# Transient and Steady-State Responses of an Asymmetric Nonlinear Oscillator

Authors: Alex Elías-Zúñiga; Oscar Martínez-Romero
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

We study the dynamical response of an asymmetric forced, damped
Helmholtz-Duffing oscillator by using Jacobi elliptic functions, the method of
elliptic balance, and Fourier series. By assuming that the modulus of the
elliptic functions is slowly varying as a function of time and by considering
the primary resonance response of the Helmholtz-Duffing oscillator, we derived
an approximate solution that provides the time-dependent amplitude-frequency
response curves. The accuracy of the derived approximate solution is evaluated
by studying the evolution of the response curves of an asymmetric Duffing
oscillator that describes the motion of a damped, forced system supported
symmetrically by simple shear springs on a smooth inclined bearing surface. We
also use the percentage overshoot value to study the influence of damping and
nonlinearity on the transient and steady-state oscillatory amplitudes.


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## Body

## 1. Introduction

Since most of the nonlinear differential equations that characterized the motion of several physical systems do not have closed-form solution, we have to use numerical or perturbation techniques to study the dynamical response of these systems. However, most of the perturbation techniques such as multiple scales, averaging, and harmonic balance, to say a few, focus on only the determination of steady-state approximate solutions because of the complexity involved in finding transient solutions of nonlinear differential equations [1, 2].The aim of this paper is to investigate the influence on the dynamical behavior of the transient and steady-state solutions of the system(1)x¨+2νx˙+Ax+Bx2+εx3=Fcos(ωft)
close to the primary resonance region. Here, x denotes the displacement of the system, A is the natural frequency, ν is the damping coefficient, ε is a dimensionless nonlinear parameter, ωf is the driving frequency, t is the running time, F is the amplitude of the driving force, and B is a system parameter. To solve the homogeneous Helmholtz-Duffing oscillator for which F=0 in (1), Hu used the harmonic balance method to calculate the first-order approximations to the periodic solutions of this equation [3]. Belhaq and Lakrad used the harmonic balance method involving Jacobian elliptic functions to obtain the approximate solution of (1) by taking ν=0 and ωf=0 [4]. Tamura [5] and Hu [6] developed the exact solution of a quadratic nonlinear oscillator that is part of (1) by using an elliptic function. Cao and coworkers investigated in [7] the various symmetry breaking phenomena associated with the Helmholtz-Duffing oscillator (1) in the case for which B=1-ε and for different values of the so-called symmetric parameter ε. They also used the second-order averaging method to investigate its local bifurcation behavior. By considering a rational form elliptic solution to (1) when F=0, Elías-Zúñiga derived its analytical solution which is similar in form to that of its exact solution when ν=0 [8].Recently, Kovacic and coworkers studied the primary resonance response of (1) by applying the harmonic balance method and derived nonlinear algebraic equations for the steady-state system response [9], while Jeyakumari et al. analyzed how the potential well of an asymmetric Duffing oscillator affects the vibrational resonance response [10].Here the approximate solution of (1) is derived by taking into account the transient and the steady-state responses without the simplifications regarding undamped and unforced system included in previously developed solutions such as [4, 6]. The approximate solution is based on trigonometric and Jacobi elliptic functions with slowly varying parameters that will help us to obtain amplitude-frequency response curves that evolve with time. Then, the influence of the nonlinear transient responses is investigated since recent studies show that transient vibrations can not only provide additional information to fully predict the system stable behavior [11], but also can be used to predict the system overshoot value [12, 13]. The determination of this value is of practical interest in understanding the importance of time in controlling the dynamical behavior of oscillatory systems [14, 15] therefore, the percentage overshoot value will be computed by considering the influence of the system parameters such as nonlinear and damping effects.In the next section, the approximate general solution of (1) is derived by using Jacobi elliptic functions.


