominator is minimum when

σ = σ_{p}

⁠, where

σ_{p} = - ε μ = - ε (\frac{μ_{0}^{2}}{2 ω} + \frac{μ_{1}^{2}}{3 ω})

(25)

Under this condition, the approximate peak response harmonic amplitude is

a_{p}^{2} = {(\frac{F}{2 μ_{0} ω^{2}})}^{2} [{(ω + \frac{ε^{2}}{12 ω} (3 μ_{0}^{2} + 2 μ_{1}^{2}))}^{2} + {(\frac{ε μ_{0}}{2})}^{2}]

(26)

A couple features can be noted. The phase angles

ϕ

and

θ

were originally cast relative to the parametric stiffness term. With

γ = 0

⁠, only the difference between these phase angles might be expected to be relevant. However, the resonant response, in the asymptotic approximation, is independent of both of these angles. The angle

θ

drops out of the particular solution when

X^{2} + Y^{2}

is carried out. The angle

ϕ

only participates in Eqs. and when

γ

is involved. The absence of a relative phase is not necessarily intuitive, as it might be expected that the phase of the cyclic damping relative to the forced response would determine whether the damping acts positively or negatively. However, without parametric stiffness, this phase relationship is not a factor.

Also, when $γ = 0$ ⁠, the cyclic damping amplitude $μ_{1}$ affects the resonance by shifting the resonant frequency, as indicated by $σ_{p}$ ⁠, which varies with $ε μ_{1}^{2}$ ⁠, but has hardly any effect on the resonant amplitude, contributing with $ε^{2} μ_{1}^{2}$ in Eq. (26). Hence, we would not expect to see much of either effect when $μ_{1}$ is small, but a growing effect when $μ_{1}$ becomes sizeable, recognizing that the asymptotic analysis is limited to order-one ranges of $μ_{1}$ ⁠. As $ε \to 0$ ⁠, $a_{p} \to \frac{F}{2 μ_{0} ω}$ in Eq. (26), which agrees with the resonance of the system without parametric excitation.

Figure 8(a) shows the half complex amplitudes (⁠ $X + i Y$ ⁠). The various values of $μ_{1}$ are not significantly different. Thus, $μ_{1}$ is not revealed as an amplifier or suppressor in the second-order analysis. Figure 8(b) shows that increasing $μ_{1}$ decreases the resonant frequency. While the perturbation analysis (solid curves) predicts very little increase in amplitude, numerical simulations for various values of $μ_{1}$ show that as $μ_{1}$ gets large enough, the peak amplitudes start to deviate from the perturbation analysis. The simulations shown are for $μ_{1} = 2.5, 3.5,$ and $5$ ⁠, which push the limits of “order-one” values acceptable in the asymptotic bookkeeping. Smaller values of $μ_{1}$ show good agreement, but also produce less significant resonance behavior.

Fig. 8

Resonance characteristics for the case of γ=0, with μ0=0.25, ε=0.1, γ=3.5, 0≤μ1≤5 in increments of 0.5, Ω=ω+εσ, and F=1. (a) Half complex amplitudes (X+iY) parameterized by −1≤σ≤1. (b) Response amplitude, a, versus Ω. The dots, triangles, and circles show numerical simulation results for μ1=2.5,3.5, and μ1=5.0, respectively. (c) Response phase ψ+θ versus Ω.

View large Download slide

Resonance characteristics for the case of $γ = 0$ ⁠, with $μ_{0} = 0.25$ ⁠, $ε = 0.1$ ⁠, $γ = 3.5$ ⁠, $0 \leq μ_{1} \leq 5$ in increments of 0.5, $Ω = ω + ε σ$ ⁠, and $F = 1$ ⁠. (⁠ $a$ ⁠) Half complex amplitudes (⁠ $X + i Y$ ⁠) parameterized by $- 1 \leq σ \leq 1$ ⁠. (⁠ $b$ ⁠) Response amplitude, $a,$ versus $Ω$ ⁠. The dots, triangles, and circles show numerical simulation results for $μ_{1} = 2.5, 3.5$ ⁠, and $μ_{1} = 5.0$ ⁠, respectively. (⁠ $c$ ⁠) Response phase $ψ + θ$ versus $Ω$ ⁠.

Figure 8(c) plots the phase of the resonating harmonic versus $Ω$ ⁠. The phase transitions through $ψ + θ = π / 2$ radians line up very well with the peak locations. These plots imply that the parameterization of curves in $Ω$ ⁠, in Fig. 8(a), is shifted clockwise with increasing $μ_{1}$ ⁠.

In comparison with the previous plots with $γ = 3.5$ ⁠, it should be noted that the plots of Fig. 8 involve values of $μ_{1}$ that are five to ten times larger to demonstrate its effects.

Figure 9 shows the peak frequency and the peak amplitude of primary resonance versus $μ_{1}$ for the case of $γ = 0$ ⁠. The curves are expressed by obtaining a numerical maximum from Eq. (24) (solid curve) and also by directly plotting the approximated peak values from Eqs. (25) and (26) (dashed curve). The overlap of the solid and dashed curves makes them indistinguishable and shows that Eq. (26) makes a very good approximation of the peak of Eq. (24).

