Manipulate optimal high-order motion parameters to construct high-speed cam curve with optimized dynamic performance

Applied Mathematics and Computation 371 (2020) 124953

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Manipulate optimal high-order motion parameters to construct high-speed cam curve with optimized dynamic performance Jianwu Yu a,b, Kaifeng Huang a,b,∗, Hong Luo a,b, Yao Wu b, Xiaobing Long b a

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Lushan South Road 2, Yuelu District, Changsha, Hunan 410082, China b National Engineering Research Center for High Efficiency Grinding, Hunan University, Changsha 410082, China

a r t i c l e

i n f o

Article history: Received 30 May 2019 Revised 4 September 2019 Accepted 1 December 2019 Available online 17 December 2019 Keywords: Construction High-speed cam curve Dynamic performance Optimization High-order parameters

a b s t r a c t Constructing a cam curve is the fundamental of designing cam mechanism. There have been many methods of constructing various cam curves in mathematical filed and mechanical engineering field. However they cannot guarantee dynamic performance and design conditions directly. In this paper, a novel method for constructing high-speed cam curves with optimized dynamic performance under any design conditions directly is proposed. First, an optimization model to determine the optimal high-order motion parameters is constructed by combing the high-order interpolation method and a feasible dynamic optimization model of cam curve. Then the solution to construct cam curve by this proposed method is presented, and a widely used single freedom dynamic model of cam mechanism is constructed to validate the dynamic performance of the constructed cam curve. Finally, taking a globoidal cam mechanism as a case, a high-speed cam curve is constructed by the proposed new method to satisfy the given design demands and the dynamic performance of constructed cam curve is evaluated. Thus the method is demonstrated to be effective and feasible in constructing high-speed cam curves. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Cam mechanism has been widely used in modern automatic machines. The curves of cam mechanism play an essential role in cam designing, and in recent years, researchers have developed many methods which are used to construct the cam curve of cam mechanism, such as the modified trapezoidal (MT) curve, the cycloidal curve, the modified sine (MS) curve, the polynomial curve, the modified constant velocity (MCV) curve [1–5]. But they would result in large vibration and dynamic inaccuracy in high-speed cams because of high-order discontinuity [6]. To avoid these disadvantages, many researchers have developed a variety of construction methods of high-speed cam curve, such as B-spline curve [7], NURBS curve [8] and Fourier series curve [9]. These methods mostly take displacement nodes as initial conditions. The high-order parameters like velocity and acceleration cannot be freely manipulated concerning the dynamic performance of cam curves, which would be some restrictions in high-speed applications. Yu et al. [10] proposed a high-order interpolation mathematical method to ∗ Corresponding author at: State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Lushan South Road 2, Yuelu District, Changsha, Hunan 410082, China. E-mail address: [email protected] (K. Huang).

https://doi.org/10.1016/j.amc.2019.124953 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.

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J. Yu, K. Huang and H. Luo et al. / Applied Mathematics and Computation 371 (2020) 124953

