Natural frequencies of multiple pendulum systems

Natural frequencies of multiple pendulum systems under free condition

Mukul K. Gupta, Nitish Sinha, Kamal Bansal & Arun K. Singh

Archive of Applied Mechanics ISSN 0939-1533 Arch Appl Mech DOI 10.1007/s00419-015-1078-4

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Author's personal copy Arch Appl Mech DOI 10.1007/s00419-015-1078-4

O R I G I NA L

Mukul K. Gupta · Nitish Sinha · Kamal Bansal · Arun K. Singh

Natural frequencies of multiple pendulum systems under free condition

Received: 31 August 2015 / Accepted: 20 October 2015 © Springer-Verlag Berlin Heidelberg 2015

Abstract In this classical article, we study natural frequencies of the multiple pendulum systems (MPSs) in a plane under the free condition. The systems of governing differential equations for the MPSs such as triple pendulum (TP) and double pendulum (DP) are derived using the Euler–Lagrangian equation of second kind to validate the Braun’s generalized expressions (Arch Appl Mech 72:899–910, 2003) for natural frequencies of multiple pendulum systems. The governing equations of the TP and DP systems are also derived in terms of angular momentum and angular displacement to confirm the basic results obtained using the aforementioned approach. The eigenvalue analysis of the pendulum systems ranging from single pendulum to quintuple indicates that natural frequency increases with degree of freedom for equal mass and length of each pendulum in a MPS. The results show that the natural frequency of a distributed pendulum system is larger than the corresponding to the point mass pendulum system. Moreover, the natural frequency of the bottom pendulum is the most sensitive to change in length or mass of either pendulum of a MPS. However, unlike mass-dependent natural frequency, the natural frequency of all pendulums of a multiple pendulum always decreases with increasing length of a pendulum in MPS. These results are, in turn, validated with Braun’s formula for natural frequency of a MPS. Keywords Multiple pendulums · Double and triple pendulums · Pendulum with point mass · Pendulum with distributed mass · Natural frequencies of multiple pendulums · Euler–Lagrange equation of second kind · Pendulum mass and length-dependent natural frequencies · Linear and nonlinear pendulum systems

Abbreviations DP DPDM DPPM DoF IP IDPDM ITPDM IMPS

Double pendulum Double pendulum with distributed mass Double pendulum with point mass Degree of freedom Inverted pendulum Inverted double pendulum with distributed mass Inverted triple pendulum with distributed mass Inverted multiple pendulum system

M. K. Gupta (B)· K. Bansal University of Petroleum and Energy Studies, Dehradun 248007, India E-mail: [email protected] N. Sinha · A. K. Singh Viswasvaraya National Institute of Technology, Nagpur 440010, India E-mail: [email protected]

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K.E. MPS P.E. QP QtP SP SPDM SPPM TPDM TPPM

Kinetic energy Multiple pendulum system Potential energy Quadruple pendulum Quintuple pendulum Single pendulum Single pendulum with distributed mass Single pendulum with point mass Triple pendulum with distributed mass Triple pendulum with point mass

1 Introduction A mechanical pendulum with a single degree of freedom (DoF) is widely used for validating control algorithms as well as dynamic models for explaining a variety of physical and mechanical phenomena [1–3]. A pendulum system with two or more DoF, generally known as a multiple pendulum system (MPS), is also important, though the mathematical modeling of a MPS is relatively complicated [1–5]. A multiple pendulum such as a triple pendulum (TP) finds applications in the laser interferometer gravitational wave observatory (LIGO) which has been built to detect and to observe gravity waves predicted by the general theory of relativity [6]. A piston-connecting rod–crankshaft system is generally modeled as the TP system under the physical constraints [7]. Awrejcewicz et al. [5,7] have studied experimentally as well as numerically the nonlinear behavior of the TP in detail considering both forced vibration and physical constraints. A MPS is also used for analyzing the gait of the human body and robotic manipulators [2,8]. Nonetheless, there are issues pertaining to linear behavior of multiple pendulums in free condition which are not properly understood. For instance, it is not known in the literature how change in DoF, mass, and length of a MPS affects its natural frequencies. What is the difference between the dynamics of a point mass pendulum and a distributed pendulum? Why does a MPS unstable in the linear regime? These are some questions which have motivated us to study the basic linear behavior of MPSs. In the present study, we have mainly used the DP and the TP systems as MPSs. However, the linear behavior of the MPSs is also discussed in the light of a single pendulum (SP), double pendulum (DP), triple pendulum (TP), quadruple pendulum (QP) and quintuple pendulum (QtP) systems for both point and distributed systems. Finally, the results are also validated with Braun’s formula for a MPS [9,10]. It is to be noted that Braun’s formula for point and distributed multiple pendulums cannot be used for evaluating natural frequency of an individual pendulum in a MPS, but his formulae are very useful for evaluating the sum of squared reciprocal frequencies of a distributed MPS and product of squared reciprocal frequencies of a point mass MPS [9].

