Modelling a uniaxial inerter in a 2D or 3D environment: Implications of centripetal acceleration.

The inerter completes the force-current analogy between mechanical and electrical components, providing the mechanical equivalent to the capacitor. As such, it is a two-terminal passive element that, when implemented ideally, is normally said to generate a force proportional to the relative acceleration between its two terminals. However, this is applicable only if the inerter does not rotate, so the only relative motion between the device ’ s terminals is axial. In many applications, this restriction is acceptable, such as in car suspension systems. However, in this paper, it is shown that the relationship between the terminal accelerations and the generated force is more complex if the inerter is used in a 2-dimensional (2D) or 3-dimensional (3D) environment, such as within a multi-bar mechanism ( e.g ., robotic arms or railway pantographs). Specifically, the inerter force is not given by simply the relative acceleration between the two terminals. The centripetal acceleration, resulting from the rotation of the inerter, needs to be accounted for to find the second derivative of the inerter length, which defines the generated force. Two case studies are presented to demonstrate the effects of this normally neglected cen-tripetal acceleration term. It is shown that when an inerter is operating in a 2D or 3D environment, significant errors may occur in evaluating the inerter force and also the system response if the centripetal acceleration term is neglected. Equations are provided for both modelling the inerter in different coordinate systems and for incorporating the inerter in 2D and 3D multibody systems.


Introduction
In 2002, Smith [1] introduced a new two-terminal ideal passive mechanical element, termed the inerter, with the property that the force F I at its two terminals is proportional to the relative acceleration between them (when the inerter does not rotate).The introduction of the inerter achieves a full analogy between mechanical and electrical networks in which force (or velocity) corresponds to current (or voltage) and a fixed point in an inertial reference frame corresponds to electrical ground.In this analogy, a spring (or damper or inerter) corresponds to an inductor (or resistor or capacitor, respectively).Since the concept of the inerter was proposed, many beneficial mechanical configurations with inerters have been explored for a wide range of engineering applications.In particular, an inerter-based device, called the 'J-damper' as a decoy name to keep the technology secret from competitors, was developed by Cambridge University and helped McLaren Racing achieve victory at the 2005 Spanish Grand Prix [2].A seismic control device, the tuned viscous mass damper (TVMD), has been investigated in structural systems [3,4] and the soil-structure interaction of a structure with an inerter system has been studied [5].Furthermore, beneficial inerter-based absorbers have been identified for a wide range of mechanical structures, such as buildings [6,7], cables [8,9], automotive vehicles [10][11][12], railway vehicles [13,14], aircraft landing gear [15], offshore wind turbines [16] and railway pantographs [17].There has been some recent work come out on the use of a pair of oblique inerters in nonlinear vibration isolators and suppression systems [18,19].In addition, inerter-combined active or semiactive vibration control has also drawn interest from researchers, e.g., [20][21][22][23].
However, the original model of the inerter has the constraint that only translational movement along the element axis between two terminals of the inerter is permitted, thus it is only suitable to model an inerter in a one-dimensional (1D) environment.For other mechanisms and devices, such as robotic arms, excavator arms or railway pantographs, they may operate in a 2-dimensional (2D) or 3dimensional (3D) environment, where the motions include both translation and rotation in general.Another example is inerter-based vibration absorbers in earthquake engineering applications.Many researchers have proposed inerter-based devices, such as tuned inerter damper [24], and tuned mass damper inerter [25,26], which have shown significant effects on the protection of building structures under earthquake conditions.In some of these studies, the inerter-based absorbers are implemented as a 'cross-bracing' element [27] where rotational movement will be involved.Meanwhile, the excitations of the earthquake contain components in multiple directions, which can also lead to rotational movement of the inter-based vibration absorber installed in building structuresthis is a field where the present work will have a significant impact.
According to the original definition of the inerter in reference [1], the uniaxial inerter force in a 1D environment can be modelled as a force that is proportional to the relative acceleration between its two terminals.However, this simple model is not generally applicable in a 2D or 3D environment.The general inerter force in a 2D or 3D environment should be modelled as a force proportional to the second time derivative of the distance between its two terminals.The difference between the simple inerter model and the general inerter model is that the latter takes into account not only the projection of the relative acceleration onto the element axis but also the centripetal acceleration term (which could be neglected intuitively).
In this paper, a model of a uniaxial inerter in a 2D or 3D environment and the methodology of applying a uniaxial inerter in a multibody system is developed.Furthermore, the effect of the centripetal acceleration correction term and the method of implementing a uniaxial inerter in a 2D multibody system are demonstrated via two examples.The sections are arranged as follows.In Section 2, the concept of a uniaxial inerter within a 2D or 3D environment is introduced, initially in a 2D polar coordinate system.Subsequently, a model of a uniaxial inerter in a 2D or 3D environment is derived in general vector form, and its expressions in Cartesian and Spherical coordinates are presented.A 2D inerter-included model is then shown as an example to demonstrate the effect of the centripetal acceleration correction on the inerter force and the system response.In Section 3, the method of adopting a uniaxial inerter in a multibody system is introduced.The proposed modelling methodology is then demonstrated using an inerter-included two-bar mechanism as an example, where the evident effect of the centripetal acceleration correction term on the system responses can be observed.The conclusions are presented in Section 4.

