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Abstract

In this article, the conditions for the asymmetric nonlinear vibrations of an unforced Watt governor were investigated and the exact analytical solutions for its time period and displacement were derived naturally from the first integral of the governing dynamic model. It was demonstrated that the unforced Watt governor can undergo asymmetric vibrations due to bifurcation and the conditions for this to happen were derived based on the dimensionless rotational speed of its spindle and the initial conditions. In contrast to previously published exact solutions that are limited in their scope of application, the present exact solutions are applicable to all conditions of asymmetric vibrations. Simulation results revealed that the unforced Watt governor produces strong nonlinear asymmetric response even for small initial amplitudes and that the expression of the nonlinear effects depended strongly on the initial conditions. The present exact solutions provide benchmark solutions that are useful for calibrating the accuracy of other approximate analytical methods especially under highly stiff nonlinear conditions. The analytical framework develop in this study can be extended to other symmetric oscillators capable of bi-stable and tri-stable vibrations.

Keywords

Watt governor bifurcation asymmetric vibrations exact periodic solution elliptic integral

Introduction

The Watt governor is a mechanical device used to achieve automatic speed control of engines. Although it has undergone several modifications since its invention, it still remains relevant today because of its pedagogical value,¹ research relevance^2,3 and historic contribution to the mathematical theory of automatic control⁴ and stability analysis.⁵ As illustrated in Figure 1, the Watt governor consists of two flyballs connected to a massless sleeve by means of arm linkages and two arms. The arms are pivoted so that the flyballs can move inward and outward as they rotate with the vertical spindle. The engine drives the spindle rotation through a bevel transmission gear. The sleeve, which is keyed to the spindle, revolves with the spindle and can slide up and down according to the spindle’s rotational speed. Two stoppers are provided to limit the vertical motion of the sleeve.

Figure 1.

Watt governor showing components and linkages (adapted from YouTube video: https://www.youtube.com/watch?v=sJDVacqQHzE).

The principle of operation of the Watt governor (Figure 1) is quite simple. When the load on the engine decreases, there is an increase in the speed of the engine and the angular velocity of the spindle. This increases the centrifugal force on the flyballs so that the flyballs move outwards and the sleeve move upwards. The upward motion of the sleeve actuates a throttle link (a bell-crank mechanism) that operates the throttle valve to decrease the fuel supply. Consequently, the power output is reduced and the engine speed is lowered. On the other hand, when the load on the engine increases, the engine speed and the centrifugal force decrease. The flyballs move inward and the sleeve moves downward. The movement actuates the throttle link in a manner that causes an opening of the throttle valve for increased fuel supply. Hence, the power output improves and engine speed increases. This way, the governor attempts to maintain a steady engine speed irrespective of the load on the engine. However, the Watt governor is insensitive to speed fluctuations at high engine speeds (above 200 rpm) and therefore, it is suitable for moderate to low speed (30–200 rpm) applications such as steam engines, marine engines, textile machinery, windmills and modern striking clocks.

From the time James Clerk Maxwell investigated the dynamics of the Watt governor in his seminal paper,⁶ several studies have been conducted on the stability^2,3,5 and vibration analysis of the Watt governor.^4,7,8 The unforced Watt governor (UWG) undergoes symmetric vibrations which are characterized by equal maximum displacement on each side of the centre of vibration. Due to this fact, the UWG model has been mostly investigated in the light of its symmetric vibrations.^9–23 Liao and Chwang⁹ obtained approximate periodic solutions for the symmetric vibration of the UWG model using the homotopy analysis method (HAM). They derived the second-order HAM solution in terms of Bessel function but the procedure is algebraically complicated for higher-order approximations. Lai et al.¹⁰ investigated the large-amplitude symmetric vibration of the UWG model. They used the Chebyserv polynomial to approximate the UWG model to a cubic-quintic Duffing oscillator, for which they obtained an approximate analytical solution using a modified Micken’s iteration method. The results of the second-order modified Micken’s iteration method were found to be more accurate than the second-order HAM solution. Ghafoori et al.¹¹ applied the differential transform method (DTM) to study the symmetric vibration response of the UWG model. Comparison with homotopy perturbation method (HPM) and variational iteration method (VIM) showed that the DTM produced more accurate results. He et al.¹² obtained an approximate analytical periodic solution for the symmetric vibration response of the UWG model using the HPM whereas Sharif et al.¹³ used the modified HPM. The HPM required approximation of the non-polynomial restoring force with a polynomial function but the modified HPM did not. Consequently the modified HPM solution showed good prediction of the large-amplitude symmetric vibration response while the HPM produced significant errors. Ismail et al.¹⁴ derived an approximate periodic solution for the symmetric vibration of the UWG model using the gamma function method (GFM). The results showed that the GFM produced accurate estimates for small- to moderate-amplitude vibrations. Moatimid and Amer¹⁵ derived an approximate analytical solution for the symmetric vibrations of the UWG model. They approximated the UWG model with a cubic-quintic Duffing oscillator using Taylor’s series expansion of the original nonlinear restoring force. The cubic-quintic Duffing oscillator was solved for specific initial conditions using the second-order Laplace Transform Homotopy Perturbation method (LTHPM). However, recent studies¹⁶ have shown that the second-order LTHPM produces significant errors in the periodic response for moderate-amplitude vibration of the UWG model. Moatimid et al.¹⁷ studied the stability and symmetric vibration of the UWG model using an equivalent linearization non-perturbative approach. However, the stability analysis based on the linearization approach could not predict the bifurcation of the UWG model. In Big-Alabo and Chuku¹⁶ and Big-Alabo and Ossia,¹⁸ the stability, bifurcation and periodic solution of the symmetric vibration of the UWG model was examined for large-amplitude cases using the continuous piecewise linearization method (CPLM) while in References 19–21 the Akbari-Ganji method (AGM) was used to derived periodic solutions for small- to moderate-amplitude cases. Most approximate analytical methods used to derive the periodic solution of the UWG model require the restoring force to be in polynomial form.^{10,12,14,15,19–21} Hence, the non-polynomial restoring force of the UWG model is normally approximated by a cubic-quintic polynomial function and this approach produces significant errors for large-amplitude vibrations.¹³ However, the CPLM works with a non-polynomial restoring force and seems to provide a more accurate periodic solution compared to the other approximate methods in References 10, 12, 14, 15, 19–21, especially for large-amplitude vibrations.