reconstruct a high-speed cam curve, which could guarantee high-order continuity and freely control the whole high-order characteristic parameters of cam curves, such as velocity, acceleration and jerk, but they did not explain how to choose proper high-order motion parameters under a group of given design conditions. Also, unlike low-speed cam mechanisms, dynamic analysis is very important for designing high-speed cam curves. When the cam is running at low speed, the motion law of the follower is basically controlled by the designed cam profile, only kinematics analysis is required. As the running speed of the cam increases, the motion law of the follower will be distorted due to the vibration of entire cam mechanism, leading to inaccurate motion and severe impact [11]. In high-speed cam mechanism, the motion accuracy of cam curves is a key factor, because it affects the whole machine performance. There are following key points should be paid attention to: (i) when dimensionless time T ≤ 1, the maximum absolute acceleration value of dynamic response of follower would be larger than its corresponding design value, which indicates that the actual inertial force will be much bigger than calculated by kinematic analysis. (ii) when dimensionless time T ≥ 1, the follower should be completely stationary according to original design demands, also the acceleration is zero. However, there is still residual vibration caused by free vibration during that time; residual vibration is not allowed in cam mechanisms that require high parking accuracy [12,13]. Above all, the dynamic performance of cam curves should be further studied when design a high-speed cam curve. The dynamic analysis of cam mechanisms is a hotspot in the field of cam mechanisms. There are many literatures published in these years, and many methods to study the dynamic analysis of a cam mechanism have been proposed [14–16]. From above literatures, the single freedom dynamic model of cam-follower system is widely used in studying the dynamic performance of a high-speed cam curve, so a single freedom dynamic model of cam mechanism will be used to verify the dynamic performance of the cam curve constructed by the new method which will be proposed in this paper. It is challenging to find an appropriate high-speed cam curve directly. When the dynamic performance of original cam curve cannot satisfy the design demands, then a dynamic optimization of cam curves is required. There are so many literatures about dynamic optimization of cam curves to improving the dynamic performance of high-speed cam mechanism. Several researchers have proposed many mathematical optimization methods. Singhose and Andresen [17] ultlized input shaping on cam profiles to reduce vibration of cam mechanism, and a good effectiveness of this method was demonstrated. Jeon et al. [18] proposed a two-step optimization technique to design an optimal cam profile which could achieve a high-speed cam mechanism with good dynamic performance in valve system. Ouyang et al. [19] presented a new method for cam profile design and optimization by integrating a single objective optimization procedure with a dynamic model of cam-follower mechanism; it achieved certainly effectiveness by this method, but this was also a limitation of this method, and the selection of an original motion law of acceleration restricted its wilder application. Acharyya and Naskar [20] proposed a polynomial mod traps (PMT), and then used genetic optimization programs to enumerate values of FPMT for synthesis of an optimal cam displacement curves. Clearly, many great achievements have been obtained in constructing cam curves with excellent dynamic performance, such as polynomials and splines, but these methods mostly take the acceleration laws as initial conditions. Almost few researchers have considered constructing cam curves with optimized dynamic performance directly by taking the displacement laws as initial conditions. Actually, compared with taking acceleration laws as initial condition to construct a cam curve, taking displacement nodes at necessary time nodes as initial conditions can directly satisfy a specific design condition. Then designers can concentrate more on the specific design requirements, rather than consider more about how to select and combine an appropriate cam curve in traditional cam curves when designing a new high-speed cam mechanism. There are many derivatives-involved interpolation methods [21–23] have been found in mathematical fields, and the highorder differential interpolation method has also been firstly attempted to apply in cam mechanism [10]. However, to the best effort, it is difficult to find any literatures about taking displacement as initial condition to construct optimized high-speed cam curves directly. As a supplement and further innovation of the previous study – reconstruction of high-speed cam curve based on high-order differential interpolation, this paper presents a new idea of directly constructing an optimized cam curve, just by using finite interpolation nodes, with excellent dynamic performance under high running speed. The proposed method is based on a combination of high-order differential interpolation model and a feasible cam mechanism dynamic optimization model. By combining these two models, a new idea of constructing high-speed cam curves is obtained. In the following, Section 2 introduces the methodology concerning determining the optimal high-order motion parameters under a group of displacement nodes. Then Section 3 presents the solution to construct a high-speed cam curve by this proposed method, and constructs a single freedom discrete dynamic model of high-speed cam-follower system. Its dynamic response of follower is obtained by analytical method to evaluate the constructed cam curves. Section 4 takes an indexing cam mechanism (ICM) in an automatic machine as an example to evaluate the feasibility and effectiveness of the proposed new method in this paper. Finally, conclusions and future works are summarized in Section 5.

J. Yu, K. Huang and H. Luo et al. / Applied Mathematics and Computation 371 (2020) 124953

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2. Methodology to determine optimal high-order motion parameters 2.1. High-order differential interpolation model [10] Based on the previous study, in order to guarantee the high fidelity and high-order continuity of constructed cam curves, a cam curve can be constructed by high-order differential interpolation model as follows:

⎡

f (T ) =

n  i=1

⎤

⎣αi (T )Si + βi (T )Vi + γi (T )Ai + δi (T )Ji ⎦             1

2

3

(1)

4

where, i is the number of sequence relating to the giving interpolation nodes, Si is corresponding dimensionless node of displacement, Vi is corresponding dimensionless node of velocity, Ai is corresponding dimensionless node of acceleration, and Ji is corresponding dimensionless node of jerk, α i (T) is corresponding displacement basis function, β i (T) is corresponding velocity basis function, γ i (T) is corresponding acceleration basis function, and δ i (T)is corresponding jerk basis function. And these basis functions are selected as follows:

⎧  T αi (T ) = [ pi3 , pi2 , pi1 , pi0 ] 1, 3T 2 − 2T 3 , (1 − T )3 , T 3 L4i (T ) ⎪ ⎪ ⎪ ⎪ ⎨β (T ) = [q , q , q , q ]1, 3T 2 − 2T 3 , (1 − T )3 , T 3 T L4 (T ) i i3 i2 i1 i0 i   2 3 3 3 T 4 ⎪ γi (T ) = [ci3 , ci2 , ci1 , ci0 ] 1, 3T − 2T , (1 − T ) , T Li (T ) ⎪ ⎪ ⎪  T ⎩ δi (T ) = [mi3 , mi2 , mi1 , mi0 ] 1, 3T 2 − 2T 3 , (1 − T )3 , T 3 L4i (T )