## 2. Approximate Solution

In order to obtain the general solution of (1) in the region of the primary resonance, we will consider that this equation can be written in equivalent form as
(2)x¨+2νx˙+Ax+Bx2+εx3=Fcn(ωft,kf2),
where cn(ωft,kf2) is the Jacobian elliptic function that has a period in ωft equal to 4K(kf2), and K(kf2) is the complete elliptic integral of the first kind for the modulus kf [16, 17]. Note that 0≤|kf2|<1 must hold, and when the modulus of cn(ωft,kf2) is zero, then cn(ωft,kf2) and the trigonometric function cos(ωft) coincide and, thus, (2) reduces to (1). Now, we make the assumption that (2) has an approximate general solution of the form
(3)x=b(t)+c(t)cn(ω1t+ϕ1,k12)+D1(t)cn(ωft,kf2)+D2(t)sn(ωft,kf2),
where ϕ1, ω1(t), b(t), D1(t), D2(t), k1(t), and kf(t) need to be determined. For the sake of simplicity, it is assumed that b(t), ω1(t), D1(t), D2(t), k1(t), and kf(t) are slowly varying as a function of time so that b˙, ω˙1, D˙1, D˙2, k˙1, and k˙f or higher order are small enough to be ignored [18]. Also, for simplicity, we will use the following notation cnf≡cnuf=cn(ωft,kf2), and cn1≡cnu1=cn(ω1t+ϕ1,k12) with similar notation for the functions sn and dn. Thus, substitution of (3) into (2) gives
(4)Ab+b2B+b3ε+cn1[D22snf2Ac+2bBc+3b2εc+cnf(2BcD1+6bεcD1)+3εcD12cnf2-ω12c+2k12ω12c+snf(2BcD2+6bεcD2)+3εcD22snf2+2νc˙+c¨]+cn12[Bc2+3bεc2]+cn13[εc3-2ω12k12c]-2ω1dn1sn1[νc+c˙]+cnfdnf[2νωfD2]+cnf[D22snf2-F+AD1+2bBD1+3b2εD1+3εc2D1cn12+2ωf2kf2D1-ωf2D1+snf(2BD1D2+6bεD1D2)+3εD1D22snf2]+cnf2[BD12+3bεD12]+cnf3[εD13-2ωf2kf2D1]-dnfsnf[2νωfD1]+cn1cnfsnf[6εcD1D2]+snf[D12D2cnf2AD2+2bBD2+3b2εD2+3εc2D2cn12+3εD12D2cnf2-kf2ωf2D2-ωf2D2]+snf2[BD22+3bεD22]+snf3[εD23+2ωf2kf2D2]=0
in which the following identities sn2+cn2=1 and dn2+kj2sn2=1 have been used [19].Notice that (4) depends on elliptic functions with different modulus. To simplify (4), we first compute its average with respect to 4Kf while keeping terms with period 4K1 constant; this yields the following equation:
(5)Ab+b2B+b3ε+(1-If)[BD22+3bεD22]+If[BD12+3bεD12]+cn1[Ac+2bBc+3b2εc+3εcD12If-ω12c+2k12ω12c+3εcD22(1-If)+2νc˙+c¨]+cn12[Bc2+3bεc2]-2ω1dn1sn1[νc+c˙]+cn13[εc3-2ω12k12c]=0.
We now take the average of (5) with respect to 4K1, to get
(6)Ab+b2B+b3ε+(1-If)[BD22+3bεD22]+If[BD12+3bεD12]+I1c2(B+3bε)=0,
where the Ij terms are defined in the appendix.Next, the average of (4) is computed with respect to 4K1 while terms with period 4Kf are held constant; this yields
(7)Ab+b2B+b3ε+I1[Bc2+3bεc2]+cnfsnf(2BD1D2+6bεD1D2)×cnf[ωf2kf2-F+AD1+2bBD1+3b2εD1+3εc2D1I1+2ωf2kf2D1-ωf2D1+3εD1D22]+cnf2[BD12+3bεD12]+cnf3[εD13-2ωf2kf2D1-3εD1D22]+cnfdnf[2νωfD2]-dnfsnf[2νωfD1]+snf2[BD22+3bεD22]+snf[ωf2kf2AD2+2bBD2+3b2εD2+3εc2D2I1+kf2ωf2D2-ωf2D2+εD23]-snfcnf2[εD23+2ωf2kf2D2-3εD12D2]=0.
Then, the average of (7) is computed with respect to 4Kf, to get the following expression:
(8)Ab+b2B+b3ε+If[BD12+3bεD12-BD22-3bεD22]+I1[Bc2+3bεc2]+(BD22+3bεD22)=0.To determine the unknown parametersϕ1, ω1(t), b(t), D1(t), D2(t), k1(t), and kf(t), Fourier series are applied to (7) to transform the terms sndn and sncn2 into sn, cnsn and cndn into cn Jacobian elliptic functions, respectively [20–22]. Therefore, (7) simplifies to
(9)cnf[ωf2kf2D22-F+AD1+2bBD1+3b2εD1+3εc2D1I1+2ωf2kf2D1-ωf2D1+3εD1D22+2νωfD2H2]+cnf2[BD12+3bεD12]+cnf3[εD13-2ωf2kf2D1-3εD1D22]+snf2[BD22+3bεD22]+snf[ωf2kf2D23AD2+2bBD2+3b2εD2+3εc2D2I1+kf2ωf2D2-ωf2D2+εD23-2νωfD1H1-(εD23+2ωf2kf2D2-3εD12D2)H3]+Ab+b2B+b3ε+I1[Bc2+3bεc2]+(higherharmonics)=0,
where H1, H2, and H3 are defined in the appendix. If we now use the amplitude for Jacobi elliptic functions so that cosφj=cn(ωjt;kj2)=cnj, and sinφj=sn(ωjt;kj2)=snj, then (5) and (9) become
