Power Analysis of Spring-Mass-Damper System

This example shows how to analyze the mechanical power and energy of a spring-mass-damper system using Powertrain Blockset™ functions. This example uses the autoblks.pwr.PlantInfo (Powertrain Blockset) object to analyze the energy of the system and the Power Accounting Bus Creator (Powertrain Blockset) block to report on system power and energy usage. Additionally, this example illustrates the power flow though the spring-mass-damper system and subsystems using the createSankeyPlot (Powertrain Blockset) function.

You can leverage insights from a power analysis to optimize performance, improve efficiency, and ensure reliability of a system.

Spring-Mass-Damper System Decomposition

The spring-mass-damper system is contains these subsystems, each of which models a separate set of components:

• Spring-Damper 1

• Mass 1

• Spring-Damper 2

• Mass 2

• Spring-Damper 3

Power Accounting Overview

The system-level power and energy accounting satisfies the conservation of energy.

$$\sum {\mathit{P}}_{\mathrm{trans}}+\sum {\mathit{P}}_{\mathrm{nottrans}}=\sum {\mathit{P}}_{\mathrm{store}}$$

Spring-Damper Power Flow

This diagram illustrates the direction of the forces for the spring and damper components.

The forces are calculated using these equations:

$$\begin{array}{l}{\mathit{F}}_{\mathit{d}}=\mathit{d}\left({\mathit{v}}_{\mathit{R}}-{\mathit{v}}_{\mathit{L}}\right)\\ {\mathit{F}}_{\mathit{k}}=\mathit{k}\left({\mathit{x}}_{\mathit{R}}-{\mathit{x}}_{\mathit{L}}\right)\\ {\mathit{F}}_{\mathit{L}}={\mathit{F}}_{\mathit{d}}+{\mathit{F}}_{\mathit{k}}\\ {\mathit{F}}_{\mathit{R}}={-\mathit{F}}_{\mathit{d}}-{\mathit{F}}_{\mathit{k}}\end{array}$$

Power Accounting Bus Creator Power Signals

This table shows the signals in the power information bus created by the Power Accounting Bus Creator block for the spring-damper components.

Signal	Description	Definition
PwrTrnsfrd — PwrFrcLeft	Power transferred at left connection	$${\mathit{F}}_{\mathit{L}}\mathit{v}$$
PwrTrnsfrd — PwrFrcRight	Power transferred at right connection	$${\mathit{F}}_{\mathit{R}}\mathit{v}$$
PwrNotTrnsFrd — PwrDampingLoss	Damper power loss	$${-\mathit{F}}_{\mathit{d}}\left({\mathit{v}}_{\mathit{R}}-{\mathit{v}}_{\mathit{L}}\right)$$
PwrStored — PwrSpringStored	Power stored by spring	$${\mathit{F}}_{\mathit{k}}\left({\mathit{v}}_{\mathit{R}}-{\mathit{v}}_{\mathit{L}}\right)$$

Mass Power Flow

This diagram illustrates the direction of the forces for the mass components.

The mass velocity and position are governed by this equation:

$$\mathit{m}\ddot{\mathit{x}}={\mathit{F}}_{\mathit{L}}+{\mathit{F}}_{\mathit{R}}+{\mathit{F}}_{\mathrm{ext}}$$

Power Accounting Bus Creator Power Signals

This table shows the signals in the power information bus created by the Power Accounting Bus Creator block for the mass components.

Signal	Description	Definition
PwrTrnsfrd — PwrFrcLeft	Power transferred at left connection	$${\mathit{F}}_{\mathit{L}}\mathit{v}$$
PwrTrnsfrd — PwrFrcRight	Power transferred at right connection	$${\mathit{F}}_{\mathit{R}}\mathit{v}$$
PwrNotTrnsFrd — PwrDampingLoss	Power from external force	$${\mathit{F}}_{\mathrm{ext}}\mathit{v}$$
PwrStored — PwrKinetic	Power stored as kinetic energy	$$\left({\mathit{F}}_{\mathit{L}}+{\mathit{F}}_{\mathit{R}}+{\mathit{F}}_{\mathrm{ext}}\right)\mathit{v}$$

For more information, see Power Signals (Powertrain Blockset).