(2)

where, p, q, c and m are corresponding parameters of each basis functions waiting to be determined. Li (T) is interpolation polynomial formulated as follows:

Li ( T ) =

n  i=1, i=k

T − Tk (T − T1 ) · · · (T − Tn−1 )(T − Tn ) = Ti − Tk (Ti − T1 ) · · · (Ti − Tn−1 )(Ti − Tn )

(i, k = 1, 2, . . . , n )

(3)

Given the whole displacement and high-order parameter nodes, each undetermined coefficients of basis function p, q, c, and m can be solved by the following equations:

f (ti ) = si , f (1) (ti ) = vi , f (2) (ti ) = ai , f (3) (ti ) = ji (i = 1, 2, . . . , n )

⎧  1, i = k ⎪ ⎪ α ( t ) = ; αi(1) (tk ) = 0; αi(2) (tk ) = 0; αi(3) (tk ) = 0 ⎪ i k ⎪ 0, i = k ⎪ ⎪  ⎪ ⎪ 1, i = k ⎪ ⎪ ⎨βi (tk ) = 0; βi(1) (tk ) = 0, i = k ; βi(2) (tk ) = 0; βi(3) (tk ) = 0  1, i = k ⎪ (1 ) (2 ) ⎪ γ ( t ) = 0 ; γ ( t ) = 0 ; γ ( t ) = ; γi(3) (tk ) = 0 ⎪ i k k k i i ⎪ 0, i = k ⎪ ⎪  ⎪ ⎪ 1, i = k ⎪ ⎪ ⎩δi (tk ) = 0; δi(1) (tk ) = 0; δi(2) (tk ) = 0; δi(3) (tk ) =

(4)

(5)

0, i = k

For instance, the group of parameters of pi , the corresponding parameters of displacement basis function on ith node, can be obtained by solving the following equations:

  T αi (Ti ) = [ pi3 , pi2 , pi1 , pi0 ] 1, 3T 2 − 2T 3 , (1 − T )3 , T 3 L4i (T ) = 1 αi(1) (Ti ) = αi(2) (Ti ) = αi(3) (Ti ) = 0

(6)

2.2. Optimization model of high-order motion parameters When designing a high-speed cam mechanism, its dynamic performance must be considered in design processes. Generally, the dynamic performance of a high-speed cam mechanism is mainly controlled by its high-order motion parameters, such as velocity, acceleration and jerk, of cam curves. Then, in order to achieve a better dynamic performance, an available optimization model, which can guarantee an overall minimization of the high-order motion parameters, can be selected as follows [24]:

min F = ω1



1 0

V 2 (T )dT + ω2



1 0

A2 (T )dT + ω3



1 0

J 2 (T )dT

(7)

where, ω1, ω2 and ω3 are positive weighting factors which is used to control the relative value among velocity, acceleration and jerk, whether the objective function can find the optimal solution mainly depends on the selection of these three parameters.

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This optimization model can ensure the velocity, acceleration and jerk in a feasible limits. From Eq. (2), it can be obtained that the corresponding velocity function of high-order interpolation is shown as follows:

⎡

V (T ) =

n  i=1

⎤

⎣αi(1) (T )Si + βi(1) (T )Vi + γi(1) (T )Ai + δi(1) (T )Ji ⎦             1

2

3

(8)

4

Then, the first integral can be expanded as:



1

0

V (T )dT =



2

1



0

n  αi(1) (T )Si

2

 dT + . . . + 2

1 0

i=1

n  n  αi(1) (T )Si βi(1) (T )Vi dT + . . .

(9)

i=1 j=i

Eq. (9) is constituted with four squared integrals of Si , Vi , Ai and Ji , and six cross product integrals of those different interpolation nodes. The squared integrals can be expanded as follows:



1



0

n  αi(1) (T )Si

2

dT =

i=1

n 

 Si2



1 0

i=1

 1 n  n  2  2Si S j αi(1) (T ) dT + αi(1) (T )α (j 1) (T ) dT

Eq. (10) can be rewritten as follows:

⎡



1



0

n  αi(1) (T )Si

2



⎢

dT =

i=1

0

1 T⎢ S ⎢ 2 ⎢

⎣  1 2

 1 T 

=

2

S

1

2

0

2  αi(1) (T ) S

(10)

0

i=1 j=i

 ⎤ α1(1) (T )αn(1) (T ) dT ] ⎥ 0 ⎥ .. ⎥S . ⎥ ⎦  1  2 2 αn(1) (T ) dT





2 αi(1) (T ) dT

···

.. .

..