2 Governing equations of multiple pendulum systems In the literature, mathematical derivations of the dynamic equations of the SP and DP systems are widely reported for both point and distributed mass systems [1,4,11–14]. These equations are hardly reported in the literature for TP or MPS with higher DOF such as QP and QtP. The nonlinear governing equations of the TP with distributed mass (TPDM) and the TP with point mass (TPPM) are derived using the Euler–Lagrange equation of second kind [1,3]. The governing differential equations are also derived in the terms of angular momentum and angular displacement aiming to confirm the correctness of the computer program. This approach of deriving governing differential equation is also basically Euler–Lagrange equation of second kind, but the equations are expressed in terms of displacement and momentum of the oscillating mass. The schematic sketch of a simple triple pendulum with distributed mass (TPDM) is shown in Fig. 1. The upper end of the top pendulum is pivoted to the fixed point O, while the lower end is connected to the middle pendulum. Moreover, m 1 , m 2 , m 3 and l1 , l2 , l3 are the mass and length of the top, middle and bottom pendulums, respectively. Further, θ1 , θ2 and θ3 represent the angular displacement of three pendulums which is measured from the axes vertical down. A pendulum oscillating about the vertically down axis passing through the point of suspension is known as a simple pendulum [1,2]. However, oscillation of the same system is about the vertically up axis, and then, it is called inverted pendulum [3]. Figure 1 presents a simple triple pendulum with distributed mass (TPDM) located in the fourth quadrant.

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O

y

x

m1,l1

P m2 ,l2

Q m3 ,l3

R

Fig. 1 A simple multipendulum system (MPS) in the form of the triple pendulum with distributed mass (TPDM) is located in the fourth quadrant of the Cartesian system

Moreover, the points P, Q and R are the center of mass of three pendulums in Fig. 1 and their coordinates are defined as     l1 l2 l2 l1 sin θ1 , − cos θ1 , Q l1 sin θ1 + sin θ2 , −l1 cos θ1 − cos θ2 P 2 2 2 2 and

  l3 l3 R l1 sin θ1 + l2 sin θ2 + sin θ3 , −l1 cos θ1 − l2 cos θ2 − cos θ3 . 2 2

Total kinetic energy (K.E.) of the TPDM is expressed in terms of polar moment of inertia Ici and linear velocity Vci of center of mass of each pendulum as   3  Ici θ˙i2 m i Vci2 + . (1) K.E. = 2 2 i=1

The total potential energy (P.E.) of the same system is also expressed in terms of mass m i of each pendulum and gravitational constant g as   l2 l2 m 1 gl1 (1 − cos θ1 ) + m 2 g l1 − l1 cos θ1 + − cos θ2 P.E. = 2 2 2   l3 l3 + m 3 g l1 − l1 cos θ1 + l2 − l2 cos θ2 + − cos θ3 . (2) 2 2 The Lagrangian (L) of the TPDM system is defined as the difference of K.E. and P.E. as follows: L = K.E. − P.E.   Ic3 θ˙32 m 1l12 θ˙12 Ic2 θ˙22 Ic1 θ˙12 m 2 2 2 l22 θ˙22 + + + + l1l2 θ˙1 θ˙2 cos (θ1 − θ2 ) + l1 θ˙1 + L= 2 8 2 2 4 2  l 2 θ˙ 2 m3 2 2 l1 θ˙1 + l22 θ˙22 + 3 3 + 2l1l2 cos (θ1 − θ2 ) θ˙1 θ˙2 + l1l3 θ˙1 θ˙3 cos (θ1 − θ3 ) + 2 4    l2 m 1 gl1 cos θ1 ˙ ˙ + m 2 g l1 cos θ1 + cos θ2 + l2 l3 θ2 θ3 cos (θ2 − θ3 ) + 2 2       l3 l2 l3 m 1 gl1 + m 3 g l1 cos θ1 + l2 cos θ2 + cos θ3 − − m 2 g l1 + − m 3 g l1 + l2 + . (3) 2 2 2 2

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where Ici = m i li2 /12 is polar moment of inertia of the each pendulum. The differential equation can be obtained using the Euler–Lagrange equation of second kind [1] as d dt



∂L ∂ θ˙i

 −

∂L = 0 where i = 1, 2, 3, . . . ., ∂θi

(4)

Using Eqs. (3) and (4), we obtain the system of coupled differential equations of the following form c11 θ¨1 + c12 θ¨2 + c13 θ¨3 + d11 = 0 c21 θ¨1 + c22 θ¨2 + c23 θ¨3 + d22 = 0. c31 θ¨1 + c32 θ¨2 + c33 θ¨3 + d33 = 0