(10)Ab+b2B+b3ε+(1-If)[BD22+3bεD22]+If[BD12+3bεD12]+cosφ1[Ac+2bBc+3b2εc+3εcD12If-ω12c+2k12ω12c+3εcD22(1-If)+2νc˙+c¨]+cos2φ1[Bc2+3bεc2]-2ω11-k12sinφ12sinφ1[νc+c˙]+cos3φ1[εc3-2ω12k12c]=0,cosφf[34-F+AD1+2bBD1+3b2εD1+3εc2D1I1+2ωf2kf2D1-ωf2D1+3εD1D22+2νωfD2H2+34(εD13-2ωf2kf2D1-3εD1D22)]+sinφf[ωf2kf2D23AD2+2bBD2+3b2εD2+3εc2D2I1+kf2ωf2D2-ωf2D2+εD23-2νωfD1H1-(εD23+2ωf2kf2D2-3εD12D2)H3]+[12(BD12+3bεD12+BD22+3bεD22)Ab+b2B+b3ε+I1[Bc2+3bεc2]+12(BD12+3bεD12+BD22+3bεD22)]+(higherharmonics)=0.
Since our original system (1) is subjected to a driving force of sinusoidal type, then the modulus kf≡0 and the following identities hold: cnf=cosf, snf=sinf, If=1/2, H1=1, and H2=1. If we also assume that 0<|k12|<1/2, then, from (A.1), I1≃1/2 and (6) becomes exactly the same as (8). This provides the following simplifications for (10):
(11)cosφ1[Ac+34(εc3-2ω12k12c)+2bBc+3b2εc+32εc(D12+D22)-ω12c+2k12ω12c+2νc˙+c¨34]-2ω1sinφ1[νc+c˙]+14cos3φ1[εc3-2ω12k12c]+(higherharmonics)=0,cosf[-F+AD1+2bBD1+3b2εD1+32εc2D1-ωf2D1+3εD1D22+2νωfD2+34(εD13-3εD1D22)]+sinf[14(εD23-3εD12D2)AD2+2bBD2+3b2εD2+32εc2D2-ωf2D2+εD23-2νωfD1-14(εD23-3εD12D2)]+(higherharmonics)=0.We then follow the harmonic balance procedure and ignore higher harmonics terms in (11), to find the following expressions:
(12)Ac+34(εc3-2ω12k12c)+2bBc+3b2εc+32εc(D12+D22)-ω12c+2k12ω12c+2νc˙+c¨=0,(13)-F+AD1+2bBD1+3b2εD1+32εc2D1-ωf2D1+34εD1D22+2νωfD2+34εD13=0,(14)AD2+2bBD2+3b2εD2+32εc2D2-ωf2D2+34εD23-2νωfD1+34εD12D2=0,(15)εc3-2ω12k12c=0,(16)νc+c˙=0.
Notice that the variable c may be determined by integration of (16); this yields
(17)c(t)=Cexp(-νt),
where C is a constant of integration. Substitution of (17) into (8), as well as (12)–(15), yields
(18)-D222(B+3bε)=Ab+b2B+b3ε+12C2exp(-2νt)(B+3bε)+D122(B+3bε),(19)ω12=A+2bB+3b2ε+32ε(D12+D22)+εC2exp(-2νt)-ν24,(20)-34ε(D12+D22)=-FD1+A+2bB+3b2ε+2νωfD2D1+32εC2exp(-2νt)-ωf2,(21)-34ε(D12+D22)=A+2bB+3b2ε+32εC2exp(-2νt)-ωf2-2νωfD1D2,(22)k12=εC2exp(-2νt)2ω12.
Then, for given parameters values of ν, ε, A, B, F, and ωf, the unknown parameters ϕ1, ω1, C, D1, D2, b, and k1 can be found from (18)–(22) and from the initial conditions (I. C.) x(0)=x10, x˙(0)=x˙10. If we assume that the value of the initial velocity x˙(0)=x˙10 is such that ϕ1=0 in (3), then our approximate solution for (2) can be written as
(23)x=b(t)+Cexp(-νt)cn(ω1t,k12)+D1(t)cos(ωft)+D2(t)sin(ωft).
Since x(0)=x10 at t=0, we get from (23) that
(24)C=x10-b(0)-D1(0).
We next use (20) and (21) to find b(t) and D2(t) as
(25)D2(t)=F∓F2-(4νωfD1)24νωf,(26)b(t)=1384D1εν2ωf2×(-2ωf2))))]1/2-128BD1ν2ωf2∓[-2ωf2))))256D1ν2ωf2×(-2ωf2)))64B2D1ν2ωf2+3ε×(-2ωf2)))32ν2ωf2(F±R1)×-D1(-2ωf2)))6εF2∓6εFR1+32ν2ωf2×(-2ωf2)))2A+3C2exp(-2νt)ε-2ωf2-2ωf2)))))))]1/2),(27)R1=(F-4D1νωf)(F+4D1νωf),
where the sign in (25) and (26) must be appropriately chosen to ensure that the higher elliptic terms in (23) have small amplitudes relative to the leading ones [23]. Substituting (25) and (26) into (18), we can easily prove, after some algebraic computations, that the values of D1(t) are determined from a ninth-order polynomial equation. Once the values of D1(t) are known, we may use (19) to determine ω1(t). Notice that when the running time t increases, the term e-νt approaches to zero, and thus (18), (19), and (22)–(26) become time-independent. In this case, the values of b, D1, D2, ω1, and k1 in (23) will remain constant. This condition provides the steady-state solution of (1).We will next discuss the accuracy of our derived approximate solution by studying the dynamical response of an asymmetric Duffing oscillator by plotting the amplitude-frequency and the percentage overshoot response curves and show how the evolution of time influences the systems behavior.