 α1(1) (T )αn(1) (T ) dT

2

.

···

1



0

Similarity, the cross product integrals can be rewritten in the form of matrix as follows:

⎡ 



1

2 0

= 12 S

⎢

⎢ n  n  1 ⎢ αi(1) (T )Si βi(1) (T )Vi dT = ST ⎢ 2 ⎢

0

1

4



(α1(1) (T )β1(1) (T ))dT .. .

⎣ 

i=1 j=i

 T

1

4

0

αi(1) (T )βi(1) (T ) V

··· ..

(α1(1) (T )βn(1) (T ))dT

(11)

4

1 0

(α1(1) (T )βn(1) (T ))dT .. .

. 

···

1

4 0

(αn(1) (T )βn(1) (T ))dT

⎤ ⎥ ⎥ ⎥ ⎥V ⎥ ⎦

(12)

The other remaining items in Eq. (9) can be expanded and rewritten in the form of Eqs. (11) and (12). Finally, Eq. (9) can be deduced as follows:

⎡ 

 0

1

V 2 (T )dT =

⎢

1  T T T T ⎢ S ,V ,A ,J ⎢ 2 ⎣



2

(αi(1) (T ))



.. . (1 )

(1 )



···

αi (T )δi (T )



..

.

···

⎤⎡ ⎤ ⎥ S ⎥⎢V⎥ .. ⎥⎣ ⎦ .   ⎦ A 2 (1 ) J (δi (T ))

αi(1) (T )δi(1) (T )

(13)

Repeat the above process, Eq. (7) can be written as follows:

⎡

3 



(k )

2



ωk (αi (T )) ⎢ ⎢ k=1 ⎢ .. 1 ⎢ F = ST , VT , AT , JT ⎢ . 2 ⎢ ⎢   ⎣ 3 ωk αi(k) (T )δi(k) (T ) k=1

··· ..

.

···

⎤  ωk αi (T )δi (T ) ⎥⎡ ⎤ ⎥ S k=1 ⎥ .. ⎥⎢V⎥ . ⎥⎣ A ⎦ ⎥ ⎥ J   3 2  ⎦ ωk (δi(k) (T ))

3 



(k )

(k )

(14)

k=1

The matrix in Eq. (14) is a positive definite matrix, because its original Eq. (7) is obviously positive for any variable of V, A and J high-order motion nodes. When giving n dimensionless time and corresponding dimensionless displacement nodes Si of cam curve as a design condition (0 ≤ Si ≤1), the corresponding velocity nodes, acceleration nodes and jerk nodes can be selected as design variables. Expanding the corresponding rows and columns in Eq. (14), the optimal highorder interpolation nodes of a high-speed cam curve under the given displacement nodes can be obtained from following

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equation:

min F (x ) =

1 T x Gx + bT x + c 2

(15)

where, x is a 3n dimension vector, whose elements are corresponding velocity, acceleration and jerk nodes of cam curve; G is a 3n × 3n principal minor of the matrix in Eq. (14), which is also a positive definite matrix; b is a 3n dimension constant vector which is correlating to given S and c is a constant which is correlated to given S. Also, the design conditions can be freely selected among displacement, velocity, acceleration and jerk according to specific requirements. Eq. (15) is a typical positive quadratic objective function. By solving Eq. (15), the optimal high-order parameters of cam curves can be obtained. 3. Solutions to construct optimized cam curve 3.1. Construction of optimized cam curve The optimization model of high-order motion parameters Eq. (15) describes an unconstrained optimization problem. In order to construct an optimized cam curve, the first step is solving this unconstraint optimization problem. Supposing the differential of Eq. (15) equals 0:

dF (x ) = x T G + b T =0 dx

(16)

Then, because the determinant of G is ’not zero (G is a positive definite matrix), it can be obtained that the sole minimum point of Eq. (15) is:

x∗ = −G−1 b

(17)

Also, this unconstrained optimization problem can be solved by numerical method such as conjugate direction method [25] and differential evolution algorithm [26,27]. From Eq. (17), the optimal high-order parameters Vi , Ai , Ji of a cam curve under given design displacement nodes have been obtained in x∗ by above process. Then use the computed x∗ in high-order differential interpolation model, the cam curve equation can be constructed:

S(T ) = [α1 (T ), . . . , αn (T )]S + [β1 (T ), . . . , βn (T ), γ1 (T ), . . . , γn (T ), δ1 (T ), . . . , δn (T )]x∗

(18)

Theoretically, it is obvious that an optimized cam curve with excellent dynamic performance under any design nodes can be obtained by this proposed method. However, if the number of interpolation nodes is too large, then the constructed cam curve is a high-order polynomial with an overlarge exponent. It will cause high-order oscillation, which is not beneficial to the dynamic performance of cam mechanism at high speeds. The relationship between maximum number of exponent and number of given nodes based on the given basis function in this study can be expressed as follows:

p = 3 + 4 × (n − 1 )