(5)

where coefficients of the variables θ¨1 , θ¨2 and θ¨3 are defined in the form of indices, of ci j which are c11 = Ic1 + (0.25m 1 + m 2 + m 3 ) l12 , c12 = 0.5m 2 l1l2 cos (θ1 − θ2 ) + m 3l1l2 cos (θ1 − θ2 ) , c13 = 0.5m 3l1l3 cos (θ3 − θ1 ) , c21 = (0.5m 2 + m 3 ) l1l2 cos (θ1 − θ2 ) , c22 = Ic2 + (0.25m 2 + m 3 ) l22 , c23 = 0.5m 3l2 l3 cos (θ2 − θ3 ) , c31 = 0.5m 3l1l3 cos (θ3 − θ1 ) , c32 = 0.5m 3l2 l3 cos (θ2 − θ3 ) , c33 = Ic3 + 0.25m 3l32 , d11 = (0.5m 2 + m 3 ) l1l2 sin (θ1 − θ2 ) θ˙22 + 0.5m 3l1l3 sin (θ1 − θ3 ) θ˙32 + (0.5m 1 + m 2 + m 3 ) gl1 sin θ1 ,

d22 = − (0.5m 2 + m 3 ) l1l2 sin (θ1 − θ2 ) θ˙12 + 0.5m 3l2 l3 sin (θ2 − θ3 ) θ˙32 + (0.5m 2 + m 3 ) gl2 sin θ2 , d33 = −0.5m 3l1l3 sin (θ1 − θ3 ) θ˙12 − 0.5m 3l2 l3 sin (θ2 − θ3 ) θ˙22 + 0.5m 3 gl3 sin θ3 .

(6)

As mentioned in the introduction of this article, in order to confirm the solutions of Eq. (5), the governing equations of the TPDM are also derived in terms of angular momentum p and angular displacement θ of each pendulum in the TPDM using the following expressions: pi =

∂L ∂L , p˙ i = ∂θ i ∂ θ˙ i

where i = 1, 2, 3 . . . n,

(7)

where n denotes the number of links in the pendulum system [11]. We note that the expressions in Eq. (7) satisfy the Euler–Lagrange equation of second kind in Eq. (4). The equation of momentum of each pendulum of the TPDM is derived in light of Eq. (7) as

m 1l12 θ˙1 m2 2 p1 = Ic1 θ˙1 + + 2l1 θ˙1 + l1l2 θ˙2 cos (θ1 − θ2 ) 4 2

m3 2 + 2l1 θ˙1 + 2l1l2 cos (θ1 − θ2 ) θ˙2 + l1l3 θ˙3 cos (θ3 − θ1 ) . 2   m 2 l22 θ˙2 p2 = Ic2 θ˙2 + + l1l2 θ˙1 cos (θ1 − θ2 ) 2 2

m3 2 + 2l2 θ˙2 + 2l1l2 cos (θ1 − θ2 ) θ˙1 + l2 l3 θ˙3 cos (θ2 − θ3 ) . 2   m 3 l32 θ˙3 ˙ ˙ ˙ p3 = Ic3 θ3 + + l1l3 θ1 cos (θ3 − θ1 ) + l2 l3 θ2 cos (θ2 − θ3 ) . 2 2

(8)

It is now Eq. (8) and may be expressed in the short form as following p1 = g11 θ˙1 + g12 θ˙2 + g13 θ˙3 p2 = g21 θ˙1 + g22 θ˙2 + g23 θ˙3 . p3 = g31 θ˙1 + g32 θ˙2 + g33 θ˙3

(9)

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where indices related with gi j are defined as m 1l12 m2 + m 2 l12 + m 3l12 , g12 = l1l2 cos(θ1 − θ2 ) + m 3l1l2 cos(θ1 − θ2 ), 4 2 = m 3l1l3 cos(θ3 − θ1 ), m 2 l22 m2 = l1l2 cos(θ1 − θ2 ) + m 3l1l2 cos(θ1 − θ2 ), g22 = Ic2 + + m 3l22 , g23 = m 3l2 l3 cos(θ2 − θ3 ), 2 4 m 3l32 m3 m3 = l1l3 cos(θ3 − θ1 ), g32 = l2 l3 cos(θ2 − θ3 ), g33 = Ic3 + . 2 2 4

g11 = Ic1 + g13 g21 g31

Equation (9) is solved for θ˙1 , θ˙2 and θ˙3 and yield the system of equations as p2 g33 − p3 g23 1 − [DC ( p1 g33 − p3 g13 ) − AD ( p2 g33 − p3 g23 )] . C (BC − AD) 2

C ( p1 g33 − p3 g13 ) − AC ( p2 g33 − p3 g23 ) = . (BC − AD)

  p3 g23 1 1 p3 − g31 p2 g33 − − g − p g g − p g p − AD p (DC ( ) ( )) 1 33 3 13 2 33 3 23 (BC−AD) . =

2C g32 g33 − (BC−AD) C ( p1 g33 − p3 g13 ) − AC ( p2 g33 − p3 g23 )         = g11 g33 − g13 g31 , B = g12 g33 − g32 g13 C = g21 g33 − g23 g31 , D = g22 g33 − g23 g32 . (10)