## 3. Numerical Simulations

In order to verify the accuracy of our proposed solution described in (23), we study the dynamical response of a rigid body supported symmetrically by a simple shear spring that is sliding over a smooth inclined bearing surface [24]. The system under consideration contains the following critical conditions: (a) finite amplitude forced vibration and (b) damping, demonstrating the general nature of the proposed solution for the resonance response of an asymmetric Duffing oscillator. In accordance with Elías-Zúñiga and Beatty [24], the equation for the damped motion of the load is a nonlinear, ordinary differential equation of the forced Duffing type with a constant static shift term of the form
(28)σ¨+2νσ˙+σ+εσ3=σe(1+εσe2)+Fcosωft,
where the dot denotes the derivative with respect to t, σ represents the amount of simple shear deformation, σe denotes the amount of static shear deflection of the load, ν is the damping ratio, F is a dimensionless driving force, and t is the dimensionless running time. If we transform (28) relative to σe by using x=σ-σe, then (28) becomes similar to the Helmholtz-Duffing equation (1) where the parameters A and B are described by
(29)A≡1+3εσe2,B≡3εσe.To find the amplitude-frequency response curves of (1), we select the following parameter values: ε=0.02, σe=0.497, F=0.1, ν=0.01, x0=1, and x˙0=0.022, and plot these curves at the values of t=1, 15, 30, 80, and 100. Figure 1 shows the evolution of these amplitude-frequency curves with time. In the curves showed in Figure 1, we have plotted Am(t)=|b(t)+D1(t)+D2(t)| versus ωf. These curves characterize the evolution of the general motion of the system for the selected set of system parameter values. In these curves, the blue dots represent unstable system behavior determined by following the bifurcation analysis described in [9, 24]. Notice from Figure 1, that when the running time t increases, that is, t≈100, the unstable regions in the amplitude-frequency response curves become smaller since the influence of time in (18), (19), and (22)–(26) is almost negligible. In Figure 1, the brown dots represent the stable steady-state amplitude-frequency response curve that characterizes the damped, forced nonlinear motion of a body supported by viscohyperelastic shear mountings [24]. Figure 2 illustrates the amplitude versus time response curve by considering the parameters values of σe=0.497, ε=0.02, ν=0.01, F=0.1, and ωf=1.1 and the initial conditions x0=0 and x˙0=0.0663 for which ϕ1=0. There, the solid lines represent the numerical integration solution of (28) while the dashed lines represent the approximate solution provided by (23). In this case, the values of b, C, D1, D2, k1, and ω1 were obtained from (18), (19), (22), (24), (25), and (26). In fact, at t=0, we get that b(0)=-0.0086, C=0.5496, D1(0)=-0.541, D2(0)=0.0653, k1(0)=0.05418, x˙(0)=0.0663, and ω1(0)=1.0144. As shown in Figures 2(a) and 2(c), the highest vibration amplitudes occur during the transient oscillations. This transient oscillatory behavior provides information to ensure that the system design constraints are not exceeded, even if the shear suspension system performs well during its steady-state behavior. Figure 2(b) shows the phase plane of the transient oscillatory motion on the time interval 0≤t≤31 plotted by considering the following initial conditions of x0=0 and x˙0=0.0663. It is clear from Figure 2(b) that the transient oscillations experience a fast increase in their amplitude values followed by a decrease when t>31, as illustrated in Figure 2(c). When the system reaches its steady-state behavior, the system phase portrait, depicted in Figure 2(d), exhibits stable behavior about the steady-state equilibria condition.Figure 1
Evolutive amplitude-frequency response curves of a rigid body supported symmetrically by a simple shear spring and sliding over a smooth inclined bearing surface with moderate static shear deflectionσe=0.497 and parameter values of ε=0.02, F=0.1, ν=0.01, and initial conditions x0=1 and x˙0=-0.0091. At t=0, the initial values of b, C, D1, D2, k1, and ω1 are given as b(0)=-0.0121, C=0.9951, D1(0)=0.0962, D2(0)=0, k1(0)=0.0902, and ω1(0)=1.0154. The blue dot curves represent unstable system response.Phase portraits and amplitude-time response curves for parameter values ofε=0.02, σe=0.497, ν=0.01, F=0.1, ωf=1.1, D1(0)=-0.541, D2(0)=0.0653, b(0)=-0.0086, k1(0)=0.05418, ω1(0)=1.0144, C=0.5496, and initial conditions x(0)=0 and x˙(0)=0.0663. (a) Amplitude-time transient curves on t[0,31.5]; (b) amplitude-time response curve on t[0,700]; (c) transient response phase portrait; (d) steady-state system response phase portrait on t[650,700]. The solid lines represent the numerical solution, and the dashed lines represent the proposed EBM solution.