(19)

where, p is the maximum exponent of constructed cam curve, n (n ≥ 3) is the number of given dimensionless time nodes. It can be realized that the number of given dimensionless time nodes under the given basis function and the exponent of constructed high-order polynomial curve is positively correlated. From available literatures [25], the exponent of a highorder polynomial dynamic cam curve should no more than 19, which means the number of given time nodes should no more than 5, otherwise a high-order oscillation would occurs. Since it is not clear if the given displacement nodes are rational, it is necessary to evaluate the dynamic performance of constructed cam curve to judge whether the constructed cam curve is feasible. If the constructed cam curve cannot satisfy specific working demand, then the given displacement nodes can be modified immediately. 3.2. Dynamic model of high speed cam curves Discretizing every part in cam mechanism and ignoring the influence of damping to emphasize the effect of cam curves, a typical single freedom cam-follower mechanism dynamic model can be described as Fig. 1. Then, its mathematical model of cam mechanism can be obtained by Newton’s second motion law:

m

d2 x + kx = ks(t ) dt 2

(20)

Introducing dimensionless displacement X = x/h and dimensionless time T = t/th or T = ø1 /ω (where h is the stroke of cam mechanism, th is the time of cam lift, ø1 is lifting angle of cam mechanism, and ω is angular velocity of cam), Eq. (19) can be deformed as:

⎧ 2 ⎨ d X + ω 2 t 2 X = ω 2 t 2 S (T ) n h n h dT 2  ⎩ ωn = k/m

where, ωn is the natural frequency of the whole cam mechanism.

(21)

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J. Yu, K. Huang and H. Luo et al. / Applied Mathematics and Computation 371 (2020) 124953

x(t)

k

s(t)

m

Fig. 1. Dynamic model of cam-follower mechanism, where, k is the equivalent stiffness of follower, m is the mass of follower, s(t) is the forced motion controlled by cam curve, x(t) is the displacement response of follower.

The solution of Eq. (20) can be solved by Duhamel’s integral [28]:

X (T ) =

1 ωn th



T 0

(ωn th )2 S(τ ) sin ωnth (T − τ )dτ

(22)

where, τ is the integral variable index without real meaning, X(T) is the forced vibration response function of follower at cam lifting stage in dimensionless time domain. When the cam mechanism runs to stopping stage, S(T) becomes a constant 1. Then the residual vibration model of cam mechanism can be expressed as follows:

d2 X + ωn2 th2 (X − 1 ) = 0 dT 2

(23)

This is a free vibration equation and its initial conditions are determined by the final motion state of its former lifting stage. So, the response of its residual vibration can be formulated as:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ A=

(X0 − 1 )2 +

V02

ωn2th2

V0 ⎪ ⎪ ψ = arctan ⎪ ⎪ X − 1 )ωn th ( 0 ⎩ R(T ) = A cos (ωn th T − ψ )

(24)

where, A is the amplitude of residual vibration, ψ is the phase of residual vibration, X0 is the final displacement response of follower at the end of its former lifting stage, V0 is the final velocity response of follower at the end of its former lifting stage, R(T) is the response function of residual vibration in dimensionless time domain. It can be concluded that the dynamic response of cam mechanism has significant relationship to the parameter ω n th from Eqs. (22)–(24), and this parameter is related to the running angular velocity and the natural frequency of cam mechanism. In order to completely characterize the dynamic performance of cam mechanism under various running angular velocities, a new parameter cam cycle ratio λ can be introduced. The cycle ratio of a cam mechanism can be defined as follows:



λ = th /t0

(25)

t0 = 2π /ωn Then the Eqs. (22) and (24) can be deformed as:

X (T ) =

1 2π λ



T 0

(2π λ )2 S(τ ) sin 2π λ(T − τ )dτ

R(T ) = A cos(2π λT − ψ )

(26) (27)

3.3. Characterization of λ in high-speed cam When designing high-speed cam mechanism, designers usually choose a material with higher stiffness and lighter mass to manufacture cam and follower, which is a basic demand in high-speed cam mechanism. From available literature [24], a

J. Yu, K. Huang and H. Luo et al. / Applied Mathematics and Computation 371 (2020) 124953

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Table 1 Classification of cam mechanism in industry and machines [29]. Type of cam Industrial low-speed cam Vehicle(overhead cam) Industrial high-speed cam Vehicle(putt)

Mass (lb)

Stiffness (lb•in−1 )

10 0.75 1.0 1.35

1.0 × 10 1.0 × 106 2.5 × 106 12.650 7

Speed (rpm) 300 3000 3000 3000

μm 2.6 2.0 1.0 2.7

μm (commonly) × × × ×

−6

10 10−5 10−4 10−2

10−6 10−5 10−4 2.7 × 10−2

START Given design displacement nodes Si Determine the optimal high-order parameters by equation (17) or differential evolutionary algorithm Modify given displacement nodes Si

Construct high-speed cam curve by equation (18) Evaluate the dynamic performance of constructed cam curve by equations (26)-(27)

No

If cam curve satisfies working demand?