Run Simulation

Click Run to create an autoblks.pwr.PlantInfo object. Use the PwrUnits and EnrgyUnits properties to set the units.

SysName = 'powerAnalysis';
open(SysName);
SysPwrAnalysis = autoblks.pwr.PlantInfo(SysName);
SysPwrAnalysis.PwrUnits = 'kW';
SysPwrAnalysis.EnrgyUnits = 'kJ';

Use the run method to turn on logging, run the simulation, and add logged data to the object.

SysPwrAnalysis.run;

Summary of Energy Values

Display the final energy values for each subsystem.

SysPwrAnalysis.dispSysSummary

 
    System Name      Efficiency    Energy Loss (kJ)    Energy Input (kJ)    Energy Output (kJ)    Energy Stored (kJ)  
----------------------------------------------------------------------------------------------------------------------
powerAnalysis             0.144               -18.8                   22                     0                  3.17  
  Mass 1                  0.755               -5.05                 20.7                 -15.6                     0  
  Mass 2                  0.594               -10.6                 26.1                 -15.5                     0  
  Spring-Damper 1         0.957               -0.48                 11.2                 -9.46                  1.26  
  Spring-Damper 2         0.993             -0.0718                  9.8                 -9.33                   0.4  
  Spring-Damper 3         0.741               -2.61                 10.1                 -5.97                   1.5  
 

Write the summary to a spreadsheet.

SysPwrAnalysis.xlsSysSummary(fullfile(fileparts( ...
    which('GenerateEnergyReport')), 'EnergySummary.xlsx'));

Generate a Sankey plot to visualize the power flow of the system using the createSankeyPlot function.

createSankeyPlot(SysPwrAnalysis);

Spring-Damper 1 Subsystem Summary

Display the energy of the Spring-Damper 1 subsystem for the autoblks.pwr.PlantInfo object using the dispSignalSummary method.

SpringDamper1Name = 'powerAnalysis/Spring-Damper 1';
SpringDamper1PwrAnalysis = SysPwrAnalysis.findChildSys(SpringDamper1Name);
SpringDamper1PwrAnalysis.dispSignalSummary;

 
powerAnalysis/Spring-Damper 1
Average Efficiency = 0.957
 
       Signal         Energy (kJ)  
-----------------------------------
Inputs                       11.2  
  Transferred                11.2  
    Left force                  0  
    Right force              11.2  
  Not transferred               0  
    Damping loss                0  
Outputs                     -9.46  
  Left force                    0  
  Right force               -9.46  
Losses                      -0.48  
  Damping loss              -0.48  
Stored                       1.26  
  Spring                     1.26  
 

Generate a Sankey plot to visualize the power flow of the Spring-Damper 1 subsystem using the createSankeyPlot function.

SpringDamper1PwrAnalysis.createSankeyPlot();

Mass 1 Subsystem Summary

Display the energy of the Mass 1 subsystem for the autoblks.pwr.PlantInfo object using the dispSignalSummary method.

Mass1Name = 'powerAnalysis/Mass 1';
Mass1PwrAnalysis = SysPwrAnalysis.findChildSys(Mass1Name);
Mass1PwrAnalysis.dispSignalSummary;

 
powerAnalysis/Mass 1
Average Efficiency = 0.755
 
            Signal              Energy (kJ)  
---------------------------------------------
Inputs                                 20.7  
  Transferred                          14.2  
    Left force                         9.46  
    Right force                        4.72  
  Not transferred                      6.47  
    Externally applied force           6.47  
Outputs                               -15.6  
  Left force                          -11.2  
  Right force                          -4.4  
Losses                                -5.05  
  Externally applied force            -5.05  
Stored                                    0  
  Kinetic                                 0  
 

Generate a Sankey plot to visualize the power flow of the Mass 1 subsystem using the createSankeyPlot function.