θ˙1 = θ˙2 θ˙3 where A

The time derivative of momentum of each pendulum of the TPDM is obtained using Eq. (7) as

m2 m3 l1l2 θ˙1 θ˙2 sin (θ2 − θ1 ) + 2l1l2 sin (θ2 − θ1 ) θ˙1 θ˙2 + l1l3 θ˙1 θ˙3 sin (θ3 − θ1 ) p˙ 1 = 2 2 m 1 gl1 sin θ1 − m 2 gl1 sin θ1 − m 3 gl1 sin θ1 . − 2

m3

m2 p˙ 2 = l1l2 θ˙1 θ˙2 sin (θ1 − θ2 ) + 2l1l2 sin (θ2 − θ3 ) θ˙1 θ˙2 − l2 l3 θ˙2 θ˙3 sin (θ2 − θ3 ) 2 2 l2 − m 2 g sin θ2 . 2

m3 p˙ 3 = l1l3 θ˙1 θ˙3 sin (θ3 − θ1 ) + l2 l3 θ˙2 θ˙3 sin (θ3 − θ2 ) + m 3 g (l1 cos θ1 + l2 cos θ2 + l3 cos θ2 /2) 2     l2 l3 m 1 gl1 − m 2 g l1 + − m 3 g l1 + l2 + . (11) − 2 2 2 Now Eqs. (10) and (11) can be used for deriving the expression for a double pendulum with distributed mass (DPDM) just by neglecting the terms related to the third (bottom) pendulum of the TPDM [9]. Likewise, the procedure can be extended for getting the governing equation of the single pendulum with distributed mass (SPDM). More interestingly, the governing differential equations can also be derived for the TP with point mass (TPPM) by ignoring the moment of inertia terms and making unity the numerical factors of remaining terms involving the equations of the TPDM. As the governing system of differential equations of pendulum system with distributed mass is being converted to the corresponding pendulum system with point mass, it is actually distributed mass being concentrated as a point mass. As a result, one has to neglect the terms related to the distributed mass, i.e., moment of inertia. Moreover, since the point mass increases in magnitude, thus the multiplying factors also increase to unity. We have listed the results of other pendulum systems in Appendix 1. 3 Results and discussions 3.1 Linear stability analysis In order to have a better understanding of the dynamics of the MPSs, linear stability analysis is carried out of the TP systems. The results are then compared with other MPSs such as the DP, QP, QtP and also with the SP.

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The linearized form of Eqs. (5–6). can now be represented in the following form   [MLD ] θ¨ + [K LD ] {θ } = 0,

(12)

where coefficients of linearized mass matrix [MLD ] and stiffness linearized matrix [K LD ] of the TPDM defined as m 1l12 m2 m3 m2 + m 2 l12 + m 3l12 , e12 = l1l2 + m 3l1l2 , e13 = l1l3 , e21 = l1l2 + m 3 l1l2 , 4 2 2 2 m 2 l22 m 3l32 m3 m3 m3 = Ic2 + + m 3l22 , e23 = l2 l3 , e31 = l1l3 , e32 = l2 l3 , e33 = Ic3 + , 4 2 2 2 4 f 11 = (m 1 + m 2 + m 3 ) gl1 , f 22 = (m 2 + m 3 ) gl2 , f 33 = m 3 gl3 , ⎡ ⎡ ⎤ ⎤ e11 , e12 , e13 f 11 , 0, 0 f 22 , 0 ⎦ . (13) [MLD ] = ⎣ e21 , e22 , e23 ⎦ , [K LD ] = ⎣ 0, f 33 0, e31 , e32 , e33 0,

e11 = Ic1 + e22

Thus, the differential equation of the TPDM can now be reduced to the TPPM by neglecting the terms related to the moment of inertia and replacing the numerical factors in the remaining coefficients of [M] and [K ] with unity. Thus, the linearized coefficients of [M L L ] and [K L L ] for the TPPM are as follows: h 11 = m 1l12 + m 2 l12 + m 3l12 , h 12 = m 2 l1l2 + m 3l1l2 , h 13 = m 3l1l3 , h 21 = m 2 l1l2 + m 3l1l2 , h 22 = m 2 l22 + m 3l22 , h 23 = m 3l2 l3 , h 31 = m 3l1l3 , h 32 = m 3l2 l3 , h 33 = m 3l32 k11 = (m 1 + m 2 + m 3 ) gl1 , k22 = (m 2 + m 3 ) gl2 , k33 = m 3 gl3

(14)