Yes Output constructed cam curve END Fig. 2. Flowchart to construct high-speed cam curve by proposed method.

parameter named dynamic coefficient of cam mechanism μm can be introduced to characterize a high-speed cam mechanism. The dynamic coefficient is defined as follows:

! ω "2 # φ $2 0 μm = = ωn ωn th

(28)

where, ω is the running angular velocity of cam mechanism. The relationship between μm and λ is:

λ=

2π

φ0 √ μm

(29)

Then, cam mechanism can be classified as follows: (i) when μm ≈ 10−6 , the cam mechanism is a low-speed cam, only kinetic analysis of cam mechanism can satisfy designing demand, (ii) when μm ≈ 10−4 , the cam mechanism can be classified as a moderate-speed cam, (iii) when μm ≈ 10−2 , it is a high-speed cam mechanism, then dynamic analysis and dynamic optimization should be studied. Specifically, classification of various cam mechanisms in modern automatic machines and commonly used in industry can be shown as Table 1. In summary, the flow of adopting this method to construct an optimized cam curve with excellent dynamic performance can be illustrated as Fig. 2. 4. Case study and discussion A high-speed ICM cam curves is constructed by the proposed method, and its dynamic performance is evaluated to verify the feasibility of this method. There are four parts in this Section, which are (i) initial design conditions of a globoidal in-

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Fig. 3. Globoidal cam mechanism.

Rotation angle of indexing plate (deg)

60 48

30 24 12 0 0

225 105 135 165 180 Rotation angle of cam (deg)

255

360

Fig. 4. Working displacement node of mechanism.

dexing cam mechanism, (ii) determination of the optimal high-order motion parameters of this cam curve and construction of optimized cam curve, (iii) dynamic validation of the constructed cam curve by proposed method in this paper and (iv) comparison on dynamic performance of this constructed cam curve and other most commonly used cam curves. 4.1. Design demand of a globoidal indexing cam mechanism Fig. 3 presents a globoidal cam mechanism which is widely used in a modern automatic machine. The rotational speed of this cam mechanism is about 1800 r/min, and the natural frequency is about 1400 rad/s; the working displacement of profile should satisfy the following nodes shown in Fig. 4, which are determined by working demand. It is difficult to construct a high performance cam curve by previous methods to satisfy these given design parameters. The proposed method is herein applied to calculate the optimal high-order motion parameters of this globoidal indexing cam mechanism, and then a highspeed cam curve can be consturcted with optimal dynamic performance under those given displacement nodes to satisfy working demand. 4.2. Construction of high-speed cam curve under design conditions From Fig. 4, it can be observed that there are six displacement nodes which should be satisfied to design this globoidal indexing mechanism. Transforming those given displacement nodes to dimensionless node, its dimensionless displacement node Si and corresponding dimensionless time nodes Ti can be calculated as the data in Table 2. Then applying the proposed method, the dimensionless velocity, dimensionless acceleration and dimensionless jerk can be assigned as design variables to construct a cam curve with optimal dynamic performance under given design condition. The start point and end point of dimensionless velocity and dimensionless acceleration should be set as 0 to ensure the stability under a high running

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Table 2 Design conditions and design variables of given design demand. T

0

1/5

2/5

1/2

4/5

1

S V A J

0.0000 0.0000 0.0000 x9

0.2000 x1 x5 x10

0.4000 x2 x6 x11

0.5000 x3 x7 x12

0.8000 x4 x8 x13

1.0000 0.0000 0.0000 x14

Table 3 The optimal results under given design conditions by using –G−1 b. T

0

1/5

2/5

1/2

4/5

1

S V A J

0.0000 0.0000 0.0000 59.9632

0.2000 0.7680 5.7603 2.3814

0.4000 1.7280 2.8800 −26.3941

0.5000 1.8750 0.0000 −30.0066

0.8000 0.7680 −5.7604 2.4062

1.0000 0.0000 0.0000 60.0271

Table 4 The finial optimal results under given design conditions by using differential evolution algorithm. T