Several studies have applied approximate analytical methods to estimate the symmetric vibration response of the UWG model and a few have also derived exact closed-form solutions. Hammad et al.²² derived an exact periodic solution for the symmetric vibration of the UWG model with non-zero initial amplitude and zero initial velocity. The solution was derived by applying an assumed ansatz solution in terms of the Weierstrass elliptic function. More general initial conditions for the symmetric vibration of the UWG model were studied by Salas et al.,²³ who transformed the vibration model from angular coordinates to Cartesian coordinates and obtained a non-natural oscillator model for the governor. The latter model was then solved exactly using an assumed ansatz solution in terms of the Jacobi cosine function from which the exact analytical solution of the symmetric vibration of the UWG model was derived. The challenge with the ansatz method is that the assumed solution may be limited in its scope of application because it was not derived naturally from the equation of motion.^24,25

On the other hand, the Watt governor can undergo asymmetric vibrations but not much attention has been given to this. The asymmetric vibrations are characterized by unequal maximum displacement on each side of the centre of vibration. Abdel-Rahman²⁶ studied the symmetry breaking phenomenon of the UWG model based on classical, semiclassical and quantum considerations. In the classical sense, rotational velocities that represent conditions where the total energy is slightly or well beyond the symmetry breaking point were considered. These conditions imply the existence of asymmetric vibrations for which the total energy is negative. A closed-form periodic solution for the asymmetric vibration was derived from the first integral in terms of the Jacobi delta amplitude function. However, the constants in the periodic solution were not completely defined and the time period solution was not derived. Furthermore, the condition for asymmetric vibration was defined in terms of the total energy and it is unclear whether the derived periodic solution is applicable to all conditions of asymmetric vibration. Markeev^7,8 studied the asymmetric vibrations of the UWG model by considering the phase portrait of the approximate set characterized by positive amplitudes. The primary and secondary conditions for asymmetric vibration were derived based on the dimensionless rotational speed and the total energy of the governor respectively. The exact periodic solution was derived naturally from the first integral of the vibration model. However, there are limitations in these studies that were addressed in the present study. First, the use of the total energy to determine the secondary conditions for asymmetric vibration is not the best approach since the energy relies on the initial conditions. A better approach involves deriving the secondary conditions in terms of the initial conditions which are input parameters. Secondly, the first argument of the Jacobi delta amplitude function in the exact displacement solution⁷ was defined only for a limited condition that does not represent all asymmetric vibrations. In the present study, the exact periodic solution derived naturally from the first integral applies to all conditions of asymmetric vibration. Furthermore, exact periodic solutions were derived for all conditions representing single-well and double-well symmetric vibrations. The goal of this paper is to establish analytically the conditions under which the UWG experiences asymmetric vibrations and to derive exact closed-form solutions for its vibration response under general initial conditions.