Noting again that the coefficients of a TP reduce to a SP by eliminating the terms related to the bottom and middle pendulums. Similarly, the linearized [M] and [K ] can also be expressed for QP and QtP systems on the basis of the DP and the TP systems. The coefficients related to the mass and stiffness matrices of these systems are provided in Appendix 1. 3.2 Effect of DoF on natural frequency Angular natural frequencies ωn of Eq. (12) can now be evaluated numerically using the characteristic equation   [K LD ] − ω2 [MLD ] = 0. The ωn of different MPSs are presented in Table 1 for equal mass of 1 kg and n length with 1 m of each pendulum in a MPS. A conclusion from Table 1 is that ωn of a distributed pendulum system is more than the corresponding point mass pendulum system. For example, ωn of SPLM and SPDM is calculated to be 3.13 and 3.83, respectively, as per Table 1. Similarly, ωn of the top and the bottom pendulums of the DPPM are estimated to be 2.40 and 5.78, respectively, while the numerical values of ωn of the DPDM are found to be 2.68 (top pendulum) and 7.18 (bottom pendulum). Moreover, all ωn of a pendulum system lies between the minimum and maximum ωn of the pendulum system with next DoF. For example, ωn of the SPPM is 3.13 which is in between the maximum and the minimum ωn of the DPPM, i.e., 2.40 and 5.78. This conclusion is valid for other pendulum systems presented in Table 1. Moreover, the results in Table 1 validate the Braun’s expression (Appendix 2) for a MPS [9]. Another important conclusion from Table 1 is that the natural frequency of the bottom pendulum is the largest to any other pendulum in a MPS. The reason for this observation may be attributed to the less constraint present at the bottom pendulum than any other pendulum in a MPS; thus, it is more free to oscillate. These observations are also valid for QP and QtP as evident from Table 1. 3.3 Effect of pendulum mass on natural frequencies of MPSs We have also studied parametrically the effect of mass of each pendulum in the TPDM as well as the DPDM on natural frequency. The results in Fig. 2 show that increasing mass m 1 of the top pendulum of the TPDM results in its gradual increase of ωn . But ωn of the remaining pendulums (middle and bottom) decreases rapidly in the same system. Now, in the case of increase in mass m 2 of the middle pendulum, ωn of the middle pendulum decreases, while ωn of the top and the bottom pendulums increases (Fig. 2). Figure 2 also shows the result that

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Table 1 The angular natural frequency (ωn ) of each pendulum of different pendulum systems comprising both lumped and distributed mass for equal mass and length of each pendulum m1 = m2 = m3 = m4 = 1 kg, l1 = l2 = l3 = l4 = 1 m

Point mass system (ωn )

Distributed mass system (ωn )

Braun’s expression (point mass) n  1 k=1

Simple pendulum (SP) Double pendulum (DP) Triple pendulum (TP) Quadruple pendulum (QP)

Quintuple pendulum (QtP)

3.13 2.40 (top) 5.78 (bottom) 2.02 (top) 4.74 (middle) 7.85 (bottom) 1.77 (top) 4.14 (second) 6.67 (third) 9.60 (bottom) 1.61 (top) 3.72 (second) 5.94 (third) 8.34 (fourth) 11.13 (bottom)

3.83 2.68 (top) 7.18 (bottom) 2.18(top) 5.42(middle) 10.24 (bottom) 1.88 (top) 4.58 (second) 7.98 (third) 13.14 (bottom) 1.68 (top) 4.00 (second) 6.78 (third) 10.45 (fourth) 15.85 (bottom)

2 ωnk

Braun formula (distributed mass) n  1 k=1

2 ωnk

0.1020 0.0052

0.2611 0.1587

1.771 × 10−4

0.2540

4.517 × 10−6

0.3511

9.219 × 10−8

0.4494

Fig. 2 Pendulum mass-dependent natural frequency ωn versus mass of top, middle and bottom pendulums of the TPDM of equal length of each pendulum, i.e., l1 = l2 = l3 = 1.0 m

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Fig. 3 Pendulum mass-dependent ωn of the top and the bottom pendulums of the DPDM of equal length of each pendulum, i.e., l1 = l2 = 1.0 m

as the mass m 3 increases, ωn of the bottom and middle pendulums increases, but ωn of the top pendulum slightly decreases. An interesting observation is that ωn of the bottom pendulum gets the most affected compared to any other pendulum of the TPDM. Figure 2 also validates the increasing sequence of ωn as top, middle and bottom pendulums, respectively. In the case of the DPDM system, as the mass m 1 of the top pendulum increases, ωn of the top pendulum increases, but ωn of the bottom pendulum decreases. On the other hand, as mass m 2 of the bottom pendulum increases, ωn of the top pendulum decreases, but ωn of the bottom pendulum increases. We note that these two observations are contradictory to each other. It is also clear from Fig. 3 that ωn of the bottom pendulum varies more rapidly than the top pendulum of the DPDM and confirms the result in Fig. 2 for the TPDM. The results in Figs. 2 and 3 are consistent with the Braun’s formula for natural frequency. For instance, natural frequencies of three pendulums of the TPDM are estimated numerically to be ωn1 = 2.23, ωn2 = 5.24 and ωn3 = 9.14 for mass m 1 = 2 kg, m 2 = m 3 = 1 kg and length l1 = l2 = l3 = 1 m. According to Braun’s  formula (Appendix 2), 3k=1 ω12 = 0.2494 is consistent with the estimated values of natural frequencies. nk