(a)
(b)
(c)
(d)We next investigate the influence of the transient solution on the dynamics response of (28) by introducing the percentage of overshoot which is defined as
(30)%overshoot=(xtran-xstxst)×100,
where xtrans and xst represent the peak transient and steady-state amplitudes of the system, respectively, that can be estimated by setting to zero the time derivative of (23). Figure 3 shows the calculated system overshoot plotted against the system angular frequency, ωf, by considering the following system parameter values: σe=0.497, F=0.1, ν=0.01, and x0=0. As we can see from Figure 3, the influence of the transient solution on the system response is evident since the percentage of overshoot values is close to a 100% at the approximate frequency values of 0.1, 0.5, 1.5, and 2. Here the blue, the red, and the purple dashed lines were calculated by considering the nonlinear parameter values of ε=0.02, 0.1, and 0.25, respectively. It is interesting to observe in Figure 3 the influence of the nonlinearity of the system in the overshoot qualitative and quantitative curves behavior. In fact, one can conclude from Figure 3 that the smallest percentage values of the overshoot occur with ε=0.02 and near to the resonance frequency. Furthermore, the predicted overshoot curves oscillate about the value of 92% on the interval 1.5≤ωf≤2.7 that corresponds to the region depicted in Figure 4 at which the ratio of the transient and the steady-state amplitudes has almost the constant value of 52%.Figure 3
Percentage overshoot versus the driving frequency for parameter values ofσe=0.497, F=0.1, ν=0.01, x0=0, and ε=0.02, 0.1, and 0.25.Figure 4
Transient and steady-state amplitude ratio(xtran/xst) versus the driving frequency. Here the parameter values are σe=0.497, F=0.1, ν=0.01, x0=0, and ε=0.02, 0.1, and 0.25.To examine damping effects on the system overshoot, we next use the parameter values ofσe=0.497, F=0.1, ε=0.02, and x0=0 and consider the damping ratio values of ν=0.01, 0.05, and 0.1. The influence of the transient response on the peak overshoot values for the curve in which ε=0.02, ν=0.01, ωf=0.5 and 1.5 is illustrated in Figure 5(a) and 5(b). Figure 5(c) shows the percentage overshoot curves plotted against the driving frequency ωf. As me can see from Figures 5(c) and for a damping value of ν=0.1, the overshoot values do not exceed 73%. It is also observed that for increasing damping values, the percentage of the system overshoot becomes smaller. Figures 5(d) and 5(e) illustrate, by using the Continuous Wavelet Transform, the influence of the transient and the driving frequencies on the nonlinear behavior of the system. It is evident from Figures 5(d) and 5(e) that at the beginning of the system motion, the transient frequency ω1 influences the fast increase of the system oscillations. Then, its effects on the system non-linear motion decrease when the running time exceeds the value of t≈200. Based on these findings, it is concluded that transient amplitude values are bigger than those of the steady-state ones as showed in Figures 3 and 5. Therefore, if one wants to reduce the impact of the transient oscillations amplitudes, we need, when possible, a fast tuning of the system to its operating bandwidth frequency region which can be achieved by using suitable control algorithms.Figure 5
System nonlinear dynamic behavior for parameter values ofσe=0.497, F=0.1, ε=0.02, x0=0. Here, (a) and (c) show the amplitude versus time response curves for which the solid black and the dashed black lines represent the numerical and the EBM solutions, respectively; (b) illustrates the percentage overshoot versus the driving frequency for damping values of ν=0.01, 0.05, and 0.1; (d) and (e) exhibit the influence of the transient and driving frequencies on the nonlinear system motion by using the Continuous Wavelet Transform.