0

1/5

2/5

1/2

4/5

1

S V A J

0.0000 0.0000 0.0000 59.2401

0.2000 0.7681 5.7607 2.2815

0.4000 1.7282 2.8844 −26.6225

0.5000 1.8745 −0.0085 −29.6253

0.8000 0.7679 −5.7613 2.3222

1.0000 0.0000 0.0000 59.6906

Table 5 The motion characteristic values of constructed curve under given design conditions. Vm

Apm

Amm

Jpm

Jmm

Qpm

Qmm

1.8745

5.7729

−5.7735

59.6906

−29.7418

6.6916

−6.7122

speed. The whole design variables and design conditions of the cam mechanism can be shown as Table 2. The key of this designing problem is to determine the remaining high-order motion parameters in Table 2. Substituting those data and variables into Eqs. (1)–(6), an optimization model like Eq. (15) can be obtained. Because the relative deviations between velocity, acceleration and jerk are not too large, setting weight factors as 1 will not make the matrix G in Eq. (15) becomes a singular matrix. Solving this optimization model by using Eq. (17), a group of optimal results under given design conditions are obtained as Table 3, and the value of objective function (15) is 746.875. Due to the existence of rounding errors, the data in Table 3 may not the exact result of this unconstrained problem. In order to obtain the exact results of this unconstrained problem, it can be resolved by using differential evolution algorithm. Setting the cross rate CR = 0.8, mutation probability F = 0.5, number of population NP = 140. After 10 0 0 iterations, the finial optimal results of Eq. (15) are shown as Table 4, and the value of objective function (15) is 738.375. Then, using the optimal data of every high-order motion parameters in Table 4 and the displacement interpolation nodes given by design conditions, a 23rd power polynomial cam curve can be constructed by Eq. (18). The motion performance of this constructed cam curve is shown as Fig. 5. It can be observed from Fig. 5, the characteristics of dimensionless velocity and acceleration have good smoothness, and the curve of jerk is continuous, which means there is no impaction in this constructed cam mechanism. The specific characteristic values of this constructed cam curve are listed in Table 5. It can be observed that the maximum dimensionless velocity is 1.8750 from Table 5, at the middle point of this curve. The maximum and minimum dimensionless acceleration are around ± 5.77. The maximum and minimum dimensionless torque Q = AV are around ± 6.7, which means this curve can be used in high-speed light-load application. 4.3. Dynamic performance verification of constructed cam curve In order to evaluate the dynamic performance of the constructed cam curve, a single freedom dynamic model of cam mechanism has been described as Fig. 1. Substituting the cam curve into Eqs. (13) and (14), the dynamic performance of constructed cam curve under various operating parameters can be shown as Fig. 6. The specific dynamic characteristics can be seen in Table 6. It can be noted from Fig. 6(a) that there is a high-order oscillation when the cycle ratio λ = 1. This means a stronger damper is required to suppress high-order oscillation when this cam mechanism operates under an ultra-high running speed

J. Yu, K. Huang and H. Luo et al. / Applied Mathematics and Computation 371 (2020) 124953

10 8 6

Dimensionless S,V and A

100

S V A J

80 60

4

40

2

20

0

0

-2

-20

-4

-40

-6

-60

-8

-80

-10

0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 Dimensionless time T

0.8

0.9

Dimensionless J

10

-100 1

Fig. 5. Motion performance of the constructed cam curve. Table 6 The dynamic characteristics of constructed curve under given design conditions.

λ

μm

Speed(rpm)

