*8.1. General Form of Step Responses*

Denote by *gj*(*t*) (*j* = 1, 2, 3) the step response to a fractional oscillator in the *j*th Class. Then, it is also the step response to the *j*th equivalent oscillator. Precisely, *gj*(*t*) is the solution to the *j*th equivalent oscillator expressed by

$$\begin{cases} m\_{eq\bar{\jmath}}\ddot{\mathbf{g}}\_{\dot{\jmath}}(t) + c\_{eq\bar{\jmath}}\dot{\mathbf{g}}\_{\dot{\jmath}}(t) + k\mathbf{g}\_{\dot{\jmath}}(t) = u(t) \\ g\_{\dot{\jmath}}(0) = 0, \dot{g}\_{\dot{\jmath}}(0) = 0 \end{cases}, j = 1, 2, 3. \tag{208}$$

The solution to the above equation is given by

$$\mathcal{G}\_{\vec{j}}(t) = \int\_0^t h\_{\vec{j}}(\tau)d\tau = \frac{1}{k} \left[ 1 - \frac{e^{-\xi\_{eq}\omega\_{eq,j}t}}{\sqrt{1 - \xi\_{eq,j}^2}} \cos\left(\omega\_{eqd,j}t - \phi\_{\vec{j}}\right) \right], j = 1, 2, 3,\tag{209}$$

where

$$\Phi\_{\vec{j}} = \tan^{-1} \frac{\mathcal{G}\_{cqj}}{\sqrt{1 - \mathcal{G}\_{cqj}^2}}, \mathbf{j} = \mathbf{1}, \mathbf{2}, \mathbf{3}. \tag{210}$$

### *8.2. Step Response to a Fractional Oscillator in Class I*

**Theorem 16** (Step response I)**.** *Let g*1(*t*) *be the unit step response to a fractional oscillator in Class I. For t* ≥ *0 and 1 < α* ≤ *2, it is given by*

$$g\_1(t) = \frac{1}{k} \left[ 1 - \frac{e^{-\frac{\omega \sin\frac{\theta \pi t}{2}}{2\left[\cos\frac{\theta \pi t}{2}\right]}t} \cos\left(\frac{\omega\_n \sqrt{1 - \frac{\omega^n \sin^2\frac{\theta \pi t}{2}}{4\omega\_n^2 \left[\cos\frac{\theta \pi t}{2}\right]}}t - \phi\_1}{\sqrt{1 - \left(\frac{\omega \sin\frac{\theta \pi t}{2}}{2\omega\_n \sqrt{1 - \cos\frac{\theta \pi t}{2}}}\right)^2}}\right)}\right],\tag{211}$$

*where*

$$\phi\_1 = \tan^{-1} \frac{\xi\_{eq1}}{\sqrt{1 - \xi\_{eq1}^2}} = \tan^{-1} \frac{\frac{\omega^2 \sin \frac{\sin \frac{\pi x}{2}}{2}}{2 \omega\_n \sqrt{|\cos \frac{\pi x}{2}|}}}{\sqrt{1 - \left(\frac{\omega^2 \sin \frac{\pi x}{2}}{2 \omega\_n \sqrt{|\cos \frac{\pi x}{2}|}}\right)^2}}. \tag{212}$$

*α*

**Proof.** Note that

$$\log\_1(t) = \frac{1}{k} \left[ 1 - \frac{e^{-\zeta\_{cq1}\omega\_{cqn,1}t}}{\sqrt{1 - \zeta\_{cq1}^2}} \cos\left(\omega\_{cqd,1}t - \phi\_1\right) \right]. \tag{213}$$