## 4. Conclusions

In this paper, we have used elliptic functions to obtain the transient and the steady-state solutions of an asymmetric Duffing oscillator and studied its influence on the region of the primary resonance response. The theoretical predictions of our derived solution given by (23) are based on the assumption that the physical parameters ω1(t), D1(t), D2(t), k1(t), and b(t) are slowly varying as a function of time. To obtain computational tractable expressions to determine the aforementioned parameters, we have computed the average of the corresponding equations with respect to the complete elliptic integral of the first kind for the moduli k1 and kf. Then, we have used Fourier series and followed the harmonic balance method to balance the lowest harmonic terms. These steps plus the assumption that 0<|k12|<1/2 provide us with the time-dependent equations to determine the unknown parameters. The effect of the running time on the shapes of the amplitude-frequency response curves becomes evident during the study of the dynamical oscillator response with no linear stiffness term and with hardening characteristics. We have also shown that when the running time increases, the frequency range at which unstable motion occurs becomes smaller. Furthermore, we have obtained the percentage overshoot charts to address the influence of the nonlinear and damping terms on the transient and steady-state system responses when the system is suddenly switched on. We have found that when the nonlinearity is small, the percentage overshoot values becomes smaller at the overshoot bandwidth regions closer to the resonance frequency with increasing values of the steady-state and the transient amplitudes ratio. The usage of Continuous Wavelet Transform helps to identify the influence of the transient response signal on the system behaviour as time evolve. We also found that by adding damping to the system, the percentage of overshoot in the system does not exceed, in the case of ν=0.1, 52% if the system is tuned into the overshoot bandwidth region of 0.5≤ωf≤1.5.Based on these findings, we can conclude that if one really wants to have a better understanding of the system dynamical behavior, transient oscillations amplitudes must be estimated to avoid undesirable system effects such as peak transient amplitude values that could violate design constraints or unstable system behavior when the excitation is suddenly switched on. In this case, the percentage overshoot charts could be used to identify the frequency bandwidth region at which the system can be tuned to have a reliable dynamical response.



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