Mass1PwrAnalysis.createSankeyPlot();

Spring-Damper 2 Subsystem Summary

Display the energy of the Spring-Damper 2 subsystem for the autoblks.pwr.PlantInfo object using the dispSignalSummary method.

SpringDamper2Name = 'powerAnalysis/Spring-Damper 2';
SpringDamper2PwrAnalysis = SysPwrAnalysis.findChildSys(SpringDamper2Name);
SpringDamper2PwrAnalysis.dispSignalSummary;

 
powerAnalysis/Spring-Damper 2
Average Efficiency = 0.993
 
       Signal         Energy (kJ)  
-----------------------------------
Inputs                        9.8  
  Transferred                 9.8  
    Left force                4.4  
    Right force              5.39  
  Not transferred               0  
    Damping loss                0  
Outputs                     -9.33  
  Left force                -4.72  
  Right force                -4.6  
Losses                    -0.0718  
  Damping loss            -0.0718  
Stored                        0.4  
  Spring                      0.4  
 

Generate a Sankey plot to visualize the power flow of the Spring-Damper 2 subsystem using the createSankeyPlot function.

SpringDamper2PwrAnalysis.createSankeyPlot();

Mass 2 Subsystem Summary

Display the energy of the Mass 2 subsystem for the autoblks.pwr.PlantInfo object using the dispSignalSummary method.

Mass2Name = 'powerAnalysis/Mass 2';
Mass2PwrAnalysis = SysPwrAnalysis.findChildSys(Mass2Name);
Mass2PwrAnalysis.dispSignalSummary;

 
powerAnalysis/Mass 2
Average Efficiency = 0.594
 
            Signal              Energy (kJ)  
---------------------------------------------
Inputs                                 26.1  
  Transferred                          10.6  
    Left force                          4.6  
    Right force                        5.97  
  Not transferred                      15.5  
    Externally applied force           15.5  
Outputs                               -15.5  
  Left force                          -5.39  
  Right force                         -10.1  
Losses                                -10.6  
  Externally applied force            -10.6  
Stored                                    0  
  Kinetic                                 0  
 

Generate a Sankey plot to visualize the power flow of the Mass 2 subsystem using the createSankeyPlot function.

Mass2PwrAnalysis.createSankeyPlot();

Spring-Damper 3 Subsystem Summary

Display the energy of the Spring-Damper 3 subsystem for the autoblks.pwr.PlantInfo object using the dispSignalSummary method.

SpringDamper3Name = 'powerAnalysis/Spring-Damper 3';
SpringDamper3PwrAnalysis = SysPwrAnalysis.findChildSys(SpringDamper3Name);
SpringDamper3PwrAnalysis.dispSignalSummary;

 
powerAnalysis/Spring-Damper 3
Average Efficiency = 0.741
 
       Signal         Energy (kJ)  
-----------------------------------
Inputs                       10.1  
  Transferred                10.1  
    Left force               10.1  
    Right force                 0  
  Not transferred               0  
    Damping loss                0  
Outputs                     -5.97  
  Left force                -5.97  
  Right force                   0  
Losses                      -2.61  
  Damping loss              -2.61  
Stored                        1.5  
  Spring                      1.5  
 

Generate a Sankey plot to visualize the power flow of the Spring-Damper 3 subsystem using the createSankeyPlot function.

SpringDamper3PwrAnalysis.createSankeyPlot();

Result Plots

Plot the powertrain energy and power analysis results for the autoblks.pwr.PlantInfo object using the sdiSummary function.

SysPwrAnalysis.sdiSummary({SpringDamper1Name, Mass1Name, ...
    SpringDamper2Name, Mass2Name, ...
    SpringDamper3Name});

For example, plot the input energy, the energy losses, and the stored energy for the system.

close('all')