Substituting *ςeq*<sup>1</sup> with the one in (141) into the above produces

$$g\_{1}(t) = \frac{1}{k} \left[ 1 - \frac{e^{-\epsilon\_{\text{eq}1}\omega\_{\text{eq}1}t} \cos(\omega\_{\text{eq}1}t - \phi\_{1})}{\sqrt{1 - \epsilon\_{\text{eq}1}^{2}}} \right] = \frac{1}{k} \left[ 1 - \frac{e^{-\frac{\omega\_{\text{eq}1}\omega\_{\text{eq}1}t}{2\omega\_{\text{eq}1}\omega\_{\text{eq}1}} \omega\_{\text{eq}1}t}{\sqrt{1 - \left(\frac{\omega\_{\text{eq}1}\omega\_{\text{eq}1}}{2\omega\_{\text{eq}1}\sqrt{-\cos\frac{\omega\_{\text{eq}1}t}{2}}}\right)^{2}}} \right]. \tag{214}$$

Replacing *<sup>ω</sup>eqn*,<sup>1</sup> and *<sup>ω</sup>eqd*,<sup>1</sup> with those in Section 5 in the above yields (211) and (212). The proof finishes. 

Figure 42 shows the unit step response *g*1(*t*) with fixed oscillation frequency *ω*. Note that *g*1(*t*) takes *ω* as an argument. Thus, we use Figure 43 to indicate *g*1(*t*) with variable *ω* in time domain. Its plots in t-ω plane are shown in Figure 44.

**Figure 42.** *Cont.*

**Figure 42.** Unit step response *g*1(*t*) to a fractional oscillator in Class I with fixed *ω* for *m* = *k* = 1. (**a**) *α* = 1.3, solid line: *ω* =1(*ςeq*<sup>1</sup> = 0.66); dot line: *ω* = 0.7 (*ςeq*<sup>1</sup> = 0.52). (**b**) *α* = 1.6, solid line: *ω* = 1 (*ςeq*<sup>1</sup> = 0.33); dot line: *ω* = 0.7 (*ςeq*<sup>1</sup> = 0.25). (**c**) *α* = 1.9, solid line: *ω* =1(*ςeq*<sup>1</sup> = 0.08); dot line: *ω* = 0.7 (*ςeq*<sup>1</sup> = 0.06). (**d**) *α* =2(*ςeq*<sup>1</sup> = 0).

**Figure 43.** Step response *g*1(*t*) to a fractional oscillator in Class I with variable *ω* for *m* = *k* = 1. (**a**) *α* = 1.3, *ω* = 1, 1.2, 1.4, ..., 5 (0.66 ≤ *ςeq*<sup>1</sup> ≤ 1.88). (**b**) *α* = 1.5, *ω* = 1, 1.2, 1.4, ..., 10 (0.66 ≤ *ςeq*<sup>1</sup> ≤ 2.95). (**c**) *α* = 1.7, *ω* = 1, 1.2, 1.4, ..., 10 (0.08 ≤ *ςeq*<sup>1</sup> ≤ 0.36). (**d**) *α* = 1.9, *ω* = 1, 1.12, 1.14, ..., 10 (0.08 ≤ *ςeq*<sup>1</sup> ≤ 0.70).

**Figure 44.** Step response *g*1(*t*) to a fractional oscillator in Class I in t-ω plane for *m* = *k* = 1, *ω* = 0, 1, ..., 5. (**a**) *α* = 1.9 (0 ≤ *ςeq*<sup>1</sup> ≤ 0.36). (**b**) *α* = 1.6 (0 ≤ *ςeq*<sup>1</sup> ≤ 1.18).

**Note 8.1:** If *α* = 2, *g*1(*t*) reduces to the conventional step response with damping free. In fact,

$$\left. \begin{aligned} \left. \begin{aligned} \left( \mathcal{G}\_{1}(t) \right) \right|\_{a=2} = \frac{1}{k} \left[ 1 - \frac{e^{-\frac{\omega n \ln \frac{\pi \mu}{4}}{2 \left[ \cos \frac{\pi \mu}{4} \right]} t} \cos \left( \frac{\omega\_{n} \sqrt{1 - \frac{\omega^{2} \sin^{2} \frac{\pi \mu}{4}}{\omega\_{n}^{2} \left[ \cos \frac{\pi \mu}{4} \right]} t}}{\sqrt{1 - \left( \frac{\omega^{2} \sin \frac{\pi \mu}{4}}{2 \omega\_{n} \sqrt{1 - \cos \frac{\pi \mu}{4}}} \right)^{2}} \right) \right] \end{aligned} \tag{215} \end{aligned} $$