Uniaxial inerter in a 2D or 3D environment
In this section, the concept of using a uniaxial inerter in a 2D or 3D environment is first introduced, using a 2D polar coordinate system.It is shown that the centripetal acceleration needs to be accounted for in modelling the inerter force when it rotates.Then the general models of all three uniaxial mechanical elementssprings, dampers and inerters -are derived in a 3D environment in general vector form, showing how inerters need to be treated differently.The general model of the inerter in 2D or 3D is then expressed in Cartesian and spherical coordinates for ease of use.Lastly, a simple 2D 1DOF inerter-included lumped mass system is used as an example to demonstrate the effect of the centripetal acceleration correction on the inerter force and the system response.

The concept of a uniaxial inerter in a 2D polar coordinate system
Uniaxial elements in this paper are defined as elements whose generated forces, i.e., the forces that the elements apply at the terminals, are only aligned along their axis, i.e., the axis across their two terminals.For example, a hydraulic damper can be modelled as a uniaxial element.A two-terminal uniaxial element, symbolised as a square block, is established in 2D polar coordinates in Fig. 1, where one terminal of the uniaxial element is fixed to the pole, point O, of the coordinate system.Without loss of generality, this symbol can represent a spring, a damper or an inerter.In this polar coordinate system, the position of the free terminal of the uniaxial element, point P, is determined by the radius ρ, i.e., a distance from the pole, and polar angle θ, i.e., the angle from the polar axis X.The motion of the free terminal, point P, is given by translation along the axis OP and rotation of the axis OP about point O.
For spring with stiffness k, the force is proportional to the change in distance between its two terminals compared with its undeformed length ρ 0 .In this 2D polar coordinate system, if tension forces are defined as positive, the spring force is given by In this 2D polar coordinate system, the relative velocity between the element's two terminals, v P , is a superposition of the radial velocity ρ along the element axis and the circumferential velocity ρ θ orthogonal to the element axis, as shown in Fig. 2. Note that the overhead '⋅' denotes the time derivative.As the circumferential velocity ρ θ is orthogonal to the element axis, it has no effect on the damper force.Hence, for a damper with damping coefficient c, the force is proportional to the component of relative velocity between the element's two terminals along its axis.That is the damper force is given by The original inerter force, defined in [1] for a 1D environment, is proportional to the relative acceleration between the inerter's two terminals.In the 2D polar coordinate system, let the relative acceleration between the element's two terminals, points O and P, be a P .Based on the definition of the inerter in [1], if there is no rotation of the inerter, i.e., the movement is restricted to be along the element axis, so θ = 0, the formula for the inerter force is where b is the inertance and a a is the component of a P along the element axis (the only component in this case).However, this definition is not applicable to the case where θ ∕ = 0, i.e., when the element is allowed to move in a 2D environment.This is because the relative acceleration between the element's two terminals, a P , consists of four components: the radial acceleration ρ along the element axis and the circumferential acceleration ρ θ orthogonal to the element axis, but also the centripetal acceleration a ce = ρ θ2 along the element axis towards the pole, and the Coriolis acceleration a co = 2 ρ θ orthogonal to the element axis (see Fig. 3).
Obviously, the circumferential acceleration ρ θ and the Coriolis acceleration a co have no effect on the inerter force, being orthogonal to the element axis.In the component of acceleration along the element axis, a a = ρ − a ce , the centripetal acceleration a ce also has no effect on the inerter force.This is because only the second time derivative of the distance between the inerter's two terminals, ρ, can generate an inerter force.For example, imagine that an inerter experiences pure rotation with a constant length between its two terminalsthe inerter force is zero in this scenario as there is no change in the distance between the two terminals.Hence, when the ineter force is modelled in a 2D (or 3D) environment, the centripetal acceleration must be removed from the relative acceleration between the terminals to find the inerter force.In this way, the correct inerter force within this 2D polar coordinate system is (4)