Conditions for asymmetric vibrations

Under real conditions, the rotational speed of the spindle undergoes small fluctuations while in operation, especially at moderate to high speeds. Also, viscous damping effects play a role in the vibration response for low rotational speeds. Therefore, if the fluctuations and damping effects are accounted for, then the resulting equations of motion will be a coupled two degree-of-freedom system as shown²⁷:

\frac{d^{2} θ}{d t^{2}} + c \frac{d θ}{dt} + ω^{2} \sin θ - \frac{1}{2} Ω^{2} \sin 2 θ = 0

(1a)

J \frac{d Ω}{dt} = n Δ T

(1b)

where $c$ is the damping coefficient per unit mass, $ω = \sqrt{g / L}$ , $g = 9.81 m / s^{2}$ , $L$ is the arm length, $Ω$ is the rotational speed of the spindle, $J$ is the moment of inertia of the governor, $Δ T = T (θ)$ is the net torque between the engine and the load and $n = Ω / σ$ is the ratio of the spindle rotational speed to the engine shaft speed ( $σ$ ). Equation (1a and b) do not produce an invariant Hamiltonian function from which the natural frequency of the Watt governor can be obtained. In order to estimate the natural frequency and obtain a completely integrable periodic solution, the rotational speed is assumed to have a constant average value and the damping effects are considered to be negligible. This reduces the UWG model to a single degree-of-freedom system as follows:

\frac{d^{2} θ}{d t^{2}} + ω^{2} \sin θ - \frac{1}{2} Ω^{2} \sin 2 θ = 0

(2)

The nonlinear restoring force of the vibration of the governor is $f (θ) = ω^{2} \sin θ - (Ω^{2} / 2) \sin 2 θ$ and the generalized initial conditions are: $θ (0) = θ_{0}$ and $d θ (0) / dt = v_{0}$ . It is noteworthy that equation (2) models the vibration of several pedagogical and practical systems such as pendulum-like systems,²⁸ magnetic spherical pendulum with uniform azimuthal velocity,^15,16 spinning top,²⁹ pendulum with rotating support¹⁸ and frictionless motion of a small mass moving round a circular ring with constant spin.¹⁹ Equation (2) can be used to investigate the conditions for the asymmetric vibrations of the Watt governor and determine the corresponding periodic response, which is the focus of this study.

By introducing time normalization, equation (2) can be rewritten as:

\frac{d^{2} θ}{d τ^{2}} + \sin θ - \frac{Λ}{2} \sin 2 θ = 0

(3)

where $τ = ω t$ is the normalized time, $Λ = {(Ω / ω)}^{2}$ is the normalized rotational speed of the spindle and $f (θ) = \sin θ - (Λ / 2) \sin 2 θ = (1 - Λ \cos θ) \sin θ$ is the normalized restoring force. Equation (3) shows that the vibration response of the UWG depends only on $Λ$ . Previous studies^16,18 have demonstrated that equation (2) produces a bifurcation when $Λ = 1.0$ , and this results to a transition from single-well potential ( $Λ < 1$ ) to double-well potential ( $Λ > 1$ ). For example, a Watt governor of arm length $0.40 m$ and spindle rotational speed of $90 rpm$ has a normalized rotational speed of $Λ = 3.62$ , which implies that it characterized by a double-well potential. Figure 2 illustrates the variation of $Λ$ for typical arm lengths and spindle speeds. It also shows the parametric conditions for single-well and double-well potentials, which confirm that under typical operating conditions the governor experiences a double-well potential more frequently than it does a single-well potential. The dashed black line in the Figure represents the parametric conditions for the bifurcation points where transition from single-well to double-well potential occurs and vice versa. The spindle rotational speed (in rpm) at which the bifurcation occurs ( $Ω_{b}$ ) is simply determined as $Ω_{b} = 30 \sqrt{g / (π^{2} L)}$ . If $Ω < Ω_{b}$ , then the governor exhibits symmetric single-well vibrations; whereas $Ω > Ω_{b}$ implies that the governor will exhibit a symmetric double-well vibration. A plot of the potential function, that is, $V (θ) = (Λ / 4) \cos 2 θ - \cos θ$ , and phase diagram for $Λ > 1$ is shown in Figure 3. Clearly, a double-well potential is visible. This happens because the system has three equilibrium points when $Λ > 1$ , of which two are stable ( $θ_{e} = \pm \cos^{- 1} [1 / Λ]$ ) and one is unstable ( $θ_{e} = 0$ ). The latter is a saddle node equilibrium.

Figure 2.

Bifurcation plot showing the conditions for transition from single-well to double-well.

Figure 3.

Potential function plot and phase diagram for $Λ = 3.0$ . The letters “C” and “S” in the phase diagram represent centre point and saddle point respectively.

A notable observation from Figure 3 is the existence of a unique energy level for $Λ > 1$ where there is no vibration. This energy level occurs specifically at the saddle node equilibrium and produces an intersecting curve in the phase diagram called separatrix. The separatrix is an asymptote representing the trajectory of all state-space points for which the potential energy of the governor is a local maximum. When the energy level exceeds the separatrix energy level, the governor undergoes a symmetric double-well (or bi-stable) vibration with its centres equidistant from the origin. On the other hand, the governor experiences a single-well vibration with its centre occurring either to the left or right of the origin when its energy level is below the separatrix energy level. Under this latter condition, the governor breaks its symmetry and undergoes asymmetric vibrations. Figure 3 shows that the asymmetric vibration on the left (where $θ < 0$ ) is a mirror image of the asymmetric vibration on the right (when $θ > 0$ ). Hence, the investigation of the asymmetric vibration was conducted based on $θ > 0$ . without loss of generality.