and

$$\left. \phi\_1 \right|\_{a=1} = \tan^{-1} \frac{\frac{\omega^{\frac{\alpha}{2}} \sin \frac{a\pi}{2}}{2\omega\_n \sqrt{|\cos \frac{a\pi}{2}|}}}{\sqrt{1 - \left(\frac{\omega^{\frac{\alpha}{2}} \sin \frac{a\pi}{2}}{2\omega\_n \sqrt{|\cos \frac{a\pi}{2}|}}\right)^2}}\bigg|\_{a=2} = 0. \tag{216}$$

### *8.3. Step Response to a Fractional Oscillator in Class II*

**Theorem 17** (Step response II)**.** *Denote by g*2(*t*) *the unit step response to a fractional oscillator in Class II. It is in the form, for t* ≥ *0 and 0 < β* ≤ *1,*

$$g\_2(t) = \frac{1}{k} \left[ 1 - \frac{e^{-\frac{\phi\omega\omega\beta - 1}{1}\sin\frac{\beta\pi t}{2}} \cos\frac{\beta\pi t}{2} t}{1 - \frac{e^{-\frac{\phi\omega\beta - 1}{1}\sin\frac{\beta\pi t}{2}} \cos\frac{\beta\pi t}{2}}{\sqrt{\left(1 - \frac{\xi}{\pi}\omega\beta^{-2}\cos\frac{\beta\pi t}{2}\right)} t - \phi\_2}}{\sqrt{1 - \frac{\xi}{1 - \frac{\xi}{\pi}\omega\beta^{-2}\cos\frac{\beta\pi t}{2}}}}}\right)\right],\tag{217}$$

*where*

$$\phi\_2 = \tan^{-1} \frac{\xi\_{eq2}}{\sqrt{1 - \xi\_{eq2}^2}} = \tan^{-1} \frac{\frac{\xi \omega^{\beta - 1} \sin \frac{\beta \pi}{2}}{\sqrt{1 - \frac{\xi}{\pi} \omega^{\beta - 2} \cos \frac{\beta \pi}{2}}}}{\sqrt{1 - \frac{\xi^2 \omega^{2(\beta - 1)} \sin^2 \frac{\beta \pi}{2}}{1 - \frac{\xi}{\pi} \omega^{\beta - 2} \cos \frac{\beta \pi}{2}}}}. \tag{218}$$

**Proof.** ; Substituting *ςeq*<sup>2</sup> with that in Section 5 into the following expression

$$\mathcal{G}\_2(t) = \frac{1}{k} \left[ 1 - \frac{e^{-\zeta\_{eq2}\omega\_{eq2}t}}{\sqrt{1 - \zeta\_{eq2}^2}} \cos\left(\omega\_{eqd,2}t - \phi\_2\right) \right] \tag{219}$$

yields

$$g\_2(t) = \frac{1}{k} \left[ 1 - \frac{e^{-\frac{\zeta \omega \vartheta - 1 \sin\frac{\beta \pi}{2}}{\sqrt{1 - \frac{\zeta}{m} \omega^{\vartheta - 2} \cos\frac{\beta \pi}{2}}} \omega\_{eq, 2} t}{\sqrt{1 - \frac{\zeta^2 \omega^{\vartheta (\beta - 1)} \sin^2 \frac{\beta \pi}{2}}{1 - \frac{\zeta}{m} \omega^{\vartheta - 2} \cos\frac{\beta \pi}{2}}}} \cos \left( \omega\_{eq d, 2} t - \phi\_2 \right) \right]. \tag{220}$$

