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Page 106 Wu. Intell Robot 2021;1(2):99-115 I http://dx.doi.org/10.20517/ir.2021.11

the damp can be ignored to determine the natural frequencies. Simplification of Equation (35) results in the
linearized elastodynamic equation below
M¥ u + Ku = 0 (36)

The rigidity of the system may be represented by the natural frequency. The higher is the frequency, the higher
is the stiffness. From Equation (36), we get

2
det(− M + K) = 0 (37)
where = /2 denotes the natural frequency.

The displacement response analysis can be carried out from Equation (35) based on the initial conditions

u 0 = u(0); ¤ u 0 = ¤ u(0) (38)

Here, the damping ratios are set to = 6% according to the manipulator structure. From Equation (37), the
displacement vector u can be represented in terms of the modal contributions, namely,

u = Q (39)
where Q and are the modal matrix and the vector of the displacements in each mode, respectively. Conse-
quently, Equation (35) can be rewritten as

(40)
¥ + ¤ + = f
with
[ ]

= Q CQ = diag 2 1 2 2 ... 2 6 (41a)
[

= Q KQ = diag 2 2 ... 2 ] (41b)
1 2 6

f = Q F (41c)
Since the mass and stiffness matrices in Equation (35) are time-varying, the common way to solve such a
problem is to divide the motion period into extremely short intervals, where the stiffness and mass matrices
are considered as constant in each interval. Let denote the complete motion period that is divided into
intervals, namely, Δ = / . In the th time interval ∈ [ −1 , ], the equation of motion in the th mode is
expressed as
2

¥ + 2 ¤ + = (42)

Thus, the th mode contributes to the displacement response [44] is

( )

( ) = Δ cos Δ + √ sin Δ ( −1 )
1 − 2
∫
1
+ ( ) − ( − ) sin ( − )d

−1
( )
1 Δ

+ sin Δ ¤ ( −1 ) (43)

where
√
= 1 − 2 (44)