Nonlinear Translational Damper

Nonlinear damper in a translational system

Libraries:
Simscape / Driveline / Couplings & Drives / Springs & Dampers

Description

The Nonlinear Translational Damper block represents a nonlinear translational damper. Polynomial and table lookup parameterizations define the nonlinear relationship between damping force and relative linear velocity. The damping force can be symmetric or asymmetric about the zero velocity point. The block applies equal and opposite damping forces on the two translational conserving ports.

The symmetric polynomial parameterization defines the damping force for both positive and negative relative velocities according to the expression:

$F = b_{1} v + s i g n (v) \cdot b_{2} v^{2} + b_{3} v^{3} + s i g n (v) \cdot b_{4} v^{4} + b_{5} v^{5},$

where:

F — Damping force
b₁, b₂,..., b₅ — Damping coefficients
v — Relative linear velocity between ports R and C, $v = v_{R} - v_{C}$
v_R — Absolute linear velocity associated with port R
v_C — Absolute linear velocity associated with port C

To avoid zero-crossings that slow simulation, eliminate the sign function from the polynomial expression by specifying an odd polynomial (b₂,b₄ = 0).

The two-sided polynomial parameterization defines the damping force for both positive and negative relative velocities according to the expression:

$F = {\begin{matrix} b_{1 e} v + b_{2 e} v^{2} + b_{3 e} v^{3} + b_{4 e} v^{4} + b_{5 e} v^{5}, & v \geq 0 \\ b_{1 c} v - b_{2 c} v^{2} + b_{3 c} v^{3} - b_{4 c} v^{4} + b_{5 c} v^{5}, & v < 0 \end{matrix},$

where:

b_1e, b_2e, ..., b_5e — Damping coefficients for positive relative velocities
b_1c, b_2c, ..., b_5c — Damping coefficients for negative relative velocities

Positive relative velocities correspond to damper extension when ports R and C move apart. Negative relative velocities correspond to damper contraction when ports R and C move together.

Both polynomial parameterizations use a fifth-order polynomial expression. To use a lower-order polynomial, set the unneeded higher-order coefficients to zero. For polynomials of order greater than five, fit to a polynomial of order smaller than or equal to five, or use the table lookup parameterization.

The table lookup parameterization defines damping force based on a set of velocity and force vectors. If not included in the vectors, the block automatically adds a data point at the origin, that is, the intersection of zero velocity and zero force.

Assumptions and Limitations

The block assumes viscous damping. The damping force depends only on velocity.

Examples

Suspension System Comparison

A testbed containing three sets of springs and dampers excited by the same oscillating velocity source. The first uses the Shock Absorber block and includes linear stiffness and damping. Optional friction and hard stops are not used. The second shock absorber uses a Nonlinear Translational Spring and a Nonlinear Translational Damper specified by symmetric polynomials. The third uses a Variable Translational Spring and Variable Translational Damper. The spring constant is varied during the simulation using open loop control and the damping coefficient adjusts to ensure critical damping is achieved. Closed loop control can be added to simulate an adaptive suspension system.

Open Model

Ports

Conserving

expand all

R — Shaft
mechanical translational

Mechanical translational conserving port associated with the shaft.

C — Case
mechanical translational

Mechanical translational conserving port associated with the case.

Parameters

expand all

Parameterization — Option to parameterize damping
`By polynomial` (default) | `By table lookup`

Whether to parameterize damping by polynomial or table lookup.

Symmetry — Option to parameterize damping symmetry
`Symmetric` (default) | `Two-sided`

Whether to use a symmetric or two-sided table lookup parameterization.

Vector of damping coefficients — Polynomial damping coefficients
`[10, 0, 1, 0, .1]` `N*s/m` (default) | vector of length 5

Damping coefficients. Physical units apply to the first element.

Dependencies

To enable this parameter, set Parameterization to By polynomial and Symmetry to Symmetric.

Vector of extension damping coefficients — Polynomial extension damping coefficients
`[10, 0, 1, 0, .1]` `N*s/m` (default) | vector of length 5

Extension damping coefficients. Physical units apply to the first element.

Dependencies

To enable this parameter, set Parameterization to By polynomial and Symmetry to Two-sided.

Vector of contraction damping coefficients — Polynomial contraction damping coefficients
`[100, -1, 10, 0, 0]` `N*s/m` (default) | vector of length 5

Contraction damping coefficients. Physical units apply to the first element.

Dependencies

To enable this parameter, set Parameterization to By polynomial and Symmetry to Two-sided.