Xm

Amplitude of R

|A m |

Vm

1 1.5 2 3

0.1736 0.0772 0.0435 0.0193

5570.4230 3713.6153 2785.2115 1856.8077

1.4634 1.0038 1.0289 1.0056

0.5177 0.0179 0.0868 0.0537

18.3808 8.2859 4.9134 6.5027

2.8225 1.0626 1.1632 1.0355

of 5570 r/min. A more powerful basis function of high-order differential interpolation which can avoid high-order exponents and guarantee excellent dynamic performance could be further studied. Then the number of given dimensionless time nodes larger than 6 can be selected to satisfy more rigorous design demands, for improving the flexibility and feasibility of the method proposed in this paper. It can be noted from Fig. 6(b) and Table 6 that the dimensionless amplitude of residual displacement is the minimum when the cycle ratio λ = 1.5, and the dimensionless error at the end of lifting stage is 0. The errors of velocity and acceleration are slightly larger than λ = 2 or 3 form Fig. 6(c) and (d), but they are not very important to a globoidal cam mechanism. It can be thus concluded that the cycle ratio λ = 1.5 is the best operating parameter of constructed cam curve. When the cycle ratio is 1.5, a predominant overall dynamic performance would be achieved under high running speed, which satisfies the classification of a high-speed cam. However, since the required running speed 1800 r/min corresponds to the cycle ratio λ = 3, the dimensionless response error of indexing plate when the globoidal cam mechanism enters into stop stage is around 0.0145 (as indicated in Fig. 6(a)), which means the indexing error of the indexing plate is around 0.870°. The amplitude of residual vibration at stop stage is 0.0537, which means the dynamic indexing error of indexing plate at stop stage is around 3.222°. Although this error will attenuate because of the existence of damping, the dynamic indexing error is too high for an indexing mechanism. When the cycle ratio is 1.5, the dimensionless amplitude of residual vibration in stop stage is 0.0179, which means the dynamic indexing error of the indexing plate is just 1.074°. Thus a feasible approach is reducing the natural frequency of the indexing plate to make the cycle ratio λ equal to 1.5 when the demanded running speed is 1800 r/min. 4.4. Comparison with other typical cam curves of globoidal cam mechanism and other construction methods The comparison on dynamic performance of this constructed cam curve and other most commonly used cam curves are shown as Fig. 7, and the nodal residuals between different cam curves and given design displacement nodes can be seen in Table 7. The dynamic response of constructed cam curve and other most commonly used cam curves in lifting stage are similar under the same operating parameters. As shown in Fig. 7(a), there are no obvious differences between these curves.

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Fig. 6. Dynamic performance of the constructed cam curve.

Table 7 Dimensionless nodal residuals between different curves and given design conditions. Node

The constructed curve

6th power polynomial

Cycloidal

MS

0 1/5 2/5 1/2 4/5 1

0 0 0 0 0 0

0 0.04096 0.13824 0.15625 0.04096 0

0 −0.00929 −0.01099 0 0.00929 0

0 0.01047 0.01043 0 −0.01047 0

However, it can be observed from Fig. 7(b) that the constructed cam curve by this method has the minimum residual vibration in stop stage, which is the most important performance of a globoidal cam mechanism. The positioning accuracy of this cam curve constructed by the proposed method is slightly better than MS curve (0.0190) and much better than other two cam curves. From Figs. 7(c) and (d), it can be observed the errors of velocity and acceleration of constructed curve are larger than MS curve, and less than cycloidal curve and asymmetrical 6th power polynomial curve. However, compared with the constructed curve by proposed method, the MS curve cannot perfectly satisfy given design conditions from Table 7. Compared with other construction methods of high-speed cam curves, there are some advantages and disadvantages can be listed as Table 8. It can be observed form Table 8 that the proposed method can freely control all characteristic parameters of a cam curve, which means the highest adaptability and flexibility.

12

J. Yu, K. Huang and H. Luo et al. / Applied Mathematics and Computation 371 (2020) 124953

Fig. 7. Dynamic performance of cam curves with different methods at λ = 1.5. Table 8 Comparison with other construction methods. Construction method Method in this paper B-Spline[7] NURBS[8] Fourier Series[9]

Displacement control √ √ √ √

Velocity control √

Acceleration control √

Jerk control √

× × ×

× × ×

× × ×

5. Conclusion This paper presents a novel idea on how to directly construct an optimized cam curve with excellent dynamic performance under high rotating speed by taking finite displacement nodes as initial conditions, and gives the methodology to achieve this goal. This idea has been demonstrated to be effective and feasible by taking an ICM cam mechanism in modern automatic machine as a design case. The following observations can be obtained. (1) The proposed method gives a new idea to construct optimized cam curves directly; it can allow designers to concentrate more on the specific design requirement, rather than consider how to select and combine an appropriate cam curve in traditional cam curves when designing a new high-speed cam mechanism. Theoretically, a designer can obtain a cam curve with excellent dynamic performance under any design conditions by the proposed method, if the given design condition is reasonable, just using some dimensionless displacement nodes or dimensionless velocity nodes or some other high-order motion nodes which are determined by design demand. This method can be further

J. Yu, K. Huang and H. Luo et al. / Applied Mathematics and Computation 371 (2020) 124953

13

extended to the fields of tool path planning in numerical control (NC) machining and surface shape adjustment in purely mathematical filed. (2) In order to achieve the best overall dynamic performance of designed cam mechanism, the optimal operating parameters should be discussed by dynamic analysis. For this globoidal cam mechanism in case study, it can be observed that the optimal operating parameter are the parameters which can make the cycle ratio λ equal to 1.5 at working. There is a limitation about this proposed method; the number of given interpolation nodes should no more than 6, otherwise a high-order oscillation would occur when the cycle ratio λ equals to 1 in this case study. It is not beneficial to the running stability of a cam mechanism under high rotating speed. A more powerful basis function can be further studied to eliminate this limitation. 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