Model of a uniaxial inerter in general vector form
In the previous subsection, the concept of the inerter force was introduced within a polar coordinate system.The discussion shows that the centripetal acceleration has to be removed from the relative acceleration between the two terminals to find the inerter force in a 2D (or 3D) environment.In this subsection, models of uniaxial elements, including inerters, in a 3D (or 2D) environment are derived in general vector form.
Let points P i and P j be the two terminals of a uniaxial element as shown in Fig. 4. Note that the superscript of P i (or P j ) indicates the body to which the point is connected (as in Section 3).These two terminals are assumed to be connected by pinned-pinned joints but constraining the rotational degree of freedom (DOF) about the P i P j axis.For point P i (or P j ), its absolute position, velocity and acceleration with respect to the global coordinates X-Y-Z are denoted as 3-element vectors r i P , ṙi P and ri P (or r j P , ṙj P and rj P ), respectively.For a spring, the force along its axis can be expressed in this 3D environment as where r ij P = r j P − r i P is the relative position vector from point P i to point P j within the global coordinates X-Y-Z, and L 0 is the undeformed length between the uniaxial element's terminals.Note that, in Eq. ( 5), the physical meaning of (r is the instantaneous distance between uniaxial element's terminals.
For a damper, by differentiating the distance between the uniaxial element's terminals, (r , the damper force along its axis can be expressed in this 3D environment as where dt .Note that Eq. ( 6) can be rewritten as where is the unit vector along P i P j and the physical meaning of ṙij P ⋅ rij P can be understood as the projection of ṙij P onto the element axis across the two terminals (see Fig. 5).
For an inerter, based on the definition of inerter in a 1D environment, i.e., only movement along its axis is allowed, F I no rotation is defined as where dt , and rij P ⋅ rij P is the projection of the relative acceleration between the two terminals onto the element axis.Based on the discussion in SubSection 2.1, in a 3D environment, the centripetal acceleration has to be taken into account in modelling the inerter force.F I in a 3D environment is defined as a force proportional to the second time derivative of the relative distance between the inerter's two terminals as In the curly braces of Eq. ( 9), the first term, rij P ⋅ rij P , is the projection of the relative acceleration onto the element axis across the two terminals (as in Eq. ( 8)), and the second term, 1 ], is the centripetal acceleration resulting from the rotation of the element.Specifically, the physical meaning of ṙij is the square of the circumferential velocity, which is denoted as |ṙ ij P⊥ | 2 (see Fig. 5).
The uniaxial inerter force model in Eq. ( 9), expressed in terms of vectors, is a general model which is valid in both 2D and 3D environments.This general inerter force model can be applied to different types of models, for example, multibody models or finite element models, and can be expressed in different coordinate systems, as in the following subsection.