Velocity vector — Velocity
`[-1, -.5, -.3, -.1, .1, .3, .5, 1]` `m/s` (default) | vector

Translational velocity. The vector requires a minimum number of elements based on the interpolation method. When you set Interpolation method to:

Linear, the minimum is two.
Smooth, the minimum is three.

The elements correspond one-to-one with the Force vector parameter.

Dependencies

To enable this parameter, set Parameterization to By table lookup.

Force vector — Damping force
`[-100, -40, -20, -5, 5, 20, 40, 100]` `N` (default) | vector

Damping force for a given translational velocity. The vector requires a minimum number of elements based on the interpolation method. When you set Interpolation method to:

Linear, the minimum is two.
Smooth, the minimum is three.

The elements correspond one-to-one with the Velocity vector parameter.

Dependencies

To enable this parameter, set Parameterization to By table lookup.

Interpolation method — Method of interpolation between breakpoint values
`Linear` (default) | `Smooth`

Method to use for lookup table breakpoint interpolation. The block uses the tablelookup function to model nonlinearity by using array data to map input values to output values:

Linear — Select this option for the lowest computational cost.
Smooth — Select this option to produce a continuous curve with continuous first-order derivatives.

For more information, see tablelookup.

Dependencies

To enable this parameter, set Parameterization to By table lookup.

Extrapolation method — Method of handling input values that fall outside range of breakpoint data set
`Linear` (default) | `Nearest` | `Error`

Method to use for lookup table breakpoint extrapolation. This method determines the output value when the input value is outside the range specified in the argument list. The block uses the tablelookup function to model nonlinearity by using array data to map input values to output values:

Linear — Select this option to produce a curve with continuous first-order derivatives in the extrapolation region and at the boundary with the interpolation region.
Nearest — Select this option to produce an extrapolation that does not go above the highest point in the data or below the lowest point in the data.
Error — Select this option to avoid extrapolating when you want your data to be within the table range. If the input signal is outside the range of the table, the simulation stops and generates an error.

Dependencies

To enable this parameter, set Parameterization to By table lookup.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2013a

Nonlinear Translational Damper

Description

Assumptions and Limitations

Examples

Suspension System Comparison

Ports

Conserving

R — Shaft mechanical translational

C — Case mechanical translational

Parameters

Parameterization — Option to parameterize damping By polynomial (default) | By table lookup

Symmetry — Option to parameterize damping symmetry Symmetric (default) | Two-sided

Vector of damping coefficients — Polynomial damping coefficients [10, 0, 1, 0, .1] N*s/m (default) | vector of length 5

Dependencies

Vector of extension damping coefficients — Polynomial extension damping coefficients [10, 0, 1, 0, .1] N*s/m (default) | vector of length 5

Dependencies

Vector of contraction damping coefficients — Polynomial contraction damping coefficients [100, -1, 10, 0, 0] N*s/m (default) | vector of length 5

Dependencies

Velocity vector — Velocity [-1, -.5, -.3, -.1, .1, .3, .5, 1] m/s (default) | vector

Dependencies

Force vector — Damping force [-100, -40, -20, -5, 5, 20, 40, 100] N (default) | vector

Dependencies

Interpolation method — Method of interpolation between breakpoint values Linear (default) | Smooth

Dependencies

Extrapolation method — Method of handling input values that fall outside range of breakpoint data set Linear (default) | Nearest | Error

Dependencies

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using Simulink® Coder™.

Version History

See Also

R — Shaft
mechanical translational

C — Case
mechanical translational

Parameterization — Option to parameterize damping
`By polynomial` (default) | `By table lookup`

Symmetry — Option to parameterize damping symmetry
`Symmetric` (default) | `Two-sided`

Vector of damping coefficients — Polynomial damping coefficients
`[10, 0, 1, 0, .1]` `N*s/m` (default) | vector of length 5

Vector of extension damping coefficients — Polynomial extension damping coefficients
`[10, 0, 1, 0, .1]` `N*s/m` (default) | vector of length 5

Vector of contraction damping coefficients — Polynomial contraction damping coefficients
`[100, -1, 10, 0, 0]` `N*s/m` (default) | vector of length 5

Velocity vector — Velocity
`[-1, -.5, -.3, -.1, .1, .3, .5, 1]` `m/s` (default) | vector

Force vector — Damping force
`[-100, -40, -20, -5, 5, 20, 40, 100]` `N` (default) | vector

Interpolation method — Method of interpolation between breakpoint values
`Linear` (default) | `Smooth`

Extrapolation method — Method of handling input values that fall outside range of breakpoint data set
`Linear` (default) | `Nearest` | `Error`

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.