Similarly, the Braun’s formula is also found to be correct for the DPDM though the results are not presented here. 3.4 Effect of pendulum length on natural frequency of MPSs We have also studied numerically the effect of change in length of each pendulum in the TPDM as well as DPDM system on its natural frequency. The results in Fig. 4 indicate that increasing length l1 of the top pendulum results in decreasing of all three natural frequencies of the TPDM. This trend is also seen in the case of change in length of the middle as well as the bottom pendulum of the TPDM (Fig. 4). An interesting observation is that ωn of the bottom pendulum gets the most affected compared to any other pendulum of the TPDM irrespective of change in length of either pendulum of the system. In the case of the DPDM also, ωn of each pendulum decreases with increase in the length of either top or bottom pendulum (Fig. 5). Like the TPDM, ωn of the bottom pendulum changes rapidly than the top pendulum of the DPDM. Thus, on the basis of the results in Figs. 4 and 5, we conclude that increase in length of a pendulum of a MPS always results in decrease of ωn of each pendulum albeit varying in magnitude. The sequence of ωn is found to be as (ωn )bottom > (ωn )middle > (ωn )top irrespective of change in length of a pendulum in a MPS. The effect of change in length, other than equal length, of a pendulum in a MPS confirms the Braun’s formula again. For example, natural frequencies of three pendulums of TPDM are found to be numerically as ωn1 = 2.46, ωn2 = 6.29, and ωn3 = 12.36 for mass m 1 = 1 kg, m 2 = m 3 = 1 kg and length l1 = 0.4 m, l2 =  l3 = 1 m. These values of natural frequency satisfy Braun’s formula [9]: 3k=1 ω12 = 0.1968 (Appendix 2). nk

Finally, we believe that the present study will be useful for design and control of a physical or mechanical systems based on multiple pendulums. Moreover, nonlinear analysis of such a multiple pendulum system will be interesting [4,5,11–14].

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Fig. 4 Pendulum length-dependent natural frequency ωn of top, middle and the bottom pendulums for equal mass of each pendulum of the TPDM, i.e., m 1 = m 2 = m 3 = 1.0 kg

Fig. 5 Pendulum length l1 and l2 dependent ωn of the top and the bottom pendulums for equal mass of each pendulum of the DPDM, i.e., m 1 = m 2 = 1.0 kg

4 Conclusions The present study establishes that natural frequency of a multiple pendulum increases with degree of freedom. Moreover, the natural frequency of a distributed pendulum is larger than corresponding to the point mass pendulum system. The natural frequency of the bottom pendulum of a multiple pendulum gets the most

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affected irrespective of change in length or mass of any pendulum in system. Moreover, increasing length of any pendulum in a multiple pendulum system results in decreasing of natural frequency of each pendulum in the system. However, in the case of increasing mass of any pendulum in multiple pendulum, natural frequency may increase or decrease depending on the location of the pendulum in the system. Nevertheless, the natural frequencies of each multiple pendulum satisfies the Braun’s formula [9] for the distributed multiple pendulum systems.

Appendix 1 The coefficients of differential equations of the DPDM can be obtained using Eq. (5) by eliminating the terms related to the bottom pendulum, i.e., third pendulum and given by   1 1 1 c11 = Ic1 + m 1 + m 2 l12 , c12 = m 2 l1l2 cos(θ1 − θ2 ), c21 = m 2 l1l2 cos(θ1 − θ2 ), 4 2 2     1 1 1 2 2 c22 = Ic2 + m 2 + m 3 l2 , d11 = m 2 l1l2 sin(θ1 − θ2 )θ2 + m 1 + m 2 gl1 sin θ1 , 4 2 2 1 1 (15) d22 = − m 2 l1l2 sin(θ1 − θ2 )θ22 + m 2 gl2 sin θ2 . 2 2 Following the similar procedure, the coefficients of mass and stiffness for the SPPM may be obtained by eliminating the terms related to the second pendulum of the simple DP system. The coefficients of [K ] and [M] matrices for the linearized quadruple pendulum with distributed mass (QPDM) are obtained on the basis of Eq. (13) as       1 1 1 d11 = m 1 + m 2 + m 3 + m 4 gl1 , d22 = m 2 + m 3 + m 4 gl2 , d33 = m 3 + m 4 gl3 , 2 2 2     1 1 1 d44 = m 4 gl4 , c11 = I c1 + m 1 + m 2 + m 3 + m 4 l12 , c12 = m 2 + m 3 + m 4 l1l2 , 2 4 2     1 1 1 c13 = m 3 + m 4 l1l3 , c14 = m 4l1l4 , c21 = m 2 + m 3 + m 4 l2 l1 , 2 2 2     1 1 1 2 m 2 + m 3 + m 4 l2 , c23 = m 3 + m 4 l2 l3 , c24 = m 4l2 l4 , c22 = I c2 + 4 2 2       1 1 1 c31 = m 3 + m 4 l3l1 , c32 = m 3 + m 4 l3l2 , c33 = Ic3 + m 3 + m 4 l32 , 2 2 4 1 1 1 1 c34 = m 4l3l4 , c41 = m 4l4l1 , c42 = m 4l4l2 , c43 = m 4l4l3 , 2 2  2 2        2 2 2 2 m m m m l l l l 1 1 2 3 4 1 2 3 4 c44 = Ic4 + m 4l42 , Ic1 = , I c2 = , Ic3 = , Ic4 = . (16) 4 12 12 12 12 And the coefficients of mass and stiffness matrices for the simple quintuple pendulum with distributed mass (QtPDM) are obtained using Eq. (16) as  d11 = d33 = c11 = c13 = c21 =