The primary condition for the asymmetric vibrations of the governor is that $Λ > 1.0$ . Other necessary conditions can be derived based on the initial conditions. At the saddle node equilibrium, the system has a maximum potential energy and no kinetic energy. Hence, the total energy or energy level of the system is equal to the maximum potential energy, which is $V_{\max} = (Λ / 4) - 1$ . This result is the same as the separatrix energy level and it is solely dependent on the normalized rotational speed.

The first integral of equation (3) gives the state-space relationship as:

\frac{d θ}{d τ} = \pm \sqrt{2 (h - (\frac{Λ}{4} \cos 2 θ - \cos θ))}

(4)

where $h = \frac{1}{2} v_{0}^{2} + \frac{Λ}{4} \cos 2 θ_{0} - \cos θ_{0}$ is the total energy of the system. Given that $h = (Λ / 4) - 1$ for the separatrix, the corresponding equation for the orbit of the separatrix can be obtained from equation (4) as:

\frac{d θ}{d τ} = \pm \sqrt{\frac{Λ (1 - \cos 2 θ) + 4 (\cos θ - 1)}{2}}

(5)

At the displacement limits of the separatrix curve where the latter intersects the horizontal axis, the kinetic energy vanishes. Thus, $Λ (1 - \cos 2 θ) + 4 (\cos θ - 1) = 0$ from which the displacement limits of the separatrix curve were derived as $θ = \pm \cos^{- 1} [(2 / Λ) - 1]$ . The implication is that the initial conditions that will give rise to asymmetric vibrations must satisfy the following constraints:

v_{0} = 0 and 0 < | θ_{0} | < \cos^{- 1} [(2 / Λ) - 1]

(6)

However, if $v_{0} > 0$ , then we can deduce from equation (5) that asymmetric vibrations will occur if the following constraints on the initial conditions are satisfied:

\begin{matrix} 0 \leq | v_{0} | < \frac{Λ - 1}{\sqrt{Λ}} and \cos^{- 1} (\frac{1 + \sqrt{Λ^{2} + 1 - Λ (v_{0}^{2} + 2)}}{Λ}) \\ < | θ_{0} | < \cos^{- 1} (\frac{1 - \sqrt{Λ^{2} + 1 - Λ (v_{0}^{2} + 2)}}{Λ}) \end{matrix}

(7)

Note that equation (7) reduces to equation (6) when $v_{0} = 0$ .

It seems from equation (7) that there are three cases of initial conditions to consider in respect of the asymmetric vibrations that is, (i) zero initial velocity and non-zero initial displacement, (ii) non-zero initial velocity and zero initial displacement and (iii) non-zero initial velocity and displacement. However, a simple examination of the three cases shows that only cases (i) and (iii) satisfy the conditions in equation (7). For case (ii), equation (7) implies that: $1 + \sqrt{Λ^{2} + 1 - Λ (v_{0}^{2} + 2)} = π Λ / 2$ . Solving for the initial velocity in this equation gives: $v_{0}^{2} = 1.1416 - 1.4674 Λ$ . Hence, real solutions for $v_{0}$ would imply that the primary condition for existence of asymmetric vibrations (i.e. $Λ > 1$ ) must be violated. In previous studies^15,16 the single-well and double-well symmetric vibrations of UWG model were investigated based on the case (ii) initial conditions. However, it has been demonstrated here that the case (ii) initial conditions does not permit asymmetric vibrations. In conclusion, equation (6) is applicable for case (i) initial conditions while equation (7) is applicable for case (iii) initial conditions.

Figure 4 shows a mapping of the parametric conditions for asymmetric vibrations to occur. The map shows that a wider range of initial conditions capable of producing asymmetric vibrations is obtained for higher values of the normalized rotational speed.

Figure 4.

Region satisfying parametric conditions for asymmetric vibrations: (a) 3-D plot for complete region and (b) 2-D projection defining the region of initial conditions for given values of $Λ$ .

Exact periodic solution

In this section, the exact periodic solution for equation (3) was derived naturally from the governing differential equation. Furthermore, arbitrary non-zero initial conditions were considered to ensure unrestricted application of the solutions.

From equation (4),

\frac{d θ}{d τ} = \pm \sqrt{Λ (Δ_{2} + Δ_{1} \cos θ - \cos^{2} θ)}

(8)

where $Δ_{1} = 2 / Λ$ and $Δ_{2} = (Λ + 4 h) / 2 Λ$ .