On the other side, replacing *<sup>ω</sup>eqn*,<sup>2</sup> by the one in (135) in the above results in

$$g\_2(t) = \frac{1}{\mathbb{E}} \left[ 1 - \frac{e^{-\frac{\zeta \omega \mathcal{S}^{-1} \sin \frac{\xi \pi}{2}}{\xi} \omega\_{\text{adj}\mathcal{S}^{-1} \sin \frac{\xi \pi}{2}} \omega\_{\text{adj}\mathcal{S}^{-1} \sin \frac{\xi \pi}{2}}}}{\sqrt{1 - \frac{\zeta^2 \omega^{2(\beta - 1)} \sin^2 \frac{\xi \pi}{2}}{1 - \frac{\zeta \omega \omega^{2 - 2} \cos \frac{\xi \pi}{2}}{\xi}}}} \right] \\ = \frac{1}{\mathbb{E}} \left[ 1 - \frac{e^{-\frac{\zeta \omega \omega \omega^{2} - 1}{1 - \frac{\zeta \omega \omega^{2} - 2 \cos \frac{\xi \pi}{2}}{\xi}} t \left( \omega\_{\text{adj}\mathcal{S}^{-1} \sin \frac{\xi \pi}{2}} \right)}{\sqrt{1 - \frac{\zeta^2 \omega^{2(\beta - 1)} \sin^2 \frac{\xi \pi}{2}}{1 - \frac{\zeta \omega \omega^{2} - 2 \cos \frac{\xi \pi}{2}}{\xi}}}} \right].$$

Finally, substituting *<sup>ω</sup>eqd*,<sup>2</sup> by that in (157) in the above produces (217) and (218). Hence, we finish the proof. 

We use Figure 45 to indicate *g*2(*t*) with fixed *ω*. When considering variable *ω*, we show *g*2(*t*) in Figure 46 in time domain and Figure 47 in t-ω plane.

**Figure 45.** Step response *g*2(*t*) to a fractional oscillator in Class II with fixed *ω* for *m* = *c* = *k* = 1. (**a**) *β* = 0.3, solid line: *ω* = 20 (*ςeq*<sup>2</sup> = 0.03); dot line: *ω* =5(*ςeq*<sup>2</sup> = 0.08). (**b**) *β* = 0.6, solid line: *ω* = 20 (*ςeq*<sup>2</sup> = 0.12); dot line: *ω* =5(*ςeq*<sup>2</sup> = 0.22). (**c**) *β* = 0.9, solid line: *ω* = 20 (*ςeq*<sup>2</sup> = 0.37); dot line: *ω* = 5 (*ςeq*<sup>2</sup> = 0.43). (**d**) *β* = 1, solid line: *ω* = 20 (*ςeq*<sup>2</sup> = 0.50); dot line: *ω* =5(*ςeq*<sup>2</sup> = 0.50).

**Figure 46.** Step response *g*2(*t*) to a fractional oscillator in Class II with variable *ω* for *m* = *c* = *k* = 1. (**a**) *β* = 0.3, *ω* = 1, 2, ..., 5 (0.08 ≤ *ςeq*<sup>2</sup> ≤ 0.69). (**b**) *β* = 0.3, *ω* = 1, 2, ..., 10 (0.22 ≤ *ςeq*<sup>2</sup> ≤ 0.63). (**c**) *β* = 0.9, *ω* = 1, 2, ..., 5 (0.43 ≤ *ςeq*<sup>2</sup> ≤ 0.54). (**d**) *β* = 0.9, *ω* = 1, 2, ..., 10 (0.40 ≤ *ςeq*<sup>2</sup> ≤ 0.54).

**Figure 47.** Step response *g*2(*t*) in t-ω plane for *m* = *c* = 1 and *ωn* = 0.3 (*k* = 0.09), with *t* = 0, 1, ..., 30, *ω* = 1, 2, 3, 4. (**a**) *β* = 0.3 (0.09 ≤ *ςeq*<sup>2</sup> ≤ 0.69). (**b**) *β* = 0.6 (0.24 ≤ *ςeq*<sup>2</sup> ≤ 0.63). (**c**) *β* = 0.9 (0.44 ≤ *ςeq*<sup>2</sup> ≤ 0.54). (**d**) *β* =1(*ςeq*<sup>2</sup> = 0.50).