Inerter force in cartesian and spherical coordinate systems
The general formula for the uniaxial inerter force, Eq. ( 9), can be applied in different 2D or 3D coordinate systems.The expressions for Cartesian and spherical coordinates are presented in this subsection.
In 3D Cartesian coordinates, defining the relative displacement vector between the inerter's two terminals, points P j and P i , as r ij P = [ x y z ] T (where superscript 'T' indicates transpose), using Eq. ( 9) the inerter force can be expressed as In 2D Cartesian coordinates, the same equation can be used, simply dropping the terms in z and its derivatives.The spherical coordinate system (ρ, θ, φ) is another common coordinate system for a 3D environment, as shown in Fig. 6.In these spherical coordinates, using Eq. ( 9), with the inerter terminals at points O and P, the inerter force can be expressed simply as i.e. the same as for 2D polar coordinates, as in Eq. (4).

Example: a 2D 1DOF inerter-included lumped mass system
The model of a uniaxial inerter in a 2D or 3D environment was introduced in the previous subsections.In this subsection, a simple 2D 1DOF inerter-included lumped mass system is developed as an example to demonstrate the effect of the centripetal acceleration correction term on the inerter force and response of the system.A 1DOF inerter-included lumped mass system in a 2D environment in the horizontal plane is defined as shown in Fig. 7.The lumped mass, m, is constrained to travel only in the Y direction.A uniaxial spring and a uniaxial inerter, in parallel, link the mass and the origin of the system, point O, with a revolute joint.Friction and damping are neglected.The offset of the line of motion of the mass from the origin, point O, in the X direction, is L a .The static equilibrium position of the mass in the Y direction is L b , that is, the undeformed length of the spring is . The position of the mass in the XY coordinate system can be expressed as a vector r = the initial position of the mass in the Y direction is set to be y 0 (y 0 ∕ = L b ), when released the mass will oscillate about the equilibrium position in the Y direction.Using Eq. ( 10), the inerter force can be expressed as Note that the second term on the right-hand side of Eq. ( 12), bLa 2 ẏ2 , is the centripetal acceleration correction term.Meanwhile, using Eq.(1) or Eq. ( 5), the spring force in this example is . Thus, the equation of motion of this 1DOF inerterincluded lumped mass system is  which can be rewritten as The second term, involving ẏ2 , is the centripetal acceleration correction term.
As an example, for a given set of parameter values, m = 5 kg, b = 100kg, k = 10, 000 Nm − 1 , L a = 1m and L b = 1 m and initial position y 0 = 1.5m, the numerical time-history responses, with and without the inclusion of the centripetal acceleration correction term, are shown in Fig. 8(a).It can be observed that the correction term affects both the magnitude and frequency of the response.With the correction term, the magnitude of the oscillations is larger and the fundamental frequency is lower than the case where the term is neglected.Also, defining the mean position offset relative to the equilibrium position as Δy = y − L b where y is the mean value of y, it can be observed that there is a negative mean position offset when the correction is included, in contrast with a small positive mean offset if it is neglected.This can be explained as follows: considering only the mean value and fundamental harmonic of the response, it can be approximated as y ≈ y + Acos(ωt), where A is the amplitude and ω the angular frequency of the fluctuating component.The centripetal correction term in the inerter force (Eq.( 13)), involving ẏ2 = A 2 ω 2 2 {1 − cos(2ωt)}, leads to a constant term proportional to A 2 in the equation of motion, which causes a negative mean offset of the displacement relative to the static equilibrium position.The inerter forces with and without the centripetal acceleration correction term are also compared in Fig. 8(b).The amplitude of the fluctuating component of the inerter force including the centripetal acceleration correction term is larger and its frequency is lower than without the correction term.Meanwhile, the mean inerter force including the centripetal acceleration correction term is smaller than without the correction term.
To explore the behaviour of the example system more broadly, numerical time-history responses of Eq. ( 14) were obtained using a series of different initial positions, with and without the inclusion of the centripetal acceleration correction term.For each initial position, the periodic response can be decomposed into a Fourier series.The amplitude and frequency of the fundamental harmonic, and the mean position offset in relation to the initial position, are shown in Fig. 9.For the response with the centripetal acceleration correction term, the fundamental frequency decreases from the natural frequency (1.518 Hz) as the initial position increases, while the mean position offset also decreases.(The natural frequency is found by linearising the equation of motion, Eq. ( 14), as shown in Appendix A).However, if the centripetal acceleration correction term is neglected, the fundamental frequency and mean position offset both show opposite trends (i.e. both increase) with increasing initial position (Fig. 9 (b&c)).It is noted that a pair of oblique inerters in nonlinear vibration isolators [18] and suppression systems [19] has been studied recently.It can be seen that Eq. ( 12) is identical to the model of the oblique inerter derived in [18,19], which is a special case of Eq. ( 10), the derived general inerter model expressed in Cartesian coordinate.The case study analysed here shown in Fig. 7 is a simplified version of the nonlinear inerter mechanism (NIM) quasi-zero-stiffness (QZS) isolator studied in [18].The results obtained in this case study can be validated and further explained with comparisons to [18,19].For example, the observed influence on the system's fundamental frequency resulting from the centripetal acceleration term is in line with the analysis in [18].
This example has demonstrated the importance of correctly modelling the inerter force in a 2D (or 3D) environment.If the centripetal acceleration correction term is neglected, as the response departs from the very low amplitude linearised behaviour, the calculated amplitude, frequency and mean position offset of the system depart significantly from the correct response, and indeed the trends of the frequency and mean position are in the opposite direction.Therefore, the correction term must be included when using a uniaxial inerter in a 2D or 3D environment when the rotation of the inerter is significant.