To derive the exact periodic solution, we need to transform the equation (8) from polar to Cartesian coordinates. Using the transformation variable $x = \cos θ$ , the time integral for the exact periodic solution was derived in Cartesian coordinates as:

τ = \mp \frac{1}{\sqrt{Λ}} \int_{x_{l}}^{x_{u}} \frac{dx}{\sqrt{(x - p) (x - q) (x - 1) (x + 1)}}

(9)

where $Λ \neq 0$ , $p = (Δ_{1} + \sqrt{Δ_{1}^{2} + 4 Δ_{2}}) / 2 = [1 + \sqrt{Λ^{2} + 1 - Λ (v_{0}^{2} + 2)}] / Λ$ , $q = (Δ_{1} - \sqrt{Δ_{1}^{2} + 4 Δ_{2}}) / 2 = [1 - \sqrt{Λ^{2} + 1 - Λ (v_{0}^{2} + 2)}] / Λ$ , $x_{u}$ is the upper integration limit and $x_{l}$ is the lower integration limit. The case of $Λ = 0$ reduces the problem to the vibration of the simple pendulum, whose exact solution is well known. Equation (9) is the fundamental integral equation on the basis of which the exact time period and displacement solutions were derived.

Exact time period for asymmetric vibration of the Watt governor

Assuming the amplitudes of the asymmetric vibrations are $A$ and $B$ where $A \neq B$ , then the integration limits for the time period are $x_{u} = \cos A$ and $x_{l} = \cos B$ . Since the potential energies at the amplitudes are equal to the total energy, the equation $V (A) = V (B) = h$ was solved to get $\cos A = p$ and $\cos B = q$ . Therefore, the exact time period is:

T_{ex} = (\frac{2}{\sqrt{Λ}}) \int_{q}^{p} \frac{dx}{\sqrt{(x - p) (x - q) (x - 1) (x + 1)}}

(10)

To evaluate the integral in equation (10), we apply formula 254.00 of Section 3.147 on page 278 of Gradshteyn and Ryzhik³⁰ given as:

\int_{c}^{u} \frac{dx}{\sqrt{(a - x) (b - x) (x - c) (x - d)}} = \frac{2}{\sqrt{(a - c) (b - d)}} F (φ, m)

(11)

where $a > b \geq u > c > d$ , $F (φ, m)$ is the incomplete elliptic integral of the first kind and the elliptic amplitude ( $φ$ ) and parameter ( $m$ ) are evaluated as follows:

φ = \sin^{- 1} (\sqrt{\frac{(b - d) (u - c)}{(b - c) (u - d)}}); m = \frac{(b - c) (a - d)}{(a - c) (b - d)}

(12a, b)

In order to compare with equation (11), the exact time period can be rearranged as:

T_{ex} = (\frac{2}{\sqrt{Λ}}) \int_{q}^{p} \frac{dx}{\sqrt{(1 - x) (p - x) (x - q) (x + 1)}}

(13)

Then comparing equations (11) and (13), $a = 1$ , $u = b = p$ , $c = q$ and $d = - 1$ . Using these results gives the exact time period as:

T_{ex} = \frac{4}{Ψ} K (m)

(14)

where $K (m) = F (π / 2, m)$ is complete elliptic integral of the first kind,

Ψ = \sqrt{Λ (1 - q) (1 + p)} = \sqrt{Λ (1 + Δ_{2} + \sqrt{Δ_{1}^{2} + 4 Δ_{2}})}

(15a)

and

m = \frac{2 (p - q)}{(1 - q) (1 + p)} = \sqrt{\frac{2 \sqrt{Δ_{1}^{2} + 4 Δ_{2}}}{1 + Δ_{2} + \sqrt{Δ_{1}^{2} + 4 Δ_{2}}}}

(15b)

Note that for $- 10^{3} \leq m \leq 0.999$ , $K (m)$ can be evaluated to an accuracy of more than 10 decimal places using an algebraic approximation composed of elementary functions.^28,31

Exact displacement history for asymmetric vibration of the Watt governor

To derive the exact displacement solution, we first define the applicable integration limits of equation (9). Given the arbitrary initial conditions, the upper and lower limits of integration are $x_{u} = x$ and $x_{l} = x_{0} = \cos θ_{0}$ respectively. Therefore,

τ = \mp \frac{1}{\sqrt{Λ}} \int_{x_{0}}^{x} \frac{dx}{\sqrt{(1 - x) (p - x) (x - q) (x + 1)}}

(16)

Assuming $x_{0} < q < x$ , then

\begin{matrix} τ = \mp \frac{1}{\sqrt{Λ}} (\int_{q}^{x} \frac{dx}{\sqrt{(1 - x) (p - x) (x - q) (x + 1)}} - \int_{q}^{x_{0}} \frac{dx}{\sqrt{(1 - x) (p - x) (x - q) (x + 1)}}) \end{matrix}

(17)

At this point we apply equations (11) and (12). Therefore,

τ = \mp \frac{2}{\sqrt{Λ (1 - q) (1 + p)}} (F (φ, m) - F (φ_{0}, m))

(18)

where $F (φ, m)$ is the incomplete elliptic integral of the first kind, $m$ is the same as derived in equation (15b) and $φ$ and $φ_{0}$ are given as:

φ = \sin^{- 1} (\sqrt{\frac{(p + 1) (x - q)}{(p - q) (x + 1)}})