Fig. 9

View large Download slide

Peak frequency and peak amplitude versus $μ_{1}$ for $γ = 0$ ⁠, with a set of curves for values of $ϕ$ through a cycle of $2 π$ in increments of $π / 4$ ⁠, for $θ = 3 π / 4 .$ The open circle symbols show peak amplitudes as obtained from numerical simulations, while open triangle symbols indicate peak frequencies from simulations.

The triangles show numerical simulations of the peak frequencies, obtained by sweeping the frequency through primary resonance at small increments, and achieving steady state at each frequency increment, for each value of $μ_{1}$ ⁠. The peak frequency was obtained from the peak of total response from each sweep. The numerical solutions of peak amplitudes $a_{p}$ of the resonating harmonic are shown by the circles and were obtained by conducting numerical solutions of Eq. (2) at the peak frequencies from $σ_{p}$ in Eq. (25) and by then finding the amplitude of the resonating harmonic from the FFT of the response. The sampling rate was chosen to eliminate leakage distortions. The peak frequencies obtained from numerical simulations match the analytical predictions very closely, even for values of $μ_{1}$ that challenge the “order-one” limit. The peak amplitudes obtained from numerical solutions match those of the theory only for the smaller values of $μ_{1}$ ⁠, although still in the “order-one” range.

In summary, the parametric damping has less effect on resonance when parametric stiffness is absent, and the relative phase between the parametric damping and direct excitation has no effect, at least in the second-order perturbation approximation. The parametric damping solo effect is to shift the resonance frequency. There is some amplification observed in simulations that is not captured by the second-order perturbation analysis and is observed when the parameter $μ_{1}$ is at values that push the notion of “order one” in the bookkeeping. Thus, in applications, parametric damping alone is expected to affect the resonance frequency more than the amplitude.

4 Conclusion

We have examined primary resonance of a weakly externally forced oscillator with parametric stiffness and parametric damping. The parametric damping, parametric stiffness, and harmonic forcing are all at the same frequency, and phases $θ$ and $ϕ$ were included in the forcing and cyclic damping terms relative to the cyclic stiffness. A second-order multiple scales analysis of primary resonance revealed parametric amplification effects, which exist in a very slow time scale (⁠ $T_{2} = ε^{2} t$ ⁠).

Without parametric damping, parametric stiffness already leads to amplification of resonance for most values of $θ$ [50]. The addition of parametric damping further accentuates these effects. A parametric damping phase near $ϕ = π / 2$ produces maximal additional amplification, while a phase near $ϕ = - π / 2$ serves to suppress the resonance. While most excitation phases $θ$ promote amplification, if $θ \approx π / 4,$ a twin-peak resonance occurs without amplification, which can be further exaggerated by parametric damping.

When parametric stiffness is not present (⁠ $γ = 0$ ⁠), the parametric damping effect is less dramatic. In such case, in the second-order analysis, the phases of direct and parametric excitations have no effect on the resonance. An increase in $μ_{1}$ leads to a shift in the frequency of the resonance, but has only a small effect on the amplitude of resonance. Numerical simulations show that analytical approximation to the frequency shift is quite accurate if $μ_{1}$ is generously within or even beyond the order-one range, while the analytical approximation to the increase in amplitude is only accurate if $μ_{1}$ is more strictly within order one, under which conditions this increase in amplitude is very small compared to the resonance of the standard (nonparametric) oscillator.

The behavior of this system may be applicable to parametric resonators or harvesters with added parametric dampers as designed enhancements, or systems with parametric excitation and cyclic aeroelastic effects, such as wind-turbine blades. The impact of parametric damping is most significant on systems that also have parametric stiffness and in such case has a strong dependence on the phases of parametric and direct forcing. These phases will determine whether the parametric damping causes amplification or suppression of the resonance. For systems with ambient or intrinsic sources of parametric excitation, the parametric effect can significantly amplify the resonance behavior relative to a model that omits the parametric terms. For parametric resonators and harvesters, the implication is that parametric damping can potentially be designed into the oscillator to enhance the amplification performance.

Further studies may examine super harmonic resonances, as the case of horizontal-axis wind turbines are designed to operate below the primary resonance. It would also be interesting to examine 2:1 parametric-to-direct excitations, as this is the more typical setting for designed parametric-stiffness resonators. Further studies could involve the addition of nonlinear terms to the stiffness or the parametric terms.

Acknowledgment

This work was based on a project supported by the National Science Foundation (Grant No. CMMI-1435126). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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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Primary Resonance in a Weakly Forced Oscillator With Both Parametric Damping and Stiffness

Abstract

1 Introduction

2 Method of Multiple Scales: Primary Resonance

3 Behavior of Primary Resonance

3.1 Combined Parametric Damping and Stiffness.

3.2 Parametric Damping Without Parametric Stiffness.

4 Conclusion

Acknowledgment

Conflict of Interest

Data Availability Statement

References

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Primary Resonance in a Weakly Forced Oscillator With Both Parametric Damping and Stiffness

Abstract

1 Introduction

2 Method of Multiple Scales: Primary Resonance

3 Behavior of Primary Resonance

3.1 Combined Parametric Damping and Stiffness.

3.2 Parametric Damping Without Parametric Stiffness.

4 Conclusion

Acknowledgment

Conflict of Interest

Data Availability Statement

References

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