Uniaxial inerter in a multibody system
The general equations for a uniaxial inerter in a 2D or 3D environment have been derived in Section 2. However, when using an inerter in a multibody system, the point where a terminal is connected to a body does not necessarily correspond to the body's reference point.In order to incorporate the inerter in a multibody system, the inerter force needs to be expressed in terms of the generalised coordinates of the bodies to which it is connected, where, for a rigid body, the generalised coordinates are the coordinates of the body reference point and the orientation of the body.In this section, the force of the uniaxial inerter is expressed in the multibody system framework, and the methodology for implementing it in a multibody model is discussed in detail.

Inerter force in the framework of a 2D multibody system
In a 2D multibody system, motions of each rigid body are described using three coordinates, i.e., two coordinates describing the translational motions of the reference point ] T , and one coordinate defining its orientation, θ i .Hence, the generalised coordinates of body i can be denoted as a 3-element vector . Assume that the two terminals of a uniaxial inerter are point P i and point P j , which are respectively located on body i and body j, as shown in Fig. 10.Note that the terminal point P i (or P j ) does not generally coincide with the body's reference point O i (or O j ).The local positions of points P i and P j with respect to the bodies' local coordinate frames X i -Y i and X j -Y j are u i P = indicates local coordinates.In this work, as all bodies are considered rigid, u i P and u j P are constant vectors.In this 2D multibody system, the relative position, velocity and acceleration vectors between the two terminals of the inerter can be written as [29, pp.107-110] where the 2D transformation matrix , and similarly for A j and A j θ .Eqs. (16&17) can be rewritten in compact matrix form as where and I 2×2 is the 2 × 2 identity matrix (similarly I 3×3 is the 3 × 3 identity matrix in SubSection 3.2).Substituting Eqs.(18&19) into Eq.( 9) yields the equation for the uniaxial inerter force expressed in terms of the generalised coordinates of the 2D multibody system, as Similarly to Eq. ( 9), the first term in the curly braces of Eq. ( 20), (H ij qij + a ij n ) ⋅ rij P , corresponds to the projection of the relative acceleration between the terminals onto the element axis, and the second term in the curly braces, ], corresponds to the term due to the centripetal acceleration.Fig. 10.A uniaxial inerter with its two terminals P i and P j respectively on body i and body j in a 2D multibody system.