**Note 8.2:** When *β* = 1, *g*2(*t*) turns to be the ordinary step response. As a matter of fact,

$$\left. \zeta\_2(t) \right|\_{\beta=1} = \frac{1}{k} \left[ 1 - \frac{e^{-\zeta \omega\_{\text{eqn},2}t}}{\sqrt{1-\zeta^2}} \cos \left( \omega\_n \sqrt{1-\zeta^2}t - \phi\_2 \right) \right],\tag{221}$$

where

$$\left. \phi\_2 \right|\_{\beta=1} = \tan^{-1} \frac{\frac{\zeta \omega^{\beta-1} \sin \frac{\beta \pi}{2}}{\sqrt{1 - \frac{\zeta}{\pi n} \omega^{\beta-2} \cos \frac{\beta \pi}{2}}}}{\sqrt{1 - \frac{\zeta^2 \omega^{2(\beta-1)} \sin^2 \frac{\beta \pi}{2}}{1 - \frac{\zeta}{\pi n} \omega^{\beta-2} \cos \frac{\beta \pi}{2}}}} \bigg|\_{\beta=1} = \tan^{-1} \frac{\xi}{\sqrt{1 - \xi^2}}.\tag{222}$$

*8.4. Step Response to a Fractional Oscillator in Class III*

**Theorem 18** (Step response III)**.** *Let g*3(*t*) *be the unit step response to a fractional oscillator in Class III. It is in the form, for t* ≥ *0, 1 < α* ≤ *2, and 0 < β* ≤ *1,*

$$g\_{3}(t) = \frac{1}{k} \left\{ 1 - \frac{\left( \begin{array}{l} -\frac{m\omega^{\beta-1}}{2(m\omega^{\beta-2}\cos\frac{\beta\pi}{2} + \cos\frac{\beta\pi}{2}) - \sin\frac{\beta\pi}{2}}t \\\\ \cos\left(\frac{\omega\_{0}}{4} \sqrt{1 - \left(\frac{m\omega^{\beta-1}}{4}\cos\frac{\omega\pi}{2} + \cos^{\beta-1}\sin\frac{\beta\pi}{2}\right)^{2}}t\right) \\\\ 1 - \frac{\left(\begin{array}{l} \left(\omega\_{0} - \left(\frac{m\omega^{\beta-1}}{4}\cos\frac{\omega\pi}{2} + \cos\frac{\beta\pi}{2}\right) - \sin\frac{\beta\pi}{2} \\\\ \sqrt{-\left(\omega^{\alpha-2}\cos\frac{\omega\pi}{2} + \frac{\omega}{\alpha}\omega^{\beta-2}\cos\frac{\beta\pi}{2}\right)^{2}}\right) \\\\ \sqrt{1 - \left[\frac{m\omega^{\alpha-1}}{2\sqrt{-\left(m\omega^{\alpha-2}\cos\frac{\omega\pi}{2} + \cos\frac{\beta\pi}{2} - \cos\frac{\beta\pi}{2}\right)^{2}}}\right]^{2} \\\\ \end{array}\right) \right\},\tag{223}$$

*where*

$$\phi\_3 = \tan^{-1} \frac{\zeta\_{eq3}}{\sqrt{1 - \zeta\_{eq3}^2}} = \tan^{-1} \frac{\frac{c\omega^{\beta - 1} \sin \frac{\beta \pi}{2}}{2\sqrt{\left(m - c\omega \beta^{-2} \cos \frac{\beta \pi}{2}\right)k}}}{\sqrt{1 - \left(\frac{c^2 \omega^{2(\beta - 1)} \sin^2 \frac{\beta \pi}{2}}{4\left(m - c\omega \beta^{-2} \cos \frac{\beta \pi}{2}\right)k}\right)}}. \tag{224}$$