   1 1 m 1 + m 2 + m 3 + m 4 + m 5 gl1 , d22 = m 2 + m 3 + m 4 + m 5 gl2 , 2 2     1 1 1 m 3 + m 4 + m 5 gl3 , d44 = m 4 + m 5 gl4 , d55 = m 5 gl5 , 2 2 2     1 1 2 I c1 + m 1 + m 2 + m 3 + m 4 + m 5 l1 , c12 = m 2 + m 3 + m 4 + m 5 l1 l2 , 4 2     1 1 1 m 3 + m 4 + m 5 l1 l3 , c14 = m 4 + m 5 l1 l4 , c15 = m 5 l1 l5 , 2 2 2       1 1 1 2 m 2 + m 3 + m 4 + m 5 l2 l1 , c22 = I c2 + m 2 + m 3 + m 4 + m 5 l2 , c23 = m 3 + m 4 + m 5 l2 l3 2 4 2

Author's personal copy Natural frequencies of multiple pendulum systems

 c24 = c33 = c42 = c51 = Ic1 =

     1 1 1 1 m 4 + m 5 l2 l4 , c25 = m 5 l2 l5 , c31 = m 3 + m 4 + m 5 l3 l1 , c32 = m 3 + m 4 + m 5 l3 l2 , 2 2 2 2       1 1 1 1 Ic3 + m 3 + m 4 + m 5 l32 , c34 = m 4 + m 5 l3 l4 , c35 = m 5 l3 l5 , c41 = m 4 + m 5 l4 l1 , 4 2 2 2       1 1 1 1 m 4 + m 5 l4 l2 , c43 = m 4 + m 5 l4 l3 , c44 = Ic4 + m 4 + m 5 l42 , c45 = m 5 l4 l5 , 2 2 4 2 1 1 1 1 1 m 5 l5 l1 , c52 = m 5 l5 l2 , c53 = m 5 l5 l3 , c54 = m 5 l5 l4 , c55 = Ic5 + m 5 l52 , 2 2 2 2   4      m 5 l52 m 1 l12 m 2 l22 m 3 l32 m 4 l42 (17) , Ic2 = , Ic3 = , Ic4 = , Ic5 = . 12 12 12 12 12

The coefficients of [K ] and [M] for the simple TPPM can be generated using Eq. (5) by eliminating the terms related to the moment of inertia and replacing the numerical factors of remaining terms with one. The following coefficients of the TPPM are obtained as c11 = (m 1 + m 2 + m 3 ) l12 , c12 = m 2 l1l2 cos (θ1 − θ2 ) + m 3l1l2 cos (θ1 − θ2 ) , c13 = m 3l1l3 cos (θ3 − θ1 ) , c21 = (m 2 + m 3 ) l1l2 cos (θ1 − θ2 ) , c22 = (m 2 + m 3 ) l22 , c23 = m 3l2 l3 cos (θ2 − θ3 ) , c31 = m 3l1l3 cos (θ3 − θ1 ) , c32 = m 3l2 l3 cos (θ2 − θ3 ) , c33 = m 3l32 , d11 = (m 2 + m 3 ) l1l2 sin (θ1 − θ2 ) θ˙22 + m 3l1l3 sin (θ1 − θ3 ) θ˙32 + (m 1 + m 2 + m 3 ) gl1 sin θ1 ,

d22 = − (m 2 + m 3 ) l1l2 sin (θ1 − θ2 ) θ˙12 + m 3l2 l3 sin (θ2 − θ3 ) θ˙32 + (m 2 + m 3 ) gl2 sin θ2 , d33 = −m 3l1l3 sin (θ1 − θ3 ) θ˙12 − m 3l2 l3 sin (θ2 − θ3 ) θ˙22 + m 3 gl3 sin θ3 .

(18)

The coefficients of [K ] and [M] for the linearized DPPM are obtained using Eq. (16) in which the coefficients related to the third pendulum are eliminated obtained as c11 = (m 1 + m 2 )l12 , c12 = m 2 l1l2 cos(θ1 − θ2 ), c21 = m 2 l1l2 cos(θ1 − θ2 ), c22 = m 2 l22 , d11 = m 2 l1l2 θ˙22 sin(θ1 − θ2 ) + (m 1 + m 2 )gl1 sin θ1 , d22 = −m 2 l1l2 θ˙12 sin(θ1 − θ2 ) + m 2 gl2 sin θ2 .