Inerter force in the framework of a 3D multibody system
The equation of the uniaxial inerter force expressed in terms of the generalised coordinates of a 2D multibody system has been derived in Eq. ( 20).This equation is also valid for a 3D multibody system.Compared with the 2D environment, the only difference is that the vectors and matrices used in Eq. ( 20), i.e., q ij , H ij , a ij n and r ij P , have different expressions for a 3D multibody system.In a 3D multibody system, motions of a rigid body are described using six coordinates, i.e., three coordinates describing the translational motions of the body and three coordinates defining the body's orientation.The translational motions of body i can be defined using the position of a reference point fixed on the body, which can be denoted using three-dimensional Cartesian coordinates In this work, Euler angles θ i = [ ϕ i θ i ψ i ] T are used to describe the body's orientation.Initially, the body's local coordinate system, X i -Y i -Z i , coincides with the global coordinate system X-Y-Z.ϕ i , θ i and ψ i represent three successive rotations which are performed about the current Z i axis, X i axis and Z i axis in turn.Hence, the generalised coordinates of body i in this 3D multibody system can be represented by a 6-element vector As in SubSection 3.1, let a uniaxial inerter be installed between point P i on body i and point P j on body j.The local positions of points P i and P j with respect to the bodies' local coordinate frames X i -Y i -Z i and X j -Y j -Z j are u i P and u j P .In a 3D multibody system, Eq. ( 15) is still valid but the transformation matrix for body i, A i , (and similarly for body j throughout this subsection) is a function of the Euler angles and its formulation in detail is shown in Appendix B. Note that A i u i P can be denoted as u i P , representing the local position vector of point P i relative to reference point O i , expressed in global coordinates.For the velocity and acceleration vectors, i.e., Eqs.
(18&19), they are still valid but with modified q ij , H ij and a ij n for the 3D multibody system.The generalised coordinates of the body i and j where the inerter is connected in this 3D multibody system are modified as The modified matrix H ij in the 3D multibody system is [29, pp. 397-398] where and the underbar denotes the skew-symmetric matrix of the corresponding vector.For example, the skew-symmetric matrix of the For the 3D multibody system, a ij n also needs to be modified as [29, pp398] With the modified r ij P , q ij , H ij and a ij n for the 3D multibody system, as in Eqs.(15, 21, 22&25), the force of the uniaxial inerter, expressed in terms of the generalised coordinates of the 3D multibody system, can be obtained using Eq.(20).