**Proof.** Replacing *ςeq*<sup>3</sup> by that in (147) on the left side of the following produces the right side in the form 

$$\begin{split} \mathcal{G}\_{3}(t) &= \frac{1}{k} \left[ 1 - \frac{e^{-\xi\_{\text{cyl}}\omega\_{\text{cyl},3}t}}{\sqrt{1 - \xi\_{\text{cyl},3}^{2}}} \cos\left(\omega\_{\text{cyl},3}t - \phi\_{3}\right) \right] \\ &= \frac{1}{k} \left[ 1 - \frac{e^{-\frac{m\omega^{\theta} - 1}\sin\frac{\theta\pi}{2} + \omega\beta^{-1} - \sin\frac{\theta\pi}{2}}{2\sqrt{-(\tan^{\alpha-2}\omega\cos\frac{\theta\pi}{2} + \omega\alpha^{\theta-2}\cos\frac{\theta\pi}{2})k}} e^{\omega\_{\text{cyl},3}t}}{\sqrt{1 - \left[ \frac{m\omega^{\alpha-1}\sin\frac{\theta\pi}{2} + \omega\alpha^{\theta-1}\sin\frac{\theta\pi}{2}}{2\sqrt{-(\tan^{\alpha-2}\omega\cos\frac{\theta\pi}{2} + \omega\alpha^{\theta-2}\cos\frac{\theta\pi}{2})k}} \right]^{2}} \right] . \end{split} \tag{225}$$

Further, replacing *<sup>ω</sup>eqn*,<sup>3</sup> with the one in (137) in the above yields

*g*3(*t*) = 1*k* ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1 − *e*− *<sup>ω</sup>n*(*mωα*−<sup>1</sup> sin *απ*2 <sup>+</sup>*cωβ*−<sup>1</sup> sin *βπ*2 ) −(*ωα*−<sup>2</sup> cos *απ*2 + *cm ωβ*−<sup>2</sup> cos *βπ*2 ) <sup>2</sup>−(*mωα*−<sup>2</sup> cos *απ*2 <sup>+</sup>*cωβ*−<sup>2</sup> cos *βπ*2 )*k t* cos(*<sup>ω</sup>eqd*,3*t*−*φ*3) "#####\$1−⎡⎢⎢⎣ *mωα*−1 sin *απ*2 <sup>+</sup>*cωβ*−<sup>1</sup> sin *βπ*2 <sup>2</sup>!− *mωα*−<sup>2</sup> cos *απ*2 <sup>+</sup>*cωβ*−<sup>2</sup> cos *βπ*2 *k* ⎤⎥⎥⎦2 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ = 1 *k* ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣1 − *e*− *mωα*−<sup>1</sup> sin *απ*2 <sup>+</sup>*cωβ*−<sup>1</sup> sin *βπ*2 <sup>2</sup>(*mωα*−<sup>2</sup>|cos *απ*2 |−*cωβ*−<sup>2</sup> cos *βπ*2 ) *t* cos(*<sup>ω</sup>eqd*,3*t*−*φ*3) "#####\$1−⎡⎢⎢⎣ *mωα*−1 sin *απ*2 <sup>+</sup>*cωβ*−<sup>1</sup> sin *βπ*2 <sup>2</sup>!− *mωα*−<sup>2</sup> cos *απ*2 <sup>+</sup>*cωβ*−<sup>2</sup> cos *βπ*2 *k* ⎤⎥⎥⎦2 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦. (226)

Finally, considering *<sup>ω</sup>eqd*,<sup>3</sup> expressed by (160), we have (223) and (224). Hence, the proof finishes. 

Figure 48 illustrates *g*3(*t*) in time with fixed *ω* while Figure 49 is with variable *ω*. Its illustrations in t-ω plane are shown in Figure 50.