(19)

The coefficients of the simple QPPM related to mass and stiffness matrices are obtained in light of Eq. (18) as d11 = (m 1 + m 2 + m 3 + m 4 )gl1 , d22 = (m 2 + m 3 + m 4 )gl2 , d33 = (m 3 + m 4 )gl3 , d44 = m 4 gl4 , c11 = (m 1 + m 2 + m 3 + m 4 )l12 , c12 = (m 2 + m 3 + m 4 )l1l2 , c13 = (m 3 + m 4 )l1l3 , c14 = m 4l1l4 , c21 = (m 2 + m 3 + m 4 )l2 l1 , c22 = (m 2 + m 3 + m 4 )l22 , c23 = (m 3 + m 4 )l2 l3 , c24 = m 4l2 l4 , c31 = (m 3 + m 4 )l3l1 , c32 = (m 3 + m 4 )l3l2 , c33 = (m 3 + m 4 )l32 , c34 = m 4l3l4 , c41 = m 4l4l1 , c42 = m 4l4l2 , c43 = m 4l4l3 , c44 = m 4l42 .

(20)

Finally, the coefficients of the linearized simple quintuple pendulum with point mass(QtPPM) on the basis of Eq. (18) are written as following c11 c14 c21 c24 c31 c35 c41

= (m 1 + m 2 + m 3 + m 4 + m 5 )l12 , c12 = (m 2 + m 3 + m 4 + m 5 )l1l2 , c13 = (m 3 + m 4 + m 5 )l1l3 , = (m 4 + m 5 )l1l4 , c15 = m 5l1l5 , = (m 2 + m 3 + m 4 + m 5 )l2 l1 , c22 = (m 2 + m 3 + m 4 + m 5 )l22 , c23 = (m 3 + m 4 + m 5 )l2 l3 , = (m 4 + m 5 )l2 l4 , c25 = m 5l2 l5 , = (m 3 + m 4 + m 5 )l3l1 , c32 = (m 3 + m 4 + m 5 )l3l2 , c33 = (m 3 + m 4 + m 5 )l32 , c34 = (m 4 + m 5 )l3l4 , = m 5l3 l5 , = (m 4 + m 5 )l4l1 , c42 = (m 4 + m 5 )l4l2 , c43 = (m 4 + m 5 )l4l3 , c44 = (m 4 + m 5 )l42 , c45 = m 5l4l5 ,

c51 = m 5l5l1 , c52 = m 5l5l2 , c53 = m 5l5l3 , c54 = m 5l5l4 , c55 = m 5l52 , d11 = (m 1 + m 2 + m 3 + m 4 + m 5 )gl1 , d22 = (m 2 + m 3 + m 4 + m 5 )gl2 , d33 = (m 3 + m 4 + m 5 )gl3 , (21) d44 = (m 4 + m 5 )gl4 , d55 = m 5 gl5 .

Author's personal copy M. K. Gupta et al.

Appendix 2 Braun formula [8]for natural frequency of continuum or distributed pendulum systems     n n n    i k2 1 sl 1 mk , where M = − s = m + mk . l + k l l 2 g ak Mk al ωnk k=1

k=l+1

k=l+1

where l total length of link, i k the radius of gyration of each member about its own pivotal joint, ak distance between two subsequent joints and sk center distance from its pivotal joints. Formula for DPDM system is given by       i 12 l1 m 1 i 22 l2 m 2 1 1 1 m1 m2 , M1 = + 2 = − + − + m 2 , M2 = . l1 + l2 + 2 g l1 2 M1 l2 2 M2 2 2 ωn1 ωn2 Similarly, the formula for the TPDM may be extended as          i 32 l3 m 3 i 12 l1 m 1 i 22 l2 m 2 1 1 1 1 + , l1 + l2 + l3 + + 2 + 2 = − + − − 2 g l1 2 M1 l2 2 M2 l3 2 M3 ωn1 ωn2 ωn3 m1 m2 m3 M1 = + m 2 + m 3 , M2 = + m 3 , M3 = . 2 2 2 Braun’s formula can be extended for the QPDM, PtDM or distributed system with higher DoF. Braun formula for natural frequency of distributed system is given by n n n−1   1 ak  m k = , ak distance between two subsequent joints. g Mk ω2 k=1 k=1 nk k=1

The expression can be extended for DPPM as     1 m1 m2 1 1 = l l , 1 2 2 ω2 g2 M1 M2 ωn1 n2

M1 = m 1 ,

TPLM as given following      m1 m2 m3 1 1 1 1 = 3 l1 l2 l3 , M1 = m 1 , 2 ω2 ω2 g M1 M2 M3 ωn1 n2 n3

M2 = m 1 + m 2 .

M2 = m 1 + m 2 ,

M3 = m 1 + m 2 + m 3 .

Similarly, the Braun formula can be extended for QPPM and PtPM.

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