(19)

\begin{matrix} φ_{0} = \sin^{- 1} (\sqrt{\frac{(p + 1) (x_{0} - q)}{(p - q) (x_{0} + 1)}}) \\ = \sin^{- 1} (\sqrt{\frac{2 (x_{0} + Δ_{2}) + Δ_{1} (x_{0} - 1) + (x_{0} + 1) \sqrt{Δ_{1}^{2} + 4 Δ_{2}}}{2 (x_{0} + 1) \sqrt{Δ_{1}^{2} + 4 Δ_{2}}}}) \end{matrix}

(20)

Equation (18) can be expressed as:

\frac{1}{2} Ψ (τ \mp τ_{0}) = F (φ, m)

(21)

where $τ_{0} = 2 F (φ_{0}, m) / Ψ$ represents the time taken from the initial displacement, $x_{0}$ , to the upper bound displacement, $q$ . Note that if $v_{0} = 0$ , $θ_{0}$ becomes the initial amplitude and $x_{0} = q$ . This implies that $φ_{0} \in {0, π / 2}$ and $τ_{0} = F (φ_{0}, m) \in {0, T / 2}$ . If $θ_{0} > θ_{e}$ then $φ_{0} = 0$ otherwise $φ_{0} = π / 2$ .

The Jacobi sine function is defined as: $\sin φ = \sin [F^{- 1} (u, m)] = sn [u, m]$ . Applying this definition to equations (21) and (19),

sn [\frac{Ψ (τ \mp τ_{0})}{2}, m] = \sqrt{\frac{(p + 1) (x - q)}{(p - q) (x + 1)}}

(22)

from which $x$ can be found as:

x (τ) = \frac{(p + 1) q + (p - q) {sn}^{2} [Ψ (τ \mp τ_{0}) / 2, m]}{(p + 1) - (p - q) {sn}^{2} [Ψ (τ \mp τ_{0}) / 2, m]}

(23)

The plus or minus sign in the first argument of the Jacobi sine function is negative if $v_{0} \geq 0$ and positive if $v_{0} < 0$ . Finally, recalling that $θ = \pm \cos^{- 1} x$ , the exact displacement solution of equation (2) is:

θ_{ex} (τ) = \pm \cos^{- 1} [\frac{(p + 1) q + (p - q) {sn}^{2} [Ψ (τ - τ_{0} sgn v_{0}) / 2, m]}{(p + 1) - (p - q) {sn}^{2} [Ψ (τ - τ_{0} sgn v_{0}) / 2, m]}]

(24)

where $sgn$ is the signum function. The solution $θ_{ex} (τ)$ has a positive displacement if $θ_{0} > 0$ and vice-versa. Since we are considering only positive displacements ( $θ > 0$ ), substituting $p$ and $q$ in equation (24) gives the exact displacement solution as:

θ_{ex} (τ) = \cos^{- 1} [\frac{(Δ_{1} - 2 Δ_{2} - \sqrt{Δ_{1}^{2} + 4 Δ_{2}}) + 2 \sqrt{Δ_{1}^{2} + 4 Δ_{2}} {sn}^{2} [Ψ (τ - τ_{0} sgn v_{0}), m]}{(2 + Δ_{1} + \sqrt{Δ_{1}^{2} + 4 Δ_{2}}) - 2 \sqrt{Δ_{1}^{2} + 4 Δ_{2}} {sn}^{2} [Ψ (τ - τ_{0} sgn v_{0}), m]}]

(25)

Results and discussions

Asymmetric vibration response

The exact periodic solutions derived in equations (14), (15) and (25) are applicable to the asymmetric vibrations of the Watt governor. The corresponding exact periodic solutions for the symmetric vibrations of the Watt governor were derived as shown in the Appendix. As discussed in Section “Conditions for asymmetric vibrations,” only cases (i) and (iii) initial conditions can be examined for the asymmetric vibrations. Figure 5(a) and (b) is for $v_{0} = 0.0$ when $θ_{0} < θ_{e}$ while Figure 5(c) and (d) is for $v_{0} = 0.0$ when $θ_{0} > θ_{e}$ ; where $θ_{e} = 70 . 529^{°}$ is the centre of vibration. Similarly, Figure 5(e, f) and (g, h) are for non-zero initial conditions when $θ_{0} < θ_{e}$ and $θ_{0} > θ_{e}$ respectively. In Figure 5(a) to (h) the separatrix amplitude was calculated as $109 . 471^{°}$ which implies that the amplitude of the asymmetric vibrations must be less than this value and the initial amplitude must satisfy this limit. For zero initial velocity (see Figure 5(a)–(d)), the range of initial amplitude that satisfies the limiting amplitude is $0^{°} < θ_{0} < 109 . 471^{°}$ except at $θ_{0} = θ_{e}$ . However, in Figure 5(e) to (h) where $v_{0} = 1.10$ , the range of initial amplitude that satisfies the limiting amplitude is $57 . 582^{°} < θ_{0} < 82 . 497^{°}$ except at $θ_{0} = θ_{e}$ .