**Figure 48.** *Cont.*

**Figure 48.** Illustrating step response *g*3(*t*) with fixed *ω* for *m* = *c* = 1, *k* = 25 (*<sup>ω</sup>n* = 5). (**a**) (*<sup>α</sup>*, *β*) = (1.8, 0.8), solid line: *ω* =1(*ςeq*<sup>3</sup> = 0.13), dot line: *ω* =2(*ςeq*<sup>3</sup> = 0.05). (**b**) (*<sup>α</sup>*, *β*) = (1.5, 0.8), solid line: *ω* =1(*ςeq*<sup>3</sup> = 0.33), dot line: *ω* =2(*ςeq*<sup>3</sup> = 0.15). (**c**) (*<sup>α</sup>*, *β*) = (1.3, 0.8), solid line: *ω* =1(*ςeq*<sup>3</sup> = 0.49), dot line: *ω* =2(*ςeq*<sup>3</sup> = 0.24). (**d**) (*<sup>α</sup>*, *β*) = (1.8, 0.5), solid line: *ω* =1(*ςeq*<sup>3</sup> = 0.09), dot line: *ω* =2(*ςeq*<sup>3</sup> = 0.03). (**e**) (*<sup>α</sup>*, *β*) = (2, 1), solid line: *ω* =1(*ςeq*<sup>3</sup> = 0), dot line: *ω* =2(*ςeq*<sup>3</sup> = 0).

**Figure 49.** Demonstrating step response *g*3(*t*) with variable *ω* ( = 1, 2, ..., 5) for *m* = *c* = 1, *k* = 25 (*<sup>ω</sup>n* = 5). (**a**) (*<sup>α</sup>*, *β*) = (1.8, 0.8) (0.13 ≤ *ςeq*<sup>3</sup> ≤ 1.22). (**b**) (*<sup>α</sup>*, *β*) = (1.5, 0.8) (0.34 ≤ *ςeq*<sup>3</sup> ≤ 0.91). (**c**) (*<sup>α</sup>*, *β*) = (1.3, 0.8) (0.49 ≤ *ςeq*<sup>3</sup> ≤ 1.14).

**Figure 50.** *Cont.*

**Figure 50.** Illustrating step response *g*3(*t*) in t-ω plane for *m* = *c* = *k* = 1, with *t* = 0, 1, ..., 30, *ω* = 1, 2, ..., 5. (**a**) (*<sup>α</sup>*, *β*) = (1.8, 0.3) (0.05 ≤ *ςeq*<sup>3</sup> ≤ 0.10). (**b**) (*<sup>α</sup>*, *β*) = (1.8, 0.5) (0.09 ≤ *ςeq*<sup>3</sup> ≤ 0.20). (**c**) (*<sup>α</sup>*, *β*) = (1.5, 0.6) (0.25 ≤ *ςeq*<sup>3</sup> ≤ 0.55). (**d**) (*<sup>α</sup>*, *β*) = (2, 1) (0.49 ≤ *ςeq*<sup>3</sup> ≤ 0.96).

**Note 8.3:** For (*<sup>α</sup>*, *β*) = (2, 1), *g*3(*t*) reduces to the conventional step response. Indeed,

$$\left. \xi g\_3(t) \right|\_{a=2, \beta=1} = \frac{1}{k} \left[ 1 - \frac{e^{-\xi \omega\_n t}}{\sqrt{1 - \xi^2}} \cos \left( \omega\_n \sqrt{1 - \xi^2} t - \phi\_3 \vert\_{a=2, \beta=1} \right) \right],\tag{227}$$

where

$$\left|\phi\_{3}\right|\_{a=2,\beta=1} = \tan^{-1}\frac{\frac{m\omega^{a-1}\sin\frac{\theta\pi}{2} + c\omega^{\beta-1}\sin\frac{\beta\pi}{2}}{2\sqrt{-\left(m\omega^{a-2}\cos\frac{\theta\pi}{2} + c\omega^{\beta-2}\cos\frac{\theta\pi}{2}\right)}k}{\sqrt{1-\left[\frac{m\omega^{a-1}\sin\frac{\theta\pi}{2} + c\omega^{\beta-1}\sin\frac{\theta\pi}{2}}{2\sqrt{-\left(m\omega^{a-2}\cos\frac{\theta\pi}{2} + c\omega^{\beta-2}\cos\frac{\theta\pi}{2}\right)}k}\right]^2}}{\left|\begin{array}{c} \\\\ \omega^{-1}\frac{-\xi}{\sqrt{1-\xi^{2}}} \end{array}\right.}\tag{228}$$