Implementation of a uniaxial inerter in a multibody model
The equation for the uniaxial inerter force expressed in terms of generalised coordinates of 2D or 3D multibody systems has been derived in SubSections 3.1 and 3.2.In this subsection, the methodology of implementing the uniaxial inerter in a multibody system is introduced.
In a centroidal coordinate system where the origin of each body's local coordinate frame is rigidly attached to its centre of mass, the equations of motion for a multibody system can be written as [29, pp. 430] where M is the system mass matrix, C q is the constraint Jacobian matrix, q s is the system generalised coordinate vector, Q e is the generalised external force vector, λ s is the vector of Lagrange multipliers which are used to calculate the reaction forces resulting from the constraints and Q d is a vector that absorbs terms that are quadratic in the velocities of the second-time differentiation of the constraint equation C(q s , t) = 0 [29, pp. 428].Assume that this model includes n b bodies with e DOFs for each (e = 6 for a 3D model and e = 3 for a 2D model) and total n c constrained DOFs due to the mechanical joints.The definitions and formulations of the above matrixes and vectors are standard and can be found in textbooks on multibody system modelling, e.g., [28,29].An inerter can be added to the multibody system by applying two equal and opposite external forces to the relevant bodies.These forces need to be implemented into the original equations of motion (Eq.( 26)) in the form of the resulting generalised external force vector associated with the generalised coordinates of body i and body j on which the inerter force acts, where L ij is the incidence matrix of the inerter indicating the connection topology between the original multibody system and the inerter.The incidence matrix L ij is comprised of two columns, each column consisting of n b e × e blocks.Apart from two e ×e identity matrices located at the ith row 1st column and jth row 2nd column corresponding to body i and body j connected to the two terminals of the inerter, the remaining blocks in L ij are filled with e × e null matrices, giving The generalised external force vector of F I associated with the generalised coordinates of body i and body j in the multibody system [29, pp. 243] is Substituting the equation for the inerter force, i.e., Eq. ( 20), into Eq.( 29), the full generalised external force vector of the uniaxial inerter can be written as Eq. ( 30) can be rewritten in a compact format as where ] which absorbs the terms related to the relative velocities.Hence, substituting Eq. (31) into Eq.( 27), the full multibody model, including the added uniaxial inerter is This matrix equation represents a set of differential-algebraic equations.For normal differential-algebraic equations of a multibody system without an inerter, numerical solutions can be obtained using direct time integration.However, including the inerter into the system, the Q ij I in the right-hand side of Eq. ( 27) or (32) is a function of the current accelerations of the generalised coordinates, qij , which makes Eq. (32) implicit.For implicit equations, direct time integration cannot be used to obtain numerical solutions.Although some integration algorithms, for example, the predictor-corrector method [30], can be applied to solve implicit equations numerically, they can be much more complicated and time-consuming.Moreover, it is inevitable that numerical errors will be introduced by using an implicit integration algorithms.In order to use a direct time integration method to solve the multibody system of equations including an inerters, a modification of Eq. ( 32) is introduced in the following.The generalised inerter force vector Q ij I consists of two parts, as in Eq. (31), i.e., the term involving the acceleration of the Noting that qij = L ijT qs , the acceleration term can be moved to the left-hand side of Eq. ( 32) and be combined with the system mass matrix M, giving where M I = bL ij H ijT rij In this way, the right-hand side of Eq. ( 33) is a function of the positions and velocities of the generalised coordinates, i.e., q s and qs , but not of any accelerations, which makes it suitable for the use of direct time integration to obtain numerical solutions.

Example: a 2D inerter-included two-bar mechanism system
The method of implementing a uniaxial inerter in a multibody system has been introduced in previous subsections.In this subsection, a 2D inerter-included two-bar mechanism is used as an example, and the effect of the centripetal acceleration correction term on the system response is demonstrated.
The example 2D inerter-included two-bar mechanism, in the horizontal plane, is defined as shown in Fig. 11.Body 1 and body 2 have masses of M 1 and M 2 and rotational inertias of J 1 and J 2 , respectively.Both bodies are 1 m long.One end of body 1 is constrained

Fig. 2 .
Fig. 2. The velocity of the free terminal, point P, in 2D polar coordinates.

Fig. 5 .
Fig. 5. Diagram of the relative displacement, velocity and acceleration vectors between the two terminals, points P i and P j .

Fig. 8 .
Fig. 8. Numerical time histories of (a) displacement responses relative to the equilibrium position (with the correction included, there is a mean displacement offset as indicated by the blue double-headed arrow) and (b) inerter forces, with and without the centripetal acceleration correction term.The dashed lines show the corresponding mean values.

Fig. 9 .
Fig. 9. Comparison of responses between the models with and without the centripetal acceleration correction term in the inerter force, for different initial positions: (a) amplitude of fundamental harmonic, (b) fundamental frequency and (c) mean position offset.

M
. Zhu et al. generalised coordinates, − bH ijT rij P rijT P H ij qij , and the term that absorbs the velocities of the generalised coordinates, Q ij Iv .

Fig. 12 .
Fig. 12.Comparison of the displacements, relative to the equilibrium position, of point C in the Y direction, between the models with and without the centripetal correction term for the example two-bar mechanism (dashed lines show the corresponding mean values).