Figure 5.

Asymmetric displacement history and corresponding relative difference of exact and numerical solutions: (a, b) $Λ = 3.0$ , $θ_{0} = 5 . 0^{°}$ and $v_{0} = 0.0$ , (c, d) $Λ = 3.0$ , $θ_{0} = 109 . 0^{°}$ and $v_{0} = 0.0$ , (e, f) $Λ = 3.0$ $θ_{0} = 58 . 0^{°}$ and $v_{0} = 1.10$ and (g, h) $Λ = 3.0$ $θ_{0} = 80 . 0^{°}$ and $v_{0} = 1.10$ . Legend for displacement plot: Exact solution (blue), numerical solution (pink) and centre of vibration (dash black line).

The results show that strong asymmetric vibrations are possible even when the initial amplitude is small (Figure 5(a)). This occurs because the oscillator experiences stronger nonlinear effects resulting in greater asymmetry as the initial amplitude approaches the limits of the applicable amplitude range. Additionally, the relative difference between the exact solution and numerical differentiation ( $ε = 1 - θ_{num} / θ_{ex}$ ) was computed and plotted as a lognormal plot in Figure 5(b), (d), (f) and (h). In general the relative difference is less than $10^{- 4}$ , which indicates a good match. The gaps in the error plot indicate regions where the relative difference is zero based on 64-bit double precision machine epsilon, which is a difference that is less than or equal to $2^{- 52}$ . However, the error analysis also highlights the deficiency of conventional numerical differentiation which produces significant errors when solving stiff nonlinear ODEs.²⁵ Therefore, the present exact solutions provide a benchmark for evaluating the accuracy of other approximate analytical or numerical solutions.

The variation of the frequency with initial displacement ( $θ_{0}$ ) and normalized rotational speed ( $Λ$ ) for $v_{0} = 0.0$ (Figure 6(a)) and $v_{0} > 0.0$ (Figure 6(b)) shows that the governor’s stiffness increased as $Λ$ increased. This implies that the system exhibits a stronger nonlinear response as $Λ$ increases. Also, the system’s stiffness increased with initial amplitude when $θ_{0} > θ_{e}$ and decreases with increase in initial amplitude when $θ_{0} < θ_{e}$ . This can be explained by the fact that stronger nonlinear responses are exhibited by the system as the initial amplitude deviates from $θ_{e}$ and approaches the limits of the applicable amplitude given by equations (5) and (6). The increase in the system’s stiffness as the initial amplitude deviates from $θ_{e}$ is more pronounced in Figure 6(a) than Figure 6(b) because the oscillator possesses a higher initial potential energy in Figure 6(a) compared to Figure 6(b). Consequently, as the initial kinetic energy of the oscillator increases the stiffening effect of the nonlinear response is less obvious and vice versa.

Figure 6.

Frequency response for varying initial displacement and system parameter: (a) $v_{0} = 0.0$ , $5^{°} \leq θ_{0} \leq 109^{°}$ and $1.1 \leq Λ \leq 10.0$ and (b) $v_{0} = 1.10$ , $58^{°} \leq θ_{0} \leq 82^{°}$ and $1.1 \leq Λ \leq 10.0$ .

Symmetric vibration response

The exact periodic solution for the symmetric vibration of the UWG model was derived in the Appendix. Although the present focus is on the asymmetric vibration, the solution of the symmetric vibration was simulated for one vibration cycle and compared with corresponding standard numerical solutions to confirm its validity. Three types of initial conditions were considered as follows: (I) $θ_{0} > 0.0$ and $v_{0} = 0.0$ ; (II) $θ_{0} = 0.0$ and $v_{0} > 0.0$ ; and (III) $θ_{0} > 0.0$ and $v_{0} > 0.0$ . Case I initial conditions are commonly used in most studies^{9–14,17–22} while Case II initial conditions have been applied in Moatimid and Amer¹⁵ and Big-Alabo and Chuku.¹⁶ On the other hand, Case III initial conditions were applied in Salas et al.²³ Hence, these initial conditions were simulated for the single-well and double-well symmetric vibrations as shown in Figures 7 and 8 respectively. The results demonstrate the presence of strong nonlinearity for the symmetric double-well vibrations as confirmed by the strong anharmonic profile of the vibration history. It can be observed that the relative difference between the present exact solution and the numerical solution is generally less than or equal to $10^{- 4}$ . This confirms the validity of the present exact solution for the symmetric vibrations. As explained earlier, the gaps in the error plot represent the regions where the relative difference between the exact and numerical solutions is zero based on machine epsilon precision.

Figure 7.

Symmetric single-well vibration and corresponding relative difference of exact and numerical solutions: (a, b) $Λ = 0.50$ , $θ_{0} = 150 . 0^{°}$ and $v_{0} = 0.0$ , (c, d) $Λ = 0.50$ , $θ_{0} = 0 . 0^{°}$ and $v_{0} = 1.85$ and (e, f) $Λ = 0.50$ , $θ_{0} = 60 . 0^{°}$ and $v_{0} = 1.75$ . Legend for displacement plot: Exact solution (blue) and numerical solution (pink).

Figure 8.

Symmetric double-well vibration and corresponding relative difference of exact and numerical solutions: (a, b) $Λ = 3.0$ , $θ_{0} = 110 . 0^{°}$ and $v_{0} = 0.0$ , (c, d) $Λ = 3.0$ , $θ_{0} = 0 . 0^{°}$ and $v_{0} = 0.75$ and (e, f) $Λ = 3.0$ , $θ_{0} = 60 . 0^{°}$ and $v_{0} = 1.15$ . Legend for displacement plot: Exact solution (blue) and numerical solution (pink).

The frequency response of the UWG model was simulated for zero initial velocity (Figure 9) and non-zero initial velocity (Figure 10). For both cases of initial conditions, the frequency response for single-well (Figures 9(a) and 10(a)) and double-well (Figures 9(b) and 10(b)) vibrations were investigated. The results show that the ranges of initial amplitude ( $θ_{0}$ ) and rotational speed ( $Λ$ ) for which stable vibrations can exist become more restricted as the initial kinetic energy of the system increases. Hence, the widest possible range of parameter values ( $θ_{0}$ , $Λ$ ) that permit the existence of stable vibrations occur when the initial kinetic energy is zero and the initial displacement is the same as the vibration amplitude.

Figure 9.

Frequency response of the symmetric vibration under varying initial displacement and system parameter when $v_{0} = 0$ : (a) $1^{°} \leq θ_{0} \leq 175^{°}$ and $0.1 \leq Λ \leq 1.0$ (b) $110^{°} \leq θ_{0} \leq 175^{°}$ and $1.1 \leq Λ \leq 10.0$ .

Figure 10.

Frequency response of the symmetric vibration under varying initial displacement and system parameter $v_{0} > 0$ : (a) $v_{0} = 1.25$ , $1^{°} \leq θ_{0} \leq 175^{°}$ and $0.1 \leq Λ \leq 1.0$ (b) $v_{0} = 1.25$ , $110^{°} \leq θ_{0} \leq 175^{°}$ and $1.1 \leq Λ \leq 10.0$ .

Conclusions

The present article has demonstrated that the UWG model, which has been mostly investigated for its symmetric vibrations, can undergo asymmetric vibrations under typical operating conditions. The conditions for the existence of the asymmetric vibrations were established and found to depend on the system’s parameter ( $Λ$ ) and the initial conditions. The primary condition for asymmetric vibrations is that $Λ > 1.0$ while the secondary conditions are given in equations (5) and (6). Analysis of the secondary conditions for asymmetric vibrations revealed that the asymmetric vibrations can only occur under the following initial conditions: $v_{0} = 0$ and $θ_{0} \neq 0$ or $v_{0} \neq 0$ and $θ_{0} \neq 0$ . Further analysis of the conditions for asymmetric vibrations revealed that range of initial conditions that can produce asymmetric vibrations increases as normalized rotational speed increases.

It was shown that exact closed-form solutions for the time period and displacement of the symmetric and asymmetric vibrations can be derived naturally from the first integral of the governing differential equation. The exact period was derived in terms of the complete elliptic integral of the first kind while the exact displacement was derived in terms of the Jacobi sine function. Comparison of the exact solutions with corresponding numerical solution for a wide range of typical operating conditions confirmed the validity of the exact solutions. A major highlight of the present exact solutions is that they are applicable to all conditions of asymmetric vibrations unlike the solutions derived in published studies.^7,26 Simulation of the vibration response revealed that the governor produces strong nonlinear asymmetric response even for small initial amplitudes and that the expression of the nonlinear effects depends strongly on the initial conditions. Also, the bi-stable vibration response produced strong anharmonic periodic histories that show evidence of highly nonlinear effects. Therefore, the present exact solutions can be used to verify the accuracy of approximate analytical solutions for strong nonlinear oscillators instead of the traditional numerical solutions that are known to produce significant errors under stiff nonlinear conditions.²⁵ This study can be extended to other symmetric oscillators capable of undergoing bi-stable and tri-stable vibrations.

Footnotes

Appendix

Handling Editor: Mahdi Mofidian

ORCID iD

Akuro Big-Alabo

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

All necessary data have been included in the manuscript.

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On the nonlinear asymmetric vibrations of the unforced Watt governor: An exact analysis

Abstract

Keywords

Introduction

Conditions for asymmetric vibrations

Exact periodic solution

Exact time period for asymmetric vibration of the Watt governor

Exact displacement history for asymmetric vibration of the Watt governor

Results and discussions

Asymmetric vibration response

Symmetric vibration response

Conclusions

Footnotes

Appendix

ORCID iD

Funding

Declaration of conflicting interests

Data availability statement

References